MCQ 511 Mark
A quadratic equation whose roots are $2+\sqrt{3}$ and $2-\sqrt{3}$ is
- ✓
$x^2-4 x+1=0$
- B
$x^2+4 x+1=0$
- C
$4 x^2-3=0$
- D
$x^2-1=0$
AnswerCorrect option: A. $x^2-4 x+1=0$
(A)$x^2-4 x+1=0$
The quadratic equation whose roots are $\alpha=2+\sqrt{3}$ and $\beta=2-\sqrt{3}$ is $x^2-(\alpha+\beta) x+\alpha \beta=0$ or, $x^2-4 x+1=0$
View full question & answer→MCQ 521 Mark
The least positive value of $k$, for which the quadratic equation $2 x^2+k x-4=0$ has rational roots, is
- A
$\pm 2 \sqrt{2}$
- ✓
- C
$\pm 2$
- D
$\sqrt{2}$
Answer(B)2
Given quadratic equation will have rational roots, if its discriminant is a perfect square.
Let $D$ be the discriminant. Then, $D=k^2+32$.
Clearly, the least positive value of $k$ for which $D$ will be a perfect square, is 2 .
View full question & answer→MCQ 531 Mark
Which of the following quadratic equations has sum of its roots as 4 ?
AnswerCorrect option: B. $-x^2+4 x+4=0$
View full question & answer→MCQ 541 Mark
The roots of the equation $x^2+3 x-10=0$ are:
AnswerCorrect option: A. $2,-5$
View full question & answer→MCQ 551 Mark
The values of $k$ for which the quadratic equation $16 x^2+4 k x+9=0$ has real and equal roots are
AnswerCorrect option: C. $6,-6$
View full question & answer→MCQ 561 Mark
If $y=1$ is a common root of the equations $a y^2+a y+3=0$ and $y^2+y+b=0$, then $a b$ equals
View full question & answer→MCQ 571 Mark
The number of real roots of the equation $(x-1)^2+(x-2)^2+(x-3)^2=0$, is
Answer(D)none of these
For any real value of $x$, we find that $(x-1)^2+(x-2)^2+(x-3)^2 \neq 0$. Hence, the given equation has no real roots.
View full question & answer→MCQ 581 Mark
If the roots of the equation $a(b-c) x^2+b(c-a) x+c(a-b)=0$ are equal, then
- ✓
$b=\frac{2 a c}{a+c}$
- B
$b=\frac{a c}{a+c}$
- C
$c=\frac{2 a b}{a+b}$
- D
$a=\frac{2 b c}{b+c}$
AnswerCorrect option: A. $b=\frac{2 a c}{a+c}$
(A)$b=\frac{2 a c}{a+c}$
We observe that $x=1$ satisfies the given equation. So, it is a root of the equation. Thus, both the roots of the given equation are equal to 1.
$
\therefore \quad \text { Product of roots }=1 \times 1 \Rightarrow \frac{c(a-b)}{a(b-c)}=1 \Rightarrow c a-c b=a b-a c \Rightarrow 2 a c=a b+b c \Rightarrow b=\frac{2 a c}{a+c}
$
View full question & answer→MCQ 591 Mark
Quadratic equation whose roots are the reciprocal of the roots of the equation $a x^2+b x+c=0$, is
- A
$a x^2+c x+b=0$
- ✓
$c x^2+b x+a=0$
- C
$c x^2-b x+a=0$
- D
$c x^2+b x-a=0$
AnswerCorrect option: B. $c x^2+b x+a=0$
(B)$c x^2+b x+a=0$
Let $\alpha, \beta$ be the roots of the equation $a x^2+b x+c=0$. Then, $\alpha+\beta=-\frac{b}{a}$ and $\alpha \beta=\frac{c}{a}$. The equation whose roots are $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is
$
x^2-x\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)+\frac{1}{\alpha \beta}=0 \text { or, } x^2-x\left(\frac{\alpha+\beta}{\alpha \beta}\right)+\frac{1}{\alpha \beta}=0 \text { or, } x^2+\frac{b x}{c}+\frac{a}{c}=0 \text { or, } c x^2+b x+a=0
$
View full question & answer→MCQ 601 Mark
If one root of the equation $a x^2+b x+c=0$ is three times the other, then
- A
$b^2=16 a c$
- B
$b^2=3 a c$
- ✓
$3 b^2=16 a c$
- D
$16 b^2=3 a c$
AnswerCorrect option: C. $3 b^2=16 a c$
(C)$3 b^2=16 a c$
Let the roots be $\alpha$ and $3 \alpha$. Then,
$\alpha+3 \alpha=-\frac{b}{a}$ and $\alpha \times 3 \alpha=\frac{c}{a} \Rightarrow \alpha=-\frac{b}{4 a}$ and $3 \alpha^2=\frac{c}{a} \Rightarrow 3 \times\left(-\frac{b}{4 a}\right)^2=\frac{c}{a} \Rightarrow 3 b^2=16 a c$
View full question & answer→MCQ 611 Mark
A quadratic equation whose one root is $1+\sqrt{2}$ and the sum of its roots is 2, is
- A
$x^2-2 x+1=0$
- ✓
$x^2-2 x-1=0$
- C
$x^2+2 x+1=0$
- D
$x^2+2 x-1=0$
AnswerCorrect option: B. $x^2-2 x-1=0$
(B) $x^2-2 x-1=0$
Let $\alpha, \beta$ be the roots of the desired equation and let $\alpha=1+\sqrt{2}$ and $\alpha+\beta=2$.
Then, $\alpha=\sqrt{2}+1$ and $\beta=1-\sqrt{2}$. So, the required equation is
$x^2-(\alpha+\beta) x+\alpha \beta=0$ or, $x^2-2 x-1=0$
View full question & answer→MCQ 621 Mark
If one root of the equation $3 x^2-8 x-(2 k+1)=0$ is seven times the other, then the value of $k$ is
- A
$\frac{7}{3}$
- B
$\frac{5}{3}$
- ✓
$-\frac{5}{3}$
- D
$-\frac{7}{3}$
AnswerCorrect option: C. $-\frac{5}{3}$
(C)$-\frac{5}{3}$
Let the roots of the given equation i.e. $3 x^2-8 x-(2 k+1)=0$ be $\alpha$ and $7 \alpha$. Then, $\alpha+7 \alpha=\frac{8}{3}$ and $\alpha \times 7 \alpha=-\frac{2 k+1}{3}$
$\Rightarrow \quad \alpha=\frac{1}{3}$ and $7 \alpha^2=-\frac{2 k+1}{3} \Rightarrow \frac{7}{9}=-\frac{2 k+1}{3} \Rightarrow 2 k+1=-\frac{7}{3} \Rightarrow k=-\frac{5}{3}$
View full question & answer→MCQ 631 Mark
If the sum and product of the roots of the equation $k x^2+6 x+4 k=0$ are equal, then the value of $k$ is
- ✓
$-\frac{3}{2}$
- B
$\frac{3}{2}$
- C
$\frac{2}{3}$
- D
$-\frac{2}{3}$
AnswerCorrect option: A. $-\frac{3}{2}$
(A)$-\frac{3}{2}$
Let $\alpha, \beta$ be the roots of the equation $k x^2+6 x+4 k=0$. Then, $\alpha+\beta=-\frac{6}{k}$ and $\alpha \beta=4$
It is given that
$
\alpha+\beta=\alpha \beta \Rightarrow-\frac{6}{k}=4 \Rightarrow k=-\frac{3}{2}
$
View full question & answer→MCQ 641 Mark
If $a, b, c$ are positive real numbers, then the number of real roots of the equation $a x^2+b|x|+c=0$, is
Answer(C)0
We find that for positive values of $a, b$ and $c ; a x^2+b|x|+c=a|x|^2+b|x|+c>0$ for all real values of $x$. Therefore, $a x^2+b|x|+c \neq 0$ for any real values of $x$.
Hence, $a x^2+b|x|+c=0$ has no real root.
View full question & answer→MCQ 651 Mark
If $c$ and $d$ are roots of the equation $(x-a)(x-b)-k=0$, then $a, b$ are roots of the equation
- A
$(x-c)(x-d)-k=0$
- ✓
$(x-c)(x-d)+k=0$
- C
$(x-a)(x-c)+k=0$
- D
$(x-b)(x-d)+k=0$
AnswerCorrect option: B. $(x-c)(x-d)+k=0$
(B)$(x-c)(x-d)+k=0$
If $c$ and $d$ are roots of the equation $(x-a)(x-b)-k=0$, then
$
\begin{array}{l}
(x-a)(x-b)-k=(x-c)(x-d) \\
\Rightarrow \quad(x-c)(x-d)+k=(x-a)(x-b) \Rightarrow a, b \text { are roots of the equation }(x-c)(x-d)+k=0
\end{array}
$
View full question & answer→MCQ 661 Mark
The number of real roots of the equation $x^2-3|x|+2=0$, is
Answer(A)4
We have, $x^2-3|x|+2=0$
$
\Rightarrow \quad|x|^2-3|x|+2=0 \quad\left[\because x^2=|x|^2\right]
$
$
\Rightarrow \quad|x|^2-2|x|-|x|+2=0 \Rightarrow(|x|-2)(|x|-1)=0 \Rightarrow|x|=1,|x|=2 \Rightarrow x= \pm 1, x= \pm 2
$
Hence, the given equation has four real roots.
View full question & answer→MCQ 671 Mark
The number of real roots of the equation $x^2+3|x|+2=0$, is
Answer(C)0
We find that $x^2+3|x|+2=|x|^2+3|x|+2 \neq 0$ for any real $x$. Hence, the given equation has no real root.
ALITER We find that $x^2+3|x|+2=|x|^2+2|x|+|x|+2=(|x|+2)(|x|+1) \neq 0$ for any $x$.
View full question & answer→MCQ 681 Mark
If $\alpha$ and $\beta$ are two roots of the quadratic equation $a x^2+b x+c=0$, then $a x^2+b x+c=$
- A
$a(x+\alpha)(x+\beta)$
- ✓
$a(x-\alpha)(x-\beta)$
- C
$b(x-\alpha)(x-\beta)$
- D
$c(x-\alpha)(x-\beta)$
AnswerCorrect option: B. $a(x-\alpha)(x-\beta)$
(B)$a(x-\alpha)(x-\beta)$
Since $\alpha$ and $\beta$ are roots of $a x^2+b x+c$.
$
\therefore \quad a x^2+b x+c=\lambda(x-\alpha)(x-\beta) \text { for some } \lambda \text { and all } x
$
$
\Rightarrow \quad a x^2+b x+c=i x^2-i x(\alpha+\beta)+\lambda \alpha \beta \text { for all } x \qquad [By comparing coefficients of x^2 ]$
$\Rightarrow \quad \lambda=a
$
Hence, $a x^2+b x+c=a(x-\alpha)(x-\beta)$.
View full question & answer→MCQ 691 Mark
Answer(C)exactly two roots
A quadratic equation has exactly two roots, say $\alpha$ and $\beta$, such that $\alpha+\beta=-\frac{b}{a}$ and $\alpha \beta=\frac{c}{a}$.
View full question & answer→MCQ 701 Mark
The equation $a x^2+b c+c=0$ is a quadratic equation for
Answer(B)all non-zero values of a
The equation $a x^2+b x+c=0, a \neq 0$ is defined as a quadratic equation for all values of $b$ and c. Hence, option (b) is correct.
View full question & answer→MCQ 711 Mark
If $a$ and $b$ are roots of the equation $x^2+a x+b=0$, then $a+b=$
View full question & answer→MCQ 721 Mark
If $\sin \alpha$ and $\cos \alpha$ are the roots of the equation $a x^2+b x+c=0$, then $b^2=$
- A
$a^2-2 a c$
- ✓
$a^2+2 a c$
- C
$a^2-a c$
- D
$a^2+a c$
AnswerCorrect option: B. $a^2+2 a c$
View full question & answer→MCQ 731 Mark
If $\left(a^2+b^2\right) x^2+2(a c+b d) x+c^2+d^2=0$ has no real roots, then
- A
$a d=b c$
- B
$a b=c d$
- C
$a c=b d$
- ✓
$a d \neq b c$
AnswerCorrect option: D. $a d \neq b c$
View full question & answer→MCQ 741 Mark
The number of quadratic equations having real roots and which do not change by squaring their roots is
MCQ 751 Mark
If $a$ and $b$ can take values $1,2,3,4$. Then the number of the equations of the form $a x^2+b x+1=0$ having real roots is
View full question & answer→MCQ 761 Mark
If the equation $x^2-b x+1=0$ does not possess real roots, then
View full question & answer→MCQ 771 Mark
If the roots of the equation $\left(a^2+b^2\right) x^2-2 b(a+c) x+\left(b^2+c^2\right)=0$ are equal, then
- A
$2 b=a+c$
- ✓
$b^2=a c$
- C
$b=\frac{2 a c}{a+c}$
- D
$b=a c$
AnswerCorrect option: B. $b^2=a c$
View full question & answer→MCQ 781 Mark
If the equation $\left(a^2+b^2\right) x^2-2(a c+b d) x+c^2+d^2=0$ has equal roots, then
- A
$a b=c d$
- ✓
$a d=b c$
- C
$a d=\sqrt{b c}$
- D
$a b=\sqrt{c d}$
AnswerCorrect option: B. $a d=b c$
View full question & answer→MCQ 791 Mark
The positive value of $k$ for which the equation $x^2+k x+64=0$ and $x^2-8 x+k=0$ will bot have real roots, is
View full question & answer→MCQ 801 Mark
If the equation $a x^2+2 x+a=0$ has two equal roots, if
- ✓
$a= \pm 1$
- B
$a=0$
- C
$a=0,1$
- D
$a=-1,0$
AnswerCorrect option: A. $a= \pm 1$
View full question & answer→MCQ 811 Mark
If the equation $9 x^2+6 k x+4=0$ has equal roots, then the roots are both equal to
- ✓
$\pm \frac{2}{3}$
- B
$\pm \frac{3}{2}$
- C
$0$
- D
$\pm 3$
AnswerCorrect option: A. $\pm \frac{2}{3}$
View full question & answer→MCQ 821 Mark
If the equation $x^2-a x+1=0$ has two distinct roots, then
- A
$|a|=2$
- B
$|a|<2$
- ✓
$|a|>2$
- D
AnswerCorrect option: C. $|a|>2$
View full question & answer→MCQ 831 Mark
If one root of the equation $4 x^2-2 x+(\lambda-4)=0$ be the reciprocal of the other, then $\lambda=$
View full question & answer→MCQ 841 Mark
If the sum of the roots of the equation $x^2-(k+6) x+2(2 k-1)=0$ is equal to half of the: product, then $k=$
View full question & answer→MCQ 851 Mark
If one root of the equation $a x^2+b x+c=0$ is three times the other, then $b^2: a c=$
- A
$3: 1$
- B
$3: 16$
- ✓
$16: 3$
- D
$16: 1$
AnswerCorrect option: C. $16: 3$
View full question & answer→MCQ 861 Mark
If $x^2+k(4 x+k-1)+2=0$ has equal roots, then $k=$
- A
$-\frac{2}{3}, 1$
- ✓
$\frac{2}{3},-1$
- C
$\frac{3}{2}, \frac{1}{3}$
- D
$-\frac{3}{2},-\frac{1}{3}$
AnswerCorrect option: B. $\frac{2}{3},-1$
View full question & answer→MCQ 871 Mark
If $p$ and $q$ are the roots of the equation $x^2+p x+q=0$, then
- ✓
$p=1, q=-2$
- B
$p=0, q=1$
- C
$p=-2, q=0$
- D
$p=-2, q=1$
AnswerCorrect option: A. $p=1, q=-2$
View full question & answer→MCQ 881 Mark
26. If 2 is a root of the equation $x^2+b x+12=0$ and the equation $x^2+b x+q=0$ has equal roots, then $q=$
View full question & answer→MCQ 891 Mark
The value of $\sqrt{6+\sqrt{6+\sqrt{6+}}} \ldots$ is
View full question & answer→MCQ 901 Mark
23. If the equation $x^2+4 x+k=0$ has real and distinct roots, then
- ✓
$k < 4$
- B
$k > 4$
- C
$k \geq 4$
- D
$k \leq 4$
AnswerCorrect option: A. $k < 4$
View full question & answer→MCQ 911 Mark
22. If $x=1$ is a common root of the equations $a x^2+a x+6=0$ and $x^2+x+b=0$, then $a b=$
View full question & answer→MCQ 921 Mark
21. If the sum of the roots of the equation $x^2-x=\lambda(2 x-1)$ is zero, then $\lambda=$
- A
- B
- ✓
$-\frac{1}{2}$
- D
$\frac{1}{2}$
AnswerCorrect option: C. $-\frac{1}{2}$
View full question & answer→MCQ 931 Mark
If the sum and product of the roots of the equation $k x^2+6 x+4 k=0$ are equal, then $k=$
- ✓
$-\frac{3}{2}$
- B
$\frac{3}{2}$
- C
$\frac{2}{3}$
- D
$-\frac{2}{3}$
AnswerCorrect option: A. $-\frac{3}{2}$
View full question & answer→MCQ 941 Mark
A quadratic equation whose one root is 2 and the sum of whose roots is zero, is
- A
$x^2+4=0$
- ✓
$x^2-4=0$
- C
$4 x^2-1=0$
- D
$x^2-2=0$
AnswerCorrect option: B. $x^2-4=0$
View full question & answer→MCQ 951 Mark
If one root of the equation $2 x^2+k x+4=0$ is 2 , then the other root is
View full question & answer→MCQ 961 Mark
If one root of the equation $x^2+a x+3=0$ is 1 , then its other root is
View full question & answer→MCQ 971 Mark
The discriminant of the quadratic equation $(x+2)^2=0$ is
View full question & answer→MCQ 981 Mark
A quadratic equation can have
MCQ 991 Mark
Which of the following equations has 3 as a root?
- ✓
$x^2-4 x+3=0$
- B
$x^2+4 x+3=0$
- C
$x^2+5 x+6=0$
- D
$x^2+7 x+12=0$
AnswerCorrect option: A. $x^2-4 x+3=0$
View full question & answer→MCQ 1001 Mark
If $-\frac{1}{2}$ is a root of the equation $x^2-k x-\frac{5}{4}=0$, then the value of $k$ is
- A
- ✓
- C
$\frac{1}{4}$
- D
$\frac{1}{2}$
View full question & answer→