Exercise 3.15 - Chapter 3 Algebra 9th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

$\mathbf{E x} 3.15$
MULTIPLE CHOICE QUESTIONS :
Question 1 .
If $x^{3}+6 x^{2}+k x+6$ is exactly divisible by $(x+2)$, then $k=?$
(1) $-6$
(2) $-7$
(3) $-8$
(4) 11
Solution:
(4) 11
Hint: $P(-2)=(-2)^{3}+6(-2)^{2}+k(-2)+6=0$
$\begin{aligned}
&\Rightarrow-8+24 \\
&\Rightarrow 22=2 \mathrm{k} \\
&\Rightarrow \mathrm{k}=11
\end{aligned}$
$\Rightarrow-8+24-2 \mathrm{k}+6=0$
 

Question $2 .$
The root of the polynomial equation $2 x+3=0$ is
(1) $\frac{1}{3}$
(2) $-\frac{1}{3}$
(3) $-\frac{3}{2}$
(4) $-\frac{2}{3}$
Solution:
(3) $-\frac{3}{2}$
 

Question $3 .$
The type of the polynomial $4-3 x^{3}$ is
(1) constant polynomial
(2) linear polynomial
(3) quadratic polynomial
(4) cubic polynomial.
Solution:
(4) cubic polynomial.

Question $4 .$
If $x^{51}+51$ is divided by $x+1$, then the remainder is
(1) 0
(2) 1
(3) 49
(4) 50
Solution:
(4) 50
Hint: $P\left(-1=(-1)^{51}+51=-1+51=50\right.$
 

Question $5 .$
The zero of the polynomial $2 x+5$ is
(1) $\frac{5}{2}$
(2) $-\frac{5}{2}$
(3) $\frac{2}{5}$
(4) $-\frac{2}{5}$
Solution:
(2) $-\frac{5}{2}$
 

Question $6 .$
The sum of the polynomials $p(x)=x^{3}-x^{2}-2, q(x)=x^{2}-3 x+1$
(1) $x^{3}-3 x-1$
(2) $x^{3}+2 x^{2}-1$
(3) $x^{3}-2 x^{2}-3 x$
(4) $x^{3}-2 x^{2}+3 x-1$
Solution:
(1) $x^{3}-3 x-1$

Question $7 .$
Degree of the polynomial $\left(y^{3}-2\right)\left(y^{3}+1\right)$ is
(1) 9
(2) 2
(3) 3
(4) 6
Solution:
(4) 6

Question 8 .
Let the polynomials be
(A) $-13 q^{5}+4 q^{2}+12 q$
(B) $\left(x^{2}+4\right)\left(x^{2}+9\right)$
(C) $4 q^{8}-q^{6}+q^{2}$
(D) $-\frac{5}{7} \mathrm{y}^{12}+\mathrm{y}^{3}+\mathrm{y}^{5}$
Then ascending order of their degree is
(1) $A, B, D, C$
(2) $A, B, C, D$
(3) B, C, D, A
(4) B, A, C, D
Solution:
(4) B, A, C, D
Hint: Degree of $(A),(B)(C) \&(D)$ are respectively be $5,4,8,12$
 

Question $9 .$
If $\mathrm{p}(a)=0$ then $(x-a)$ is a of $p(x)$
(1) divisor
(2) quotient
(3) remainder
(4) factor
Solution:
(4) factor

Question $10 .$
Zeros of $(2-3 x)$ is
(1) 3
(2) 2
(3) $\frac{2}{3}$
(4) $\frac{3}{2}$
Solution:
(3) $\frac{2}{3}$

Question $11 .$
Which of the following has $x-1$ as a factor?
(1) $2 \mathrm{x}-1$
(2) $3 x-3$
(3) $4 x-3$
(4) $3 x-4$
Solution:
(2) $3 x-3$
Hint: $p(x)=3 x-3$
$P(1)=3(1)-3=0$
$\therefore(x-1)$ is a factor of $p(x)$

Question 12 .
If $x-3$ is a factor of $p$ (x), then the remainder is
(1) 3
(2) $-3$
(3) $\mathrm{p}(3)$
(4) $\mathrm{p}(-3)$
Solution:
(3) p(3)

Question $13 .$
$(x+y)\left(x^{2}-x y+y^{2}\right)$ is equal to
(1) $(x+y)^{3}$
(2) $(x-y)^{3}$
(3) $x^{3}+y^{3}$
(4) $x^{3}-y^{3}$
Solution:
(3) $x^{3}+y^{3}$
 

Question $14 .$
$(a+b-c)^{2}$ is equal to
(1) $(a-b+c)^{2}$
(2) $(-a-b+c)^{2}$
(3) $(a+b+c)^{2}$
$(4)(a-b-c)^{2}$
Solution:
(2) $(-a-b+c)^{2}$
Hint: $(a+b-c)^{2}=[-(-a-b+c)]^{2}=(-a-b+c)^{2}$

Question $15 .$
In an expression $\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}$ the sum and product of the factors respectively,
(1) $a, b c$
(2) b, ac

(3) ac, b
(4) bc, a
Solution:
(2) b, ac
 

Question $16 .$
If $(x+5)$ and $(x-3)$ are the factors of $a x^{2}+b x+c$, then values of $a, b$ and $c$ are
(1) $1,2,3$
(2) $1,2,15$
(3) $1,2,-15$
(4) $1,-2,15$
Solution:
(3) $1,2,-15$
Hint: $\mathrm{p}(-5)=\mathrm{a}\left(-5^{2}\right)+\mathrm{b}(-5)+\mathrm{c}=25 \mathrm{a}-5 \mathrm{~b}+\mathrm{c}=0 \ldots \ldots \ldots$ (1)
$\mathrm{p}(3)=\mathrm{a}\left(3^{2}\right)+\mathrm{bc}+3+\mathrm{c}=9+3 \mathrm{~b}+\mathrm{c}=0 \ldots \ldots .(2)$
$25 a-5 b=9 a+3 b$
$25 a-9 a=3 b+5 b$
$16 a=8 b$
$\frac{a}{b}=\frac{8}{16}=\frac{1}{2}$
Substitute $a=1, b=2$ in (1)
$25(1)-5(2)=-\mathrm{c}$
$25-10=15=-\mathrm{c}$
$\mathrm{c}=-15$

Question $17 .$
Cubic polynomial may have a maximum of
(1) 1
(2) 2
(3) 3
(4) 4
Solution:
(3) 3

Question $18 .$
Degree of the constant polynomial is
(1) 3
(2) 2
(3) 1
(4) 0
Solution:
(4) 0

Question $19 .$
Find the value of $m$ from the equation $2 x+3 y=m$. If its one solution is $x=2$ and $y=-2$.
(1) 2
(2) $-2$
(3) 10
(4) 0
Solution:
(2) - 2

Question $20 .$
Which of the following is a linear equation?
(1) $x+\frac{1}{x}=2 x$
(2) $x(x-1)=2$
(3) $3 x+5=\frac{2}{3}$
(4) $x^{3}-x=5$
Solution:
(3) $3 x+5=\frac{2}{3}$
Hint: $x+\frac{1}{x}=2 \Rightarrow x^{2}-2 x+1=0 ; x(x-1)=2 \Rightarrow x^{2}-x-2=0$
 

Question $21 .$
Which of the following is a solution of the equation $2 x-y=6$ ?
(1) $(2,4)$
(2) $(4,2)$
(3) $(3,-1)$
(4) $(0,6)$
Solution:
(2) $(4,2)$
Hint: $2 x-y=6$
$2(4)-2=8-2=6=\mathrm{RHS}$
 

Question $22 .$
If $(2,3)$ is a solution of linear equation $2 x+3 y=k$ then, the value of $k$ is
(1) 12
(2) 6
(3) 0
(4) 13
Solution:
(4) 13
Hint: $2 x+3 y=k$
$2(2)+3(3)=4+9=13$

Question $23 .$
Which condition does not satisfy the linear equation $a x+b y+c=0$
(1) $a \neq 0, b=0$
(2) $a=0, b \neq 0$
(3) $\mathrm{a}=0, \mathrm{~b}=0, \mathrm{c} \neq 0$
(4) $a \neq 0, b \neq 0$
Solution:
(3) $\mathrm{a}=0, \mathrm{~b}=0, \mathrm{c} \neq 0$
Hint: $a=0, b=0, c \neq 0 \Rightarrow(0) x+(0) y+c=0$ False
 

Question $24 .$
Which of the following is not a linear equation in two variable
(1) $a x+b y+c=0$
(2) $0 x+0 y+c=0$
(3) $0 x+b y+c=0$
(4) $a x+0 y+c=0$
Solution:
(2) $0 \mathrm{x}+0 \mathrm{y}+\mathrm{c}=0$
Hint: $a$ and $b$ both can not be zero

Question $25 .$
The value of $k$ for which the pair of linear equations $4 x+6 y-1=0$ and $2 x+k y-7=0$ represents parallel lines is
(1) $\mathrm{k}=3$
(2) $k=2$
(3) $\mathrm{k}=4$
(4) $k=-3$

Solution:
(1) $k=3$

Question $27 .$
If $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ where $\mathrm{a}_{1} \mathrm{x}+\mathrm{b}_{1} \mathrm{y}+\mathrm{c}_{1}=0$ and $\mathrm{a}_{2} \mathrm{x}+\mathrm{b}_{2} \mathrm{y}+\mathrm{c}_{2}=0$ then the given pair of linear equation has solution(s)
(1) no solution
(2) two solutions
(3) unique
(4) infinite
Solution:
(3) unique
Hint: $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} ;$ unique solution
 

Question $28 .$
If $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ where $a_{1} x+b_{1} y+c_{1}=0$ and $a_{2} x+b_{2} y+c_{2}=0$ then the given pair of linear equation has
(1) no solution
(2) two solutions
(3) infinite
(4) unique
Solution:
(1) no solution
Hint: $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ : parallel

Question $29 .$
GCD of any two prime numbers is
(1) -1
(2) 0
(3) 1
(4) 2
Solution:
(3) 1

Question $30 .$
The GCD of $x^{4}-y^{4}$ and $x^{2}-y^{2}$ is
(1) $x^{4}-y^{4}$
(2) $x^{2}-y^{2}$
(3) $(x+y)^{2}$
(4) $(x+y)^{4}$
Solution:
(2) $x^{2}-y^{2}$
Hint:
$\begin{aligned}
&x^{4}-y^{4}=\left(x^{2}\right)^{2}-\left(y^{2}\right)^{2}=\left(x^{2}+y^{2}\right)\left(x^{2}-y^{2}\right) \\
&x^{2}-y^{2}=x^{2}-y^{2} \\
&\text { G.C.D. is }=x^{2}-y^{2}
\end{aligned}$