Samacheer Kalvi 10th Maths Exercise 3.5 Solutions

10th Maths Exercise 3.5 solutions (Chapter 3 Algebra) with step-by-step answers for all 11 questions, as per the 2026-27 Samacheer Kalvi textbook.

Answers at a glance11 answers · full steps below

Q1 (i) $\dfrac{3x^3z}{5y^3}$
Q1 (ii) $p+4$
Q1 (iii) $\dfrac{3t^2}{4}$
Q2 (i) $\dfrac{3x-4y}{2x-5}$
Q2 (ii) $\dfrac{x^2+xy+y^2}{3(x+2y)}$
Q3 (i) $-5$
Q3 (ii) $\dfrac{b-4}{b+2}$
Q3 (iii) $\dfrac{3y}{x-3}$
Q3 (iv) $\dfrac{4(2t-1)}{3}$
Q4 $\dfrac{4}{9}$
Q5 $q(x)=(x+2)^2=x^2+4x+4$

Question 1

(i)

Simplify $\dfrac{4x^2y}{2z^2} \times \dfrac{6xz^3}{20y^4}$

$\dfrac{4x^2y}{2z^2}\times\dfrac{6xz^3}{20y^4}=\dfrac{4\times 6\,x^3yz^3}{2\times 20\,y^4z^2}$
$=\dfrac{24x^3z}{40y^3}$
Divide numerator and denominator by $8$ .
$=\dfrac{3x^3z}{5y^3}$

Answer

$\dfrac{3x^3z}{5y^3}$

(ii)

Simplify $\dfrac{p^2 - 10p + 21}{p - 7} \times \dfrac{p^2 + p - 12}{(p-3)^2}$

$p^2-10p+21=(p-7)(p-3)$
$p^2+p-12=(p+4)(p-3)$
$\dfrac{(p-7)(p-3)}{p-7}\times\dfrac{(p+4)(p-3)}{(p-3)^2}$
Cancel $(p-7)$ and $(p-3)^2$ .
$=p+4$

Answer

$p+4$

(iii)

Simplify $\dfrac{5t^3}{4t - 8} \times \dfrac{6t - 12}{10t}$

$4t-8=4(t-2)$ and $6t-12=6(t-2)$
$\dfrac{5t^3}{4(t-2)}\times\dfrac{6(t-2)}{10t}$
Cancel $(t-2)$ .
$=\dfrac{30t^3}{40t}=\dfrac{3t^2}{4}$

Answer

$\dfrac{3t^2}{4}$

Question 2

(i)

Simplify $\dfrac{x+4}{3x+4y} \times \dfrac{9x^2 - 16y^2}{2x^2 + 3x - 20}$

$9x^2-16y^2=(3x+4y)(3x-4y)$
$2x^2+3x-20=(2x-5)(x+4)$
$\dfrac{x+4}{3x+4y}\times\dfrac{(3x+4y)(3x-4y)}{(2x-5)(x+4)}$
Cancel $(x+4)$ and $(3x+4y)$ .
$=\dfrac{3x-4y}{2x-5}$

Answer

$\dfrac{3x-4y}{2x-5}$

(ii)

Simplify $\dfrac{x^3 - y^3}{3x^2 + 9xy + 6y^2} \times \dfrac{x^2 + 2xy + y^2}{x^2 - y^2}$

Show steps (6)

$x^3-y^3=(x-y)(x^2+xy+y^2)$
$3x^2+9xy+6y^2=3(x^2+3xy+2y^2)=3(x+y)(x+2y)$
$x^2+2xy+y^2=(x+y)^2$ and $x^2-y^2=(x+y)(x-y)$
$\dfrac{(x-y)(x^2+xy+y^2)}{3(x+y)(x+2y)}\times\dfrac{(x+y)^2}{(x+y)(x-y)}$
Cancel $(x-y)$ and $(x+y)^2$ .
$=\dfrac{x^2+xy+y^2}{3(x+2y)}$

Answer

$\dfrac{x^2+xy+y^2}{3(x+2y)}$

Question 3

(i)

Simplify $\dfrac{2a^2 + 5a + 3}{2a^2 + 7a + 6} \div \dfrac{a^2 + 6a + 5}{-5a^2 - 35a - 50}$

Show steps (7)

$2a^2+5a+3=(2a+3)(a+1)$ , $2a^2+7a+6=(2a+3)(a+2)$
$a^2+6a+5=(a+1)(a+5)$
$-5a^2-35a-50=-5(a+2)(a+5)$
Change $\div$ into $\times$ by inverting:
$\dfrac{(2a+3)(a+1)}{(2a+3)(a+2)}\times\dfrac{-5(a+2)(a+5)}{(a+1)(a+5)}$
Cancel all common factors.
$=-5$

Answer

$-5$

(ii)

Simplify $\dfrac{b^2 + 3b - 28}{b^2 + 4b + 4} \div \dfrac{b^2 - 49}{b^2 - 5b - 14}$

$b^2+3b-28=(b+7)(b-4)$ , $b^2+4b+4=(b+2)^2$
$b^2-49=(b+7)(b-7)$ , $b^2-5b-14=(b-7)(b+2)$
$\dfrac{(b+7)(b-4)}{(b+2)^2}\times\dfrac{(b-7)(b+2)}{(b+7)(b-7)}$
Cancel $(b+7)$ , $(b-7)$ and one $(b+2)$ .
$=\dfrac{b-4}{b+2}$

Answer

$\dfrac{b-4}{b+2}$

(iii)

Simplify $\dfrac{x+2}{4y} \div \dfrac{x^2 - x - 6}{12y^2}$

$x^2-x-6=(x-3)(x+2)$
$\dfrac{x+2}{4y}\times\dfrac{12y^2}{(x-3)(x+2)}$
Cancel $(x+2)$ and $4y$ .
$=\dfrac{3y}{x-3}$

Answer

$\dfrac{3y}{x-3}$

(iv)

Simplify $\dfrac{12t^2 - 22t + 8}{3t} \div \dfrac{3t^2 + 2t - 8}{2t^2 + 4t}$

$12t^2-22t+8=2(6t^2-11t+4)=2(3t-4)(2t-1)$
$3t^2+2t-8=(3t-4)(t+2)$ , $2t^2+4t=2t(t+2)$
$\dfrac{2(3t-4)(2t-1)}{3t}\times\dfrac{2t(t+2)}{(3t-4)(t+2)}$
Cancel $(3t-4)$ , $(t+2)$ and $t$ .
$=\dfrac{4(2t-1)}{3}=\dfrac{8t-4}{3}$

Answer

$\dfrac{4(2t-1)}{3}$

Q4. If $x = \dfrac{a^2 + 3a - 4}{3a^2 - 3}$ and $y = \dfrac{a^2 + 2a - 8}{2a^2 - 2a - 4}$ find the value of $x^2 y^{-2}$ .

$x=\dfrac{a^2+3a-4}{3a^2-3}=\dfrac{(a+4)(a-1)}{3(a-1)(a+1)}=\dfrac{a+4}{3(a+1)}$
$y=\dfrac{a^2+2a-8}{2a^2-2a-4}=\dfrac{(a+4)(a-2)}{2(a-2)(a+1)}=\dfrac{a+4}{2(a+1)}$
$x^2y^{-2}=\left(\dfrac{x}{y}\right)^2$
$\dfrac{x}{y}=\dfrac{a+4}{3(a+1)}\times\dfrac{2(a+1)}{a+4}=\dfrac{2}{3}$
$\left(\dfrac{2}{3}\right)^2=\dfrac{4}{9}$

Answer

$\dfrac{4}{9}$

Q5. If a polynomial $p(x) = x^2 - 5x - 14$ is divided by another polynomial $q(x)$ we get …

If a polynomial $p(x) = x^2 - 5x - 14$ is divided by another polynomial $q(x)$ we get $\dfrac{x-7}{x+2}$ , find $q(x)$ .

$\dfrac{p(x)}{q(x)}=\dfrac{x-7}{x+2}$ , so $q(x)=p(x)\times\dfrac{x+2}{x-7}$
$p(x)=x^2-5x-14=(x-7)(x+2)$
$q(x)=\dfrac{(x-7)(x+2)(x+2)}{x-7}$
$q(x)=(x+2)^2=x^2+4x+4$

Answer

$q(x)=(x+2)^2=x^2+4x+4$

Quick check

Graph of a linear equation is a _______
1. (1) straight line
2. (2) circle
3. (3) parabola
4. (4) hyperbola
Show answer
Answer: (1) straight line
1. A linear equation in two variables has degree one, and its graph is a straight line.
Question 1 of 5
$\dfrac{3y - 3}{y} \div \dfrac{7y - 7}{3y^2}$ 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 2 of 5
Transpose of a column matrix is
1. (1) unit matrix
2. (2) diagonal matrix
3. (3) column matrix
4. (4) row matrix
Show answer
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 3 of 5
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 4 of 5
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 5 of 5

Chapter 3 formulas

Roots of a quadratic equation
$ax^2 + bx + c = 0,\; (a \neq 0) \;\Rightarrow\; x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Sum of the roots
$\alpha + \beta = \dfrac{-b}{a} = \dfrac{-\text{Co-efficient of } x}{\text{Co-efficient of } x^2}$
Product of the roots
$\alpha\beta = \dfrac{c}{a} = \dfrac{\text{Constant term}}{\text{Co-efficient of } x^2}$
Quadratic equation formed from its roots
$x^2 - (\alpha + \beta)x + \alpha\beta = 0$
Discriminant
$\Delta = b^2 - 4ac$
Nature of roots: real and unequal
$\Delta > 0 \;\Rightarrow\; \text{the roots are real and unequal}$
Nature of roots: real and equal
$\Delta = 0 \;\Rightarrow\; \text{the roots are real and equal}$
Nature of roots: no real roots
$\Delta < 0 \;\Rightarrow\; \text{there are no real roots}$
Cases for a system of linear equations in three variables
$\text{(i) Unique solution}\quad \text{(ii) Infinitely many solutions}\quad \text{(iii) No solution}$
LCM of algebraic expressions
$\text{LCM} = \text{expression of lowest degree that each given expression exactly divides}$
Quadratic polynomial and its zeroes
$p(x) = ax^2 + bx + c \;(a \neq 0) \;\Rightarrow\; p(x) \text{ has atmost two zeroes, which lie on the } x\text{-axis}$
Order of a matrix
$\text{Order of } A = (\text{Number of rows}) \times (\text{Number of columns}) = m \times n$
Row matrix (row vector)
$A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \end{pmatrix}_{1 \times n}$
Column matrix (column vector)
$A = \begin{pmatrix} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{pmatrix}_{m \times 1}$
Square matrix
$\text{Number of rows} = \text{Number of columns},\; \text{order } n \times n$
Zero matrix (null matrix)
$a_{ij} = 0 \text{ for all } i,\, j$
Transpose of a matrix
$\left(A^{T}\right)_{ij} = a_{ji} \quad (\text{rows and columns of } A \text{ interchanged})$
Diagonal matrix
$a_{ij} = 0 \text{ for } i \neq j \quad (\text{square matrix})$
Scalar matrix
$a_{ij} = \begin{cases} k, & i = j \\ 0, & i \neq j \end{cases}$
Identity matrix (unit matrix)
$a_{ij} = \begin{cases} 1, & i = j \\ 0, & i \neq j \end{cases}$
Lower triangular matrix
$a_{ij} = 0 \text{ for } i < j \quad (\text{all entries above the leading diagonal are zero})$
Upper triangular matrix
$a_{ij} = 0 \text{ for } i > j \quad (\text{all entries below the leading diagonal are zero})$
Equality of matrices
$A = B \iff \text{same order and } a_{ij} = b_{ij} \text{ for all } i,\, j$

Answers checked against the Samacheer Kalvi textbook · Updated 28 September 2026

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Question 1

(i)

(ii)

(iii)

Question 2

(i)

(ii)

Question 3

(i)

(ii)

(iii)

(iv)

Q4. If x=a2+3a−43a2−3x = \dfrac{a^2 + 3a - 4}{3a^2 - 3}x=3a2−3a2+3a−4​ and y=a2+2a−82a2−2a−4y = \dfrac{a^2 + 2a - 8}{2a^2 - 2a - 4}y=2a2−2a−4a2+2a−8​ find the value of x2y−2x^2 y^{-2}x2y−2 .

Q5. If a polynomial p(x)=x2−5x−14p(x) = x^2 - 5x - 14p(x)=x2−5x−14 is divided by another polynomial q(x)q(x)q(x) we get …

Quick check

Q4. If $x = \dfrac{a^2 + 3a - 4}{3a^2 - 3}$ and $y = \dfrac{a^2 + 2a - 8}{2a^2 - 2a - 4}$ find the value of $x^2 y^{-2}$ .

Q5. If a polynomial $p(x) = x^2 - 5x - 14$ is divided by another polynomial $q(x)$ we get …