MCQ 11 Mark
Zero of the zero polynomial is
MCQ 21 Mark
Zero of the polynomial $f(x)=3 x+7$ is
- A
$\frac{7}{3}$
- B
$\frac{-3}{7}$
- ✓
$-\frac{7}{3}$
- D
$-7$
AnswerCorrect option: C. $-\frac{7}{3}$
View full question & answer→MCQ 31 Mark
$(x+1)$ is a factor of $x^n+1$ only if
AnswerCorrect option: A. $n$ is an odd integer
View full question & answer→MCQ 41 Mark
$x+1$ is a factor of the polynomial
- A
$x^3+x^2-x+1$
- ✓
$x^3+x^2+x+1$
- C
$x^4+x^3+x^2+1$
- D
$x^4+3 x^3+3 x^2+x+1$
AnswerCorrect option: B. $x^3+x^2+x+1$
View full question & answer→MCQ 51 Mark
Which one of the following is a polynomial?
AnswerD. $p(x)=2 x^2+\frac{5 x^{3 / 2}+4 \sqrt{x}}{\sqrt{x}}$
$f(x)=x+\frac{1}{x}$ may be written as $f(x)=x+x^{-1}$. We find that in one term the exponent of $x$ is -1 , which is a negative integer. So, $f(x)$ is not a polynomial.
$\ln g(x)=\frac{(x-1)(x-3)}{x}=\frac{x^2-4 x+3}{x}=x-4+3 x^{-1}$, there is one term in which the exponent of $x$ is -1 . So, it is not a polynomial.
In $h(x)=\frac{x+2}{x+1}=\frac{(x+1)+1}{x+1}=1+(x+1)^{-1}$, one term contains $(x+1)^{-1}$. So, it is not a polynomial.
$p(x)=2 x^2+\frac{5 x^{3 / 2}+4 \sqrt{x}}{\sqrt{x}}=2 x^2+5 x+4$, clearly, it is a polynomial.
View full question & answer→MCQ 61 Mark
Which one of the following is a polynomial?
AnswerCorrect option: C. $x^2+\frac{3 x^{3 / 2}}{\sqrt{x}}$
View full question & answer→MCQ 71 Mark
Which of the following is a polynomial?
- A
$x^{-2}+2 x^{-1}+3$
- B
$x+x^{-1}+5$
- C
$2 x^{-1}$
AnswerD. 0
Expressions in first three options contain negative powers of variable $x$. So, none of them is a polynomial. In option (d), 0 is zero polynomial whose degree is not defined.
View full question & answer→MCQ 81 Mark
When $x^3-2 x^2+a x-b$ is divided by $x^2-2 x-3$, the remainder is $x-6$. The values of $a$ and $b$ are respectively
View full question & answer→MCQ 91 Mark
When the polynomial $p(x)=a x^2+b x+c$ is divided by $x, x-2$ and $x+3$, the remainders obtained are 7,9 and 49 respectively. The value of $3 a+5 b+2 c$ is
View full question & answer→MCQ 101 Mark
When the polynomial $p(x)=a x^2+b x+c$ is divided by $(x-1)$ and $(x+1)$, the remainders obtained are 6 and 10 respectively. If the value of $p(x)$ at $x=0$ is 5 , then $5 a-2 b+5 c=$
Answer D. 44
It is given that
$ \begin{array}{ll} & p(1)=6, p(-1)=10 \text { and } p(0)=5 \\ \Rightarrow & a+b+c=6, a-b+c=10 \text { and } c=5 \Rightarrow a+b=1, a-b=5 \text { and } c=5 \\ \Rightarrow & a=3, b=-2 \text { and } c=5 \\ \therefore & 5 a-2 b+5 c=15+4+24=44 \end{array} $
View full question & answer→MCQ 111 Mark
The value of $k$ for which $x-1$ is a factor of $4 x^3+3 x^2-4 x+k$, is
View full question & answer→MCQ 121 Mark
The remainder when $f(x)=x^5$ is divided by $g(x)=x^2-9$, is
- A
$81 x$
- B
$81 x+10$
- C
$243 x+81$
AnswerA. 81x
Since $g(x)=x^2-9$ is a quadratic polynomial. Therefore, when $f(x)$ is divided by $g(x)$ the remainder is a linear polynomial. Let $f(x)=a x+b$ be the remainder and $q(x)$ be the quotient.
$ f(x)=q(x) g(x)+r(x) \Rightarrow x^5=\left(x^2-9\right) q(x)+a x+b $
Zeroes of $g(x)$ are given by
$ g(x)=0 \Rightarrow x^2-9=0 \Rightarrow(x-3)(x+3)=0 \Rightarrow x=-3,3 $
Putting $x=-3$ and $x=3$ successively in (i), we obtain
$ -243=-3 a+b \text { and } 243=3 a+b \Rightarrow a=81 \text { and } b=0 $
Hence, $r(x)=81 x$.
View full question & answer→MCQ 131 Mark
The remainder when $f(x)=x^{45}+x^{25}+x^{14}+x^9+x$ is divided by $g(x)=x^2-1$, is
- A
$4 x-1$
- B
$4 x+2$
- ✓
$4 x+1$
- D
$4 x-2$
AnswerCorrect option: C. $4 x+1$
View full question & answer→MCQ 141 Mark
The remainder when $f(x)=x^3+a x^2+6 x+a$ is divided by $(x+a)$, is
Answerd. 10
If $f(x)$ is divided by $(x+a)$, the remainder is $f(-a)$.\[\therefore \quad \text { Remainder }=f(-a)=(-a)^3+a(-a)^2+6(-a)+a=-a^3+a^3-6 a+a=-5 a\]
View full question & answer→MCQ 151 Mark
The remainder when $f(x)=x^3-2 x^2+6 x-2$ is divided by $(x-2)$, is
AnswerD. 10
$f(2)$ is the remainder when $f(x)$ is divided by $x+2$.
Hence, remainder $=f(2)=2^3-2 \times 2^2+6 \times 2-2=10$
View full question & answer→MCQ 161 Mark
The ratio of the remainders when $f(x)-x^2+a x+b$ is divided by $(x-2)$ and $(x-1$ respectively is 4 : 3. If $(1+1)$ is a factor of $f(x)$, then
- A
$a=9, b=-10$
- B
$a=-9, b=10$
- C
$a-9, b=10$
- D
$a=-9, b=-10$
AnswerD. a=-9, b=-10
It is given that
$\frac{f(2)}{f(1)}=\frac{t}{3}$ and $f(-1)=0$ $\Rightarrow \frac{4+2 a+b}{1+a+b}=\frac{4}{3}$ and $1-a+b=0$
$\Rightarrow$ 2a - b + 8 =0 and -a + b + 1 = 0 $\Rightarrow a=-9, b=-10$.
View full question & answer→MCQ 171 Mark
The ratio of remainders when $f(x)=x^2+a x+b$ is divided by $(x-2)$ and $(x-3)$ respectively is $5: 4$. If $(x-1)$ is a factor of $f(x)$, then
- A
$a=-\frac{11}{3}, b=\frac{14}{3}$
- ✓
$a=-\frac{14}{3}, b=\frac{11}{3}$
- C
$a=\frac{14}{3}, b=-\frac{11}{3}$
- D
$a=-\frac{14}{3}, b=-\frac{11}{3}$
AnswerCorrect option: B. $a=-\frac{14}{3}, b=\frac{11}{3}$
View full question & answer→MCQ 181 Mark
$\sqrt{3}$ is a polynomial of degree
AnswerB. 0
$f(x)=\sqrt{3}$ is a constant polynomial. The degree of a non-zero constant polynomial is zero.
View full question & answer→MCQ 191 Mark
$\sqrt{2}$ is a polynomial of degree
View full question & answer→MCQ 201 Mark
One of the zeros of the polynomial $f(x)=2 x^2+7 x-4$ is
- A
- ✓
$\frac{1}{2}$
- C
$-\frac{1}{2}$
- D
AnswerCorrect option: B. $\frac{1}{2}$
View full question & answer→MCQ 211 Mark
One factor of $x^4+x^2-20$ is $x^2+5$. The other factor is
AnswerCorrect option: A. $x^2-4$
View full question & answer→MCQ 221 Mark
Let $f(x)$ be a polynomial such that $f\left(-\frac{1}{2}\right)=0$, then a factor of $f(x)$ is
- A
$2 x-1$
- ✓
$2 x+1$
- C
$x-1$
- D
$x+1$
AnswerCorrect option: B. $2 x+1$
View full question & answer→MCQ 231 Mark
If $x+a$ is a factor of $x^4-a^2 x^2+3 x-6 a$, then $a=$
View full question & answer→MCQ 241 Mark
If $x-a$ is a factor of $x^3-3 x^2 a+2 a^2 x+b$, then the value of $b$ is
View full question & answer→MCQ 251 Mark
If $(x-a)$ is a factor of the polynomial $p(x)=x^3-a x^2+2 x+a-6$, then the value of $a$ is
AnswerC. 2
If $(x-a)$ is a factor of $p(x)$, then
\[p(a)=0 \Rightarrow a^3-a^3+2 a+a-6=0 \Rightarrow 3 a-6=0 \Rightarrow a=2\]
View full question & answer→MCQ 261 Mark
If $(x-a)$ and $(x-b)$ are factors of $x^2+a x-b$, then
AnswerCorrect option: D. $a=-1, b=2$
View full question & answer→MCQ 271 Mark
If $(x-a)$ and $(x-b)$ are factors of $x^2+a x+b$, then
AnswerCorrect option: A. $a=1, b=-2$
View full question & answer→MCQ 281 Mark
If $x^{51}+51$ is divided by $x+1$, the remainder is
View full question & answer→MCQ 291 Mark
If $x-3$ is a factor of $x^2-a x-15$, then $a=$
View full question & answer→MCQ 301 Mark
If $(x-3)$ is a factor of $f(x)=x^2+a$, then the remainder when $f(x)$ is divided by $(x-2)$ is
AnswerB. -5
Given that $(x-3)$ is a factor of $f(x)=x^2+a$
$
\therefore \quad f(3)=0 \Rightarrow 3^2+a=0 \Rightarrow a=-9
$
Thus, $f(x)=x^2-9$
The remainder when $f(x)$ is divided by $(x-2)$ is: $f(2)=2^2-9=-5$.
View full question & answer→MCQ 311 Mark
If $x^3+6 x^2+4 x+k$ is exactly divisible by $x+2$, then $k=$
View full question & answer→MCQ 321 Mark
If $x^2+x+1$ is a factor of the polynomial $3 x^3+8 x^2+8 x+3+5 k$, then the value of $k$ is
AnswerCorrect option: B. $2 / 5$
View full question & answer→MCQ 331 Mark
If $x^2+k x+6=(x+2)(x+3)$ for all $x$, then the value of $k$ is
View full question & answer→MCQ 341 Mark
If $x+2$ is a factor of $x^2+m x+14$, then $m=$
View full question & answer→MCQ 351 Mark
If $x-2$ is a factor of $x^2+3 a x-2 a$, then $a=$
View full question & answer→MCQ 361 Mark
If $(x-2)$ is a factor of $f(x)=x^2+a x+1$, then the remainder when $x^2+a x+1$ is divided by $(2 x+3)$, is
AnswerA. 7
If $(x-2)$ is a factor of $f(x)$, then
$ f(2)=0 \Rightarrow 2^2+2 a+1=0 \Rightarrow a=-\frac{5}{2} $
The remainder when $f(x)$ is divided by $2 x+3$ is
$ f\left(-\frac{3}{2}\right)=\left(-\frac{3}{2}\right)^2+a\left(-\frac{3}{2}\right)+1=\frac{9}{4}-\frac{3}{2} \times-\frac{5}{2}+1=7 $
View full question & answer→MCQ 371 Mark
If $x+2$ and $x-1$ are the factors of $x^3+10 x^2+m x+n$, then the values of $m$ and $n$ are respectively
View full question & answer→MCQ 381 Mark
If $x^2-1$ is a factor of $a x^4+b x^3+c x^2+d x+e$, then
View full question & answer→MCQ 391 Mark
If $x+1$ is a factor of the polynomial $2 x^2+k x$, then $k=$
View full question & answer→MCQ 401 Mark
If $(x-1)$ is a factor of polynomial $f(x)$ but not of $g(x)$, then it must be a factor of
View full question & answer→MCQ 411 Mark
If $(x+1)$ and $(x-1)$ are factors of $f(x)=a x^3+b x^2+c x+d$, then
- A
$a+b=0$
- B
$b+c=0$
- C
$b+d=0$
- D
$a+d=0$
Answer C. $b+d=0$
Given that $(x-1)$ and $(x+1)$ are factors of $f(x)$. $ \begin{array}{ll} \therefore & f(1)=0 \text { and } f(-1)=0 \\ \Rightarrow & a+b+c+d=0 \text { and }-a+b-c+d=0 \\ \Rightarrow & 2(b+d)=0 \text { and } 2(a+c)=0 \\ \Rightarrow & b+d=0 \text { and } a+c=0 \end{array} $
View full question & answer→MCQ 421 Mark
If $x^{140}+2 x^{151}+k$ is divisible by $x+1$, then the value of $k$ is
View full question & answer→MCQ 431 Mark
If $x^{101}+101$ is divided by $x+1$, then the remainder is
AnswerB. 100
Let $f(x)=x^{101}+101$. If $f(x)$ is divided by $x+1$, then
\[\text { Remainder }=f(-1)=(-1)^{101}+101=-1+101=100\]
View full question & answer→MCQ 441 Mark
If $f(x)=x+3$, then $f(x)+f(-x)$ is equal to
View full question & answer→MCQ 451 Mark
If $f(x)=x^2-2 \sqrt{2} x+1$, then $f(2 \sqrt{2})$ is equal to
- A
$0$
- ✓
- C
$4 \sqrt{2}$
- D
$8 \sqrt{2}+1$
View full question & answer→MCQ 461 Mark
If $f(x)=x^{100}+2 x^{99}+k$ is divisible by $(x+1)$, then the value of $k$ is
AnswerA. 1
If $f(x)$ is divisible by $(x+1)$, then
\[f(-1)=0 \Rightarrow(-1)^{100}+2(-1)^{99}+k=0 \Rightarrow 1-2+k=0 \Rightarrow k=1\]
View full question & answer→MCQ 471 Mark
If $f(x+3)=x^2+x-6$, then one of the factors of $f(x)$ is
AnswerC. $x-5$
We have, $f(x+3)=x^2+x-6$
Let $x+3=u$. Then $x=u-3$. Putting $x=u-3$ in $f(x)=x^2+x-6$, we obtain$
f(u)=(u-3)^2+(u-3)-6 \Rightarrow f(u)=u^2-5 u \text { or, } f(u)=u(u-5)
$
Thus, we obtain $f(x)=x(x-5)$.
Hence, $x$ and $x-5$ are factors of $f(x)$.
We have, $f(x+3)=1^2+x-6$ or, $f(x+3)=(x+3)(x-2)$
Replacing x by $x-3$, we obtain: $f(x)-x(x-5)$
View full question & answer→MCQ 481 Mark
If $f(x+3)=x^2-7 x+2$, then the remainder when $f(x)$ is divided by $(x+1)$, is
View full question & answer→MCQ 491 Mark
If $f(x+3)=x^2-7 x+2$, then the remainder when $f(x)$ is divided by $(x+1)$ is
AnswerD. 46
We have, $f(x+3)=x^2-7 x+2$. Let $x+3=\alpha$. Then, $x=\alpha-3$.
Replacing $x$ by $\alpha-3$, we obtain
$
f(\alpha)=(\alpha-3)^2-7(\alpha-3)+2 \text { or, } f(\alpha)=\alpha^2-13 \alpha+32
$
Thus, we obtain $f(x)=x^2-13 x+32$.
The remainder when $f(x)$ is divided by $x+1$ is $f(-1)=1+13+32=46$.
ALITER Putting $x+3=-1$ i.e. $x=-4$ in $f(x+3)=x^2-7 x+2$, we obtain: $f(-1)=16+28+2=46$.
View full question & answer→MCQ 501 Mark
If $f(x-2)=2 x^2-3 x+4$, then the remainder when $f(x)$ is divided by $(x-1)$, is
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