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Maths M.C.Q

Factorization Of Polynomials [NEW] · Gujarat Board (GSEB) STD 9 (GSEB - English Medium). 60 questions.

Questions · Page 2 of 2

M.C.Q

MCQ 511 Mark
If $f(x+1)=2 x^2+7 x+5$, then one of the factors of $f(x)$ is
  • A
    $2 x+3$
  • B
    $3 x+2$
  • C
    $2 x-3$
  • D
    $3 x-2$
Answer
A. $2 x+3$
We have,
\[\begin{array}{ll} & f(x+1)=2 x^2+7 x+5 \\\Rightarrow \quad & f(u)=2(u-1)^2+7(u-1)+5, \text { where } x+1=u \text { or, } x=u-1 \\\Rightarrow \quad & f(u)=2 u^2+3 u \Rightarrow f(u)=u(2 u+3) \Rightarrow f(x)=x(2 x+3)\end{array}\]
Hence, $2 x+3$ is a factor of $f(x)$.
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MCQ 521 Mark
If $f\left(x+\frac{1}{x}\right)=x^2+\frac{1}{x^2}$, the remainder when $f(x)$ is divided by $x-3$ is
  • A
    10
  • B
    11
  • C
    7
  • D
    5
Answer
C. 7
We have, $f\left(x+\frac{1}{x}\right)=x^2+\frac{1}{x^2}$
or, $\quad f\left(x+\frac{1}{x}\right)=\left(x+\frac{1}{x}\right)^2-2$ i.e. $f(u)=u^2-2$, where $u=x+\frac{1}{x}$
The remainder when $f(u)=u^2-2$ is divided by $u-3$ is $f(3)=3^2-2=7$.
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MCQ 531 Mark
If $f\left(x-\frac{1}{x}\right)=x^2+\frac{1}{x^2}$, then the remainder when $f(x)$ is divided by $(x-3)$, is
  • A
    10
  • ✓
    11
  • C
    7
  • D
    $\frac{82}{9}$
Answer
Correct option: B.
11
b
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MCQ 541 Mark
If $f\left(x+\frac{1}{x}\right)=x^2+\frac{1}{x^2}$, then the remainder when $f(x)$ is divided by $(2 x+1)$, is
  • ✓
    $-\frac{7}{4}$
  • B
    $-\frac{9}{4}$
  • C
    $\frac{9}{4}$
  • D
    $\frac{11}{4}$
Answer
Correct option: A.
$-\frac{7}{4}$
a
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MCQ 551 Mark
If both $x-2$ and $x-\frac{1}{2}$ are factors of $p x^2+5 x+r$, then
  • ✓
    p = r
  • B
    p + r = 0
  • C
    2p + r = 0
  • D
    p + 2r = 0
Answer
Correct option: A.
p = r
a
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MCQ 561 Mark
If a quadratic polynomial $f(x)$ leares remainders 4,4 and 0 respectively when divided by $(x-1),(x-2)$ and $(x-3)$ respectively, then $f(x)=$
  • A
    $-2 x^2+6 x+3$
  • B
    $-2 x^2+6 x$
  • C
    $-2 x^2+6 x+5$
  • D
    $-2 x^2+6 x-5$
Answer
B. $-2 x^2+6 x$
Let $f(x)=a x^2+b x+c$. It is given that $ \begin{array}{ll} & f(1)=4, f(2)=4 \text { and } f(3)=0 \\ \Rightarrow \quad & a+b+c=4,4 a+2 b+c=4 \text { and } 9 a+3 b+c=0 \Rightarrow a=-2, b=6, c=0 \end{array} $ Hence, $f(x)=-2 x^2+6 x$.
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MCQ 571 Mark
If $(3 x-1)^7=a_7 x^7+a_6 x^6+a_5 x^5+\cdots+a_1 x+a_0$, then $a_7+a_6+a_5+\cdots+a_1+a_0=$
  • A
    $0$
  • B
    1
  • ✓
    128
  • D
    64
Answer
Correct option: C.
128
c
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MCQ 581 Mark
If $(2 x-1)$ is a factor of $f(x)=2 z^2+a x-2$, then the other factor of $f(x)$ is
  • A
    $x-2$
  • B
    $x+2$
  • C
    $x-1$
  • D
    $x+1$
Answer
B. $x+2$
Given that $(2 x-1)$ is a factor of $f(x)$
$ \begin{array}{ll} \therefore & f\left(\frac{1}{2}\right)=0 \Rightarrow 2\left(\frac{1}{2}\right)^2+\frac{a}{2}-2=0 \Rightarrow a=3 \\ \therefore & f(x)=2 x^2+3 x-2 \Rightarrow f(x)=(2 x-1)(x+2) \end{array} $
Hence, $(x+2)$ is the other factor of $f(x)$.
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MCQ 591 Mark
Degree of the zero polynomial is
  • A
    $0$
  • B
    1
  • C
    any natural number
  • ✓
    not defined
Answer
Correct option: D.
not defined
d
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