10th Maths Guide – Samacheer Kalvi Book Solutions
10th Maths guide with Samacheer Kalvi book back solutions, chapter by chapter, as per the 2026-27 textbook.
This page has step-by-step answers for every exercise in the 2026-27 Samacheer Kalvi Class 10 Maths textbook, from Relations and Functions and Numbers and Sequences through Algebra, Geometry, Coordinate Geometry, Trigonometry, Mensuration, and Statistics and Probability. Use it as a 10th maths guide while you practise at home. All samacheer kalvi 10th maths book solutions here are free to read.
Chapter 1: Relations and Functions
Ex 1.1: 12 questions
- Q1(i). Find A × B, A × A and B × A (i) A={2,-2,3} and B={1,-4}
- Q1(ii). Find A × B, A × A and B × A (ii) A = B = {p, q}
- Q1(iii). Find A × B, A × A and B × A (iii) A = {m, n}; B = φ
- Q2. Let A = {1,2,3} and B = {x ∣ x is a prime number less than 10}. Find A × B and B × A.
- Q3. If B × A = {(-2,3),(-2,4),(0,3),(0,4),(3,3),(3,4)} find A and B.
- Q4. If A = {5,6}, B = {4,5,6}, C = {5,6,7}, Show that A × A = (B × B) ∩ (C × C).
- Q5. Given A={1,2,3}, B = {2,3,5}, C = {3,4} and D = {1,3,5}, check if (A ∩ C) × (B ∩ D) = (A × B) ∩ (C × D) is true?
- Q6(i). Let A = {x ∈ W ∣ x < 2}, B = {x ∈ N ∣ 1 < x ≤ 4} and C = {3,5}. Verify that (i) A × (B ∪ C) = (A × B) ∪ (A × C)
- Q6(ii). Let A = {x ∈ W ∣ x < 2}, B = {x ∈ N ∣ 1 < x ≤ 4} and C = {3,5}. Verify that (ii) A × (B ∩ C) = (A × B) ∩ (A × C)
- Q6(iii). Let A = {x ∈ W ∣ x < 2}, B = {x ∈ N ∣ 1 < x ≤ 4} and C = {3,5}. Verify that (iii) (A ∪ B) × C = (A × C) ∪ (B × C)
- Q7(i). Let A = The set of all natural numbers less than 8, B = The set of all prime numbers less than 8, C = The set of even p…
- Q7(ii). Let A = The set of all natural numbers less than 8, B = The set of all prime numbers less than 8, C = The set of even p…
Ex 1.2: 9 questions
- Q1(i). Let A = {1,2,3,7} and B = {3,0,-1,7}, which of the following are relation from A to B? (i) R₁ = {(2,1), (7,1)}
- Q1(ii). Let A = {1,2,3,7} and B = {3,0,-1,7}, which of the following are relation from A to B? (ii) R₂ = {(-1,1)}
- Q1(iii). Let A = {1,2,3,7} and B = {3,0,-1,7}, which of the following are relation from A to B? (iii) R₃ = {(2,-1), (7,7), (1,3)}
- Q1(iv). Let A = {1,2,3,7} and B = {3,0,-1,7}, which of the following are relation from A to B? (iv) R₄ = {(7,-1), (0,3), (3,3),…
- Q2. Let A = {1,2,3,4,...,45} and R be the relation defined as “square is of a number” on A. Write R as a subset of A × A. A…
- Q3. A Relation R is given by the set {(x,y) / y = x + 3,\ x ∈ {0,1,2,3,4,5}}. Determine its domain and range.
- Q4(i). Represent each of the given relations by (a) an arrow diagram, (b) a graph and (c) a set in roster form, wherever possi…
- Q4(ii). Represent each of the given relations by (a) an arrow diagram, (b) a graph and (c) a set in roster form, wherever possi…
- Q5. A company has four categories of employees given by Assistants (A), Clerks (C), Managers (M) and an Executive Officer (…
Ex 1.3: 25 questions
- Q1. Let f = {(x,y) ∣ x, y ∈ N and y = 2x} be a relation on N. Find the domain, co-domain and range. Is this relation a func…
- Q2. Let X = {3, 4, 6, 8}. Determine whether the relation R = {(x, f(x)) ∣ x ∈ X, f(x) = x² + 1} is a function from X to N?
- Q3(i). Given the function f: x → x² - 5x + 6, evaluate (i) f(-1)
- Q3(ii). Given the function f: x → x² - 5x + 6, evaluate (ii) f(2a)
- Q3(iii). Given the function f: x → x² - 5x + 6, evaluate (iii) f(2)
- Q3(iv). Given the function f: x → x² - 5x + 6, evaluate (iv) f(x - 1)
- Q4(i)(a). A graph representing the function f(x) is given in Fig.1.16 it is clear that f(9) = 2. (i) Find the following values of…
- Q4(i)(b). A graph representing the function f(x) is given in Fig.1.16 it is clear that f(9) = 2. (i) Find the following values of…
- Q4(i)(c). A graph representing the function f(x) is given in Fig.1.16 it is clear that f(9) = 2. (i) Find the following values of…
- Q4(i)(d). A graph representing the function f(x) is given in Fig.1.16 it is clear that f(9) = 2. (i) Find the following values of…
- Q4(ii). A graph representing the function f(x) is given in Fig.1.16 it is clear that f(9) = 2. (ii) For what value of x is f(x)…
- Q4(iii). A graph representing the function f(x) is given in Fig.1.16 it is clear that f(9) = 2. (iii) Describe the following (i)…
- Q4(iv). A graph representing the function f(x) is given in Fig.1.16 it is clear that f(9) = 2. (iv) What is the image of 6 unde…
- Q5. Let f(x) = 2x + 5. If x ≠ 0 then find (f(x+2) - f(2))/x.
- Q6(i). A function f is defined by f(x) = 2x - 3. (i) find (f(0) + f(1))/2.
- Q6(ii). A function f is defined by f(x) = 2x - 3. (ii) find x such that f(x) = 0.
- Q6(iii). A function f is defined by f(x) = 2x - 3. (iii) find x such that f(x) = x.
- Q6(iv). A function f is defined by f(x) = 2x - 3. (iv) find x such that f(x) = f(1 - x).
- Q7. An open box is to be made from a square piece of material, 24 cm on a side, by cutting equal squares from the corners a…
- Q8. A function f is defined by f(x) = 3 - 2x. Find x such that f(x²) = (f(x))².
- Q9. A plane is flying at a speed of 500 km per hour. Express the distance 'd' travelled by the plane as function of time t…
- Q10(i). The data in the adjacent table depicts the length of a person forehand and their corresponding height. Based on this da…
- Q10(ii). The data in the adjacent table depicts the length of a person forehand and their corresponding height. Based on this da…
- Q10(iii). The data in the adjacent table depicts the length of a person forehand and their corresponding height. Based on this da…
- Q10(iv). The data in the adjacent table depicts the length of a person forehand and their corresponding height. Based on this da…
Ex 1.4: 32 questions
- Q1(i). Determine whether the graph given below represent functions. Give reason for your answers concerning each graph. (i)
- Q1(ii). Determine whether the graph given below represent functions. Give reason for your answers concerning each graph. (ii)
- Q1(iii). Determine whether the graph given below represent functions. Give reason for your answers concerning each graph. (iii)
- Q1(iv). Determine whether the graph given below represent functions. Give reason for your answers concerning each graph. (iv)
- Q2(i). Let f: A → B be a function defined by f(x) = x/2 - 1, where A = {2, 4, 6, 10, 12}, B = {0, 1, 2, 4, 5, 9}. Represent f…
- Q2(ii). Let f: A → B be a function defined by f(x) = x/2 - 1, where A = {2, 4, 6, 10, 12}, B = {0, 1, 2, 4, 5, 9}. Represent f…
- Q2(iii). Let f: A → B be a function defined by f(x) = x/2 - 1, where A = {2, 4, 6, 10, 12}, B = {0, 1, 2, 4, 5, 9}. Represent f…
- Q2(iv). Let f: A → B be a function defined by f(x) = x/2 - 1, where A = {2, 4, 6, 10, 12}, B = {0, 1, 2, 4, 5, 9}. Represent f…
- Q3(i). Represent the function f = {(1,2),(2,2),(3,2),(4,3),(5,4)} through (i) an arrow diagram
- Q3(ii). Represent the function f = {(1,2),(2,2),(3,2),(4,3),(5,4)} through (ii) a table form
- Q3(iii). Represent the function f = {(1,2),(2,2),(3,2),(4,3),(5,4)} through (iii) a graph
- Q4. Show that the function f: ℕ → ℕ defined by f(x) = 2x - 1 is one-one but not onto.
- Q5. Show that the function f: ℕ → ℕ defined by f(m) = m² + m + 3 is one-one function.
- Q6(i). Let A = {1,2,3,4} and B = ℕ. Let f: A → B be defined by f(x) = x³ then, (i) find the range of f
- Q6(ii). Let A = {1,2,3,4} and B = ℕ. Let f: A → B be defined by f(x) = x³ then, (ii) identify the type of function
- Q7(i). In each of the following cases state whether the function is bijective or not. Justify your answer. (i) f: ℝ → ℝ define…
- Q7(ii). In each of the following cases state whether the function is bijective or not. Justify your answer. (ii) f: ℝ → ℝ defin…
- Q8. Let A = {-1, 1} and B = {0, 2}. If the function f: A → B defined by f(x) = ax + b is an onto function? Find a and b.
- Q9(i). If the function f is defined by f(x) = x + 2;, x 1; 2;, -1 ≤ x ≤ 1; x - 1;, -3 < x < -1 find the values of (i) f(3)
- Q9(ii). If the function f is defined by f(x) = x + 2;, x 1; 2;, -1 ≤ x ≤ 1; x - 1;, -3 < x < -1 find the values of (ii) f(0)
- Q9(iii). If the function f is defined by f(x) = x + 2;, x 1; 2;, -1 ≤ x ≤ 1; x - 1;, -3 < x < -1 find the values of (iii) f(-1 ·…
- Q9(iv). If the function f is defined by f(x) = x + 2;, x 1; 2;, -1 ≤ x ≤ 1; x - 1;, -3 < x < -1 find the values of (iv) f(2) +…
- Q10(i). A function f: [-5, 9] → ℝ is defined as follows: f(x) = 6x + 1;, -5 ≤ x < 2; 5x² - 1;, 2 ≤ x < 6; 3x - 4;, 6 ≤ x ≤ 9 Fi…
- Q10(ii). A function f: [-5, 9] → ℝ is defined as follows: f(x) = 6x + 1;, -5 ≤ x < 2; 5x² - 1;, 2 ≤ x < 6; 3x - 4;, 6 ≤ x ≤ 9 Fi…
- Q10(iii). A function f: [-5, 9] → ℝ is defined as follows: f(x) = 6x + 1;, -5 ≤ x < 2; 5x² - 1;, 2 ≤ x < 6; 3x - 4;, 6 ≤ x ≤ 9 Fi…
- Q10(iv). A function f: [-5, 9] → ℝ is defined as follows: f(x) = 6x + 1;, -5 ≤ x < 2; 5x² - 1;, 2 ≤ x < 6; 3x - 4;, 6 ≤ x ≤ 9 Fi…
- Q11. The distance S an object travels under the influence of gravity in time t seconds is given by S(t) = 1/2gt² + at + b wh…
- Q12(i). The function 't' which maps temperature in Celsius (C) into temperature in Fahrenheit (F) is defined by t(C) = F where…
- Q12(ii). The function 't' which maps temperature in Celsius (C) into temperature in Fahrenheit (F) is defined by t(C) = F where…
- Q12(iii). The function 't' which maps temperature in Celsius (C) into temperature in Fahrenheit (F) is defined by t(C) = F where…
- Q12(iv). The function 't' which maps temperature in Celsius (C) into temperature in Fahrenheit (F) is defined by t(C) = F where…
- Q12(v). The function 't' which maps temperature in Celsius (C) into temperature in Fahrenheit (F) is defined by t(C) = F where…
Ex 1.5: 18 questions
- Q1(i). Using the functions f and g given below, find f ° g and g ° f. Check whether f ° g = g ° f. (i) f(x) = x - 6, g(x) = x²
- Q1(ii). Using the functions f and g given below, find f ° g and g ° f. Check whether f ° g = g ° f. (ii) f(x) = 2/x, g(x) = 2x²…
- Q1(iii). Using the functions f and g given below, find f ° g and g ° f. Check whether f ° g = g ° f. (iii) f(x) = (x+6)/3, g(x)…
- Q1(iv). Using the functions f and g given below, find f ° g and g ° f. Check whether f ° g = g ° f. (iv) f(x) = 3 + x, g(x) = x…
- Q1(v). Using the functions f and g given below, find f ° g and g ° f. Check whether f ° g = g ° f. (v) f(x) = 4x² - 1, g(x) =…
- Q2(i). Find the value of k, such that f ° g = g ° f (i) f(x) = 3x + 2, g(x) = 6x - k
- Q2(ii). Find the value of k, such that f ° g = g ° f (ii) f(x) = 2x - k, g(x) = 4x + 5
- Q3. If f(x) = 2x - 1, g(x) = (x+1)/2, show that f ° g = g ° f = x
- Q4. If f(x) = x² - 1, g(x) = x - 2 find a, if g ° f(a) = 1.
- Q5. Let A, B, C ⊆ ℕ and a function f: A → B be defined by f(x) = 2x + 1 and g: B → C be defined by g(x) = x². Find the rang…
- Q6(i). Let f(x) = x² - 1. Find (i) f ° f
- Q6(ii). Let f(x) = x² - 1. Find (ii) f ° f ° f
- Q7. If f: ℝ → ℝ and g: ℝ → ℝ are defined by f(x) = x⁵ and g(x) = x⁴ then check if f, g are one-one and f ° g is one-one?
- Q8(i). Consider the functions f(x), g(x), h(x) as given below. Show that (f ° g) ° h = f ° (g ° h) in each case. (i) f(x) = x…
- Q8(ii). Consider the functions f(x), g(x), h(x) as given below. Show that (f ° g) ° h = f ° (g ° h) in each case. (ii) f(x) = x…
- Q8(iii). Consider the functions f(x), g(x), h(x) as given below. Show that (f ° g) ° h = f ° (g ° h) in each case. (iii) f(x) =…
- Q9. Let f = {(-1,3),(0,-1),(2,-9)} be a linear function from ℤ into ℤ. Find f(x).
- Q10. In electrical circuit theory, a circuit C(t) is called a linear circuit if it satisfies the superposition principle giv…
Ex 1.6: 15 questions
- Q1. If n(A × B) = 6 and A = {1,3} then n(B) is
- Q2. A = {a,b,p}, B = {2,3}, C = {p,q,r,s} then n[(A ∪ C) × B] is
- Q3. If A = {1,2}, B = {1,2,3,4}, C = {5,6} and D = {5, 6, 7, 8} then state which of the following statement is true.
- Q4. If there are 1024 relations from a set A = {1, 2, 3, 4, 5} to a set B, then the number of elements in B is
- Q5. The range of the relation R = {(x, x²) ∣ x is a prime number less than 13} is
- Q6. If the ordered pairs (a + 2, 4) and (5, 2a + b) are equal then (a,b) is
- Q7. Let n(A) = m and n(B) = n then the total number of non-empty relations that can be defined from A to B is
- Q8. If {(a,8),(6,b)} represents an identity function, then the value of a and b are respectively
- Q9. Let A = {1,2,3,4} and B = {4,8,9,10}. A function f: A → B given by f = {(1,4),(2,8),(3,9),(4,10)} is a
- Q10. If f(x) = 2x² and g(x) = 1/3x, then f ° g is
- Q11. If f: A → B is a bijective function and if n(B) = 7, then n(A) is equal to
- Q12. Let f and g be two functions given by f = {(0,1),(2,0),(3,-4),(4,2),(5,7)}, g = {(0,2),(1,0),(2,4),(-4,2),(7,0)} then t…
- Q13. Let f(x) = √(1 + x²) then
- Q14. If g = {(1,1),(2,3),(3,5),(4,7)} is a function given by g(x) = α x + β then the values of α and β are
- Q15. f(x) = (x + 1)³ - (x - 1)³ represents a function which is
Unit Exercise 1: 16 questions
- Q1. If the ordered pairs (x²-3x,\ y²+4y) and (-2,5) are equal, then find x and y.
- Q2. The cartesian product A × A has 9 elements among which (-1, 0) and (0,1) are found. Find the set A and the remaining el…
- Q3(i). Given that f(x) = √(x-1), x ≥ 1; 4, x < 1. Find (i) f(0)
- Q3(ii). Given that f(x) = √(x-1), x ≥ 1; 4, x < 1. Find (ii) f(3)
- Q3(iii). Given that f(x) = √(x-1), x ≥ 1; 4, x < 1. Find (iii) f(a+1) in terms of a. (Given that a ≥ 0)
- Q4. Let A = {9,10,11,12,13,14,15,16,17} and let f: A → N be defined by f(n) = the highest prime factor of n ∈ A. Write f as…
- Q5. Find the domain of the function f(x) = √(1 + √(1 - √(1 - x²)))
- Q6. If f(x) = x², g(x) = 3x and h(x) = x-2, Prove that (f ° g) ° h = f ° (g ° h).
- Q7. Let A = {1,2} and B = {1,2,3,4}, C = {5,6} and D = {5,6,7,8}. Verify whether A × C is a subset of B × D?
- Q8. If f(x) = (x-1)/(x+1),\ x ≠ -1 show that f(f(x)) = -1/x, provided x ≠ 0.
- Q9(i). The functions f and g are defined by f(x) = 6x + 8; g(x) = (x-2)/3. (i) Calculate the value of gg(1/2)
- Q9(ii). The functions f and g are defined by f(x) = 6x + 8; g(x) = (x-2)/3. (ii) Write an expression for gf(x) in its simplest…
- Q10(i). Write the domain of the following real functions (i) f(x) = (2x+1)/(x-9)
- Q10(ii). Write the domain of the following real functions (ii) p(x) = (-5)/(4x²+1)
- Q10(iii). Write the domain of the following real functions (iii) g(x) = √(x-2)
- Q10(iv). Write the domain of the following real functions (iv) h(x) = x + 6
Chapter 2: Numbers and Sequences
Ex 2.1: 13 questions
- Q1. Find all positive integers, when divided by 3 leaves remainder 2.
- Q2. A man has 532 flower pots. He wants to arrange them in rows such that each row contains 21 flower pots. Find the number…
- Q3. Prove that the product of two consecutive positive integers is divisible by 2.
- Q4. When the positive integers a, b and c are divided by 13, the respective remainders are 9,7 and 10. Show that a+b+c is d…
- Q5. Prove that square of any integer leaves the remainder either 0 or 1 when divided by 4.
- Q6(i). Use Euclid's Division Algorithm to find the Highest Common Factor (HCF) of (i) 340 and 412
- Q6(ii). Use Euclid's Division Algorithm to find the Highest Common Factor (HCF) of (ii) 867 and 255
- Q6(iii). Use Euclid's Division Algorithm to find the Highest Common Factor (HCF) of (iii) 10224 and 9648
- Q6(iv). Use Euclid's Division Algorithm to find the Highest Common Factor (HCF) of (iv) 84, 90 and 120
- Q7. Find the largest number which divides 1230 and 1926 leaving remainder 12 in each case.
- Q8. If d is the Highest Common Factor of 32 and 60, find x and y satisfying d = 32x + 60y.
- Q9. A positive integer when divided by 88 gives the remainder 61. What will be the remainder when the same number is divide…
- Q10. Prove that two consecutive positive integers are always coprime.
Ex 2.2: 9 questions
- Q1. For what values of natural number n, 4ⁿ can end with the digit 6?
- Q2. If m, n are natural numbers, for what values of m, does 2ⁿ × 5m ends in 5?
- Q3. Find the HCF of 252525 and 363636.
- Q4. If 13824 = 2a × 3b then find a and b.
- Q5. If p₁(x₁) × p₂(x₂) × p₃(x₃) × p₄(x₄) = 113400 where p₁, p₂, p₃, p₄ are primes in ascending order and x₁, x₂, x₃, x₄ are…
- Q6. Find the LCM and HCF of 408 and 170 by applying the fundamental theorem of arithmetic.
- Q7. Find the greatest number consisting of 6 digits which is exactly divisible by 24,15,36?
- Q8. What is the smallest number that when divided by three numbers such as 35, 56 and 91 leaves remainder 7 in each case?
- Q9. Find the least number that is divisible by the first ten natural numbers.
Ex 2.3: 14 questions
- Q1(i). Find the least positive value of x such that 71 ≡ x pmod 8
- Q1(ii). Find the least positive value of x such that 78 + x ≡ 3 pmod 5
- Q1(iii). Find the least positive value of x such that 89 ≡ (x + 3) pmod 4
- Q1(iv). Find the least positive value of x such that 96 ≡ x/7 pmod 5
- Q1(v). Find the least positive value of x such that 5x ≡ 4 pmod 6
- Q2. If x is congruent to 13 modulo 17 then 7x - 3 is congruent to which number modulo 17?
- Q3. Solve 5x ≡ 4 pmod 6
- Q4. Solve 3x - 2 ≡ 0 pmod11
- Q5. What is the time 100 hours after 7 a.m.?
- Q6. What is the time 15 hours before 11 p.m.?
- Q7. Today is Tuesday. My uncle will come after 45 days. In which day my uncle will be coming?
- Q8. Prove that 2ⁿ + 6 × 9ⁿ is always divisible by 7 for any positive integer n.
- Q9. Find the remainder when 2⁸¹ is divided by 17.
- Q10. The duration of flight travel from Chennai to London through British Airlines is approximately 11 hours. The airplane b…
Ex 2.4: 13 questions
- Q1(i). Find the next three terms of the following sequence. (i) 8,\ 24,\ 72,\ …
- Q1(ii). Find the next three terms of the following sequence. (ii) 5,\ 1,\ -3,\ …
- Q1(iii). Find the next three terms of the following sequence. (iii) 1/4,\ 2/9,\ 3/16,\ …
- Q2(i). Find the first four terms of the sequences whose n(th) terms are given by (i) aₙ = n³ - 2
- Q2(ii). Find the first four terms of the sequences whose n(th) terms are given by (ii) aₙ = (-1)ⁿ⁺¹ n(n+1)
- Q2(iii). Find the first four terms of the sequences whose n(th) terms are given by (iii) aₙ = 2n² - 6
- Q3(i). Find the n(th) term of the following sequences (i) 2, 5, 10, 17, …
- Q3(ii). Find the n(th) term of the following sequences (ii) 0, 1/2, 2/3, …
- Q3(iii). Find the n(th) term of the following sequences (iii) 3, 8, 13, 18, …
- Q4(i). Find the indicated terms of the sequences whose n(th) terms are given by (i) aₙ = 5n/(n+2); a₆ and a₁₃
- Q4(ii). Find the indicated terms of the sequences whose n(th) terms are given by (ii) aₙ = -(n² - 4); a₄ and a₁₁
- Q5. Find a₈ and a₁₅ whose n(th) term is aₙ = (n²-1)/(n+3),;\ n is even,\ n ∈ ℕ; [2ex] (n²)/(2n+1),;\ n is odd,\ n ∈ ℕ
- Q6. If a₁ = 1, a₂ = 1 and aₙ = 2aₙ₋₁ + aₙ₋₂, n ≥ 3, n ∈ ℕ, then find the first six terms of the sequence.
Ex 2.5: 21 questions
- Q1(i). Check whether the following sequences are in A.P. (i) a - 3,\ a - 5,\ a - 7,...
- Q1(ii). Check whether the following sequences are in A.P. (ii) 1/2, 1/3, 1/4, 1/5,...
- Q1(iii). Check whether the following sequences are in A.P. (iii) 9,\ 13,\ 17,\ 21,\ 25,...
- Q1(iv). Check whether the following sequences are in A.P. (iv) (-1)/3, 0, 1/3, 2/3,...
- Q1(v). Check whether the following sequences are in A.P. (v) 1, -1, 1, -1, 1, -1,...
- Q2(i). First term a and common difference d are given below. Find the corresponding A.P. (i) a = 5, d = 6
- Q2(ii). First term a and common difference d are given below. Find the corresponding A.P. (ii) a = 7, d = -5
- Q2(iii). First term a and common difference d are given below. Find the corresponding A.P. (iii) a = 3/4, d = 1/2
- Q3(i). Find the first term and common difference of the Arithmetic Progressions whose n(th) terms are given below (i) tₙ = -3…
- Q3(ii). Find the first term and common difference of the Arithmetic Progressions whose n(th) terms are given below (ii) tₙ = 4…
- Q4. Find the 19(th) term of an A.P. -11, -15, -19,...
- Q5. Which term of an A.P. 16, 11, 6, 1,... is -54?
- Q6. Find the middle term(s) of an A.P. 9, 15, 21, 27,…,183.
- Q7. If nine times ninth term is equal to the fifteen times fifteenth term, show that six times twenty fourth term is zero.
- Q8. If 3 + k,\ 18 - k,\ 5k + 1 are in A.P. then find k.
- Q9. Find x, y and z, given that the numbers x, 10, y, 24, z are in A.P.
- Q10. In a theatre, there are 20 seats in the front row and 30 rows were allotted. Each successive row contains two additiona…
- Q11. The sum of three consecutive terms that are in A.P. is 27 and their product is 288. Find the three terms.
- Q12. The ratio of 6(th) and 8(th) term of an A.P. is 7:9. Find the ratio of 9(th) term to 13(th) term.
- Q13. In a winter season let us take the temperature of Ooty from Monday to Friday to be in A.P. The sum of temperatures from…
- Q14. Priya earned ₹15,000 in the first month. Thereafter her salary increased by ₹1500 per year. Her expenses are ₹13,000 du…
Ex 2.6: 16 questions
- Q1(i). Find the sum of the following: (i) 3, 7, 11, … up to 40 terms.
- Q1(ii). Find the sum of the following: (ii) 102, 97, 92, … up to 27 terms.
- Q1(iii). Find the sum of the following: (iii) 6 + 13 + 20 + … + 97
- Q2. How many consecutive odd integers beginning with 5 will sum to 480?
- Q3. Find the sum of first 28 terms of an A.P. whose n(th) term is 4n - 3.
- Q4. The sum of first n terms of a certain series is given as 2n² - 3n. Show that the series is an A.P.
- Q5. The 104(th) term and 4(th) term of an A.P. are 125 and 0. Find the sum of first 35 terms.
- Q6. Find the sum of all odd positive integers less than 450.
- Q7. Find the sum of all natural numbers between 602 and 902 which are not divisible by 4.
- Q8(i). Raghu wish to buy a laptop. He can buy it by paying ₹40,000 cash or by giving it in 10 installments as ₹4800 in the fir…
- Q8(ii). Raghu wish to buy a laptop. He can buy it by paying ₹40,000 cash or by giving it in 10 installments as ₹4800 in the fir…
- Q9. A man repays a loan of ₹65,000 by paying ₹400 in the first month and then increasing the payment by ₹300 every month. H…
- Q10(i). A brick staircase has a total of 30 steps. The bottom step requires 100 bricks. Each successive step requires two brick…
- Q10(ii). A brick staircase has a total of 30 steps. The bottom step requires 100 bricks. Each successive step requires two brick…
- Q11. If S₁, S₂, S₃, …, Sm are the sums of n terms of m A.P.'s whose first terms are 1, 2, 3, …, m and whose common differenc…
- Q12. Find the sum [ (a-b)/(a+b) + (3a-2b)/(a+b) + (5a-3b)/(a+b) + … to 12 terms ].
Ex 2.7: 21 questions
- Q1(i). Which of the following sequences are in G.P.? (i) 3, 9, 27, 81,…
- Q1(ii). Which of the following sequences are in G.P.? (ii) 4, 44, 444, 4444,...
- Q1(iii). Which of the following sequences are in G.P.? (iii) 0.5, 0.05, 0.005,…
- Q1(iv). Which of the following sequences are in G.P.? (iv) 1/3, 1/6, 1/12,...
- Q1(v). Which of the following sequences are in G.P.? (v) 1, -5, 25, -125,…
- Q1(vi). Which of the following sequences are in G.P.? (vi) 120, 60, 30, 18,…
- Q1(vii). Which of the following sequences are in G.P.? (vii) 16, 4, 1, 1/4,...
- Q2(i). Write the first three terms of the G.P. whose first term and the common ratio are given below. (i) a = 6, r = 3
- Q2(ii). Write the first three terms of the G.P. whose first term and the common ratio are given below. (ii) a = √2, r = √2
- Q2(iii). Write the first three terms of the G.P. whose first term and the common ratio are given below. (iii) a = 1000, r = 2/5
- Q3. In a G.P. 729, 243, 81,… find t₇.
- Q4. Find x so that x + 6, x + 12 and x + 15 are consecutive terms of a Geometric Progression.
- Q5(i). Find the number of terms in the following G.P. (i) 4, 8, 16,…, 8192?
- Q5(ii). Find the number of terms in the following G.P. (ii) 1/3, 1/9, 1/27,..., 1/2187
- Q6. In a G.P. the 9(th) term is 32805 and 6(th) term is 1215. Find the 12(th) term.
- Q7. Find the 10(th) term of a G.P. whose 8(th) term is 768 and the common ratio is 2.
- Q8. If a, b, c are in A.P. then show that 3a, 3b, 3c are in G.P.
- Q9. In a G.P. the product of three consecutive terms is 27 and the sum of the product of two terms taken at a time is 57/2.…
- Q10. A man joined a company as Assistant Manager. The company gave him a starting salary of ₹60,000 and agreed to increase h…
- Q11. Sivamani is attending an interview for a job and the company gave two offers to him. Offer A: ₹20,000 to start with fol…
- Q12. If a, b, c are three consecutive terms of an A.P. and x, y, z are three consecutive terms of a G.P. then prove that x(b…
Ex 2.8: 13 questions
- Q1(i). Find the sum of first n terms of the G.P. (i) 5, -3, 9/5, -27/25, …
- Q1(ii). Find the sum of first n terms of the G.P. (ii) 256, 64, 16, …
- Q2. Find the sum of first six terms of the G.P. 5, 15, 45, …
- Q3. Find the first term of the G.P. whose common ratio 5 and whose sum to first 6 terms is 46872.
- Q4(i). Find the sum to infinity of (i) 9 + 3 + 1 + …
- Q4(ii). Find the sum to infinity of (ii) 21 + 14 + 28/3 + …
- Q5. If the first term of an infinite G.P. is 8 and its sum to infinity is 32/3 then find the common ratio.
- Q6(i). Find the sum to n terms of the series (i) 0.4 + 0.44 + 0.444 + … to n terms
- Q6(ii). Find the sum to n terms of the series (ii) 3 + 33 + 333 + … to n terms
- Q7. Find the sum of the Geometric series 3 + 6 + 12 + … + 1536.
- Q8. Kumar writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different pe…
- Q9. Find the rational form of the number 0.123.
- Q10. If Sₙ = (x+y) + (x² + xy + y²) + (x³ + x²y + xy² + y³) + … n terms then prove that (x-y)Sₙ = [(x²(xⁿ - 1))/(x-1) - (y²(…
Ex 2.9: 14 questions
- Q1(i). Find the sum of the following series: (i) 1 + 2 + 3 + … + 60
- Q1(ii). Find the sum of the following series: (ii) 3 + 6 + 9 + … + 96
- Q1(iii). Find the sum of the following series: (iii) 51 + 52 + 53 + … + 92
- Q1(iv). Find the sum of the following series: (iv) 1 + 4 + 9 + 16 + … + 225
- Q1(v). Find the sum of the following series: (v) 6² + 7² + 8² + … + 21²
- Q1(vi). Find the sum of the following series: (vi) 10³ + 11³ + 12³ + … + 20³
- Q1(vii). Find the sum of the following series: (vii) 1 + 3 + 5 + … + 71
- Q2. If 1 + 2 + 3 + … + k = 325, then find 1³ + 2³ + 3³ + … + k³.
- Q3. If 1³ + 2³ + 3³ + … + k³ = 44100 then find 1 + 2 + 3 + … + k.
- Q4. How many terms of the series 1³ + 2³ + 3³ + … should be taken to get the sum 14400?
- Q5. The sum of the cubes of the first n natural numbers is 2025, then find the value of n.
- Q6. Rekha has 15 square colour papers of sizes 10 cm, 11 cm, 12 cm,…, 24 cm. How much area can be decorated with these colo…
- Q7(i). Find the sum of the series (2³ - 1³) + (4³ - 3³) + (6³ - 5³) + … to (i) n terms
- Q7(ii). Find the sum of the series (2³ - 1³) + (4³ - 3³) + (6³ - 5³) + … to (ii) 8 terms
Ex 2.10: 15 questions
- Q1. Euclid's division lemma states that for positive integers a and b, there exist unique integers q and r such that a = bq…
- Q2. Using Euclid's division lemma, if the cube of any positive integer is divided by 9 then the possible remainders are
- Q3. If the HCF of 65 and 117 is expressible in the form of 65m - 117, then the value of m is
- Q4. The sum of the exponents of the prime factors in the prime factorization of 1729 is
- Q5. The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is
- Q6. 7(4k) ≡ ____ (mod 100)
- Q7. Given F₁ = 1, F₂ = 3 and Fₙ = Fₙ₋₁ + Fₙ₋₂ then F₅ is
- Q8. The first term of an arithmetic progression is unity and the common difference is 4. Which of the following will be a t…
- Q9. If 6 times of 6(th) term of an A.P. is equal to 7 times the 7(th) term, then the 13(th) term of the A.P. is
- Q10. An A.P. consists of 31 terms. If its 16(th) term is m, then the sum of all the terms of this A.P. is
- Q11. In an A.P., the first term is 1 and the common difference is 4. How many terms of the A.P. must be taken for their sum…
- Q12. If A = 2⁶⁵ and B = 2⁶⁴ + 2⁶³ + 2⁶² + … + 2⁰ which of the following is true?
- Q13. The next term of the sequence 3/16, 1/8, 1/12, 1/18, … is
- Q14. If the sequence t₁, t₂, t₃, … are in A.P. then the sequence t₆, t₁₂, t₁₈, … is
- Q15. The value of (1³ + 2³ + 3³ + … + 15³) - (1 + 2 + 3 + … + 15) is
Unit Exercise 2: 12 questions
- Q1. Prove that n² - n divisible by 2 for every positive integer n.
- Q2(i). A milk man has 175 litres of cow's milk and 105 litres of buffalow's milk. He wishes to sell the milk by filling the tw…
- Q2(ii). A milk man has 175 litres of cow's milk and 105 litres of buffalow's milk. He wishes to sell the milk by filling the tw…
- Q2(iii). A milk man has 175 litres of cow's milk and 105 litres of buffalow's milk. He wishes to sell the milk by filling the tw…
- Q3. When the positive integers a, b and c are divided by 13 the respective remainders are 9, 7 and 10. Find the remainder w…
- Q4. Show that 107 is of the form 4q + 3 for any integer q.
- Q5. If (m + 1)(th) term of an A.P. is twice the (n + 1)(th) term, then prove that (3m+1)(th) term is twice the (m + n + 1)(…
- Q6. Find the 12(th) term from the last term of the A. P -2, -4, -6, … -100.
- Q7. Two A.P.'s have the same common difference. The first term of one A.P. is 2 and that of the other is 7. Show that the d…
- Q8. A man saved ₹16500 in ten years. In each year after the first he saved ₹100 more than he did in the preceding year. How…
- Q9. Find the G.P. in which the 2(nd) term is √6 and the 6(th) term is 9√6.
- Q10. The value of a motor cycle depreciates at the rate of 15\% per year. What will be the value of the motor cycle 3 year h…
Chapter 3: Algebra
- Ex 3.1
- Ex 3.2
- Ex 3.3
- Ex 3.4
- Ex 3.5
- Ex 3.6
- Ex 3.7
- Ex 3.8
- Ex 3.9
- Ex 3.10
- Ex 3.11
- Ex 3.12
- Ex 3.13
- Ex 3.14
- Ex 3.15
- Ex 3.16
- Ex 3.17
- Ex 3.18
- Ex 3.19
- Ex 3.20
- Unit Exercise 3
Ex 3.1: 9 questions
- Q1(i). Solve the following system of linear equations in three variables (i) x + y + z = 5; 2x - y + z = 9; x - 2y + 3z = 16
- Q1(ii). Solve the following system of linear equations in three variables (ii) 1/x - 2/y + 4 = 0; 1/y - 1/z + 1 = 0; 2/z + 3/x…
- Q1(iii). Solve the following system of linear equations in three variables (iii) x + 20 = 3y/2 + 10 = 2z + 5 = 110 - (y + z)
- Q2(i). Discuss the nature of solutions of the following system of equations (i) x + 2y - z = 6; -3x - 2y + 5z = -12; x - 2z = 3
- Q2(ii). Discuss the nature of solutions of the following system of equations (ii) 2y + z = 3(-x + 1); -x + 3y - z = -4; 3x + 2y…
- Q2(iii). Discuss the nature of solutions of the following system of equations (iii) (y+z)/4 = (z+x)/3 = (x+y)/2; x + y + z = 27
- Q3. Vani, her father and her grand father have an average age of 53. One-half of her grand father's age plus one-third of h…
- Q4. The sum of the digits of a three-digit number is 11. If the digits are reversed, the new number is 46 more than five ti…
- Q5. There are 12 pieces of five, ten and twenty rupee currencies whose total value is ₹105. When first 2 sorts are intercha…
Ex 3.2: 10 questions
- Q1(i). Find the GCD of the given polynomials: x⁴ + 3x³ - x - 3, x³ + x² - 5x + 3
- Q1(ii). Find the GCD of the given polynomials: x⁴ - 1, x³ - 11x² + x - 11
- Q1(iii). Find the GCD of the given polynomials: 3x⁴ + 6x³ - 12x² - 24x, 4x⁴ + 14x³ + 8x² - 8x
- Q1(iv). Find the GCD of the given polynomials: 3x³ + 3x² + 3x + 3, 6x³ + 12x² + 6x + 12
- Q2(i). Find the LCM of the given expressions: 4x²y, 8x³y²
- Q2(ii). Find the LCM of the given expressions: 9a³b², 12a²b²c
- Q2(iii). Find the LCM of the given expressions: 16m, 12m²n², 8n²
- Q2(iv). Find the LCM of the given expressions: p² - 3p + 2, p² - 4
- Q2(v). Find the LCM of the given expressions: 2x² - 5x - 3, 4x² - 36
- Q2(vi). Find the LCM of the given expressions: (2x² - 3xy)², (4x - 6y)³, 8x³ - 27y³
Ex 3.3: 9 questions
- Q1(i). Find the LCM and GCD for the following and verify that f(x) × g(x) = LCM × GCD (i) 21x²y,\ 35xy²
- Q1(ii). Find the LCM and GCD for the following and verify that f(x) × g(x) = LCM × GCD (ii) (x³-1)(x+1),\ (x³+1)
- Q1(iii). Find the LCM and GCD for the following and verify that f(x) × g(x) = LCM × GCD (iii) (x²y+xy²),\ (x²+xy)
- Q2(i). Find the LCM of each pair of the following polynomials (i) a²+4a-12,\ a²-5a+6 whose GCD is a-2
- Q2(ii). Find the LCM of each pair of the following polynomials (ii) x⁴-27a³x,\ (x-3a)² whose GCD is (x-3a)
- Q3(i). Find the GCD of each pair of the following polynomials (i) 12(x⁴-x³),\ 8(x⁴-3x³+2x²) whose LCM is 24x³(x-1)(x-2)
- Q3(ii). Find the GCD of each pair of the following polynomials (ii) (x³+y³),\ (x⁴+x²y²+y⁴) whose LCM is (x³+y³)(x²+xy+y²)
- Q4(i). Given the LCM and GCD of the two polynomials p(x) and q(x) find the unknown polynomial in the following table (i) LCM =…
- Q4(ii). Given the LCM and GCD of the two polynomials p(x) and q(x) find the unknown polynomial in the following table (ii) LCM…
Ex 3.4: 8 questions
- Q1(i). Reduce each of the following rational expressions to its lowest form. (i) (x²-1)/(x²+x)
- Q1(ii). Reduce each of the following rational expressions to its lowest form. (ii) (x²-11x+18)/(x²-4x+4)
- Q1(iii). Reduce each of the following rational expressions to its lowest form. (iii) (9x²+81x)/(x³+8x²-9x)
- Q1(iv). Reduce each of the following rational expressions to its lowest form. (iv) (p²-3p-40)/(2p³-24p²+64p)
- Q2(i). Find the excluded values, if any of the following expressions. (i) y/(y²-25)
- Q2(ii). Find the excluded values, if any of the following expressions. (ii) t/(t²-5t+6)
- Q2(iii). Find the excluded values, if any of the following expressions. (iii) (x²+6x+8)/(x²+x-2)
- Q2(iv). Find the excluded values, if any of the following expressions. (iv) (x³-27)/(x³+x²-6x)
Ex 3.5: 11 questions
- Q1(i). Simplify (4x²y)/(2z²) × (6xz³)/(20y⁴)
- Q1(ii). Simplify (p² - 10p + 21)/(p - 7) × (p² + p - 12)/((p-3)²)
- Q1(iii). Simplify (5t³)/(4t - 8) × (6t - 12)/10t
- Q2(i). Simplify (x+4)/(3x+4y) × (9x² - 16y²)/(2x² + 3x - 20)
- Q2(ii). Simplify (x³ - y³)/(3x² + 9xy + 6y²) × (x² + 2xy + y²)/(x² - y²)
- Q3(i). Simplify (2a² + 5a + 3)/(2a² + 7a + 6) ÷ (a² + 6a + 5)/(-5a² - 35a - 50)
- Q3(ii). Simplify (b² + 3b - 28)/(b² + 4b + 4) ÷ (b² - 49)/(b² - 5b - 14)
- Q3(iii). Simplify (x+2)/4y ÷ (x² - x - 6)/(12y²)
- Q3(iv). Simplify (12t² - 22t + 8)/3t ÷ (3t² + 2t - 8)/(2t² + 4t)
- Q4. If x = (a² + 3a - 4)/(3a² - 3) and y = (a² + 2a - 8)/(2a² - 2a - 4) find the value of x² y⁻².
- Q5. If a polynomial p(x) = x² - 5x - 14 is divided by another polynomial q(x) we get (x-7)/(x+2), find q(x).
Ex 3.6: 11 questions
- Q1(i). Simplify (i) (x(x+1))/(x-2)+(x(1-x))/(x-2)
- Q1(ii). Simplify (ii) (x+2)/(x+3)+(x-1)/(x-2)
- Q1(iii). Simplify (iii) (x³)/(x-y)+(y³)/(y-x)
- Q2(i). Simplify (i) ((2x+1)(x-2))/(x-4)-((2x²-5x+2))/(x-4)
- Q2(ii). Simplify (ii) 4x/(x²-1)-(x+1)/(x-1)
- Q3. Subtract 1/(x²+2) from (2x³+x²+3)/((x²+2)²)
- Q4. Which rational expression should be subtracted from (x²+6x+8)/(x³+8) to get 3/(x²-2x+4)
- Q5. If A=(2x+1)/(2x-1), B=(2x-1)/(2x+1) find 1/(A-B)-2B/(A²-B²)
- Q6. If A=x/(x+1), B=1/(x+1), prove that ((A+B)²+(A-B)²)/(A÷ B)=(2(x²+1))/(x(x+1)²)
- Q7. Pari needs 4 hours to complete a work. His friend Yuvan needs 6 hours to complete the same work. How long will it take…
- Q8. Iniya bought 50 kg of fruits consisting of apples and bananas. She paid twice as much per kg for the apple as she did f…
Ex 3.7: 7 questions
- Q1(i). Find the square root of the following rational expressions. (i) (400x⁴y¹²z¹⁶)/(100x⁸y⁴z⁴)
- Q1(ii). Find the square root of the following rational expressions. (ii) (7x²+2√14x+2)/(x²-1/2x+1/16)
- Q1(iii). Find the square root of the following rational expressions. (iii) (121(a+b)⁸(x+y)⁸(b-c)⁸)/(81(b-c)⁴(a-b)¹²(b-c)⁴)
- Q2(i). Find the square root of the following (i) 4x²+20x+25
- Q2(ii). Find the square root of the following (ii) 9x²-24xy+30xz-40yz+25z²+16y²
- Q2(iii). Find the square root of the following (iii) (4x²-9x+2)(7x²-13x-2)(28x²-3x-1)
- Q2(iv). Find the square root of the following (iv) (2x²+17/6x+1)(3/2x²+4x+2)(4/3x²+11/3x+2)
Ex 3.8: 8 questions
- Q1(i). Find the square root of the following polynomials by division method: (i) x⁴ - 12x³ + 42x² - 36x + 9
- Q1(ii). Find the square root of the following polynomials by division method: (ii) 37x² - 28x³ + 4x⁴ + 42x + 9
- Q1(iii). Find the square root of the following polynomials by division method: (iii) 16x⁴ + 8x² + 1
- Q1(iv). Find the square root of the following polynomials by division method: (iv) 121x⁴ - 198x³ - 183x² + 216x + 144
- Q2(i). Find the values of a and b if the following polynomials are perfect squares: (i) 4x⁴ - 12x³ + 37x² + bx + a
- Q2(ii). Find the values of a and b if the following polynomials are perfect squares: (ii) ax⁴ + bx³ + 361x² + 220x + 100
- Q3(i). Find the values of m and n if the following polynomials are perfect sqaures: (i) 36x⁴ - 60x³ + 61x² - mx + n
- Q3(ii). Find the values of m and n if the following polynomials are perfect sqaures: (ii) x⁴ - 8x³ + mx² + nx + 16
Ex 3.9: 8 questions
- Q1(i). Determine the quadratic equations, whose sum and product of roots are (i) -9,\ 20
- Q1(ii). Determine the quadratic equations, whose sum and product of roots are (ii) 5/3,\ 4
- Q1(iii). Determine the quadratic equations, whose sum and product of roots are (iii) (-3)/2,\ -1
- Q1(iv). Determine the quadratic equations, whose sum and product of roots are (iv) -(2-a)²,\ (a+5)²
- Q2(i). Find the sum and product of the roots for each of the following quadratic equations (i) x² + 3x - 28 = 0
- Q2(ii). Find the sum and product of the roots for each of the following quadratic equations (ii) x² + 3x = 0
- Q2(iii). Find the sum and product of the roots for each of the following quadratic equations (iii) 3 + 1/a = 10/(a²)
- Q2(iv). Find the sum and product of the roots for each of the following quadratic equations (iv) 3y² - y - 4 = 0
Ex 3.10: 6 questions
- Q1(i). Solve the following quadratic equations by factorization method: 4x² - 7x - 2 = 0
- Q1(ii). Solve the following quadratic equations by factorization method: 3(p² - 6) = p(p + 5)
- Q1(iii). Solve the following quadratic equations by factorization method: √(a(a - 7)) = 3√2
- Q1(iv). Solve the following quadratic equations by factorization method: √2x² + 7x + 5√2 = 0
- Q1(v). Solve the following quadratic equations by factorization method: 2x² - x + 1/8 = 0
- Q2. The number of volleyball games that must be scheduled in a league with n teams is given by G(n) = (n² - n)/2 where each…
Ex 3.11: 7 questions
- Q1(i). Solve the following quadratic equations by completing the square method (i) 9x² - 12x + 4 = 0
- Q1(ii). Solve the following quadratic equations by completing the square method (ii) (5x + 7)/(x - 1) = 3x + 2
- Q2(i). Solve the following quadratic equations by formula method (i) 2x² - 5x + 2 = 0
- Q2(ii). Solve the following quadratic equations by formula method (ii) √2f² - 6f + 3√2 = 0
- Q2(iii). Solve the following quadratic equations by formula method (iii) 3y² - 20y - 23 = 0
- Q2(iv). Solve the following quadratic equations by formula method (iv) 36y² - 12ay + (a² - b²) = 0
- Q3. A ball rolls down a slope and travels a distance d = t² - 0.75t feet in t seconds. Find the time when the distance trav…
Ex 3.12: 9 questions
- Q1. If the difference between a number and its reciprocal is 24/5, find the number.
- Q2. A garden measuring 12m by 16m is to have a pedestrian pathway that is 'w' meters wide installed all the way around so t…
- Q3. A bus covers a distance of 90 km at a uniform speed. Had the speed been 15 km/hour more it would have taken 30 minutes…
- Q4. A girl is twice as old as her sister. Five years hence, the product of their ages (in years) will be 375. Find their pr…
- Q5. A pole has to be erected at a point on the boundary of a circular ground of diameter 20 m in such a way that the differ…
- Q6. From a group of 2x² black bees, square root of half of the group went to a tree. Again eight-ninth of the bees went to…
- Q7. Music is been played in two opposite galleries with certain group of people. In the first gallery a group of 4 singers…
- Q8. There is a square field whose side is 10 m. A square flower bed is prepared in its centre leaving a gravel path all rou…
- Q9. The hypotenuse of a right angled triangle is 25 cm and its perimeter 56 cm. Find the length of the smallest side.
Ex 3.13: 10 questions
- Q1(i). Determine the nature of the roots for the following quadratic equations: (i) 15x² + 11x + 2 = 0
- Q1(ii). Determine the nature of the roots for the following quadratic equations: (ii) x² - x - 1 = 0
- Q1(iii). Determine the nature of the roots for the following quadratic equations: (iii) √2t² - 3t + 3√2 = 0
- Q1(iv). Determine the nature of the roots for the following quadratic equations: (iv) 9y² - 6√2y + 2 = 0
- Q1(v). Determine the nature of the roots for the following quadratic equations: (v) 9a²b²x² - 24abcdx + 16c²d² = 0, a ≠ 0, b ≠…
- Q2(i). Find the value(s) of 'k' for which the roots of the following equations are real and equal. (i) (5k - 6)x² + 2kx + 1 = 0
- Q2(ii). Find the value(s) of 'k' for which the roots of the following equations are real and equal. (ii) kx² + (6k + 2)x + 16 =…
- Q3. If the roots of (a - b)x² + (b - c)x + (c - a) = 0 are real and equal, then prove that b, a, c are in arithmetic progre…
- Q4. If a, b are real then show that the roots of the equation (a - b)x² - 6(a + b)x - 9(a - b) = 0 are real and unequal.
- Q5. If the roots of the equation (c² - ab)x² - 2(a² - bc)x + b² - ac = 0 are real and equal prove that either a = 0 (or) a³…
Ex 3.14: 13 questions
- Q1(i). Write each of the following expression in terms of α + β and αβ. (i) α/3β + β/3α
- Q1(ii). Write each of the following expression in terms of α + β and αβ. (ii) 1/(α²β) + 1/(β²α)
- Q1(iii). Write each of the following expression in terms of α + β and αβ. (iii) (3α - 1)(3β - 1)
- Q1(iv). Write each of the following expression in terms of α + β and αβ. (iv) (α + 3)/β + (β + 3)/α
- Q2(i). The roots of the equation 2x² - 7x + 5 = 0 are α and β. Without solving for the roots, find (i) 1/α + 1/β
- Q2(ii). The roots of the equation 2x² - 7x + 5 = 0 are α and β. Without solving for the roots, find (ii) α/β + β/α
- Q2(iii). The roots of the equation 2x² - 7x + 5 = 0 are α and β. Without solving for the roots, find (iii) (α + 2)/(β + 2) + (β…
- Q3(i). The roots of the equation x² + 6x - 4 = 0 are α, β. Find the quadratic equation whose roots are (i) α² and β²
- Q3(ii). The roots of the equation x² + 6x - 4 = 0 are α, β. Find the quadratic equation whose roots are (ii) 2/α and 2/β
- Q3(iii). The roots of the equation x² + 6x - 4 = 0 are α, β. Find the quadratic equation whose roots are (iii) α²β and β²α
- Q4. If α, β are the roots of 7x² + ax + 2 = 0 and if β - α = (-13)/7. Find the values of a.
- Q5. If one root of the equation 2y² - ay + 64 = 0 is twice the other then find the values of a.
- Q6. If one root of the equation 3x² + kx + 81 = 0 (having real roots) is the square of the other then find k.
Ex 3.15: 12 questions
- Q1(i). A garment shop announces a flat 50% discount on every purchase of items for their customers. Draw the graph for the rel…
- Q1(ii). A garment shop announces a flat 50% discount on every purchase of items for their customers. Draw the graph for the rel…
- Q2(i). Draw the graph of xy = 24, x, y 0. Using the graph find, (i) y when x = 3.
- Q2(ii). Draw the graph of xy = 24, x, y 0. Using the graph find, (ii) x when y = 6.
- Q3(i). Graph the following linear function y = 1/2x. Identify the constant of variation and verify it with the graph. Also (i)…
- Q3(ii). Graph the following linear function y = 1/2x. Identify the constant of variation and verify it with the graph. Also (ii…
- Q4(i). The following table shows the data about the number of pipes and the time taken to fill the same tank. Draw the graph f…
- Q4(ii). The following table shows the data about the number of pipes and the time taken to fill the same tank. Draw the graph f…
- Q5(i). A school announces that for a certain competitions, the cash price will be distributed for all the participants equally…
- Q5(ii). A school announces that for a certain competitions, the cash price will be distributed for all the participants equally…
- Q6(i). A two wheeler parking zone near bus stand charges as below. Check if the amount charged are in direct variation or in i…
- Q6(ii). A two wheeler parking zone near bus stand charges as below. Check if the amount charged are in direct variation or in i…
Ex 3.16: 13 questions
- Q1(i). Graph the following quadratic equations and state their nature of solutions. (i) x² - 9x + 20 = 0
- Q1(ii). Graph the following quadratic equations and state their nature of solutions. (ii) x² - 4x + 4 = 0
- Q1(iii). Graph the following quadratic equations and state their nature of solutions. (iii) x² + x + 7 = 0
- Q1(iv). Graph the following quadratic equations and state their nature of solutions. (iv) x² - 9 = 0
- Q1(v). Graph the following quadratic equations and state their nature of solutions. (v) x² - 6x + 9 = 0
- Q1(vi). Graph the following quadratic equations and state their nature of solutions. (vi) (2x - 3)(x + 2) = 0
- Q2. Draw the graph of y = x² - 4 and hence solve x² - x - 12 = 0
- Q3. Draw the graph of y = x² + x and hence solve x² + 1 = 0
- Q4. Draw the graph of y = x² + 3x + 2 and use it to solve x² + 2x + 1 = 0
- Q5. Draw the graph of y = x² + 3x - 4 and hence use it to solve x² + 3x - 4 = 0
- Q6. Draw the graph of y = x² - 5x - 6 and hence solve x² - 5x - 14 = 0
- Q7. Draw the graph of y = 2x² - 3x - 5 and hence solve 2x² - 4x - 6 = 0
- Q8. Draw the graph of y = (x - 1)(x + 3) and hence solve x² - x - 6 = 0
Ex 3.17: 12 questions
- Q1(i). In the matrix A = 8, 9, 4, 3; -1, √7, (√3)/2, 5; 1, 4, 3, 0; 6, 8, -11, 1, write (i) The number of elements
- Q1(ii). In the matrix A = 8, 9, 4, 3; -1, √7, (√3)/2, 5; 1, 4, 3, 0; 6, 8, -11, 1, write (ii) The order of the matrix
- Q1(iii). In the matrix A = 8, 9, 4, 3; -1, √7, (√3)/2, 5; 1, 4, 3, 0; 6, 8, -11, 1, write (iii) Write the elements a₂₂, a₂₃, a₂₄…
- Q2. If a matrix has 18 elements, what are the possible orders it can have? What if it has 6 elements?
- Q3(i). Construct a 3 × 3 matrix whose elements are given by (i) a(ij) = ∣i - 2j∣
- Q3(ii). Construct a 3 × 3 matrix whose elements are given by (ii) a(ij) = ((i+j)³)/3
- Q4. If A = 5, 4, 3; 1, -7, 9; 3, 8, 2 then find the transpose of A.
- Q5. If A = √7, -3; -√5, 2; √3, -5 then find the transpose of -A.
- Q6. If A = 5, 2, 2; -√17, 0.7, 5/2; 8, 3, 1 then verify (AT)T = A
- Q7(i). Find the values of x, y and z from the following equations (i) 12, 3; x, 5/3 = y, z; 3, 5
- Q7(ii). Find the values of x, y and z from the following equations (ii) x+y, 2; 5+z, xy = 6, 2; 5, 8
- Q7(iii). Find the values of x, y and z from the following equations (iii) x+y+z; x+z; y+z = 9; 5; 7
Ex 3.18: 11 questions
- Q1(i). If A= 1, 9; 3, 4; 8, -3, B= 5, 7; 3, 3; 1, 0 then verify that (i) A+B=B+A
- Q1(ii). If A= 1, 9; 3, 4; 8, -3, B= 5, 7; 3, 3; 1, 0 then verify that (ii) A+(-A)=(-A)+A=O.
- Q2. If A= 4, 3, 1; 2, 3, -8; 1, 0, -4, B= 2, 3, 4; 1, 9, 2; -7, 1, -1 and C= 8, 3, 4; 1, -2, 3; 2, 4, -1 then verify that A…
- Q3. Find X and Y if X+Y= 7, 0; 3, 5 and X-Y= 3, 0; 0, 4
- Q4(i). If A= 0, 4, 9; 8, 3, 7, B= 7, 3, 8; 1, 4, 9 find the value of (i) B-5A
- Q4(ii). If A= 0, 4, 9; 8, 3, 7, B= 7, 3, 8; 1, 4, 9 find the value of (ii) 3A-9B
- Q5(i). Find the values of x,\ y,\ z if (i) x-3, 3x-z; x+y+7, x+y+z = 1, 0; 1, 6
- Q5(ii). Find the values of x,\ y,\ z if (ii) (x y-z z+3)+(y 4 3)=(4 8 16)
- Q6. Find x and y if x 4; -3 +y -2; 3 = 4; 6
- Q7. Find the non-zero values of x satisfying the matrix equation x 2x, 2; 3, x +2 8, 5x; 4, 4x =2 x²+8, 24; 10, 6x
- Q8. Solve for x,\ y: x²; y² +2 -2x; -y = 5; 8
Ex 3.19: 19 questions
- Q1(i). Find the order of the product matrix AB if (i) Order of A is 3×3 and order of B is 3×3. · (i) · (ii) · (iii) · (iv) · (…
- Q1(ii). Find the order of the product matrix AB if (ii) Order of A is 4×3 and order of B is 3×2. · (i) · (ii) · (iii) · (iv) ·…
- Q1(iii). Find the order of the product matrix AB if (iii) Order of A is 4×2 and order of B is 2×2. · (i) · (ii) · (iii) · (iv) ·…
- Q1(iv). Find the order of the product matrix AB if (iv) Order of A is 4×5 and order of B is 5×1. · (i) · (ii) · (iii) · (iv) ·…
- Q1(v). Find the order of the product matrix AB if (v) Order of A is 1×1 and order of B is 1×3. · (i) · (ii) · (iii) · (iv) · (…
- Q2. If A is of order p × q and B is of order q × r what is the order of AB and BA?
- Q3. A has 'a' rows and 'a + 3' columns. B has 'b' rows and '17-b' columns, and if both products AB and BA exist, find a,\ b?
- Q4. If A = 2, 5; 4, 3, B = 1, -3; 2, 5 find AB,\ BA and verify AB = BA?
- Q5. Given that A = 1, 3; 5, -1, B = 1, -1, 2; 3, 5, 2, C = 1, 3, 2; -4, 1, 3 verify that A(B + C) = AB + AC.
- Q6. Show that the matrices A = 1, 2; 3, 1, B = 1, -2; -3, 1 satisfy commutative property AB = BA
- Q7(i). Let A = 1, 2; 1, 3, B = 4, 0; 1, 5, C = 2, 0; 1, 2 Show that (i) A(BC) = (AB)C
- Q7(ii). Let A = 1, 2; 1, 3, B = 4, 0; 1, 5, C = 2, 0; 1, 2 Show that (ii) (A - B)C = AC - BC
- Q7(iii). Let A = 1, 2; 1, 3, B = 4, 0; 1, 5, C = 2, 0; 1, 2 Show that (iii) (A - B)T = AT - BT
- Q8. If A = cosθ, 0; 0, cosθ, B = sinθ, 0; 0, sinθ then show that A² + B² = I.
- Q9. If A = cosθ, sinθ; -sinθ, cosθ prove that AAT = I.
- Q10. Verify that A² = I when A = 5, -4; 6, -5
- Q11. If A = a, b; c, d and I = 1, 0; 0, 1 show that A² - (a + d)A = (bc - ad)I₂
- Q12. If A = 5, 2, 9; 1, 2, 8, B = 1, 7; 1, 2; 5, -1 verify that (AB)T = BT AT
- Q13. If A = 3, 1; -1, 2 show that A² - 5A + 7I₂ = 0
Ex 3.20: 20 questions
- Q1. A system of three linear equations in three variables is inconsistent if their planes
- Q2. The solution of the system x + y - 3z = -6, -7y + 7z = 7, 3z = 9 is
- Q3. If (x - 6) is the HCF of x² - 2x - 24 and x² - kx - 6 then the value of k is
- Q4. (3y - 3)/y ÷ (7y - 7)/(3y²) is
- Q5. y² + 1/(y²) is not equal to
- Q6. x/(x² - 25) - 8/(x² + 6x + 5) gives
- Q7. The square root of (256 x⁸ y⁴ z¹⁰)/(25 x⁶ y⁶ z⁶) is equal to
- Q8. Which of the following should be added to make x⁴ + 64 a perfect square
- Q9. The solution of (2x - 1)² = 9 is equal to
- Q10. The values of a and b if 4x⁴ - 24x³ + 76x² + ax + b is a perfect square are
- Q11. If the roots of the equation q²x² + p²x + r² = 0 are the squares of the roots of the equation qx² + px + r = 0, then q,…
- Q12. Graph of a linear equation is a ____
- Q13. The number of points of intersection of the quadratic polynomial x² + 4x + 4 with the X axis is
- Q14. For the given matrix A = 1, 3, 5, 7; 2, 4, 6, 8; 9, 11, 13, 15 the order of the matrix AT is
- Q15. If A is a 2 × 3 matrix and B is a 3 × 4 matrix, how many columns does AB have
- Q16. If number of columns and rows are not equal in a matrix then it is said to be a
- Q17. Transpose of a column matrix is
- Q18. Find the matrix X if 2X + 1, 3; 5, 7 = 5, 7; 9, 5
- Q19. Which of the following can be calculated from the given matrices A = 1, 2; 3, 4; 5, 6, B = 1, 2, 3; 4, 5, 6; 7, 8, 9, (…
- Q20. If A = 1, 2, 3; 3, 2, 1, B = 1, 0; 2, -1; 0, 2 and C = 0, 1; -2, 5. Which of the following statements are correct? (i)…
Unit Exercise 3: 23 questions
- Q1. Solve 1/3(x + y - 5) = y - z = 2x - 11 = 9 - (x + 2z)
- Q2. One hundred and fifty students are admitted to a school. They are distrbuted over three sections A, B and C. If 6 stude…
- Q3. In a three-digit number, when the tens and the hundreds digit are interchanged the new number is 54 more than three tim…
- Q4. Find the least common multiple of xy(k² + 1) + k(x² + y²) and xy(k² - 1) + k(x² - y²)
- Q5. Find the GCD of the following by division algorithm 2x⁴ + 13x³ + 27x² + 23x + 7, x³ + 3x² + 3x + 1, x² + 2x + 1
- Q6(i). Reduce the given Rational expressions to its lowest form (i) (x(3a) - 8)/(x(2a) + 2xa + 4)
- Q6(ii). Reduce the given Rational expressions to its lowest form (ii) (10x³ - 25x² + 4x - 10)/(-4 - 10x²)
- Q7. Simplify (1/p + 1/(q+r))/(1/p - 1/(q+r)) × (1 + (q² + r² - p²)/2qr)
- Q8. Arul, Madan and Ram working together can clean a store in 6 hours. Working alone, Madan takes twice as long to clean th…
- Q9. Find the square root of 289x⁴ - 612x³ + 970x² - 684x + 361.
- Q10. Solve √(y+1) + √(2y-5) = 3
- Q11. A boat takes 1.6 hours longer to go 36 kms up a river than down the river. If the speed of the water current is 4 km pe…
- Q12. Is it possible to design a rectangular park of perimeter 320 m and area 4800 m²? If so find its length and breadth.
- Q13. At t minutes past 2 pm, the time needed to 3 pm is 3 minutes less than (t²)/4. Find t.
- Q14. The number of seats in a row is equal to the total number of rows in a hall. The total number of seats in the hall will…
- Q15(i). If α and β are the roots of the polynomial f(x) = x² - 2x + 3, find the polynomial whose roots are (i) α + 2,\ β + 2
- Q15(ii). If α and β are the roots of the polynomial f(x) = x² - 2x + 3, find the polynomial whose roots are (ii) (α - 1)/(α + 1)…
- Q16. If -4 is a root of the equation x² + px - 4 = 0 and if the equation x² + px + q = 0 has equal roots, find the values of…
- Q17(i). Two farmers Thilagan and Kausigan cultivates three varieties of grains namely rice, wheat and ragi. If the sale (in ₹)…
- Q17(ii). Two farmers Thilagan and Kausigan cultivates three varieties of grains namely rice, wheat and ragi. If the sale (in ₹)…
- Q18. If cosθ cosθ, sinθ; -sinθ, cosθ + sinθ x, -cosθ; cosθ, x = I₂, find x.
- Q19. Given A = p, 0; 0, 2,\ B = 0, -q; 1, 0,\ C = 2, -2; 2, 2 and if BA = C², find p and q.
- Q20. A = 3, 0; 4, 5,\ B = 6, 3; 8, 5,\ C = 3, 6; 1, 1 find the matrix D, such that CD - AB = 0
Chapter 4: Geometry
Ex 4.1: 17 questions
- Q1(i). In the given triangles, check which triangles are similar? Also find the value of x. (i)
- Q1(ii). In the given triangles, check which triangles are similar? Also find the value of x. (ii)
- Q2. A girl looks the reflection of the top of the lamp post on the mirror which is 6.6 m away from the foot of the lamppost…
- Q3. A vertical stick of length 6 m casts a shadow 400 cm long on the ground and at the same time a tower casts a shadow 28…
- Q4. Two triangles QPR and QSR, right angled at P and S respectively are drawn on the same base QR and on the same side of Q…
- Q5. In the adjacent figure, Δ ABC is right angled at C and DE ⊥ AB. Prove that Δ ABC ~ Δ ADE and hence find the lengths of…
- Q6. In the adjacent figure, Δ ACB ~ Δ APQ. If BC = 8 cm, PQ = 4 cm, BA = 6.5 cm and AP = 2.8 cm, find CA and AQ.
- Q7(i). If figure OPRQ is a square and ∠ MLN = 90°. Prove that (i) Δ LOP ~ Δ QMO
- Q7(ii). If figure OPRQ is a square and ∠ MLN = 90°. Prove that (ii) Δ LOP ~ Δ RPN
- Q7(iii). If figure OPRQ is a square and ∠ MLN = 90°. Prove that (iii) Δ QMO ~ Δ RPN
- Q7(iv). If figure OPRQ is a square and ∠ MLN = 90°. Prove that (iv) QR² = MQ × RN
- Q8. If Δ ABC ~ Δ DEF such that area of Δ ABC is 9cm² and the area of Δ DEF is 16cm² and BC = 2.1 cm. Find the length of EF.
- Q9. Two vertical poles of heights 6 m and 3 m are erected above a horizontal ground AC. Find the value of y.
- Q10. Construct a triangle similar to a given triangle PQR with its sides equal to 2/3 of the corresponding sides of the tria…
- Q11. Construct a triangle similar to a given triangle LMN with its sides equal to 4/5 of the corresponding sides of the tria…
- Q12. Construct a triangle similar to a given triangle ABC with its sides equal to 6/5 of the corresponding sides of the tria…
- Q13. Construct a triangle similar to a given triangle PQR with its sides equal to 7/3 of the corresponding sides of the tria…
Ex 4.2: 19 questions
- Q1(i). In Δ ABC, D and E are points on the sides AB and AC respectively such that DE ∥ BC. (i) If AD/DB = 3/4 and AC = 15 cm f…
- Q1(ii). In Δ ABC, D and E are points on the sides AB and AC respectively such that DE ∥ BC. (ii) If AD = 8x - 7, DB = 5x - 3, A…
- Q2. ABCD is a trapezium in which AB ∥ DC and P, Q are points on AD and BC respectively, such that PQ ∥ DC if PD = 18 cm, BQ…
- Q3. In Δ ABC, D and E are points on the sides AB and AC respectively. Show that DE ∥ BC if AB = 12 cm, AD = 8 cm, AE = 12 c…
- Q4(i). In fig. if PQ ∥ BC and PR ∥ CD prove that (i) AR/AD = AQ/AB.
- Q4(ii). In fig. if PQ ∥ BC and PR ∥ CD prove that (ii) QB/AQ = DR/AR.
- Q5. Rhombus PQRB is inscribed in Δ ABC such that ∠ B is one of its angle. P, Q and R lie on AB, AC and BC respectively. If…
- Q6. In trapezium ABCD, AB ∥ DC, E and F are points on non-parallel sides AD and BC respectively, such that EF ∥ AB. Show th…
- Q7. In figure DE ∥ BC and CD ∥ EF. Prove that AD² = AB × AF.
- Q8(i). Check whether AD is bisector of ∠ A of Δ ABC in each of the following (i) AB = 5 cm, AC = 10 cm, BD = 1.5 cm and CD = 3…
- Q8(ii). Check whether AD is bisector of ∠ A of Δ ABC in each of the following (ii) AB = 4 cm, AC = 6 cm, BD = 1.6 cm and CD = 2…
- Q9. In figure ∠ QPR = 90°, PS is its bisector. If ST ⊥ PR, prove that ST × (PQ + PR) = PQ × PR.
- Q10. ABCD is a quadrilateral in which AB = AD, the bisector of ∠ BAC and ∠ CAD intersect the sides BC and CD at the points E…
- Q11. Construct a Δ PQR which the base PQ = 4.5 cm, ∠ R = 35° and the median RG from R to PQ is 6 cm.
- Q12. Construct a Δ PQR in which QR = 5 cm, ∠ P = 40° and the median PG from P to QR is 4.4 cm. Find the length of the altitu…
- Q13. Construct a Δ PQR such that QR = 6.5 cm, ∠ P = 60° and the altitude from P to QR is of length 4.5 cm.
- Q14. Construct a Δ ABC such that AB = 5.5 cm, ∠ C = 25° and the altitude from C to AB is 4 cm.
- Q15. Draw a triangle ABC of base BC = 5.6 cm, ∠ A = 40° and the bisector of ∠ A meets BC at D such that CD = 4 cm.
- Q16. Draw Δ PQR such that PQ = 6.8 cm, vertical angle is 50° and the bisector of the vertical angle meets the base at D wher…
Ex 4.3: 8 questions
- Q1. A man goes 18 m due east and then 24 m due north. Find the distance of his current position from the starting point?
- Q2. There are two paths that one can choose to go from Sarah's house to James house. One way is to take C street, and the o…
- Q3. To get from point A to point B you must avoid walking through a pond. You must walk 34 m south and 41 m east. To the ne…
- Q4. In the rectangle WXYZ, XY+YZ=17 cm, and XZ+YW=26 cm. Calculate the length and breadth of the rectangle?
- Q5. The hypotenuse of a right triangle is 6 m more than twice of the shortest side. If the third side is 2 m less than the…
- Q6. 5 m long ladder is placed leaning towards a vertical wall such that it reaches the wall at a point 4m high. If the foot…
- Q7. The perpendicular PS on the base QR of a Δ PQR intersects QR at S, such that QS = 3\ SR. Prove that 2PQ² = 2PR² + QR²
- Q8. In the adjacent figure, ABC is a right angled triangle with right angle at B and points D, E trisect BC. Prove that 8AE…
Ex 4.4: 16 questions
- Q1. The length of the tangent to a circle from a point P, which is 25 cm away from the centre is 24 cm. What is the radius…
- Q2. △ LMN is a right angled triangle with ∠ L = 90°. A circle is inscribed in it. The lengths of the sides containing the r…
- Q3. A circle is inscribed in Δ ABC having sides 8 cm, 10 cm and 12 cm as shown in figure, Find AD, BE and CF.
- Q4. PQ is a tangent drawn from a point P to a circle with centre O and QOR is a diameter of the circle such that ∠ POR = 12…
- Q5. A tangent ST to a circle touches it at B. AB is a chord such that ∠ ABT = 65°. Find ∠ AOB, where “O” is the centre of t…
- Q6. In figure, O is the centre of the circle with radius 5 cm. T is a point such that OT = 13 cm and OT intersects the circ…
- Q7. In two concentric circles, a chord of length 16 cm of larger circle becomes a tangent to the smaller circle whose radiu…
- Q8. Two circles with centres O and O' of radii 3 cm and 4 cm, respectively intersect at two points P and Q, such that OP an…
- Q9. Show that the angle bisectors of a triangle are concurrent.
- Q10. An artist has created a triangular stained glass window and has one strip of small length left before completing the wi…
- Q11. Draw a tangent at any point R on the circle of radius 3.4 cm and centre at P?
- Q12. Draw a circle of radius 4.5 cm. Take a point on the circle. Draw the tangent at that point using the alternate segment…
- Q13. Draw the two tangents from a point which is 10 cm away from the centre of a circle of radius 5 cm. Also, measure the le…
- Q14. Take a point which is 11 cm away from the centre of a circle of radius 4 cm and draw the two tangents to the circle fro…
- Q15. Draw the two tangents from a point which is 5 cm away from the centre of a circle of diameter 6 cm. Also, measure the l…
- Q16. Draw a tangent to the circle from the point P having radius 3.6 cm, and centre at O. Point P is at a distance 7.2 cm fr…
Ex 4.5: 15 questions
- Q1. If in triangles ABC and EDF, AB/DE = BC/FD then they will be similar, when
- Q2. In Δ LMN, ∠ L = 60°, ∠ M = 50°. If Δ LMN ~ Δ PQR then the value of ∠ R is
- Q3. If Δ ABC is an isosceles triangle with ∠ C = 90° and AC = 5 cm, then AB is
- Q4. In a given figure ST ∥ QR, PS = 2 cm and SQ = 3 cm. Then the ratio of the area of Δ PQR to the area of Δ PST is
- Q5. The perimeters of two similar triangles Δ ABC and Δ PQR are 36 cm and 24 cm respectively. If PQ = 10 cm, then the lengt…
- Q6. If in Δ ABC, DE ∥ BC. AB = 3.6 cm, AC = 2.4 cm and AD = 2.1 cm then the length of AE is
- Q7. In a Δ ABC, AD is the bisector of ∠ BAC. If AB = 8 cm, BD = 6 cm and DC = 3 cm. The length of the side AC is
- Q8. In the adjacent figure ∠ BAC = 90° and AD ⊥ BC then
- Q9. Two poles of heights 6 m and 11 m stand vertically on a plane ground. If the distance between their feet is 12 m, what…
- Q10. In the given figure, PR = 26 cm, QR = 24 cm, ∠ PAQ = 90°, PA = 6 cm and QA = 8 cm. Find ∠ PQR
- Q11. A tangent is perpendicular to the radius at the
- Q12. How many tangents can be drawn to the circle from an exterior point?
- Q13. The two tangents from an external points P to a circle with centre at O are PA and PB. If ∠ APB = 70° then the value of…
- Q14. In figure CP and CQ are tangents to a circle with centre at O. ARB is another tangent touching the circle at R. If CP =…
- Q15. In figure if PR is tangent to the circle at P and O is the centre of the circle, then ∠ POQ is
Unit Exercise 4: 13 questions
- Q1(i). In the figure, if BD ⊥ AC and CE ⊥ AB, prove that (i) Δ AEC ~ Δ ADB
- Q1(ii). In the figure, if BD ⊥ AC and CE ⊥ AB, prove that (ii) CA/AB = CE/DB
- Q2. In the given figure AB ∥ CD ∥ EF. If AB = 6 cm, CD = x cm, EF = 4 cm, BD = 5 cm and DE = y cm. Find x and y.
- Q3. O is any point inside a triangle ABC. The bisector of ∠ AOB, ∠ BOC and ∠ COA meet the sides AB, BC and CA in point D, E…
- Q4. In the figure, ABC is a triangle in which AB = AC. Points D and E are points on the side AB and AC respectively such th…
- Q5. Two trains leave a railway station at the same time. The first train travels due west and the second train due north. T…
- Q6(i). D is the mid point of side BC and AE ⊥ BC. If BC = a, AC = b, AB = c, ED = x, AD = p and AE = h, prove that (i) b² = p²…
- Q6(ii). D is the mid point of side BC and AE ⊥ BC. If BC = a, AC = b, AB = c, ED = x, AD = p and AE = h, prove that (ii) c² = p…
- Q6(iii). D is the mid point of side BC and AE ⊥ BC. If BC = a, AC = b, AB = c, ED = x, AD = p and AE = h, prove that (iii) b² +…
- Q7. A man whose eye-level is 2 m above the ground wishes to find the height of a tree. He places a mirror horizontally on t…
- Q8. An Emu which is 8 feet tall is standing at the foot of a pillar which is 30 feet high. It walks away from the pillar. T…
- Q9. Two circles intersect at A and B. From a point P on one of the circles lines PAC and PBD are drawn intersecting the sec…
- Q10. Let ABC be a triangle and D,E,F are points on the respective sides AB, BC, AC (or their extensions). Let AD: DB = 5: 3,…
Chapter 5: Coordinate Geometry
Ex 5.1: 18 questions
- Q1(i). Find the area of the triangle formed by the points (i) (1, -1), (-4, 6) and (-3, -5)
- Q1(ii). Find the area of the triangle formed by the points (ii) (-10, -4), (-8, -1) and (-3, -5)
- Q2(i). Determine whether the sets of points are collinear? (i) (-1/2, 3), (-5, 6) and (-8, 8)
- Q2(ii). Determine whether the sets of points are collinear? (ii) (a, b+c), (b, c+a) and (c, a+b)
- Q3(i). Vertices of given triangles are taken in order and their areas are provided aside. In each case, find the value of 'p'.…
- Q3(ii). Vertices of given triangles are taken in order and their areas are provided aside. In each case, find the value of 'p'.…
- Q4(i). In each of the following, find the value of 'a' for which the given points are collinear. (i) (2, 3), (4, a) and (6, -3)
- Q4(ii). In each of the following, find the value of 'a' for which the given points are collinear. (ii) (a, 2-2a), (-a+1, 2a) an…
- Q5(i). Find the area of the quadrilateral whose vertices are at (i) (-9, -2), (-8, -4), (2, 2) and (1, -3)
- Q5(ii). Find the area of the quadrilateral whose vertices are at (ii) (-9, 0), (-8, 6), (-1, -2) and (-6, -3)
- Q6. Find the value of k, if the area of a quadrilateral is 28 sq.units, whose vertices are taken in the order (-4, -2), (-3…
- Q7. If the points A(-3, 9), B(a, b) and C(4, -5) are collinear and if a + b = 1, then find a and b.
- Q8. Let P(11, 7), Q(13.5, 4) and R(9.5, 4) be the mid-points of the sides AB, BC and AC respectively of Δ ABC. Find the coo…
- Q9. In the figure, the quadrilateral swimming pool shown is surrounded by concrete patio. Find the area of the patio.
- Q10. A triangular shaped glass with vertices at A(-5, -4), B(1, 6) and C(7, -4) has to be painted. If one bucket of paint co…
- Q11(i). In the figure, find the area of (i) triangle AGF.
- Q11(ii). In the figure, find the area of (ii) triangle FED.
- Q11(iii). In the figure, find the area of (iii) quadrilateral BCEG.
Ex 5.2: 17 questions
- Q1(i). What is the slope of a line whose inclination with positive direction of x-axis is (i) 90°
- Q1(ii). What is the slope of a line whose inclination with positive direction of x-axis is (ii) 0°
- Q2(i). What is the inclination of a line whose slope is (i) 0
- Q2(ii). What is the inclination of a line whose slope is (ii) 1
- Q3(i). Find the slope of a line joining the points (i) (5,√5) with the origin
- Q3(ii). Find the slope of a line joining the points (ii) (sinθ, -cosθ) and (-sinθ, cosθ)
- Q4. What is the slope of a line perpendicular to the line joining A(5,1) and P where P is the mid-point of the segment join…
- Q5. Show that the given points are collinear: (-3,-4), (7,2) and (12,5)
- Q6. If the three points (3,-1), (a,3) and (1,-3) are collinear, find the value of a.
- Q7. The line through the points (-2,a) and (9,3) has slope -1/2. Find the value of a.
- Q8. The line through the points (-2,6) and (4,8) is perpendicular to the line through the points (8,12) and (x,24). Find th…
- Q9(i). Show that the given points form a right angled triangle. (i) A(1,-4), B(2,-3) and C(4,-7)
- Q9(ii). Show that the given points form a right angled triangle. (ii) L(0,5), M(9,12) and N(3,14)
- Q10. Show that the given points form a parallelogram: A(2.5, 3.5), B(10,-4), C(2.5,-2.5) and D(-5,5)
- Q11. If the points A(2,2), B(-2,-3), C(1,-3) and D(x, y) form a parallelogram then find the value of x and y.
- Q12. Let A(3,-4), B(9,-4), C(5,-7) and D(7,-7). Show that ABCD is a trapezium.
- Q13. A quadrilateral has vertices at A(-4,-2), B(5,-1), C(6,5) and D(-7,6). Show that the mid-points of its sides form a par…
Ex 5.3: 22 questions
- Q1(i). Find the equation of a straight line passing through the mid-point of a line segment joining the points (1,-5), (4,2) a…
- Q1(ii). Find the equation of a straight line passing through the mid-point of a line segment joining the points (1,-5), (4,2) a…
- Q2. The equation of a straight line is 2(x-y)+5=0. Find its slope, inclination and intercept on the Y axis.
- Q3. Find the equation of a line whose inclination is 30° and making an intercept -3 on the Y axis.
- Q4. Find the slope and y intercept of √3x+(1-√3)y=3.
- Q5. Find the value of 'a', if the line through (-2,3) and (8,5) is perpendicular to y=ax+2
- Q6. The hill in the form of a right triangle has its foot at (19,3). The inclination of the hill to the ground is 45°. Find…
- Q7(i). Find the equation of a line through the given pair of points (i) (2,2/3) and ((-1)/2,-2)
- Q7(ii). Find the equation of a line through the given pair of points (ii) (2,3) and (-7,-1)
- Q8. A cat is located at the point (-6,-4) in xy plane. A bottle of milk is kept at (5,11). The cat wish to consume the milk…
- Q9(i). If the vertices of a Δ ABC are A(6,2), B(-5,-1) and C(1,9); through the vertex A (i) find the equation of median
- Q9(ii). If the vertices of a Δ ABC are A(6,2), B(-5,-1) and C(1,9); through the vertex A (ii) find the equation of altitude
- Q10. Find the equation of a straight line which has slope (-5)/4 and passing through the point (-1,2).
- Q11(i). You are downloading a song. The percent y (in decimal form) of mega bytes remaining to get downloaded in x seconds is g…
- Q11(ii). You are downloading a song. The percent y (in decimal form) of mega bytes remaining to get downloaded in x seconds is g…
- Q11(iii). You are downloading a song. The percent y (in decimal form) of mega bytes remaining to get downloaded in x seconds is g…
- Q12(i). Find the equation of a line whose intercepts on the x and y axes are given below. (i) 4,\ -6
- Q12(ii). Find the equation of a line whose intercepts on the x and y axes are given below. (ii) -5,3/4
- Q13(i). Find the intercepts made by the following lines on the coordinate axes. (i) 3x-2y-6=0
- Q13(ii). Find the intercepts made by the following lines on the coordinate axes. (ii) 4x+3y+12=0
- Q14(i). Find the equation of a straight line (i) passing through (1,-4) and has intercepts which are in the ratio 2:5
- Q14(ii). Find the equation of a straight line (ii) passing through (-8,4) and making equal intercepts on the coordinate axes
Ex 5.4: 15 questions
- Q1(i). Find the slope of the following straight lines (i) 5y - 3 = 0
- Q1(ii). Find the slope of the following straight lines (ii) 7x - 3/17 = 0
- Q2(i). Find the slope of the line which is (i) parallel to y = 0.7x - 11
- Q2(ii). Find the slope of the line which is (ii) perpendicular to the line x = -11
- Q3(i). Check whether the given lines are parellel or perpendicular (i) x/3 + y/4 + 1/7 = 0 and 2x/3 + y/2 + 1/10 = 0
- Q3(ii). Check whether the given lines are parellel or perpendicular (ii) 5x + 23y + 14 = 0 and 23x - 5y + 9 = 0
- Q4. If the straight lines 12y = -(p+3)x + 12, 12x - 7y = 16 are perpendicular then find 'p'.
- Q5. Find the equation of a straight line passing through the point P(-5,2) and parallel to the line joining the points Q(3,…
- Q6. Find the equation of a line passing through (6,-2) and perpendicular to the line joining the points (6,7) and (2,-3).
- Q7. A(-3,0) B(10,-2) and C(12,3) are the vertices of Δ ABC. Find the equation of the altitude through A and B.
- Q8. Find the equation of the perpendicular bisector of the line joining the points A(-4,2) and B(6,-4).
- Q9. Find the equation of a straight line through the intersection of lines 7x + 3y = 10, 5x - 4y = 1 and parallel to the li…
- Q10. Find the equation of a straight line through the intersection of lines 5x - 6y = 2, 3x + 2y = 10 and perpendicular to t…
- Q11. Find the equation of a straight line joining the point of intersection of 3x + y + 2 = 0 and x - 2y - 4 = 0 to the poin…
- Q12. Find the equation of a straight line through the point of intersection of the lines 8x + 3y = 18, 4x + 5y = 9 and bisec…
Ex 5.5: 15 questions
- Q1. The area of triangle formed by the points (-5,0), (0,-5) and (5,0) is
- Q2. A man walks near a wall, such that the distance between him and the wall is 10 units. Consider the wall to be the Y axi…
- Q3. The straight line given by the equation x = 11 is
- Q4. If (5,7), (3,p) and (6,6) are collinear, then the value of p is
- Q5. The point of intersection of 3x - y = 4 and x + y = 8 is
- Q6. The slope of the line joining (12,3), (4,a) is 1/8. The value of 'a' is
- Q7. The slope of the line which is perpendicular to a line joining the points (0,0) and (-8,8) is
- Q8. If slope of the line PQ is 1/(√3) then slope of the perpendicular bisector of PQ is
- Q9. If A is a point on the Y axis whose ordinate is 8 and B is a point on the X axis whose abscissae is 5 then the equation…
- Q10. The equation of a line passing through the origin and perpendicular to the line 7x - 3y + 4 = 0 is
- Q11. Consider four straight lines (i) l₁;\ 3y = 4x + 5 (ii) l₂;\ 4y = 3x - 1 (iii) l₃;\ 4y + 3x = 7 (iv) l₄;\ 4x + 3y = 2 Wh…
- Q12. A straight line has equation 8y = 4x + 21. Which of the following is true
- Q13. When proving that a quadrilateral is a trapezium, it is necessary to show
- Q14. When proving that a quadrilateral is a parallelogram by using slopes you must find
- Q15. (2,\ 1) is the point of intersection of two lines.
Unit Exercise 5: 10 questions
- Q1. PQRS is a rectangle formed by joining the points P(-1,-1), Q(-1,4), R(5,4) and S(5,-1). A, B, C and D are the mid-point…
- Q2. The area of a triangle is 5 sq.units. Two of its vertices are (2,1) and (3, -2). The third vertex is (x, y) where y = x…
- Q3. Find the area of a triangle formed by the lines 3x + y - 2 = 0, 5x + 2y - 3 = 0 and 2x - y - 3 = 0
- Q4. If vertices of a quadrilateral are at A(-5,7), B(-4,k), C(-1,-6) and D(4,5) and its area is 72 sq.units. Find the value…
- Q5. Without using distance formula, show that the points (-2,-1), (4,0), (3,3) and (-3,2) are vertices of a parallelogram.
- Q6. Find the equations of the lines, whose sum and product of intercepts are 1 and -6 respectively.
- Q7. The owner of a milk store finds that, he can sell 980 litres of milk each week at ₹14/litre and 1220 litres of milk eac…
- Q8. Find the image of the point (3,8) with respect to the line x + 3y = 7 assuming the line to be a plane mirror.
- Q9. Find the equation of a line passing through the point of intersection of the lines 4x + 7y - 3 = 0 and 2x - 3y + 1 = 0…
- Q10. A person standing at a junction (crossing) of two straight paths represented by the equations 2x - 3y + 4 = 0 and 3x +…
Chapter 6: Trigonometry
Ex 6.1: 19 questions
- Q1(i). Prove the following identity. cotθ + tanθ = secθ cosec θ
- Q1(ii). Prove the following identity. tan⁴θ + tan²θ = sec⁴θ - sec²θ
- Q2(i). Prove the following identity. (1-tan²θ)/(cot²θ-1) = tan²θ
- Q2(ii). Prove the following identity. (cosθ)/(1+sinθ) = secθ - tanθ
- Q3(i). Prove the following identity. √((1+sinθ)/(1-sinθ)) = secθ + tanθ
- Q3(ii). Prove the following identity. √((1+sinθ)/(1-sinθ)) + √((1-sinθ)/(1+sinθ)) = 2secθ
- Q4(i). Prove the following identity. sec⁶θ = tan⁶θ + 3tan²θsec²θ + 1
- Q4(ii). Prove the following identity. (sinθ + secθ)² + (cosθ + cosec θ)² = 1 + (secθ + cosec θ)²
- Q5(i). Prove the following identity. sec⁴θ(1-sin⁴θ) - 2tan²θ = 1
- Q5(ii). Prove the following identity. (cotθ - cosθ)/(cotθ + cosθ) = (cosec θ - 1)/(cosec θ + 1)
- Q6(i). Prove the following identity. (sin A - sin B)/(cos A + cos B) + (cos A - cos B)/(sin A + sin B) = 0
- Q6(ii). Prove the following identity. (sin³ A + cos³ A)/(sin A + cos A) + (sin³ A - cos³ A)/(sin A - cos A) = 2
- Q7(i). If sinθ + cosθ = √3, then prove that tanθ + cotθ = 1.
- Q7(ii). If √3sinθ - cosθ = 0, then show that tan 3θ = (3tanθ - tan³θ)/(1 - 3tan²θ)
- Q8(i). If (cosα)/(cosβ) = m and (cosα)/(sinβ) = n, then prove that (m² + n²)cos²β = n²
- Q8(ii). If cotθ + tanθ = x and secθ - cosθ = y, then prove that (x²y)(2/3) - (xy²)(2/3) = 1
- Q9(i). If sinθ + cosθ = p and secθ + cosec θ = q, then prove that q(p² - 1) = 2p
- Q9(ii). If sinθ(1 + sin²θ) = cos²θ, then prove that cos⁶θ - 4cos⁴θ + 8cos²θ = 4
- Q10. If (cosθ)/(1+sinθ) = 1/a, then prove that (a²-1)/(a²+1) = sinθ
Ex 6.2: 7 questions
- Q1. Find the angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of a tow…
- Q2. A road is flanked on either side by continuous rows of houses of height 4√3 m with no space in between them. A pedestri…
- Q3. To a man standing outside his house, the angles of elevation of the top and bottom of a window are 60° and 45° respecti…
- Q4. A statue 1.6 m tall stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of t…
- Q5(i). A flag pole of height 'h' metres is on the top of the hemispherical dome of radius 'r' metres. A man is standing 7 m aw…
- Q5(ii). A flag pole of height 'h' metres is on the top of the hemispherical dome of radius 'r' metres. A man is standing 7 m aw…
- Q6. The top of a 15 m high tower makes an angle of elevation of 60° with the bottom of an electronic pole and angle of elev…
Ex 6.3: 6 questions
- Q1. From the top of a rock 50√3 m high, the angle of depression of a car on the ground is observed to be 30°. Find the dist…
- Q2. The horizontal distance between two buildings is 70 m. The angle of depression of the top of the first building when se…
- Q3. From the top of the tower 60 m high the angles of depression of the top and bottom of a vertical lamp post are observed…
- Q4. An aeroplane at an altitude of 1800 m finds that two boats are sailing towards it in the same direction. The angles of…
- Q5. From the top of a lighthouse, the angle of depression of two ships on the opposite sides of it are observed to be 30° a…
- Q6. A lift in a building of height 90 feet with transparent glass walls is descending from the top of the building. At the…
Ex 6.4: 9 questions
- Q1. From the top of a tree of height 13 m the angle of elevation and depression of the top and bottom of another tree are 4…
- Q2. A man is standing on the deck of a ship, which is 40 m above water level. He observes the angle of elevation of the top…
- Q3. If the angle of elevation of a cloud from a point 'h' metres above a lake is θ₁ and the angle of depression of its refl…
- Q4. The angle of elevation of the top of a cell phone tower from the foot of a high apartment is 60° and the angle of depre…
- Q5(i). The angles of elevation and depression of the top and bottom of a lamp post from the top of a 66 m high apartment are 6…
- Q5(ii). The angles of elevation and depression of the top and bottom of a lamp post from the top of a 66 m high apartment are 6…
- Q5(iii). The angles of elevation and depression of the top and bottom of a lamp post from the top of a 66 m high apartment are 6…
- Q6(i). Three villagers A, B and C can see each other using telescope across a valley. The horizontal distance between A and B…
- Q6(ii). Three villagers A, B and C can see each other using telescope across a valley. The horizontal distance between A and B…
Ex 6.5: 15 questions
- Q1. The value of sin²θ + 1/(1+tan²θ) is equal to
- Q2. tanθ cosec²θ - tanθ is equal to
- Q3. If (sinα + cosec α)² + (cosα + secα)² = k + tan²α + cot²α, then the value of k is equal to
- Q4. If sinθ + cosθ = a and secθ + cosec θ = b, then the value of b(a²-1) is equal to
- Q5. If 5x = secθ and 5/y = tanθ, then x² - 1/(y²) is equal to
- Q6. If sinθ = cosθ, then 2tan²θ + sin²θ - 1 is equal to
- Q7. If x = atanθ and y = bsecθ then
- Q8. (1+tanθ+secθ)(1+cotθ-cosec θ) is equal to
- Q9. acotθ + b cosec θ = p and bcotθ + a cosec θ = q then p² - q² is equal to
- Q10. If the ratio of the height of a tower and the length of its shadow is √3:1, then the angle of elevation of the sun has…
- Q11. The electric pole subtends an angle of 30° at a point on the same level as its foot. At a second point 'b' metres above…
- Q12. A tower is 60 m heigh. Its shadow reduces by x metres when the angle of elevation of the sun increases from 30° to 45°…
- Q13. The angle of depression of the top and bottom of 20 m tall building from the top of a multistoried building are 30° and…
- Q14. Two persons are standing 'x' metres apart from each other and the height of the first person is double that of the othe…
- Q15. The angle of elevation of a cloud from a point h metres above a lake is β. The angle of depression of its reflection in…
Unit Exercise 6: 13 questions
- Q1(i). Prove that cot² A((sec A-1)/(1+sin A))+sec² A((sin A-1)/(1+sec A))=0
- Q1(ii). Prove that (tan²θ-1)/(tan²θ+1)=1-2cos²θ
- Q2. Prove that ((1+sinθ-cosθ)/(1+sinθ+cosθ))²=(1-cosθ)/(1+cosθ)
- Q3. If xsin³θ+ycos³θ=sinθcosθ and xsinθ=ycosθ, then prove that x²+y²=1.
- Q4. If acosθ-bsinθ=c, then prove that (asinθ+bcosθ)=±√(a²+b²-c²).
- Q5. A bird is sitting on the top of a 80 m high tree. From a point on the ground, the angle of elevation of the bird is 45°…
- Q6. An aeroplane is flying parallel to the Earth's surface at a speed of 175 m/sec and at a height of 600 m. The angle of e…
- Q7(i). A bird is flying from A towards B at an angle of 35°, a point 30 km away from A. At B it changes its course of flight a…
- Q7(ii). A bird is flying from A towards B at an angle of 35°, a point 30 km away from A. At B it changes its course of flight a…
- Q7(iii). A bird is flying from A towards B at an angle of 35°, a point 30 km away from A. At B it changes its course of flight a…
- Q7(iv). A bird is flying from A towards B at an angle of 35°, a point 30 km away from A. At B it changes its course of flight a…
- Q8. Two ships are sailing in the sea on either side of the lighthouse. The angles of depression of two ships as observed fr…
- Q9. A building and a statue are in opposite side of a street from each other 35 m apart. From a point on the roof of buildi…
Chapter 7: Mensuration
Ex 7.1: 10 questions
- Q1. The radius and height of a cylinder are in the ratio 5:7 and its curved surface area is 5500 sq.cm. Find its radius and…
- Q2. A solid iron cylinder has total surface area of 1848 sq.cm. Its curved surface area is five – sixth of its total surfac…
- Q3. The external radius and the length of a hollow wooden log are 16 cm and 13 cm respectively. If its thickness is 4 cm th…
- Q4. A right angled triangle PQR where ∠ Q = 90° is rotated about QR and PQ as central axes. If QR=16 cm and PR=20 cm, compa…
- Q5. 4 persons live in a conical tent whose slant height is 19 m. If each person require 22 m² of the floor area, then find…
- Q6. A girl wishes to prepare birthday caps in the form of right circular cones for her birthday party, using a sheet of pap…
- Q7. The ratio of the radii of two right circular cones of same height is 1:3. Find the ratio of their curved surface area w…
- Q8. The radius of a sphere increases by 25\%. Find the percentage increase in its surface area.
- Q9. The internal and external diameters of a hollow hemispherical vessel are 20 cm and 28 cm respectively. Find the cost to…
- Q10. The frustum shaped outer portion of the table lamp has to be painted including the top part. Find the total cost of pai…
Ex 7.2: 10 questions
- Q1. A 14 m deep well with inner diameter 10 m is dug and the earth taken out is evenly spread all around the well to form a…
- Q2. A cylindrical glass with diameter 20 cm has water to a height of 9 cm. A small cylindrical metal of radius 5 cm and hei…
- Q3. If the circumference of a conical wooden piece is 484 cm then find its volume when its height is 105 cm.
- Q4. A conical container is fully filled with petrol. The radius is 10m and the height is 15 m. If the container can release…
- Q5. A right angled triangle whose sides are 6 cm, 8 cm and 10 cm is revolved about the sides containing the right angle in…
- Q6. The volumes of two cones of same base radius are 3600 cm³ and 5040 cm³. Find the ratio of heights.
- Q7. If the ratio of radii of two spheres is 4:7, find the ratio of their volumes.
- Q8. A solid sphere and a solid hemisphere have equal total surface area. Prove that the ratio of their volume is 3√3: 4.
- Q9. The outer and the inner surface areas of a spherical copper shell are 576π cm² and 324π cm² respectively. Find the volu…
- Q10. A container open at the top is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper end…
Ex 7.3: 9 questions
- Q1. A vessel is in the form of a hemispherical bowl mounted by a hollow cylinder. The diameter is 14 cm and the height of t…
- Q2. Nathan, an engineering student was asked to make a model shaped like a cylinder with two cones attached at its two ends…
- Q3. From a solid cylinder whose height is 2.4 cm and the diameter 1.4 cm, a cone of the same height and same diameter is ca…
- Q4. A solid consisting of a right circular cone of height 12 cm and radius 6 cm standing on a hemisphere of radius 6 cm is…
- Q5. A capsule is in the shape of a cylinder with two hemisphere stuck to each of its ends. If the length of the entire caps…
- Q6. As shown in figure a cubical block of side 7 cm is surmounted by a hemisphere. Find the surface area of the solid.
- Q7(i). A right circular cylinder just enclose a sphere of radius r units. Calculate (i) the surface area of the sphere.
- Q7(ii). A right circular cylinder just enclose a sphere of radius r units. Calculate (ii) the curved surface area of the cylind…
- Q7(iii). A right circular cylinder just enclose a sphere of radius r units. Calculate (iii) the ratio of the areas obtained in (…
Ex 7.4: 8 questions
- Q1. An aluminium sphere of radius 12 cm is melted to make a cylinder of radius 8 cm. Find the height of the cylinder.
- Q2. Water is flowing at the rate of 15 km per hour through a pipe of diameter 14 cm into a rectangular tank which is 50 m l…
- Q3. A conical flask is full of water. The flask has base radius r units and height h units, the water is poured into a cyli…
- Q4. A solid right circular cone of diameter 14 cm and height 8 cm is melted to form a hollow sphere. If the external diamet…
- Q5. Seenu's house has an overhead tank in the shape of a cylinder. This is filled by pumping water from a sump (underground…
- Q6. The internal and external diameter of a hollow hemispherical shell are 6 cm and 10 cm respectively. If it is melted and…
- Q7. A solid sphere of radius 6 cm is melted into a hollow cylinder of uniform thickness. If the external radius of the base…
- Q8. A hemispherical bowl is filled to the brim with juice. The juice is poured into a cylindrical vessel whose radius is 50…
Ex 7.5: 15 questions
- Q1. The curved surface area of a right circular cone of height 15 cm and base diameter 16 cm is
- Q2. If two solid hemispheres of same base radius r units are joined together along their bases, then curved surface area of…
- Q3. The height of a right circular cone whose radius is 5 cm and slant height is 13 cm will be
- Q4. If the radius of the base of a right circular cylinder is halved keeping the same height, then the ratio of the volume…
- Q5. The total surface area of a cylinder whose radius is 1/3 of its height is
- Q6. In a hollow cylinder, the sum of the external and internal radii is 14 cm and the width is 4 cm. If its height is 20 cm…
- Q7. If the radius of the base of a cone is tripled and the height is doubled then the volume is
- Q8. The total surface area of a hemi-sphere is how much times the square of its radius.
- Q9. A solid sphere of radius x cm is melted and cast into a shape of a solid cone of same radius. The height of the cone is
- Q10. A frustum of a right circular cone is of height 16cm with radii of its ends as 8cm and 20cm. Then, the volume of the fr…
- Q11. A shuttle cock used for playing badminton has the shape of the combination of
- Q12. A spherical ball of radius r₁ units is melted to make 8 new identical balls each of radius r₂ units. Then r₁: r₂ is
- Q13. The volume (in cm³) of the greatest sphere that can be cut off from a cylindrical log of wood of base radius 1 cm and h…
- Q14. The height and radius of the cone of which the frustum is a part are h₁ units and r₁ units respectively. Height of the…
- Q15. The ratio of the volumes of a cylinder, a cone and a sphere, if each has the same diameter and same height is
Unit Exercise 7: 10 questions
- Q1. The barrel of a fountain-pen cylindrical in shape, is 7 cm long and 5 mm in diameter. A full barrel of ink in the pen w…
- Q2. A hemi-spherical tank of radius 1.75 m is full of water. It is connected with a pipe which empties the tank at the rate…
- Q3. Find the maximum volume of a cone that can be carved out of a solid hemisphere of radius r units.
- Q4. An oil funnel of tin sheet consists of a cylindrical portion 10 cm long attached to a frustum of a cone. If the total h…
- Q5. Find the number of coins, 1.5 cm in diameter and 2 mm thick, to be melted to form a right circular cylinder of height 1…
- Q6. A hollow metallic cylinder whose external radius is 4.3 cm and internal radius is 1.1 cm and whole length is 4 cm is me…
- Q7. The slant height of a frustum of a cone is 4 m and the perimeter of circular ends are 18 m and 16 m. Find the cost of p…
- Q8. A hemi-spherical hollow bowl has material of volume 436π/3 cubic cm. Its external diameter is 14 cm. Find its thickness.
- Q9. The volume of a cone is 10055/7 cu. cm. The area of its base is 2011/7 sq. cm. Find the slant height of the cone.
- Q10. A metallic sheet in the form of a sector of a circle of radius 21 cm has central angle of 216°. The sector is made into…
Chapter 8: Statistics and Probability
Ex 8.1: 16 questions
- Q1(i). Find the range and coefficient of range of the following data. (i) 63, 89, 98, 125, 79, 108, 117, 68
- Q1(ii). Find the range and coefficient of range of the following data. (ii) 43.5, 13.6, 18.9, 38.4, 61.4, 29.8
- Q2. If the range and the smallest value of a set of data are 36.8 and 13.4 respectively, then find the largest value.
- Q3. Calculate the range of the following data. · Income · 400-450 · 450-500 · 500-550 · 550-600 · 600-650 · Number of worke…
- Q4. A teacher asked the students to complete 60 pages of a record note book. Eight students have completed only 32, 35, 37,…
- Q5. Find the variance and standard deviation of the wages of 9 workers given below: ₹310, ₹290, ₹320, ₹280, ₹300, ₹290, ₹32…
- Q6. A wall clock strikes the bell once at 1 o' clock, 2 times at 2 o' clock, 3 times at 3 o' clock and so on. How many time…
- Q7. Find the standard deviation of first 21 natural numbers.
- Q8. If the standard deviation of a data is 4.5 and if each value of the data is decreased by 5, then find the new standard…
- Q9. If the standard deviation of a data is 3.6 and each value of the data is divided by 3, then find the new variance and n…
- Q10. The rainfall recorded in various places of five districts in a week are given below. Find its standard deviation. · Rai…
- Q11. In a study about viral fever, the number of people affected in a town were noted as Find its standard deviation. · Age…
- Q12. The measurements of the diameters (in cms) of the plates prepared in a factory are given below. Find its standard devia…
- Q13. The time taken by 50 students to complete a 100 meter race are given below. Find its standard deviation. · Time taken (…
- Q14. For a group of 100 candidates the mean and standard deviation of their marks were found to be 60 and 15 respectively. L…
- Q15. The mean and variance of seven observations are 8 and 16 respectively. If five of these are 2, 4, 10, 12 and 14, then f…
Ex 8.2: 8 questions
- Q1. The standard deviation and mean of a data are 6.5 and 12.5 respectively. Find the coefficient of variation.
- Q2. The standard deviation and coefficient of variation of a data are 1.2 and 25.6 respectively. Find the value of mean.
- Q3. If the mean and coefficient of variation of a data are 15 and 48 respectively, then find the value of standard deviatio…
- Q4. If n = 5, x = 6, Σ x² = 765, then calculate the coefficient of variation.
- Q5. Find the coefficient of variation of 24, 26, 33, 37, 29, 31.
- Q6. The time taken (in minutes) to complete a homework by 8 students in a day are given by 38, 40, 47, 44, 46, 43, 49, 53.…
- Q7. The total marks scored by two students Sathya and Vidhya in 5 subjects are 460 and 480 with standard deviation 4.6 and…
- Q8. The mean and standard deviation of marks obtained by 40 students of a class in three subjects Mathematics, Science and…
Ex 8.3: 29 questions
- Q1. Write the sample space for tossing three coins using tree diagram.
- Q2. Write the sample space for selecting two balls at a time from a bag containing 6 balls numbered 1 to 6 (using tree diag…
- Q3(i). If A is an event of a random experiment such that P(A): P(A) = 17:15 and n(S)=640 then find (i) P(A).
- Q3(ii). If A is an event of a random experiment such that P(A): P(A) = 17:15 and n(S)=640 then find (ii) n(A).
- Q4. A coin is tossed thrice. What is the probability of getting two consecutive tails?
- Q5(i). At a fete, cards bearing numbers 1 to 1000, one number on one card are put in a box. Each player selects one card at ra…
- Q5(ii). At a fete, cards bearing numbers 1 to 1000, one number on one card are put in a box. Each player selects one card at ra…
- Q6(i). A bag contains 12 blue balls and x red balls. If one ball is drawn at random (i) what is the probability that it will b…
- Q6(ii). A bag contains 12 blue balls and x red balls. If one ball is drawn at random (ii) If 8 more red balls are put in the ba…
- Q7(i). Two unbiased dice are rolled once. Find the probability of getting (i) a doublet (equal numbers on both dice).
- Q7(ii). Two unbiased dice are rolled once. Find the probability of getting (ii) the product as a prime number.
- Q7(iii). Two unbiased dice are rolled once. Find the probability of getting (iii) the sum as a prime number.
- Q7(iv). Two unbiased dice are rolled once. Find the probability of getting (iv) the sum as 1.
- Q8(i). Three fair coins are tossed together. Find the probability of getting (i) all heads.
- Q8(ii). Three fair coins are tossed together. Find the probability of getting (ii) atleast one tail.
- Q8(iii). Three fair coins are tossed together. Find the probability of getting (iii) atmost one head.
- Q8(iv). Three fair coins are tossed together. Find the probability of getting (iv) atmost two tails.
- Q9(i). A bag contains 5 red balls, 6 white balls, 7 green balls, 8 black balls. One ball is drawn at random from the bag. Find…
- Q9(ii). A bag contains 5 red balls, 6 white balls, 7 green balls, 8 black balls. One ball is drawn at random from the bag. Find…
- Q9(iii). A bag contains 5 red balls, 6 white balls, 7 green balls, 8 black balls. One ball is drawn at random from the bag. Find…
- Q9(iv). A bag contains 5 red balls, 6 white balls, 7 green balls, 8 black balls. One ball is drawn at random from the bag. Find…
- Q10. In a box there are 20 non-defective and some defective bulbs. If the probability that a bulb selected at random from th…
- Q11. Some boys are playing a game, in which the stone thrown by them landing in a circular region (given in the figure) is c…
- Q12(i). Two customers Priya and Amuthan are visiting a particular shop in the same week (Monday to Saturday). Each is equally l…
- Q12(ii). Two customers Priya and Amuthan are visiting a particular shop in the same week (Monday to Saturday). Each is equally l…
- Q12(iii). Two customers Priya and Amuthan are visiting a particular shop in the same week (Monday to Saturday). Each is equally l…
- Q13(i). In a game, the entry fee is ₹150. The game consists of tossing a coin 3 times. Dhana bought a ticket for entry. If one…
- Q13(ii). In a game, the entry fee is ₹150. The game consists of tossing a coin 3 times. Dhana bought a ticket for entry. If one…
- Q13(iii). In a game, the entry fee is ₹150. The game consists of tossing a coin 3 times. Dhana bought a ticket for entry. If one…
Ex 8.4: 15 questions
- Q1. If P(A) = 2/3, P(B) = 2/5, P(A ∪ B) = 1/3 then find P(A ∩ B).
- Q2(i). A and B are two events such that, P(A) = 0.42, P(B) = 0.48, and P(A ∩ B) = 0.16. Find (i) P(not A)
- Q2(ii). A and B are two events such that, P(A) = 0.42, P(B) = 0.48, and P(A ∩ B) = 0.16. Find (ii) P(not B)
- Q2(iii). A and B are two events such that, P(A) = 0.42, P(B) = 0.48, and P(A ∩ B) = 0.16. Find (iii) P(A or B)
- Q3. If A and B are two mutually exclusive events of a random experiment and P(not A) = 0.45, P(A ∪ B) = 0.65, then find P(B…
- Q4. The probability that atleast one of A and B occur is 0.6. If A and B occur simultaneously with probability 0.2, then fi…
- Q5. The probability of happening of an event A is 0.5 and that of B is 0.3. If A and B are mutually exclusive events, then…
- Q6. Two dice are rolled once. Find the probability of getting an even number on the first die or a total of face sum 8.
- Q7. A box contains cards numbered 3,\ 5,\ 7,\ 9,\ …\ 35,\ 37. A card is drawn at random from the box. Find the probability…
- Q8. Three unbiased coins are tossed once. Find the probability of getting atmost 2 tails or atleast 2 heads.
- Q9. The probability that a person will get an electrification contract is 3/5 and the probability that he will not get plum…
- Q10. In a town of 8000 people, 1300 are over 50 years and 3000 are females. It is known that 30\% of the females are over 50…
- Q11. A coin is tossed thrice. Find the probability of getting exactly two heads or atleast one tail or two consecutive heads.
- Q12. If A,\ B,\ C are any three events such that probability of B is twice as that of probability of A and probability of C…
- Q13. In a class of 35, students are numbered from 1 to 35. The ratio of boys to girls is 4:3. The roll numbers of students b…
Ex 8.5: 15 questions
- Q1. Which of the following is not a measure of dispersion?
- Q2. The range of the data 8, 8, 8, 8, 8... 8 is
- Q3. The sum of all deviations of the data from its mean is
- Q4. The mean of 100 observations is 40 and their standard deviation is 3. The sum of squares of all observations is
- Q5. Variance of first 20 natural numbers is
- Q6. The standard deviation of a data is 3. If each value is multiplied by 5 then the new variance is
- Q7. If the standard deviation of x,\ y,\ z is p then the standard deviation of 3x+5, 3y+5, 3z+5 is
- Q8. If the mean and coefficient of variation of a data are 4 and 87.5\% then the standard deviation is
- Q9. Which of the following is incorrect?
- Q10. The probability a red marble selected at random from a jar containing p red, q blue and r green marbles is
- Q11. A page is selected at random from a book. The probability that the digit at units place of the page number chosen is le…
- Q12. The probability of getting a job for a person is x/3. If the probability of not getting the job is 2/3 then the value o…
- Q13. Kamalam went to play a lucky draw contest. 135 tickets of the lucky draw were sold. If the probability of Kamalam winni…
- Q14. If a letter is chosen at random from the English alphabets {a,b,…,z}, then the probability that the letter chosen prece…
- Q15. A purse contains 10 notes of ₹2000, 15 notes of ₹500, and 25 notes of ₹200. One note is drawn at random. What is the pr…
Unit Exercise 8: 11 questions
- Q1. The mean of the following frequency distribution is 62.8 and the sum of all frequencies is 50. Compute the missing freq…
- Q2. The diameter of circles (in mm) drawn in a design are given below. Calculate the standard deviation. · Diameters · 33-3…
- Q3. The frequency distribution is given below. In the table, k is a positive integer, has a varience of 160. Determine the…
- Q4. The standard deviation of some temperature data in degree celsius (oC) is 5. If the data were converted into degree Far…
- Q5. If for a distribution, sum (x-5) = 3, sum (x-5)² = 43, and total number of observations is 18, find the mean and standa…
- Q6. Prices of peanut packets in various places of two cities are given below. In which city, prices were more stable? · Pri…
- Q7. If the range and coefficient of range of the data are 20 and 0.2 respectively, then find the largest and smallest value…
- Q8. If two dice are rolled, then find the probability of getting the product of face value 6 or the difference of face valu…
- Q9. In a two children family, find the probability that there is at least one girl in a family.
- Q10. A bag contains 5 white and some black balls. If the probability of drawing a black ball from the bag is twice the proba…
- Q11. The probability that a student will pass the final examination in both English and Tamil is 0.5 and the probability of…