10th Maths One Mark Questions with Answers

10th Maths One Mark Questions with answers: every one-mark question from the Samacheer Kalvi textbook, chapter by chapter.

Chapter 1: Relations and Functions

If $n(A \times B) = 6$ and $A = \{1,3\}$ then $n(B)$ is
1. (1) 1
2. (2) 2
3. (3) 3
4. (4) 6
Show answer
Answer: (3) 3
1. $n(A\times B)=n(A)\times n(B)$ .
2. $6=2\times n(B)$ .
3. $n(B)=3$ .
Question 1 of 15
$A = \{a,b,p\}$ , $B = \{2,3\}$ , $C = \{p,q,r,s\}$ then $n[(A \cup C) \times B]$ is
1. (1) 8
2. (2) 20
3. (3) 12
4. (4) 16
Show answer
Answer: (3) 12
1. $A\cup C=\{a,b,p,q,r,s\}$ , so $n(A\cup C)=6$ .
2. $n(B)=2$ .
3. $n[(A\cup C)\times B]=6\times 2=12$ .
Question 2 of 15
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.
1. (1) $(A \times C) \subset (B \times D)$
2. (2) $(B \times D) \subset (A \times C)$
3. (3) $(A \times B) \subset (A \times D)$
4. (4) $(D \times A) \subset (B \times A)$
Show answer
Answer: (1) $(A \times C) \subset (B \times D)$
1. $A=\{1,2\}\subset B=\{1,2,3,4\}$ and $C=\{5,6\}\subset D=\{5,6,7,8\}$ .
2. So every pair of $A\times C$ is also a pair of $B\times D$ .
3. Hence $(A\times C)\subset (B\times D)$ .
Question 3 of 15
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
1. (1) 3
2. (2) 2
3. (3) 4
4. (4) 8
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Answer: (2) 2
1. Number of relations from $A$ to $B$ is $2^{n(A)\cdot n(B)}$ .
2. $2^{5\,n(B)}=1024=2^{10}$ .
3. $5\,n(B)=10\Rightarrow n(B)=2$ .
Question 4 of 15
The range of the relation $R = \{(x, x^2) \mid x$ is a prime number less than 13 $\}$ is
1. (1) $\{2,3,5,7\}$
2. (2) $\{2,3,5,7,11\}$
3. (3) $\{4,9,25,49,121\}$
4. (4) $\{1,4,9,25,49,121\}$
Show answer
Answer: (3) $\{4,9,25,49,121\}$
1. Primes less than 13 are $2,3,5,7,11$ .
2. Range consists of the squares $x^{2}$ .
3. Range $=\{4,9,25,49,121\}$ .
Question 5 of 15
If the ordered pairs $(a + 2, 4)$ and $(5, 2a + b)$ are equal then $(a,b)$ is
1. (1) $(2, -2)$
2. (2) $(5,1)$
3. (3) $(2,3)$
4. (4) $(3, -2)$
Show answer
Answer: (4) $(3, -2)$
1. Equal ordered pairs: $a+2=5$ and $2a+b=4$ .
2. $a=3$ .
3. $2(3)+b=4\Rightarrow b=-2$ .
Question 6 of 15
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
1. (1) $m^n$
2. (2) $n^m$
3. (3) $2^{mn} - 1$
4. (4) $2^{mn}$
Show answer
Answer: (3) $2^{mn} - 1$
1. Total number of relations from $A$ to $B$ is $2^{mn}$ .
2. Remove the empty relation.
3. Non-empty relations $=2^{mn}-1$ .
Question 7 of 15
If $\{(a,8),(6,b)\}$ represents an identity function, then the value of $a$ and $b$ are respectively
1. (1) $(8,6)$
2. (2) $(8,8)$
3. (3) $(6,8)$
4. (4) $(6,6)$
Show answer
Answer: (1) $(8,6)$
1. For an identity function $f(x)=x$ , each pair is of the form $(x,x)$ .
2. $(a,8)\Rightarrow a=8$ .
3. $(6,b)\Rightarrow b=6$ .
Question 8 of 15
Let $A = \{1,2,3,4\}$ and $B = \{4,8,9,10\}$ . A function $f : A \to B$ given by $f = \{(1,4),(2,8),(3,9),(4,10)\}$ is a
1. (1) Many-one function
2. (2) Identity function
3. (3) One-to-one function
4. (4) Into function
Show answer
Answer: (3) One-to-one function
1. Images are $4,8,9,10$ — all different.
2. Different elements of $A$ have different images.
3. So $f$ is one-to-one.
Question 9 of 15
If $f(x) = 2x^2$ and $g(x) = \dfrac{1}{3x}$ , then $f \circ g$ is
1. (1) $\dfrac{3}{2x^2}$
2. (2) $\dfrac{2}{3x^2}$
3. (3) $\dfrac{2}{9x^2}$
4. (4) $\dfrac{1}{6x^2}$
Show answer
Answer: (3) $\dfrac{2}{9x^2}$
1. $(f\circ g)(x)=f\big(g(x)\big)=f\left(\dfrac{1}{3x}\right)$ .
2. $=2\left(\dfrac{1}{3x}\right)^{2}$ .
3. $=\dfrac{2}{9x^{2}}$ .
Question 10 of 15
If $f : A \to B$ is a bijective function and if $n(B) = 7$ , then $n(A)$ is equal to
1. (1) 7
2. (2) 49
3. (3) 1
4. (4) 14
Show answer
Answer: (1) 7
1. A bijection pairs each element of $A$ with exactly one element of $B$ .
2. So $n(A)=n(B)$ .
3. $n(A)=7$ .
Question 11 of 15
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 the range of $f \circ g$ is
1. (1) $\{0,2,3,4,5\}$
2. (2) $\{-4,1,0,2,7\}$
3. (3) $\{1,2,3,4,5\}$
4. (4) $\{0,1,2\}$
Show answer
Answer: (4) $\{0,1,2\}$
1. $(f\circ g)(0)=f(2)=0$ , $(f\circ g)(1)=f(0)=1$ .
2. $(f\circ g)(2)=f(4)=2$ , $(f\circ g)(-4)=f(2)=0$ , $(f\circ g)(7)=f(0)=1$ .
3. Range $=\{0,1,2\}$ .
Question 12 of 15
Let $f(x) = \sqrt{1 + x^2}$ then
1. (1) $f(xy) = f(x).f(y)$
2. (2) $f(xy) \geq f(x).f(y)$
3. (3) $f(xy) \leq f(x).f(y)$
4. (4) None of these
Show answer
Answer: (3) $f(xy) \leq f(x).f(y)$
1. $f(xy)=\sqrt{1+x^{2}y^{2}}$ .
2. $f(x)\cdot f(y)=\sqrt{(1+x^{2})(1+y^{2})}=\sqrt{1+x^{2}+y^{2}+x^{2}y^{2}}$ .
3. Since $x^{2}+y^{2}\ge 0$ , $\;1+x^{2}y^{2}\le 1+x^{2}+y^{2}+x^{2}y^{2}$ .
4. Hence $f(xy)\le f(x)\cdot f(y)$ .
Question 13 of 15
If $g = \{(1,1),(2,3),(3,5),(4,7)\}$ is a function given by $g(x) = \alpha x + \beta$ then the values of $\alpha$ and $\beta$ are
1. (1) $(-1,2)$
2. (2) $(2, -1)$
3. (3) $(-1, -2)$
4. (4) $(1,2)$
Show answer
Answer: (2) $(2, -1)$
1. $g(1)=1\Rightarrow \alpha+\beta=1$ .
2. $g(2)=3\Rightarrow 2\alpha+\beta=3$ .
3. Subtracting, $\alpha=2$ , then $\beta=-1$ .
Question 14 of 15
$f(x) = (x + 1)^3 - (x - 1)^3$ represents a function which is
1. (1) linear
2. (2) cubic
3. (3) reciprocal
4. (4) quadratic
Show answer
Answer: (4) quadratic
1. $(x+1)^{3}=x^{3}+3x^{2}+3x+1$ , $(x-1)^{3}=x^{3}-3x^{2}+3x-1$ .
2. Subtracting, $f(x)=6x^{2}+2$ .
3. Highest power is $2$ , so it is quadratic.
Question 15 of 15

Chapter 2: Numbers and Sequences

Euclid's division lemma states that for positive integers $a$ and $b$ , there exist unique integers $q$ and $r$ such that $a = bq + r$ , where $r$ must satisfy.
1. (1) $1 < r < b$
2. (2) $0 < r < b$
3. (3) $0 \leq r < b$
4. (4) $0 < r \leq b$
Show answer
Answer: (3) $0 \leq r < b$
1. In $a = bq + r$ , the remainder $r$ can be zero (when $b$ divides $a$ ) but must be less than the divisor $b$ .
2. So $0 \le r < b$ .
Question 1 of 15
Using Euclid's division lemma, if the cube of any positive integer is divided by $9$ then the possible remainders are
1. (1) $0, 1, 8$
2. (2) $1, 4, 8$
3. (3) $0, 1, 3$
4. (4) $1, 3, 5$
Show answer
Answer: (1) $0, 1, 8$
1. Any positive integer is of the form $3q$ , $3q+1$ or $3q+2$ .
2. $(3q)^3 = 27q^3 = 9(3q^3) \Rightarrow$ remainder $0$ .
3. $(3q+1)^3 = 27q^3+27q^2+9q+1 = 9(3q^3+3q^2+q)+1 \Rightarrow$ remainder $1$ .
4. $(3q+2)^3 = 27q^3+54q^2+36q+8 = 9(3q^3+6q^2+4q)+8 \Rightarrow$ remainder $8$ .
Question 2 of 15
If the HCF of $65$ and $117$ is expressible in the form of $65m - 117$ , then the value of $m$ is
1. (1) $4$
2. (2) $2$
3. (3) $1$
4. (4) $3$
Show answer
Answer: (2) $2$
1. $117 = 65 \times 1 + 52$ , $65 = 52 \times 1 + 13$ , $52 = 13 \times 4 + 0$ .
2. So HCF $(65,117) = 13$ .
3. $65m - 117 = 13 \Rightarrow 65m = 130 \Rightarrow m = 2$ .
Question 3 of 15
The sum of the exponents of the prime factors in the prime factorization of $1729$ is
1. (1) $1$
2. (2) $2$
3. (3) $3$
4. (4) $4$
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Answer: (3) $3$
1. $1729 = 7 \times 13 \times 19 = 7^1 \times 13^1 \times 19^1$ .
2. Sum of exponents $= 1 + 1 + 1 = 3$ .
Question 4 of 15
The least number that is divisible by all the numbers from $1$ to $10$ (both inclusive) is
1. (1) $2025$
2. (2) $5220$
3. (3) $5025$
4. (4) $2520$
Show answer
Answer: (4) $2520$
1. The least such number is the LCM of $1,2,\ldots,10$ .
2. $\text{LCM} = 2^3 \times 3^2 \times 5 \times 7 = 2520$ .
Question 5 of 15
$7^{4k} \equiv$ _____ $(\text{mod } 100)$
1. (1) $1$
2. (2) $2$
3. (3) $3$
4. (4) $4$
Show answer
Answer: (1) $1$
1. $7^2 = 49$ , so $7^4 = 49^2 = 2401$ .
2. $2401 = 24 \times 100 + 1 \Rightarrow 7^4 \equiv 1 \pmod{100}$ .
3. Hence $7^{4k} = (7^4)^k \equiv 1^k \equiv 1 \pmod{100}$ .
Question 6 of 15
Given $F_1 = 1$ , $F_2 = 3$ and $F_n = F_{n-1} + F_{n-2}$ then $F_5$ is
1. (1) $3$
2. (2) $5$
3. (3) $8$
4. (4) $11$
Show answer
Answer: (4) $11$
1. $F_3 = F_2 + F_1 = 3 + 1 = 4$ .
2. $F_4 = F_3 + F_2 = 4 + 3 = 7$ .
3. $F_5 = F_4 + F_3 = 7 + 4 = 11$ .
Question 7 of 15
The first term of an arithmetic progression is unity and the common difference is $4$ . Which of the following will be a term of this A.P.
1. (1) $4551$
2. (2) $10091$
3. (3) $7881$
4. (4) $13531$
Show answer
Answer: (3) $7881$
1. $a = 1$ , $d = 4$ , so $t_n = 1 + (n-1)4 = 4n - 3$ .
2. Every term leaves remainder $1$ when divided by $4$ .
3. $7881 = 4 \times 1971 - 3$ , so $7881$ is the $1971^{\text{st}}$ term.
Question 8 of 15
If $6$ times of $6^{\text{th}}$ term of an A.P. is equal to $7$ times the $7^{\text{th}}$ term, then the $13^{\text{th}}$ term of the A.P. is
1. (1) $0$
2. (2) $6$
3. (3) $7$
4. (4) $13$
Show answer
Answer: (1) $0$
1. $6t_6 = 7t_7 \Rightarrow 6(a+5d) = 7(a+6d)$ .
2. $6a + 30d = 7a + 42d \Rightarrow a + 12d = 0$ .
3. $t_{13} = a + 12d = 0$ .
Question 9 of 15
An A.P. consists of $31$ terms. If its $16^{\text{th}}$ term is $m$ , then the sum of all the terms of this A.P. is
1. (1) $16\,m$
2. (2) $62\,m$
3. (3) $31\,m$
4. (4) $\dfrac{31}{2}\,m$
Show answer
Answer: (3) $31\,m$
1. $t_{16} = a + 15d = m$ .
2. $S_{31} = \dfrac{31}{2}\,[2a + 30d] = \dfrac{31}{2} \times 2(a + 15d)$ .
3. $= 31(a+15d) = 31m$ .
Question 10 of 15
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 to be equal to $120$ ?
1. (1) $6$
2. (2) $7$
3. (3) $8$
4. (4) $9$
Show answer
Answer: (3) $8$
1. $S_n = \dfrac{n}{2}[2(1) + (n-1)4] = \dfrac{n}{2}(4n - 2) = n(2n-1)$ .
2. $n(2n-1) = 120 \Rightarrow 2n^2 - n - 120 = 0$ .
3. $(n-8)(2n+15) = 0 \Rightarrow n = 8$ .
Question 11 of 15
If $A = 2^{65}$ and $B = 2^{64} + 2^{63} + 2^{62} + \cdots + 2^{0}$ which of the following is true?
1. (1) $B$ is $2^{64}$ more than $A$
2. (2) $A$ and $B$ are equal
3. (3) $B$ is larger than $A$ by $1$
4. (4) $A$ is larger than $B$ by $1$
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Answer: (4) $A$ is larger than $B$ by $1$
1. $B = 2^0 + 2^1 + \cdots + 2^{64}$ is a G.P. with $a = 1$ , $r = 2$ , $n = 65$ .
2. $B = \dfrac{1(2^{65} - 1)}{2 - 1} = 2^{65} - 1$ .
3. $A - B = 2^{65} - (2^{65} - 1) = 1$ .
Question 12 of 15
The next term of the sequence $\dfrac{3}{16}, \dfrac{1}{8}, \dfrac{1}{12}, \dfrac{1}{18}, \cdots$ is
1. (1) $\dfrac{1}{24}$
2. (2) $\dfrac{1}{27}$
3. (3) $\dfrac{2}{3}$
4. (4) $\dfrac{1}{81}$
Show answer
Answer: (2) $\dfrac{1}{27}$
1. $\dfrac{1/8}{3/16} = \dfrac{16}{24} = \dfrac{2}{3}$ , $\dfrac{1/12}{1/8} = \dfrac{2}{3}$ , $\dfrac{1/18}{1/12} = \dfrac{2}{3}$ .
2. It is a G.P. with $r = \dfrac{2}{3}$ .
3. Next term $= \dfrac{1}{18} \times \dfrac{2}{3} = \dfrac{1}{27}$ .
Question 13 of 15
If the sequence $t_1, t_2, t_3, \ldots$ are in A.P. then the sequence $t_6, t_{12}, t_{18}, \ldots$ is
1. (1) a Geometric Progression
2. (2) an Arithmetic Progression
3. (3) neither an Arithmetic Progression nor a Geometric Progression
4. (4) a constant sequence
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Answer: (2) an Arithmetic Progression
1. Let the A.P. have first term $a$ and common difference $d$ .
2. $t_6 = a+5d$ , $t_{12} = a+11d$ , $t_{18} = a+17d$ .
3. Difference $= 6d$ , a constant, so the new sequence is an A.P.
Question 14 of 15
The value of $(1^3 + 2^3 + 3^3 + \cdots + 15^3) - (1 + 2 + 3 + \cdots + 15)$ is
1. (1) $14400$
2. (2) $14200$
3. (3) $14280$
4. (4) $14520$
Show answer
Answer: (3) $14280$
1. $1^3+2^3+\cdots+15^3 = \left[\dfrac{15 \times 16}{2}\right]^2 = 120^2 = 14400$ .
2. $1+2+\cdots+15 = \dfrac{15 \times 16}{2} = 120$ .
3. $14400 - 120 = 14280$ .
Question 15 of 15

Chapter 3: Algebra

A system of three linear equations in three variables is inconsistent if their planes
1. (1) intersect only at a point
2. (2) intersect in a line
3. (3) coincides with each other
4. (4) do not intersect
Show answer
Answer: (4) do not intersect
1. Inconsistent means the system has no solution.
2. No solution happens only when the three planes have no common point.
3. So the planes do not intersect.
Question 1 of 20
The solution of the system $x + y - 3z = -6$ , $-7y + 7z = 7$ , $3z = 9$ is
1. (1) $x = 1,\ y = 2,\ z = 3$
2. (2) $x = -1,\ y = 2,\ z = 3$
3. (3) $x = -1,\ y = -2,\ z = 3$
4. (4) $x = 1,\ y = -2,\ z = 3$
Show answer
Answer: (1) $x = 1,\ y = 2,\ z = 3$
1. From $3z = 9$ , $z = 3$ .
2. Put $z=3$ in $-7y + 7z = 7$ : $-7y + 21 = 7 \Rightarrow y = 2$ .
3. Put $y=2,\ z=3$ in $x + y - 3z = -6$ : $x + 2 - 9 = -6 \Rightarrow x = 1$ .
Question 2 of 20
If $(x - 6)$ is the HCF of $x^2 - 2x - 24$ and $x^2 - kx - 6$ then the value of $k$ is
1. (1) $3$
2. (2) $5$
3. (3) $6$
4. (4) $8$
Show answer
Answer: (2) $5$
1. $x^2 - 2x - 24 = (x-6)(x+4)$ , so $(x-6)$ is a factor.
2. Since $(x-6)$ is the HCF, it must divide $x^2 - kx - 6$ .
3. So $x = 6$ makes it zero: $36 - 6k - 6 = 0$ .
4. $6k = 30 \Rightarrow k = 5$ .
Question 3 of 20
$\dfrac{3y - 3}{y} \div \dfrac{7y - 7}{3y^2}$ is
1. (1) $\dfrac{9y}{7}$
2. (2) $\dfrac{9y^3}{(21y - 21)}$
3. (3) $\dfrac{21y^2 - 42y + 21}{3y^3}$
4. (4) $\dfrac{7(y^2 - 2y + 1)}{y^2}$
Show answer
Answer: (1) $\dfrac{9y}{7}$
1. $\dfrac{3y-3}{y} \div \dfrac{7y-7}{3y^2} = \dfrac{3(y-1)}{y} \times \dfrac{3y^2}{7(y-1)}$
2. Cancel $(y-1)$ and one $y$ .
3. $= \dfrac{9y}{7}$
Question 4 of 20
$y^2 + \dfrac{1}{y^2}$ is not equal to
1. (1) $\dfrac{y^4 + 1}{y^2}$
2. (2) $\left(y + \dfrac{1}{y}\right)^2$
3. (3) $\left(y - \dfrac{1}{y}\right)^2 + 2$
4. (4) $\left(y + \dfrac{1}{y}\right)^2 - 2$
Show answer
Answer: (2) $\left(y + \dfrac{1}{y}\right)^2$
1. $\left(y + \dfrac{1}{y}\right)^2 = y^2 + 2 + \dfrac{1}{y^2}$ , which is $2$ more than $y^2 + \dfrac{1}{y^2}$ .
2. Options (A), (C), (D) all simplify to $y^2 + \dfrac{1}{y^2}$ .
3. So (B) is the one that is not equal.
Question 5 of 20
$\dfrac{x}{x^2 - 25} - \dfrac{8}{x^2 + 6x + 5}$ gives
1. (1) $\dfrac{x^2 - 7x + 40}{(x - 5)(x + 5)}$
2. (2) $\dfrac{x^2 + 7x + 40}{(x - 5)(x + 5)(x + 1)}$
3. (3) $\dfrac{x^2 - 7x + 40}{(x^2 - 25)(x + 1)}$
4. (4) $\dfrac{x^2 + 10}{(x^2 - 25)(x + 1)}$
Show answer
Answer: (3) $\dfrac{x^2 - 7x + 40}{(x^2 - 25)(x + 1)}$
1. $x^2 - 25 = (x+5)(x-5)$ and $x^2 + 6x + 5 = (x+5)(x+1)$ .
2. LCM $= (x+5)(x-5)(x+1)$ .
3. $\dfrac{x(x+1) - 8(x-5)}{(x+5)(x-5)(x+1)} = \dfrac{x^2 + x - 8x + 40}{(x^2-25)(x+1)}$
4. $= \dfrac{x^2 - 7x + 40}{(x^2-25)(x+1)}$
Question 6 of 20
The square root of $\dfrac{256 x^8 y^4 z^{10}}{25 x^6 y^6 z^6}$ is equal to
1. (1) $\dfrac{16}{5}\left|\dfrac{x^2 z^4}{y^2}\right|$
2. (2) $16\left|\dfrac{y^2}{x^2 z^4}\right|$
3. (3) $\dfrac{16}{5}\left|\dfrac{y}{xz^2}\right|$
4. (4) $\dfrac{16}{5}\left|\dfrac{xz^2}{y}\right|$
Show answer
Answer: (4) $\dfrac{16}{5}\left|\dfrac{xz^2}{y}\right|$
1. $\dfrac{256x^8y^4z^{10}}{25x^6y^6z^6} = \dfrac{256\,x^2 z^4}{25\,y^2}$
2. Taking square root: $\dfrac{16}{5}\left|\dfrac{x z^2}{y}\right|$
Question 7 of 20
Which of the following should be added to make $x^4 + 64$ a perfect square
1. (1) $4x^2$
2. (2) $16x^2$
3. (3) $8x^2$
4. (4) $-8x^2$
Show answer
Answer: (2) $16x^2$
1. For $(x^2 + 8)^2 = x^4 + 16x^2 + 64$ .
2. Given expression has $x^4 + 64$ , so the missing middle term is $16x^2$ .
Question 8 of 20
The solution of $\left(2x - 1\right)^2 = 9$ is equal to
1. (1) $-1$
2. (2) $2$
3. (3) $-1,\ 2$
4. (4) None of these
Show answer
Answer: (3) $-1,\ 2$
1. $(2x-1)^2 = 9 \Rightarrow 2x - 1 = \pm 3$ .
2. $2x - 1 = 3 \Rightarrow x = 2$ .
3. $2x - 1 = -3 \Rightarrow x = -1$ .
Question 9 of 20
The values of $a$ and $b$ if $4x^4 - 24x^3 + 76x^2 + ax + b$ is a perfect square are
1. (1) $100,120$
2. (2) $10,12$
3. (3) $-120,100$
4. (4) $12,10$
Show answer
Answer: (3) $-120,100$
1. Let $4x^4 - 24x^3 + 76x^2 + ax + b = (2x^2 - 6x + c)^2$ .
2. Expanding: $4x^4 - 24x^3 + (36 + 4c)x^2 - 12cx + c^2$ .
3. Compare $x^2$ terms: $36 + 4c = 76 \Rightarrow c = 10$ .
4. So $a = -12(10) = -120$ and $b = 10^2 = 100$ .
Question 10 of 20
If the roots of the equation $q^2x^2 + p^2x + r^2 = 0$ are the squares of the roots of the equation $qx^2 + px + r = 0$ , then $q$ , $p$ , $r$ are in _______
1. (1) $A.P$
2. (2) $G.P$
3. (3) Both $A.P$ and $G.P$
4. (4) none of these
Show answer
Answer: (2) $G.P$
1. Let $\alpha,\beta$ be roots of $qx^2 + px + r = 0$ : $\alpha+\beta = -\dfrac{p}{q}$ , $\alpha\beta = \dfrac{r}{q}$ .
2. For $q^2x^2 + p^2x + r^2 = 0$ : $\alpha^2 + \beta^2 = -\dfrac{p^2}{q^2}$ , $\alpha^2\beta^2 = \dfrac{r^2}{q^2}$ .
3. Now $\alpha^2+\beta^2 = (\alpha+\beta)^2 - 2\alpha\beta = \dfrac{p^2}{q^2} - \dfrac{2r}{q}$ .
4. So $\dfrac{p^2}{q^2} - \dfrac{2r}{q} = -\dfrac{p^2}{q^2} \Rightarrow \dfrac{2p^2}{q^2} = \dfrac{2r}{q} \Rightarrow p^2 = qr$ .
5. Hence $q, p, r$ are in G.P.
Question 11 of 20
Graph of a linear equation is a _______
1. (1) straight line
2. (2) circle
3. (3) parabola
4. (4) hyperbola
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Answer: (1) straight line
1. A linear equation in two variables has degree one, and its graph is a straight line.
Question 12 of 20
The number of points of intersection of the quadratic polynomial $x^2 + 4x + 4$ with the $X$ axis is
1. (1) $0$
2. (2) $1$
3. (3) $0$ or $1$
4. (4) $2$
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Answer: (2) $1$
1. $x^2 + 4x + 4 = (x+2)^2$ .
2. It has equal roots $x = -2, -2$ .
3. So the curve touches the $X$ axis at exactly one point.
Question 13 of 20
For the given matrix $A = \begin{pmatrix} 1 & 3 & 5 & 7 \\ 2 & 4 & 6 & 8 \\ 9 & 11 & 13 & 15 \end{pmatrix}$ the order of the matrix $A^T$ is
1. (1) $2 \times 3$
2. (2) $3 \times 2$
3. (3) $3 \times 4$
4. (4) $4 \times 3$
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Answer: (4) $4 \times 3$
1. $A$ has $3$ rows and $4$ columns, so order of $A$ is $3 \times 4$ .
2. Transpose interchanges rows and columns, so $A^T$ is $4 \times 3$ .
Question 14 of 20
If $A$ is a $2 \times 3$ matrix and $B$ is a $3 \times 4$ matrix, how many columns does $AB$ have
1. (1) $3$
2. (2) $4$
3. (3) $2$
4. (4) $5$
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Answer: (2) $4$
1. $A_{2\times 3} B_{3\times 4}$ gives order $2 \times 4$ .
2. So $AB$ has $4$ columns.
Question 15 of 20
If number of columns and rows are not equal in a matrix then it is said to be a
1. (1) diagonal matrix
2. (2) rectangular matrix
3. (3) square matrix
4. (4) identity matrix
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Answer: (2) rectangular matrix
1. A matrix with number of rows $\ne$ number of columns is a rectangular matrix.
Question 16 of 20
Transpose of a column matrix is
1. (1) unit matrix
2. (2) diagonal matrix
3. (3) column matrix
4. (4) row matrix
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Answer: (4) row matrix
1. A column matrix has order $m \times 1$ .
2. Its transpose has order $1 \times m$ , which is a row matrix.
Question 17 of 20
Find the matrix $X$ if $2X + \begin{pmatrix} 1 & 3 \\ 5 & 7 \end{pmatrix} = \begin{pmatrix} 5 & 7 \\ 9 & 5 \end{pmatrix}$
1. (1) $\begin{pmatrix} -2 & -2 \\ 2 & -1 \end{pmatrix}$
2. (2) $\begin{pmatrix} 2 & 2 \\ 2 & -1 \end{pmatrix}$
3. (3) $\begin{pmatrix} 1 & 2 \\ 2 & 2 \end{pmatrix}$
4. (4) $\begin{pmatrix} 2 & 1 \\ 2 & 2 \end{pmatrix}$
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Answer: (2) $\begin{pmatrix} 2 & 2 \\ 2 & -1 \end{pmatrix}$
1. $2X = \begin{pmatrix} 5 & 7 \\ 9 & 5 \end{pmatrix} - \begin{pmatrix} 1 & 3 \\ 5 & 7 \end{pmatrix} = \begin{pmatrix} 4 & 4 \\ 4 & -2 \end{pmatrix}$
2. $X = \dfrac{1}{2}\begin{pmatrix} 4 & 4 \\ 4 & -2 \end{pmatrix} = \begin{pmatrix} 2 & 2 \\ 2 & -1 \end{pmatrix}$
Question 18 of 20
Which of the following can be calculated from the given matrices $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{pmatrix}$ , $B = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$ , (i) $A^2$ (ii) $B^2$ (iii) $AB$ (iv) $BA$
1. (1) (i) and (ii) only
2. (2) (ii) and (iii) only
3. (3) (ii) and (iv) only
4. (4) all of these
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Answer: (3) (ii) and (iv) only
1. Order of $A$ is $3 \times 2$ and order of $B$ is $3 \times 3$ .
2. $A^2 = A\,A$ : $(3\times2)(3\times2)$ — columns of first $\ne$ rows of second, not defined.
3. $B^2$ : $(3\times3)(3\times3)$ — defined.
4. $AB$ : $(3\times2)(3\times3)$ — not defined.
5. $BA$ : $(3\times3)(3\times2)$ — defined.
Question 19 of 20
If $A = \begin{pmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \end{pmatrix}$ , $B = \begin{pmatrix} 1 & 0 \\ 2 & -1 \\ 0 & 2 \end{pmatrix}$ and $C = \begin{pmatrix} 0 & 1 \\ -2 & 5 \end{pmatrix}$ . Which of the following statements are correct? (i) $AB + C = \begin{pmatrix} 5 & 5 \\ 5 & 5 \end{pmatrix}$ (ii) $BC = \begin{pmatrix} 0 & 1 \\ 2 & -3 \\ -4 & 10 \end{pmatrix}$ (iii) $BA + C = \begin{pmatrix} 2 & 5 \\ 3 & 0 \end{pmatrix}$ (iv) $(AB)C = \begin{pmatrix} -8 & 20 \\ -8 & 13 \end{pmatrix}$
1. (1) (i) and (ii) only
2. (2) (ii) and (iii) only
3. (3) (iii) and (iv) only
4. (4) all of these
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Answer: (1) (i) and (ii) only
1. $AB = \begin{pmatrix} 5 & 4 \\ 7 & 0 \end{pmatrix}$ , so $AB + C = \begin{pmatrix} 5 & 5 \\ 5 & 5 \end{pmatrix}$ — (i) is correct.
2. $BC = \begin{pmatrix} 0 & 1 \\ 2 & -3 \\ -4 & 10 \end{pmatrix}$ — (ii) is correct.
3. $BA$ is of order $3\times3$ and $C$ is $2\times2$ , so $BA + C$ is not defined — (iii) is wrong.
4. $(AB)C = \begin{pmatrix} 5 & 4 \\ 7 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ -2 & 5 \end{pmatrix} = \begin{pmatrix} -8 & 25 \\ 0 & 7 \end{pmatrix}$ — (iv) is wrong.
Question 20 of 20

Chapter 4: Geometry

If in triangles $ABC$ and $EDF$ , $\dfrac{AB}{DE} = \dfrac{BC}{FD}$ then they will be similar, when
1. (1) $\angle B = \angle E$
2. (2) $\angle A = \angle D$
3. (3) $\angle B = \angle D$
4. (4) $\angle A = \angle F$
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Answer: (3) $\angle B = \angle D$
1. $\dfrac{AB}{DE}=\dfrac{BC}{FD}$
2. In $\Delta ABC$ the sides $AB$ , $BC$ include $\angle B$ .
3. In $\Delta EDF$ the sides $DE$ , $FD$ include $\angle D$ .
4. By SAS similarity, the included angles must be equal.
5. So $\angle B=\angle D$ .
Question 1 of 15
In $\Delta LMN$ , $\angle L = 60^\circ$ , $\angle M = 50^\circ$ . If $\Delta LMN \sim \Delta PQR$ then the value of $\angle R$ is
1. (1) $40^\circ$
2. (2) $70^\circ$
3. (3) $30^\circ$
4. (4) $110^\circ$
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Answer: (2) $70^\circ$
1. In $\Delta LMN$ , $\angle N = 180^\circ-(60^\circ+50^\circ)=70^\circ$ .
2. $\Delta LMN \sim \Delta PQR$ gives $\angle R = \angle N$ .
3. $\angle R = 70^\circ$ .
Question 2 of 15
If $\Delta ABC$ is an isosceles triangle with $\angle C = 90^\circ$ and $AC = 5$ cm, then $AB$ is
1. (1) $2.5$ cm
2. (2) $5$ cm
3. (3) $10$ cm
4. (4) $5\sqrt{2}$ cm
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Answer: (4) $5\sqrt{2}$ cm
1. $\angle C=90^\circ$ , so $AB$ is the hypotenuse and the equal sides are $AC=BC=5$ cm.
2. $AB^2=AC^2+BC^2=5^2+5^2=50$
3. $AB=\sqrt{50}=5\sqrt{2}$ cm.
Question 3 of 15
In a given figure $ST \parallel QR$ , $PS = 2$ cm and $SQ = 3$ cm. Then the ratio of the area of $\Delta PQR$ to the area of $\Delta PST$ is
1. (1) $25 : 4$
2. (2) $25 : 7$
3. (3) $25 : 11$
4. (4) $25 : 13$
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Answer: (1) $25 : 4$
1. $PQ=PS+SQ=2+3=5$ cm.
2. $ST\parallel QR \Rightarrow \Delta PST \sim \Delta PQR$ .
3. $\dfrac{[\Delta PQR]}{[\Delta PST]}=\dfrac{PQ^2}{PS^2}=\dfrac{5^2}{2^2}=\dfrac{25}{4}$ .
Question 4 of 15
The perimeters of two similar triangles $\Delta ABC$ and $\Delta PQR$ are $36$ cm and $24$ cm respectively. If $PQ = 10$ cm, then the length of $AB$ is
1. (1) $6\dfrac{2}{3}$ cm
2. (2) $\dfrac{10\sqrt{6}}{3}$ cm
3. (3) $66\dfrac{2}{3}$ cm
4. (4) $15$ cm
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Answer: (4) $15$ cm
1. In similar triangles, ratio of perimeters = ratio of corresponding sides.
2. $\dfrac{AB}{PQ}=\dfrac{36}{24}$
3. $AB=10\times\dfrac{36}{24}=15$ cm.
Question 5 of 15
If in $\Delta ABC$ , $DE \parallel BC$ . $AB = 3.6$ cm, $AC = 2.4$ cm and $AD = 2.1$ cm then the length of $AE$ is
1. (1) $1.4$ cm
2. (2) $1.8$ cm
3. (3) $1.2$ cm
4. (4) $1.05$ cm
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Answer: (1) $1.4$ cm
1. $DE\parallel BC$ , so by Thales theorem $\dfrac{AD}{AB}=\dfrac{AE}{AC}$ .
2. $\dfrac{2.1}{3.6}=\dfrac{AE}{2.4}$
3. $AE=\dfrac{2.1\times 2.4}{3.6}=1.4$ cm.
Question 6 of 15
In a $\Delta ABC$ , $AD$ is the bisector of $\angle BAC$ . If $AB = 8$ cm, $BD = 6$ cm and $DC = 3$ cm. The length of the side $AC$ is
1. (1) $6$ cm
2. (2) $4$ cm
3. (3) $3$ cm
4. (4) $8$ cm
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Answer: (2) $4$ cm
1. $AD$ bisects $\angle BAC$ , so $\dfrac{AB}{AC}=\dfrac{BD}{DC}$ .
2. $\dfrac{8}{AC}=\dfrac{6}{3}=2$
3. $AC=\dfrac{8}{2}=4$ cm.
Question 7 of 15
In the adjacent figure $\angle BAC = 90^\circ$ and $AD \perp BC$ then
1. (1) $BD \cdot CD = BC^2$
2. (2) $AB.AC = BC^2$
3. (3) $BD \cdot CD = AD^2$
4. (4) $AB \cdot AC = AD^2$
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Answer: (3) $BD \cdot CD = AD^2$
1. $\angle BAC=90^\circ$ and $AD\perp BC$ , so $AD$ is the altitude to the hypotenuse.
2. Then $\Delta ABD \sim \Delta CAD$ .
3. $\dfrac{BD}{AD}=\dfrac{AD}{CD}$
4. $BD\cdot CD = AD^2$ .
Question 8 of 15
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 is the distance between their tops?
1. (1) $13$ m
2. (2) $14$ m
3. (3) $15$ m
4. (4) $12.8$ m
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Answer: (1) $13$ m
1. Difference of heights $=11-6=5$ m.
2. Distance between feet $=12$ m.
3. Distance between tops $=\sqrt{12^2+5^2}=\sqrt{144+25}=\sqrt{169}$
4. $=13$ m.
Question 9 of 15
In the given figure, $PR = 26$ cm, $QR = 24$ cm, $\angle PAQ = 90^\circ$ , $PA = 6$ cm and $QA = 8$ cm. Find $\angle PQR$
1. (1) $80^\circ$
2. (2) $85^\circ$
3. (3) $75^\circ$
4. (4) $90^\circ$
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Answer: (4) $90^\circ$
1. In right $\Delta PAQ$ , $PQ^2=PA^2+QA^2=6^2+8^2=100$ , so $PQ=10$ cm.
2. Now $PQ^2+QR^2=10^2+24^2=100+576=676$ .
3. $PR^2=26^2=676$ .
4. Since $PQ^2+QR^2=PR^2$ , by converse of Pythagoras theorem $\angle PQR=90^\circ$ .
Question 10 of 15
A tangent is perpendicular to the radius at the
1. (1) centre
2. (2) point of contact
3. (3) infinity
4. (4) chord
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Answer: (2) point of contact
1. The radius drawn to the point where the tangent touches the circle is perpendicular to the tangent.
2. So it is perpendicular at the point of contact.
Question 11 of 15
How many tangents can be drawn to the circle from an exterior point?
1. (1) one
2. (2) two
3. (3) infinite
4. (4) zero
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Answer: (2) two
1. From a point outside a circle exactly two tangent lines can be drawn.
2. (From a point on the circle only one, from an interior point none.)
Question 12 of 15
The two tangents from an external points $P$ to a circle with centre at $O$ are $PA$ and $PB$ . If $\angle APB = 70^\circ$ then the value of $\angle AOB$ is
1. (1) $100^\circ$
2. (2) $110^\circ$
3. (3) $120^\circ$
4. (4) $130^\circ$
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Answer: (2) $110^\circ$
1. $OA\perp PA$ and $OB\perp PB$ , so $\angle OAP=\angle OBP=90^\circ$ .
2. In quadrilateral $OAPB$ , $\angle AOB+\angle APB+90^\circ+90^\circ=360^\circ$ .
3. $\angle AOB=360^\circ-180^\circ-70^\circ=110^\circ$ .
Question 13 of 15
In figure $CP$ and $CQ$ are tangents to a circle with centre at $O$ . $ARB$ is another tangent touching the circle at $R$ . If $CP = 11$ cm and $BC = 7$ cm, then the length of $BR$ is
1. (1) $6$ cm
2. (2) $5$ cm
3. (3) $8$ cm
4. (4) $4$ cm
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Answer: (4) $4$ cm
1. Tangents from $C$ : $CQ=CP=11$ cm.
2. $BQ=CQ-BC=11-7=4$ cm.
3. Tangents from $B$ : $BR=BQ$ .
4. $BR=4$ cm.
Question 14 of 15
In figure if $PR$ is tangent to the circle at $P$ and $O$ is the centre of the circle, then $\angle POQ$ is
1. (1) $120^\circ$
2. (2) $100^\circ$
3. (3) $110^\circ$
4. (4) $90^\circ$
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Answer: (1) $120^\circ$
1. $OP\perp PR$ (radius $\perp$ tangent), so $\angle OPR=90^\circ$ .
2. $\angle OPQ=90^\circ-60^\circ=30^\circ$ .
3. $OP=OQ$ (radii), so $\angle OQP=\angle OPQ=30^\circ$ .
4. $\angle POQ=180^\circ-(30^\circ+30^\circ)=120^\circ$ .
Question 15 of 15

Chapter 5: Coordinate Geometry

The area of triangle formed by the points $(-5,0)$ , $(0,-5)$ and $(5,0)$ is
1. (1) $0$ sq.units
2. (2) $25$ sq.units
3. (3) $5$ sq.units
4. (4) none of these
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Answer: (2) $25$ sq.units
1. The points $(-5,0)$ and $(5,0)$ lie on the X axis, so the base $=10$ units.
2. The third point $(0,-5)$ is at a distance $5$ from the X axis, so height $=5$ units.
3. Area $=\dfrac{1}{2}\times 10\times 5=25$ sq.units.
Question 1 of 15
A man walks near a wall, such that the distance between him and the wall is 10 units. Consider the wall to be the $Y$ axis. The path travelled by the man is
1. (1) $x = 10$
2. (2) $y = 10$
3. (3) $x = 0$
4. (4) $y = 0$
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Answer: (1) $x = 10$
1. The wall is the Y axis, and his distance from it is always $10$ units.
2. So his $x$ coordinate is always $10$ .
3. Path is the line $x=10$ .
Question 2 of 15
The straight line given by the equation $x = 11$ is
1. (1) parallel to $X$ axis
2. (2) parallel to $Y$ axis
3. (3) passing through the origin
4. (4) passing through the point $(0,11)$
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Answer: (2) parallel to $Y$ axis
1. $x=11$ means every point on it has $x$ coordinate $11$ , whatever be $y$ .
2. Such a line is vertical, hence parallel to the Y axis.
Question 3 of 15
If $(5,7)$ , $(3,p)$ and $(6,6)$ are collinear, then the value of $p$ is
1. (1) $3$
2. (2) $6$
3. (3) $9$
4. (4) $12$
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Answer: (3) $9$
1. Slope of the line joining $(5,7)$ and $(6,6)$ is $\dfrac{6-7}{6-5}=-1$ .
2. For collinearity, slope of $(5,7)$ and $(3,p)$ is also $-1$ .
3. $\dfrac{p-7}{3-5}=-1\Rightarrow p-7=2$
4. $p=9$
Question 4 of 15
The point of intersection of $3x - y = 4$ and $x + y = 8$ is
1. (1) $(5,3)$
2. (2) $(2,4)$
3. (3) $(3,5)$
4. (4) $(4,4)$
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Answer: (3) $(3,5)$
1. $3x-y=4$ and $x+y=8$ .
2. Adding: $4x=12\Rightarrow x=3$ .
3. Then $3+y=8\Rightarrow y=5$ .
4. Point of intersection is $(3,5)$ .
Question 5 of 15
The slope of the line joining $(12,3)$ , $(4,a)$ is $\dfrac{1}{8}$ . The value of ' $a$ ' is
1. (1) $1$
2. (2) $4$
3. (3) $-5$
4. (4) $2$
Show answer
Answer: (4) $2$
1. Slope $=\dfrac{a-3}{4-12}=\dfrac{a-3}{-8}$ .
2. $\dfrac{a-3}{-8}=\dfrac{1}{8}\Rightarrow a-3=-1$
3. $a=2$
Question 6 of 15
The slope of the line which is perpendicular to a line joining the points $(0,0)$ and $(-8,8)$ is
1. (1) $-1$
2. (2) $1$
3. (3) $\dfrac{1}{3}$
4. (4) $-8$
Show answer
Answer: (2) $1$
1. Slope of the line joining $(0,0)$ and $(-8,8)$ is $\dfrac{8-0}{-8-0}=-1$ .
2. Product of slopes of perpendicular lines $=-1$ .
3. Required slope $=\dfrac{-1}{-1}=1$ .
Question 7 of 15
If slope of the line $PQ$ is $\dfrac{1}{\sqrt{3}}$ then slope of the perpendicular bisector of $PQ$ is
1. (1) $\sqrt{3}$
2. (2) $-\sqrt{3}$
3. (3) $\dfrac{1}{\sqrt{3}}$
4. (4) $0$
Show answer
Answer: (2) $-\sqrt{3}$
1. The perpendicular bisector of $PQ$ is perpendicular to $PQ$ .
2. Slope of $PQ=\dfrac{1}{\sqrt{3}}$ .
3. Required slope $=-\dfrac{1}{(1/\sqrt{3})}=-\sqrt{3}$ .
Question 8 of 15
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 of the line $AB$ is
1. (1) $8x + 5y = 40$
2. (2) $8x - 5y = 40$
3. (3) $x = 8$
4. (4) $y = 5$
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Answer: (1) $8x + 5y = 40$
1. $A$ is on the Y axis with ordinate $8$ , so $A(0,8)$ ; $B$ is on the X axis with abscissa $5$ , so $B(5,0)$ .
2. Intercept form: $\dfrac{x}{5}+\dfrac{y}{8}=1$ .
3. Multiplying by $40$ : $8x+5y=40$ .
Question 9 of 15
The equation of a line passing through the origin and perpendicular to the line $7x - 3y + 4 = 0$ is
1. (1) $7x - 3y + 4 = 0$
2. (2) $3x - 7y + 4 = 0$
3. (3) $3x + 7y = 0$
4. (4) $7x - 3y = 0$
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Answer: (3) $3x + 7y = 0$
1. Slope of $7x-3y+4=0$ is $\dfrac{-7}{-3}=\dfrac{7}{3}$ .
2. Slope of the perpendicular line $=-\dfrac{3}{7}$ .
3. Through the origin: $y=-\dfrac{3}{7}x\Rightarrow 3x+7y=0$ .
Question 10 of 15
Consider four straight lines (i) $l_1;\ 3y = 4x + 5$ (ii) $l_2;\ 4y = 3x - 1$ (iii) $l_3;\ 4y + 3x = 7$ (iv) $l_4;\ 4x + 3y = 2$ Which of the following statement is true ?
1. (1) $l_1$ and $l_2$ are perpendicular
2. (2) $l_1$ and $l_4$ are parallel
3. (3) $l_2$ and $l_4$ are perpendicular
4. (4) $l_2$ and $l_3$ are parallel
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Answer: (3) $l_2$ and $l_4$ are perpendicular
1. Slopes: $l_1=\dfrac{4}{3}$ , $l_2=\dfrac{3}{4}$ , $l_3=-\dfrac{3}{4}$ , $l_4=-\dfrac{4}{3}$ .
2. $m_2\times m_4=\dfrac{3}{4}\times\left(-\dfrac{4}{3}\right)=-1$ .
3. So $l_2$ and $l_4$ are perpendicular.
Question 11 of 15
A straight line has equation $8y = 4x + 21$ . Which of the following is true
1. (1) The slope is 0.5 and the $y$ intercept is 2.6
2. (2) The slope is 5 and the $y$ intercept is 1.6
3. (3) The slope is 0.5 and the $y$ intercept is 1.6
4. (4) The slope is 5 and the $y$ intercept is 2.6
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Answer: (1) The slope is 0.5 and the $y$ intercept is 2.6
1. $8y=4x+21\Rightarrow y=\dfrac{4}{8}x+\dfrac{21}{8}$ .
2. $y=0.5x+2.625$ .
3. Slope $=0.5$ , $y$ intercept $=2.625\approx 2.6$ .
Question 12 of 15
When proving that a quadrilateral is a trapezium, it is necessary to show
1. (1) Two sides are parallel.
2. (2) Two parallel and two non-parallel sides.
3. (3) Opposite sides are parallel.
4. (4) All sides are of equal length.
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Answer: (2) Two parallel and two non-parallel sides.
1. A trapezium has exactly one pair of parallel sides.
2. So we must show two sides are parallel and the other two are not parallel.
Question 13 of 15
When proving that a quadrilateral is a parallelogram by using slopes you must find
1. (1) The slopes of two sides
2. (2) The slopes of two pair of opposite sides
3. (3) The lengths of all sides
4. (4) Both the lengths and slopes of two sides
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Answer: (2) The slopes of two pair of opposite sides
1. In a parallelogram both pairs of opposite sides are parallel.
2. Using slopes, we find the slopes of the two pairs of opposite sides and show each pair is equal.
Question 14 of 15
$(2,\ 1)$ is the point of intersection of two lines.
1. (1) $x - y - 3 = 0;\ 3x - y - 7 = 0$
2. (2) $x + y = 3;\ 3x + y = 7$
3. (3) $3x + y = 3;\ x + y = 7$
4. (4) $x + 3y - 3 = 0;\ x - y - 7 = 0$
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Answer: (2) $x + y = 3;\ 3x + y = 7$
1. Substitute $x=2,\ y=1$ in option (B).
2. $x+y=2+1=3$ ✓
3. $3x+y=6+1=7$ ✓
4. Both are satisfied, so $(2,1)$ is their point of intersection.
Question 15 of 15

Chapter 6: Trigonometry

The value of $\sin^2\theta + \dfrac{1}{1+\tan^2\theta}$ is equal to
1. (1) $\tan^2\theta$
2. (2) $1$
3. (3) $\cot^2\theta$
4. (4) $0$
Show answer
Answer: (2) $1$
1. $1+\tan^2\theta=\sec^2\theta$ , so $\dfrac{1}{1+\tan^2\theta}=\cos^2\theta$ .
2. $\sin^2\theta+\cos^2\theta=1$ .
Question 1 of 15
$\tan\theta\,\mathrm{cosec}^2\theta - \tan\theta$ is equal to
1. (1) $\sec\theta$
2. (2) $\cot^2\theta$
3. (3) $\sin\theta$
4. (4) $\cot\theta$
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Answer: (4) $\cot\theta$
1. $\tan\theta\,\mathrm{cosec}^2\theta-\tan\theta=\tan\theta(\mathrm{cosec}^2\theta-1)$ .
2. $\mathrm{cosec}^2\theta-1=\cot^2\theta$ .
3. $=\tan\theta\cdot\cot^2\theta=\cot\theta$ .
Question 2 of 15
If $(\sin\alpha + \mathrm{cosec}\,\alpha)^2 + (\cos\alpha + \sec\alpha)^2 = k + \tan^2\alpha + \cot^2\alpha$ , then the value of $k$ is equal to
1. (1) $9$
2. (2) $7$
3. (3) $5$
4. (4) $3$
Show answer
Answer: (2) $7$
1. $(\sin\alpha+\mathrm{cosec}\,\alpha)^2=\sin^2\alpha+2+\mathrm{cosec}^2\alpha$ .
2. $(\cos\alpha+\sec\alpha)^2=\cos^2\alpha+2+\sec^2\alpha$ .
3. Sum $=(\sin^2\alpha+\cos^2\alpha)+4+(1+\cot^2\alpha)+(1+\tan^2\alpha)$ .
4. $=7+\tan^2\alpha+\cot^2\alpha$ , so $k=7$ .
Question 3 of 15
If $\sin\theta + \cos\theta = a$ and $\sec\theta + \mathrm{cosec}\,\theta = b$ , then the value of $b(a^2-1)$ is equal to
1. (1) $2a$
2. (2) $3a$
3. (3) $0$
4. (4) $2ab$
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Answer: (1) $2a$
1. $a^2-1=(\sin\theta+\cos\theta)^2-1=2\sin\theta\cos\theta$ .
2. $b=\sec\theta+\mathrm{cosec}\,\theta=\dfrac{\sin\theta+\cos\theta}{\sin\theta\cos\theta}=\dfrac{a}{\sin\theta\cos\theta}$ .
3. $b(a^2-1)=\dfrac{a}{\sin\theta\cos\theta}\times 2\sin\theta\cos\theta=2a$ .
Question 4 of 15
If $5x = \sec\theta$ and $\dfrac{5}{y} = \tan\theta$ , then $x^2 - \dfrac{1}{y^2}$ is equal to
1. (1) $25$
2. (2) $\dfrac{1}{25}$
3. (3) $5$
4. (4) $1$
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Answer: (2) $\dfrac{1}{25}$
1. $x=\dfrac{\sec\theta}{5}$ and $\dfrac{1}{y}=\dfrac{\tan\theta}{5}$ .
2. $x^2-\dfrac{1}{y^2}=\dfrac{\sec^2\theta-\tan^2\theta}{25}=\dfrac{1}{25}$ .
Question 5 of 15
If $\sin\theta = \cos\theta$ , then $2\tan^2\theta + \sin^2\theta - 1$ is equal to
1. (1) $\dfrac{-3}{2}$
2. (2) $\dfrac{3}{2}$
3. (3) $\dfrac{2}{3}$
4. (4) $\dfrac{-2}{3}$
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Answer: (2) $\dfrac{3}{2}$
1. $\sin\theta=\cos\theta\Rightarrow\tan\theta=1\Rightarrow\theta=45^\circ$ .
2. $2\tan^2 45^\circ+\sin^2 45^\circ-1=2(1)+\dfrac{1}{2}-1=\dfrac{3}{2}$ .
Question 6 of 15
If $x = a\tan\theta$ and $y = b\sec\theta$ then
1. (1) $\dfrac{y^2}{b^2} - \dfrac{x^2}{a^2} = 1$
2. (2) $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$
3. (3) $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$
4. (4) $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 0$
Show answer
Answer: (1) $\dfrac{y^2}{b^2} - \dfrac{x^2}{a^2} = 1$
1. $\dfrac{x}{a}=\tan\theta$ , $\dfrac{y}{b}=\sec\theta$ .
2. $\sec^2\theta-\tan^2\theta=1\Rightarrow\dfrac{y^2}{b^2}-\dfrac{x^2}{a^2}=1$ .
Question 7 of 15
$(1+\tan\theta+\sec\theta)(1+\cot\theta-\mathrm{cosec}\,\theta)$ is equal to
1. (1) $0$
2. (2) $1$
3. (3) $2$
4. (4) $-1$
Show answer
Answer: (3) $2$
1. Write the second bracket as $1+\dfrac{\cos\theta}{\sin\theta}-\dfrac{1}{\sin\theta}=\dfrac{\sin\theta+\cos\theta-1}{\sin\theta}$ .
2. First bracket $=\dfrac{\cos\theta+\sin\theta+1}{\cos\theta}$ .
3. Product $=\dfrac{(\sin\theta+\cos\theta)^2-1}{\sin\theta\cos\theta}=\dfrac{2\sin\theta\cos\theta}{\sin\theta\cos\theta}=2$ .
Question 8 of 15
$a\cot\theta + b\,\mathrm{cosec}\,\theta = p$ and $b\cot\theta + a\,\mathrm{cosec}\,\theta = q$ then $p^2 - q^2$ is equal to
1. (1) $a^2 - b^2$
2. (2) $b^2 - a^2$
3. (3) $a^2 + b^2$
4. (4) $b - a$
Show answer
Answer: (2) $b^2 - a^2$
1. $p^2-q^2=(a\cot\theta+b\,\mathrm{cosec}\,\theta)^2-(b\cot\theta+a\,\mathrm{cosec}\,\theta)^2$ .
2. $=(a^2-b^2)\cot^2\theta-(a^2-b^2)\mathrm{cosec}^2\theta$ (the $2ab$ terms cancel).
3. $=(a^2-b^2)(\cot^2\theta-\mathrm{cosec}^2\theta)=(a^2-b^2)(-1)=b^2-a^2$ .
Question 9 of 15
If the ratio of the height of a tower and the length of its shadow is $\sqrt{3}:1$ , then the angle of elevation of the sun has measure
1. (1) $45^\circ$
2. (2) $30^\circ$
3. (3) $90^\circ$
4. (4) $60^\circ$
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Answer: (4) $60^\circ$
1. $\tan\theta=\dfrac{\text{height}}{\text{shadow}}=\dfrac{\sqrt3}{1}=\sqrt3$ .
2. $\theta=60^\circ$ .
Question 10 of 15
The electric pole subtends an angle of $30^\circ$ at a point on the same level as its foot. At a second point ' $b$ ' metres above the first, the depression of the foot of the pole is $60^\circ$ . The height of the pole (in metres) is equal to
1. (1) $\sqrt{3}\,b$
2. (2) $\dfrac{b}{3}$
3. (3) $\dfrac{b}{2}$
4. (4) $\dfrac{b}{\sqrt{3}}$
Show answer
Answer: (2) $\dfrac{b}{3}$
1. Let $d$ be the horizontal distance from the pole and $h$ its height.
2. From the point $b$ m high, depression of the foot is $60^\circ$ : $\tan 60^\circ=\dfrac{b}{d}\Rightarrow d=\dfrac{b}{\sqrt3}$ .
3. From the first point, elevation of the top is $30^\circ$ : $\tan 30^\circ=\dfrac{h}{d}$ .
4. $h=\dfrac{1}{\sqrt3}\times\dfrac{b}{\sqrt3}=\dfrac{b}{3}$ .
Question 11 of 15
A tower is 60 m heigh. Its shadow reduces by $x$ metres when the angle of elevation of the sun increases from $30^\circ$ to $45^\circ$ then $x$ is equal to
1. (1) 41.92 m
2. (2) 43.92 m
3. (3) 43 m
4. (4) 45.6 m
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Answer: (2) 43.92 m
1. Shadow at $30^\circ$ : $\dfrac{60}{\tan 30^\circ}=60\sqrt3$ m.
2. Shadow at $45^\circ$ : $\dfrac{60}{\tan 45^\circ}=60$ m.
3. $x=60\sqrt3-60=60(1.732-1)=43.92$ m.
Question 12 of 15
The angle of depression of the top and bottom of 20 m tall building from the top of a multistoried building are $30^\circ$ and $60^\circ$ respectively. The height of the multistoried building and the distance between two buildings (in metres) is
1. (1) $20,\ 10\sqrt{3}$
2. (2) $30,\ 5\sqrt{3}$
3. (3) $20,\ 10$
4. (4) $30,\ 10\sqrt{3}$
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Answer: (4) $30,\ 10\sqrt{3}$
1. Let $h$ be the height of the multistoried building and $d$ the distance between them.
2. $\tan 60^\circ=\dfrac{h}{d}\Rightarrow h=d\sqrt3$ .
3. $\tan 30^\circ=\dfrac{h-20}{d}\Rightarrow h-20=\dfrac{d}{\sqrt3}$ .
4. $d\sqrt3-\dfrac{d}{\sqrt3}=20\Rightarrow \dfrac{2d}{\sqrt3}=20\Rightarrow d=10\sqrt3$ .
5. $h=10\sqrt3\times\sqrt3=30$ m.
Question 13 of 15
Two persons are standing ' $x$ ' metres apart from each other and the height of the first person is double that of the other. If from the middle point of the line joining their feet an observer finds the angular elevations of their tops to be complementary, then the height of the shorter person (in metres) is
1. (1) $\sqrt{2}\,x$
2. (2) $\dfrac{x}{2\sqrt{2}}$
3. (3) $\dfrac{x}{\sqrt{2}}$
4. (4) $2x$
Show answer
Answer: (2) $\dfrac{x}{2\sqrt{2}}$
1. Let the shorter height be $h$ ; the other is $2h$ . The observer is at the midpoint, $\dfrac{x}{2}$ from each.
2. $\tan\theta=\dfrac{2h}{x/2}=\dfrac{4h}{x}$ and $\tan(90^\circ-\theta)=\cot\theta=\dfrac{h}{x/2}=\dfrac{2h}{x}$ .
3. $\tan\theta\cdot\cot\theta=1\Rightarrow\dfrac{4h}{x}\times\dfrac{2h}{x}=1$ .
4. $8h^2=x^2\Rightarrow h=\dfrac{x}{2\sqrt2}$ .
Question 14 of 15
The angle of elevation of a cloud from a point $h$ metres above a lake is $\beta$ . The angle of depression of its reflection in the lake is $45^\circ$ . The height of location of the cloud from the lake is
1. (1) $\dfrac{h(1+\tan\beta)}{1-\tan\beta}$
2. (2) $\dfrac{h(1-\tan\beta)}{1+\tan\beta}$
3. (3) $h\tan(45^\circ-\beta)$
4. (4) none of these
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Answer: (1) $\dfrac{h(1+\tan\beta)}{1-\tan\beta}$
1. Let the cloud be at height $H$ above the lake and $d$ the horizontal distance.
2. Elevation: $\tan\beta=\dfrac{H-h}{d}$ ; depression of reflection: $\tan 45^\circ=\dfrac{H+h}{d}$ .
3. So $d=H+h$ and $(H-h)=(H+h)\tan\beta$ .
4. $H(1-\tan\beta)=h(1+\tan\beta)\Rightarrow H=\dfrac{h(1+\tan\beta)}{1-\tan\beta}$ .
Question 15 of 15

Chapter 7: Mensuration

The curved surface area of a right circular cone of height $15$ cm and base diameter $16$ cm is
1. (1) $60\pi$ cm $^2$
2. (2) $68\pi$ cm $^2$
3. (3) $120\pi$ cm $^2$
4. (4) $136\pi$ cm $^2$
Show answer
Answer: (4) $136\pi$ cm $^2$
1. $r=\dfrac{16}{2}=8$ cm, $h=15$ cm.
2. $l=\sqrt{r^2+h^2}=\sqrt{64+225}=\sqrt{289}=17$ cm.
3. CSA $=\pi r l=\pi\times 8\times 17=136\pi$ cm $^2$ .
Question 1 of 15
If two solid hemispheres of same base radius $r$ units are joined together along their bases, then curved surface area of this new solid is
1. (1) $4\pi r^2$ sq. units
2. (2) $6\pi r^2$ sq. units
3. (3) $3\pi r^2$ sq. units
4. (4) $8\pi r^2$ sq. units
Show answer
Answer: (1) $4\pi r^2$ sq. units
1. Two hemispheres joined along their bases form a sphere of radius $r$ .
2. CSA of sphere $=4\pi r^2$ sq. units.
Question 2 of 15
The height of a right circular cone whose radius is $5$ cm and slant height is $13$ cm will be
1. (1) $12$ cm
2. (2) $10$ cm
3. (3) $13$ cm
4. (4) $5$ cm
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Answer: (1) $12$ cm
1. $l^2=r^2+h^2$
2. $13^2=5^2+h^2\Rightarrow h^2=169-25=144$
3. $h=12$ cm.
Question 3 of 15
If the radius of the base of a right circular cylinder is halved keeping the same height, then the ratio of the volume of the cylinder thus obtained to the volume of original cylinder is
1. (1) $1:2$
2. (2) $1:4$
3. (3) $1:6$
4. (4) $1:8$
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Answer: (2) $1:4$
1. New volume $=\pi\left(\dfrac{r}{2}\right)^2h=\dfrac{\pi r^2h}{4}$ .
2. Ratio $=\dfrac{\pi r^2h/4}{\pi r^2h}=\dfrac{1}{4}$ .
Question 4 of 15
The total surface area of a cylinder whose radius is $\dfrac{1}{3}$ of its height is
1. (1) $\dfrac{9\pi h^2}{8}$ sq.units
2. (2) $24\pi h^2$ sq.units
3. (3) $\dfrac{8\pi h^2}{9}$ sq.units
4. (4) $\dfrac{56\pi h^2}{9}$ sq.units
Show answer
Answer: (3) $\dfrac{8\pi h^2}{9}$ sq.units
1. Given $r=\dfrac{h}{3}$ .
2. TSA $=2\pi r(r+h)=2\pi\dfrac{h}{3}\left(\dfrac{h}{3}+h\right)$
3. $=2\pi\dfrac{h}{3}\times\dfrac{4h}{3}=\dfrac{8\pi h^2}{9}$ .
Question 5 of 15
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, the volume of the material in it is
1. (1) $5600\pi$ cm $^3$
2. (2) $1120\pi$ cm $^3$
3. (3) $56\pi$ cm $^3$
4. (4) $3600\pi$ cm $^3$
Show answer
Answer: (2) $1120\pi$ cm $^3$
1. $R+r=14$ , width $R-r=4$ , $h=20$ .
2. Volume of material $=\pi(R^2-r^2)h=\pi(R+r)(R-r)h$
3. $=\pi\times 14\times 4\times 20=1120\pi$ cm $^3$ .
Question 6 of 15
If the radius of the base of a cone is tripled and the height is doubled then the volume is
1. (1) made $6$ times
2. (2) made $18$ times
3. (3) made $12$ times
4. (4) unchanged
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Answer: (2) made $18$ times
1. New volume $=\dfrac{1}{3}\pi(3r)^2(2h)=\dfrac{1}{3}\pi\cdot 9r^2\cdot 2h$
2. $=18\times\dfrac{1}{3}\pi r^2h$ .
3. So the volume is made $18$ times.
Question 7 of 15
The total surface area of a hemi-sphere is how much times the square of its radius.
1. (1) $\pi$
2. (2) $4\pi$
3. (3) $3\pi$
4. (4) $2\pi$
Show answer
Answer: (3) $3\pi$
1. TSA of hemisphere $=3\pi r^2$ .
2. So it is $3\pi$ times the square of the radius.
Question 8 of 15
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
1. (1) $3x$ cm
2. (2) $x$ cm
3. (3) $4x$ cm
4. (4) $2x$ cm
Show answer
Answer: (3) $4x$ cm
1. Volume of sphere $=$ volume of cone
2. $\dfrac{4}{3}\pi x^3=\dfrac{1}{3}\pi x^2h$
3. $h=4x$ cm.
Question 9 of 15
A frustum of a right circular cone is of height $16$ cm with radii of its ends as $8$ cm and $20$ cm. Then, the volume of the frustum is
1. (1) $3328\pi$ cm $^3$
2. (2) $3228\pi$ cm $^3$
3. (3) $3240\pi$ cm $^3$
4. (4) $3340\pi$ cm $^3$
Show answer
Answer: (1) $3328\pi$ cm $^3$
1. $V=\dfrac{1}{3}\pi h(R^2+Rr+r^2)$
2. $=\dfrac{1}{3}\pi(16)(400+160+64)$
3. $=\dfrac{16}{3}\pi(624)=3328\pi$ cm $^3$ .
Question 10 of 15
A shuttle cock used for playing badminton has the shape of the combination of
1. (1) a cylinder and a sphere
2. (2) a hemisphere and a cone
3. (3) a sphere and a cone
4. (4) frustum of a cone and a hemisphere
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Answer: (4) frustum of a cone and a hemisphere
1. The cork base is a hemisphere and the feather part widens like a frustum of a cone.
2. So it is frustum of a cone and a hemisphere.
Question 11 of 15
A spherical ball of radius $r_1$ units is melted to make $8$ new identical balls each of radius $r_2$ units. Then $r_1 : r_2$ is
1. (1) $2:1$
2. (2) $1:2$
3. (3) $4:1$
4. (4) $1:4$
Show answer
Answer: (1) $2:1$
1. Volume is unchanged: $\dfrac{4}{3}\pi r_1^3=8\times\dfrac{4}{3}\pi r_2^3$
2. $r_1^3=8r_2^3\Rightarrow r_1=2r_2$
3. $r_1:r_2=2:1$ .
Question 12 of 15
The volume (in cm $^3$ ) of the greatest sphere that can be cut off from a cylindrical log of wood of base radius $1$ cm and height $5$ cm is
1. (1) $\dfrac{4}{3}\pi$
2. (2) $\dfrac{10}{3}\pi$
3. (3) $5\pi$
4. (4) $\dfrac{20}{3}\pi$
Show answer
Answer: (1) $\dfrac{4}{3}\pi$
1. The greatest sphere has radius equal to the base radius $=1$ cm.
2. $V=\dfrac{4}{3}\pi(1)^3=\dfrac{4}{3}\pi$ cm $^3$ .
Question 13 of 15
The height and radius of the cone of which the frustum is a part are $h_1$ units and $r_1$ units respectively. Height of the frustum is $h_2$ units and radius of the smaller base is $r_2$ units. If $h_2 : h_1 = 1 : 2$ then $r_2 : r_1$ is
1. (1) $1 : 3$
2. (2) $1 : 2$
3. (3) $2 : 1$
4. (4) $3 : 1$
Show answer
Answer: (2) $1 : 2$
1. The small cone cut off has height $h_1-h_2$ and radius $r_2$ .
2. By similar triangles $\dfrac{r_2}{r_1}=\dfrac{h_1-h_2}{h_1}$ .
3. Given $h_2=\dfrac{h_1}{2}$ , so $\dfrac{r_2}{r_1}=\dfrac{h_1-\frac{h_1}{2}}{h_1}=\dfrac{1}{2}$ .
Question 14 of 15
The ratio of the volumes of a cylinder, a cone and a sphere, if each has the same diameter and same height is
1. (1) $1:2:3$
2. (2) $2:1:3$
3. (3) $1:3:2$
4. (4) $3:1:2$
Show answer
Answer: (4) $3:1:2$
1. Same diameter and same height $\Rightarrow h=2r$ .
2. Cylinder $=\pi r^2(2r)=2\pi r^3$
3. Cone $=\dfrac{1}{3}\pi r^2(2r)=\dfrac{2}{3}\pi r^3$ , Sphere $=\dfrac{4}{3}\pi r^3$
4. Ratio $=2:\dfrac{2}{3}:\dfrac{4}{3}=6:2:4=3:1:2$ .
Question 15 of 15

Chapter 8: Statistics and Probability

Which of the following is not a measure of dispersion?
1. (1) Range
2. (2) Standard deviation
3. (3) Arithmetic mean
4. (4) Variance
Show answer
Answer: (3) Arithmetic mean
1. Dispersion measures how spread out the data is: range, standard deviation, variance.
2. Arithmetic mean is a measure of central tendency, not dispersion.
Question 1 of 15
The range of the data $8, 8, 8, 8, 8. . . 8$ is
1. (1) $0$
2. (2) $1$
3. (3) $8$
4. (4) $3$
Show answer
Answer: (1) $0$
1. Largest value $= 8$ , smallest value $= 8$ .
2. Range $= L - S = 8 - 8 = 0$ .
Question 2 of 15
The sum of all deviations of the data from its mean is
1. (1) Always positive
2. (2) always negative
3. (3) zero
4. (4) non-zero integer
Show answer
Answer: (3) zero
1. $\sum (x_i-\bar{x}) = \sum x_i - n\bar{x}$ .
2. Since $\bar{x}=\dfrac{\sum x_i}{n}$ , we get $n\bar{x}=\sum x_i$ .
3. So the sum of deviations $=0$ .
Question 3 of 15
The mean of $100$ observations is $40$ and their standard deviation is $3$ . The sum of squares of all observations is
1. (1) $40000$
2. (2) $160900$
3. (3) $160000$
4. (4) $30000$
Show answer
Answer: (2) $160900$
1. $\sigma^2=\dfrac{\sum x^2}{n}-\left(\bar{x}\right)^2$
2. $9=\dfrac{\sum x^2}{100}-1600$
3. $\dfrac{\sum x^2}{100}=1609$
4. $\sum x^2=160900$
Question 4 of 15
Variance of first $20$ natural numbers is
1. (1) $32.25$
2. (2) $44.25$
3. (3) $33.25$
4. (4) $30$
Show answer
Answer: (3) $33.25$
1. Variance of first $n$ natural numbers $=\dfrac{n^2-1}{12}$ .
2. $=\dfrac{20^2-1}{12}=\dfrac{399}{12}=33.25$
Question 5 of 15
The standard deviation of a data is $3$ . If each value is multiplied by $5$ then the new variance is
1. (1) $3$
2. (2) $15$
3. (3) $5$
4. (4) $225$
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Answer: (4) $225$
1. Multiplying each value by $5$ multiplies the S.D. by $5$ .
2. New S.D. $=5\times 3=15$ .
3. New variance $=15^2=225$ .
Question 6 of 15
If the standard deviation of $x,\ y,\ z$ is $p$ then the standard deviation of $3x+5$ , $3y+5$ , $3z+5$ is
1. (1) $3p+5$
2. (2) $3p$
3. (3) $p+5$
4. (4) $9p+15$
Show answer
Answer: (2) $3p$
1. Adding a constant does not change the S.D.; multiplying by $3$ multiplies the S.D. by $3$ .
2. New standard deviation $=3p$ .
Question 7 of 15
If the mean and coefficient of variation of a data are $4$ and $87.5\%$ then the standard deviation is
1. (1) $3.5$
2. (2) $3$
3. (3) $4.5$
4. (4) $2.5$
Show answer
Answer: (1) $3.5$
1. C.V. $=\dfrac{\sigma}{\bar{x}}\times 100$
2. $87.5=\dfrac{\sigma}{4}\times 100$
3. $\sigma=\dfrac{87.5\times 4}{100}=3.5$
Question 8 of 15
Which of the following is incorrect?
1. (1) $P(A) > 1$
2. (2) $0 \leq P(A) \leq 1$
3. (3) $P(\phi) = 0$
4. (4) $P(A) + P(\bar{A}) = 1$
Show answer
Answer: (1) $P(A) > 1$
1. Probability of any event always lies between $0$ and $1$ .
2. So $P(A)>1$ is impossible.
Question 9 of 15
The probability a red marble selected at random from a jar containing $p$ red, $q$ blue and r green marbles is
1. (1) $\dfrac{q}{p+q+r}$
2. (2) $\dfrac{p}{p+q+r}$
3. (3) $\dfrac{p+q}{p+q+r}$
4. (4) $\dfrac{p+r}{p+q+r}$
Show answer
Answer: (2) $\dfrac{p}{p+q+r}$
1. Total marbles $=p+q+r$ ; red marbles $=p$ .
2. $P(\text{red})=\dfrac{p}{p+q+r}$
Question 10 of 15
A page is selected at random from a book. The probability that the digit at units place of the page number chosen is less than $7$ is
1. (1) $\dfrac{3}{10}$
2. (2) $\dfrac{7}{10}$
3. (3) $\dfrac{3}{9}$
4. (4) $\dfrac{7}{9}$
Show answer
Answer: (2) $\dfrac{7}{10}$
1. Units digit can be $0,1,2,\ldots,9$ , so $n(S)=10$ .
2. Digits less than $7$ : $0,1,2,3,4,5,6$ , so $n(A)=7$ .
3. $P(A)=\dfrac{7}{10}$
Question 11 of 15
The probability of getting a job for a person is $\dfrac{x}{3}$ . If the probability of not getting the job is $\dfrac{2}{3}$ then the value of $x$ is
1. (1) $2$
2. (2) $1$
3. (3) $3$
4. (4) $1.5$
Show answer
Answer: (2) $1$
1. $P(A)+P(\bar{A})=1$
2. $\dfrac{x}{3}+\dfrac{2}{3}=1$
3. $\dfrac{x}{3}=\dfrac{1}{3}\Rightarrow x=1$
Question 12 of 15
Kamalam went to play a lucky draw contest. $135$ tickets of the lucky draw were sold. If the probability of Kamalam winning is $\dfrac{1}{9}$ , then the number of tickets bought by Kamalam is
1. (1) $5$
2. (2) $10$
3. (3) $15$
4. (4) $20$
Show answer
Answer: (3) $15$
1. Let the number of tickets bought be $n$ .
2. $\dfrac{n}{135}=\dfrac{1}{9}$
3. $n=\dfrac{135}{9}=15$
Question 13 of 15
If a letter is chosen at random from the English alphabets $\{a,b,\ldots,z\}$ , then the probability that the letter chosen precedes $x$
1. (1) $\dfrac{12}{13}$
2. (2) $\dfrac{1}{13}$
3. (3) $\dfrac{23}{26}$
4. (4) $\dfrac{3}{26}$
Show answer
Answer: (3) $\dfrac{23}{26}$
1. Total letters $n(S)=26$ .
2. Letters before $x$ are $a$ to $w$ , that is $23$ letters.
3. $P=\dfrac{23}{26}$
Question 14 of 15
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 probability that the note is either a ₹ $500$ note or ₹ $200$ note?
1. (1) $\dfrac{1}{5}$
2. (2) $\dfrac{3}{10}$
3. (3) $\dfrac{2}{3}$
4. (4) $\dfrac{4}{5}$
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Answer: (4) $\dfrac{4}{5}$
1. Total notes $=10+15+25=50$ .
2. Favourable notes $=15+25=40$ .
3. $P=\dfrac{40}{50}=\dfrac{4}{5}$
Question 15 of 15

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