MCQ 11 Mark
Statement-1 (A): If $x+7$ is a factor of $f(x)=x^2+11 x-2 a$, then $a=-14$
Statcment-2 $(R)$ : If $x+a$ is a factor of a polynomial, then $f(a)=0$.
- A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-2
- B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is True, Statement-2 is False.
- D
Statement-1 is False, Statement-2 is True.
AnswerC. If $x+7$ is a factor of $f(x)=x^2+11 x-2 a$, then 12$f(-7)=0$.
$\therefore$ $(-7)^2+11(-7)-2 a=0 \Rightarrow 49-77-2 a=0 \Rightarrow-28-2 a=0 \Rightarrow a=-14$
Thus statement- 1 is true but statement- 2 is not true. Hence, option (c) is correct.
View full question & answer→MCQ 21 Mark
Statement-1 (A): If $x+2 a$ is a factor of $f(x)=x^5-4 a^2 x^3+2 x+2 a+3$, then $2 a-3=0$
Statement-2 (R): If $f(x)$ is divisible by $(a x+b)$, then $f\left(-\frac{b}{a}\right)=0$.
- ✓
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-5
- C
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
AnswerCorrect option: A. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
View full question & answer→MCQ 31 Mark
Statement-1 (A): If $x+2 a$ is a factor of $f(x)=x^5-4 a^2 x^3+2 x+2 a+3$, then $2 a-3=0$.
Statement-2 (R): If $f(x)$ is divisible by $(a x+b)$, then $f\left(-\frac{b}{a}\right)=0$.
- ✓
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
AnswerCorrect option: A. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
View full question & answer→MCQ 41 Mark
Statement-1 (A): If $x+1$ is a factor of $f(x)=p x^2+5 x+r$, then $p+r+5=0$.
Statement-2 (R): If $x-2$ and $2 x-1$ are factors of $f(x)=p x^2+5 x+r$, then $p=r$.
- A
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-4
- C
Statement-1 is true, Statement-2 is false.
- ✓
Statement-1 is false, Statement-2 is true.
AnswerCorrect option: D. Statement-1 is false, Statement-2 is true.
View full question & answer→MCQ 51 Mark
Statement-1 (A): If $x+1$ is a factor of $f(x)=p x^2+5 x+r$ then $p+r+5=0$.
Statement-2 (R): If $x-2$ and $2 x-1$ are factors of $f(x)=p x^2+5 x+r$, then $p=r$.
- A
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is true, Statement-2 is false.
- ✓
Statement-1 is false, Statement-2 is true.
AnswerCorrect option: D. Statement-1 is false, Statement-2 is true.
View full question & answer→MCQ 61 Mark
Statement-1 (A): If the polynomial $p(x)=x^3+a x^2-2 x+a+4$ has $(x+a)$ as one of its factors, then $a=-\frac{4}{3}$.
Statement-2 (R): If $f(x)=a x^2+b+c$ is exactly divisible by $2 x-3$ then $4 a+6 b+9 c=0$.
- A
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- ✓
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
AnswerCorrect option: B. Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
View full question & answer→MCQ 71 Mark
Statement-1 (A): If the polynomial $p(x)=x^3+a x^2-2 x+a+4$ has $(x+a)$ as one of its factors, then $a=-\frac{4}{3}$.
Statement-2 (R): If $f(x)=a x^2+b+c$ is exactly divisible by $2 x-3$ then $4 a+6 b+9 c=0$
- A
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- ✓
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1
- C
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
AnswerCorrect option: B. Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1
View full question & answer→MCQ 81 Mark
Statement-1 (A): If the polynomial $f(x)=3 x^4-11 x^2+6 x+k$ when divided by $(x-3)$ leaves remainder 7 , then $k=-155$.
Statement-2 $( R )$ : If a polynomial is divided by $(x-a)$, the remainder is $f(a)$.
- ✓
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-2
- C
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
AnswerCorrect option: A. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
View full question & answer→MCQ 91 Mark
Statement-1 (A): If the polynomial $f(x)=3 x^4-11 x^2+6 x+k$ when divided by $(x-3)$ leaves remainder 7 , then $k=-155$.
Statement-2 (R): If a polynomial is divided by $(x-a)$, the remainder is $f(a)$.
- ✓
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
AnswerCorrect option: A. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
View full question & answer→MCQ 101 Mark
Statement-1 (A): If the polynomial $f(x)=2 x^3+3 x^2-5 x+a$ when divided by $x+2$ leaves the remainder $3 a+2$, then $a=2$.
Statement $-2(R)$ : The remainder when a polynomial $p(x)$ is divided by $(x-a)$ is given by $p(a)$.
- A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-3
- B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is True, Statement-2 is False.
- D
Statement-1 is False, Statement-2 is True.
AnswerA. Statement-2 is the remainder theorem. So, it is true. Using statement-2, the remainder when $f(x)=2 x^3+3 x^2-5 x+a$ is divided by $x+2$, is $f(-2)$. But, it is given that the remainder is 3a + 2
$
\begin{array}{ll}
\therefore & f(-2)=3 a+2 \\
\Rightarrow & 2 \times(-2)^3+3(-2)^2-5(-2)+a=3 a+2 \\
\Rightarrow & -16+12+10+a=3 a+2 \Rightarrow a+6=3 a+2 \Rightarrow a=2
\end{array}
$
Thus, both the statements are true and statement-2 is a correct explanation for statement-1.
View full question & answer→MCQ 111 Mark
Statement-1 (A): If sum of all the coefficients, including the constant term, of a polynomial is zero, then $(x-1)$ is one of its factors.
Statement-2 $(R)$ : If a polynomial $f(x)$ is divisible by $(x-\alpha)$, then $f(\alpha)=0$.
- A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-4
- B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is True, Statement-2 is False.
- D
Statement-1 is False, Statement-2 is True.
AnswerA. If a polynomial $f(x)$ is divisible by $(x-\alpha)$, then $(x-\alpha)$ is one of its factors. Therefore, $f(\alpha)=0$. So, statement-2 is true.
Let $f(x)=a_0 x^n+a_1 x^{n-1}+\ldots+a_{n-2} x^2+a_{n-1} x+a_n$ be a polynomial such that the sum of all the coefficients is zero.
i.e.$
a_0+a_1+a_2+\ldots+a_{n-2}+a_{n-1}+a_n=0
$
$
\Rightarrow \quad f(1)=0
$
$\Rightarrow \quad(x-1)$ is a factor of $f(x)$ or, $f(x)$ is divisible by $(x-1)$.
So, statement-1 is true.
Thus, both the statements are true and statement-2 is a correct explanation for statement- 1 . Hence, option (a) is correct.
View full question & answer→MCQ 121 Mark
Statement-1 (A): If $f(x+2)=2 x^2+x-3$ is divided by $(x-1)$, the remainder is 2.
Statement-2 $(R)$ : If $f(x)$ is divided by $(2-3 x)$, the remainder is $f(2 / 3)$.
- A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-5
- B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is True, Statement-2 is False.
- D
Statement-1 is False, Statement-2 is True.
AnswerD. Using remainder theorem, we find that if $f(x)$ is divided by $(2-3 x)$, the remainder is $f(2 / 3)$. So, statement-2 is true.
$
\begin{array}{ll}
\text { Now, } & f(x+2)=2 x^2+x-3 \\
\Rightarrow & f(u)=2(u-2)^2+(u-2)-3, \text { where } u=x+2 \\
\Rightarrow & f(u)=2 u^2-7 u+3 \Rightarrow f(x)=2 x^2-7 x+3
\end{array}
$
So, the remainder when $f(x)$ is divided by $(x-1)$ is $f(1)=2-7+3=-2$.
View full question & answer→MCQ 131 Mark
Statement-1 (A): If $f(x+2)=2 x^2+7 x+5$, then the remainder when $f(x)$ is divided by $(x-$ $1)$ is 0 .
Statement-2 (R): If a polynomial $f(x)$ is divided by $(a x+b)$, then the remainder is $f(b / a)$.
- A
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-3
- ✓
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
AnswerCorrect option: C. Statement-1 is true, Statement-2 is false.
View full question & answer→MCQ 141 Mark
Statement-1 (A): If $f(x+2)=2 x^2+7 x+5$, then the remainder when $f(x)$ is divided by (x - 1) is 0.
Statement-2 (R): If a polynomial $f(x)$ is divided by $(a x+b)$, then the remainder is $f(b / a)$.
- A
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- ✓
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
AnswerCorrect option: C. Statement-1 is true, Statement-2 is false.
View full question & answer→MCQ 151 Mark
Statement-1 (A): If $a \neq 0$ and $a x^2+b x+a$ is exactly divisible by $(x-a)$, then $a^2+b+1=0$
Statement-2 $(R)$ : If $(x-a)$ is a factor of a polynomial $f(x)$, then $f(a)=0$.
- A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1
- B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is True, Statement-2 is False.
- D
Statement-1 is False, Statement-2 is True.
AnswerA. Statement-2 is true (see Factor Theorem). Using statement-2, if $f(x)=a x^2+b x+a$ is evacth divisible by $(x-a)$, then
$
f(a)=0 \Rightarrow a^3+b a+a=0 \Rightarrow a^2+b+1=0 \quad[\because a \neq 0]
$
So, statement- 1 is also true and statement- 2 is a correct explanation for statement- 1 .
Hence, option (a) is correct.
View full question & answer→