Samacheer Kalvi 10th Maths Exercise 3.7 Solutions

10th Maths Exercise 3.7 solutions (Chapter 3 Algebra) with step-by-step answers for all 7 questions, as per the 2026-27 Samacheer Kalvi textbook.

Answers at a glance7 answers · full steps below

Q1 (i) $\dfrac{2y^4z^6}{x^2}$
Q1 (ii) $\left|\dfrac{\sqrt7x+\sqrt2}{x-\dfrac14}\right|$
Q1 (iii) $\dfrac{11(a+b)^4(x+y)^4}{9(a-b)^6}$
Q2 (i) $|2x+5|$
Q2 (ii) $|3x-4y+5z|$
Q2 (iii) $|(4x-1)(x-2)(7x+1)|$
Q2 (iv) $\dfrac16\,|(3x+2)(4x+3)(x+2)|$

Q1. Find the square root of the following rational expressions.

(i)

$\dfrac{400x^4y^{12}z^{16}}{100x^8y^4z^4}$

Simplify the rational expression first.
$\dfrac{400x^4y^{12}z^{16}}{100x^8y^4z^4}=\dfrac{4y^8z^{12}}{x^4}$
Take the square root of each factor.
$\sqrt{\dfrac{4y^8z^{12}}{x^4}}=\dfrac{2y^4z^6}{x^2}$

Answer

$\dfrac{2y^4z^6}{x^2}$

(ii)

$\dfrac{7x^2+2\sqrt{14}x+2}{x^2-\dfrac{1}{2}x+\dfrac{1}{16}}$

Show steps (6)

Factorise the numerator as a perfect square.
$7x^2+2\sqrt{14}\,x+2=(\sqrt7 x)^2+2(\sqrt7 x)(\sqrt2)+(\sqrt2)^2=(\sqrt7x+\sqrt2)^2$
Factorise the denominator.
$x^2-\dfrac12x+\dfrac{1}{16}=x^2-2(x)\left(\dfrac14\right)+\left(\dfrac14\right)^2=\left(x-\dfrac14\right)^2$
So the expression is $\dfrac{(\sqrt7x+\sqrt2)^2}{\left(x-\frac14\right)^2}$ .
Square root $=\left|\dfrac{\sqrt7x+\sqrt2}{x-\frac14}\right|=\left|\dfrac{4(\sqrt7x+\sqrt2)}{4x-1}\right|$

Answer

$\left|\dfrac{\sqrt7x+\sqrt2}{x-\dfrac14}\right|$

(iii)

$\dfrac{121(a+b)^8(x+y)^8(b-c)^8}{81(b-c)^4(a-b)^{12}(b-c)^4}$

Cancel the common $(b-c)$ factors.
$\dfrac{121(a+b)^8(x+y)^8(b-c)^8}{81(b-c)^4(a-b)^{12}(b-c)^4}=\dfrac{121(a+b)^8(x+y)^8}{81(a-b)^{12}}$
Take the square root of each factor.
$=\dfrac{11(a+b)^4(x+y)^4}{9(a-b)^6}$

Answer

$\dfrac{11(a+b)^4(x+y)^4}{9(a-b)^6}$

Q2. Find the square root of the following

(i)

$4x^2+20x+25$

Write as a perfect square.
$4x^2+20x+25=(2x)^2+2(2x)(5)+5^2=(2x+5)^2$
$\sqrt{4x^2+20x+25}=|2x+5|$

Answer

$|2x+5|$

(ii)

$9x^2-24xy+30xz-40yz+25z^2+16y^2$

Arrange the terms: $9x^2+16y^2+25z^2-24xy+30xz-40yz$ .
Compare with $(a+b+c)^2$ taking $a=3x,\ b=-4y,\ c=5z$ .
$2ab=2(3x)(-4y)=-24xy$ , $2bc=2(-4y)(5z)=-40yz$ , $2ca=2(5z)(3x)=30xz$ — all match.
So the expression $=(3x-4y+5z)^2$ .

Answer

$|3x-4y+5z|$

(iii)

$(4x^2-9x+2)(7x^2-13x-2)(28x^2-3x-1)$

Show steps (6)

Factorise each polynomial.
$4x^2-9x+2=(4x-1)(x-2)$
$7x^2-13x-2=(7x+1)(x-2)$
$28x^2-3x-1=(7x+1)(4x-1)$
Product $=(4x-1)^2(x-2)^2(7x+1)^2$
Square root $=|(4x-1)(x-2)(7x+1)|$

Answer

$|(4x-1)(x-2)(7x+1)|$

(iv)

$\left(2x^2+\dfrac{17}{6}x+1\right)\left(\dfrac{3}{2}x^2+4x+2\right)\left(\dfrac{4}{3}x^2+\dfrac{11}{3}x+2\right)$

Show steps (6)

Take out the fractions and factorise.
$2x^2+\dfrac{17}{6}x+1=\dfrac16(12x^2+17x+6)=\dfrac16(3x+2)(4x+3)$
$\dfrac32x^2+4x+2=\dfrac12(3x^2+8x+4)=\dfrac12(3x+2)(x+2)$
$\dfrac43x^2+\dfrac{11}{3}x+2=\dfrac13(4x^2+11x+6)=\dfrac13(4x+3)(x+2)$
Product $=\dfrac{1}{36}(3x+2)^2(4x+3)^2(x+2)^2$
Square root $=\dfrac16\,|(3x+2)(4x+3)(x+2)|$

Answer

$\dfrac16\,|(3x+2)(4x+3)(x+2)|$

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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