MCQ 11 Mark
Statement-1 (A): Degree of a zero polynomial is not defined.
Statement-2 (R): Degree of a non-zero constant polynomial is zero.
- 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 21 Mark
Statement-1 (A): If the graph of a polynomial touches the $x$-axis at only one point, then the polynomial cannot be a quadratic polynomial.
Statement-2 (R) : A polynomial of degree $n(n>1)$ can have at most $n$ real zeroes.
- 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 31 Mark
Statement-1 (A): The polynomial $p(x)=x^2+3 x+3$ has two real zeroes.
Statement-2 (R): A quadratic polynomial can have at most two real zeroes.
- 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 41 Mark
Statement-1 (A): If $\alpha, \beta$ and $\gamma$ are the zeroes of the polynomial $6 x^3+3 x^2-5 x+1$, then $\alpha^{-1}+\beta^{-1}+\gamma^{-1}=5$.
Statement-2 (R): If $\alpha, \beta, \gamma$ are the zeroes of the cubic polynomial $a x^3+b x^2+c x+d$, then $\alpha+\beta+\gamma=-\frac{b}{a}$.
- 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 51 Mark
Statement-1 (A): If $\alpha$ and $\beta$ are zeroes of the quadratic polynomial $x^2+7 x+12$, then
$
\frac{12}{\alpha}+\frac{12}{\beta}-24 \alpha \beta=395
$
Statement-2 (R): If $\alpha$ and $\beta$ are zeroes of the quadratic polynomial $a x^2+b x+c$, then
$
\alpha+\beta=-\frac{b}{a} \text { and } \alpha \beta=\frac{c}{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.
- 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 one root of the quadratic polynomial $f(x)=(k-1) x^2-10 x+3, k \neq 1$ is reciprocal of the other, then $k=4$.
Statement-2 (R): The product of roots of the quadratic polynomial $a x^2+b x+c, a \neq 0$ is $\frac{a}{c}$
- 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 71 Mark
Statement-1 (A): A quadratic polynomial having $\frac{1}{2}$ and $\frac{1}{3}$ as its zeroes is $6 x^2-5 x+1$
Statement-2 (R): Quadratic polynomials having $\alpha$ and $\beta$ as zeroes are given by $f(x)=k\left|x^2-(\alpha+\beta) x+\alpha \beta\right|$, where $k$ is a non-zero constant.
- ✓
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 81 Mark
Statement-1 (A): The polynomial $f(x)=x^2-2 x+2$ has two real zeros.
Statement-2 (R): A quadratic polynomial can have at most two real zeroes.
- 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 91 Mark
Statement-1 (A): If $\alpha, \beta$ are roots of the polynomial $2 x^2+6 x+c, c<0$, then $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}$ is less than -2 .
Statement-2 (R): The sum of a negative number and its reciprocal is always less than or equal to -2 .
- ✓
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.
(A)Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
Let $x$ be a negative real number. Then, $x=-y$ for some $y >0$. We know that
$
\left(\sqrt{y}-\frac{1}{\sqrt{y}}\right)^2 \geq 0 \Rightarrow y+\frac{1}{y}-2 \geq 0 \Rightarrow-x-\frac{1}{x}-2 \geq 0 \Rightarrow x+\frac{1}{x}+2 \leq 0 \Rightarrow x+\frac{1}{x} \leq-2
$
So, statement-2 is true.
It is given that $\alpha, \beta$ are roots of $2 x^2+6 x+c, c< 0$.
$
\therefore \quad \alpha \beta=\frac{c}{2} < 0
$
$\Rightarrow \quad \alpha$ and $\beta$ are of opposite signs.
$\Rightarrow \quad \frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ both are negative $\Rightarrow \frac{\alpha}{\beta}+\frac{\beta}{\alpha} < -2$
View full question & answer→MCQ 101 Mark
Statement- 1(A) : If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $k x^2+4 x+4$, where $k$ is an integer such that $(\alpha+\beta)^2-2 \alpha \beta=24$, then $k=1$.
Statement-2(R): If $\alpha$ and $\beta$ are zeroes of the polynomial $a x^2+b x+c, a \neq 0$, then $\alpha+\beta=-\frac{b}{a}$ and $\alpha \beta=\frac{c}{a}$.
- 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.
- 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.
(D)Statement- 1 is false, Statement- 2 is true.
Given that $\alpha, \beta$ are zeroes of $k x^2+4 x+4$. Therefore, $\alpha+\beta=-\frac{4}{k}$ and $\alpha \beta=\frac{4}{k}$. Now, $\quad(\alpha+\beta)^2-2 \alpha \beta=24$
$
\Rightarrow \quad \frac{16}{k^2}-\frac{8}{k}=24 \Rightarrow 16-8 k=24 k^2 \Rightarrow 3 k^2+k-2=0 \Rightarrow(3 k-2)(k+1)=0 \Rightarrow k=\frac{2}{3},-1
$
So, statement- 1 is false. Clearly, statement- 2 is true. Hence, option (d) is correct.
View full question & answer→MCQ 111 Mark
Statement-1 (A) : If the graph of the polynomial $f(x)=a x^2+b x+c, a > 0$ touches $x$-axis at its lowest point, then its zeroes are equal to $-\frac{b}{2 a}$.
Statement-2 (R): If the graph of a quadratic polynomial touches $x$-axis, then its zeroes are equal.
- ✓
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.
(A)Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
If the graph of the quadratic polynomial touches $x$-axis at its vertex, then its zeroes are equal. Let each zero be $\alpha$. Then,
$
\alpha+\alpha=-\frac{b}{a} \Rightarrow \alpha=-\frac{b}{2 a}
$
So, statement- 1 is true. Clearly, statement- 2 is also true and it is a correct explanation for statement1. Hence, option (a) is correct.
View full question & answer→MCQ 121 Mark
Statement-1 (A) : If zeroes of the polynomial $f(x)=5 x^2-11 x-(k-3)$ are reciprocal of each other, then $k=-2$.
Statement-2 (R) : The product of the zeroes of the polynomial $a x^2+b x+c$ is $-\frac{c}{a}$.
- 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.
(C)Statement- 1 is true, Statement-2 is false.
It is given that the zeroes of $f(x)=5 x^2-11 x-(k-3)$ are reciprocal of each other. Therefore, product of its zeroes is 1 .
i.e.$\quad
-\frac{(k-3)}{5}=1 \Rightarrow k-3=-5 \Rightarrow k=-2
$
So, statement 1 is true. The product of the zeroes of the polynomial $a x^2+b x+c$ is $\frac{c}{a}$. So, statement-2 is false. Hence, option (c) is correct.
View full question & answer→MCQ 131 Mark
Statement-1 (A): If the sum of the zeroes of the quadratic polynomial $f(x)=3 x^2+k x+5$ is $-\frac{2}{3}$, Then the value of k is 2 .
Statement-2 (R): The product of zeroes of the polynomial $a x^2+b x+c$ is $\frac{c}{a}$.
- 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.
(B)Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
The sum of the zeroes of $f(x)=3 x^2+k x+5$ is $-\frac{2}{3}$.
$
\therefore \quad-\frac{k}{3}=-\frac{2}{3} \Rightarrow k=2 \quad\left[\because \alpha+\beta=-\frac{b}{a}\right]
$
So, statement-1 is true. Statement-2 is also true. But, statement-2 is not a correct explanation for statement-1. Hence, option (b) is correct.
View full question & answer→MCQ 141 Mark
Statement-1 (A): A quadratic polynomial having $2+\sqrt{3}$ and $2-\sqrt{3}$ as its zeroes is given by $f(x)=x^2-4 x+1$.
Statement-2 (R): Quadratic polynomials whose two zeroes are $\alpha$ and $\beta$ are given by $f(x)=k\left\{x^2-x(\alpha+\beta)+\alpha \beta\right\}$, where $k$ is any non-zero real number
- ✓
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.
(A)Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
The Statement-2 is true. So, quadratic polynomials having $2+\sqrt{3}$ and $2-\sqrt{3}$ as its zeroes are given by $f(x)=k\left\{x^2-x(2+\sqrt{3}+2-\sqrt{3}+(2+\sqrt{3})(2-\sqrt{3})\} \quad\right.$ or $f(x)=k\left(x^2-4 x+1\right)$. Therefore, $x^2-4 x+1$ is a quadratic polynomial whose two zeroes are $2+\sqrt{3}$ and $2-\sqrt{3}$. Thus, statement -1 is also true and statement 2 is a correct explanation for statement-1. Hence, option (a) is correct.
View full question & answer→MCQ 151 Mark
Statement-1 (A): The polynomial $f(x)=x^2-2 x+2$ has two real zeros.
Statement-2 (R): A quadratic polynomial can have at most two real zeroes.
- 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→