Question 1: Palette 1/160

JEE Mains 2026 MCQ

If $\alpha, \beta$, where $\alpha<\beta$, are the roots of the equation $\lambda x^2-(\lambda+3) x+3=0$ such that $\frac{1}{\alpha}-\frac{1}{\beta}=\frac{1}{3}$, then the sum of all possible values of $\lambda$ is

A

2

B

6

C

8

D

4

Question 2: Palette 2/160

JEE Mains 2026 MCQ

Let $\mathrm{S}=\left\{x^3+a x^2+b x+c: a, b, c \in \mathrm{~N}\right.$ and $\left.a, b, c \leq 20\right\}$ be a set of polynomials. Then the number of polynomials in S , which are divisible by $x^2+2$, is

A

6

B

120

C

20

D

10

Question 3: Palette 3/160

JEE Mains 2026 MCQ

The smallest positive integral value of $a$, for which all the roots of $x^4-a x^2+9=0$ are real and distinct, is equal to

A

7

B

3

C

4

D

9

Question 4: Palette 4/160

JEE Mains 2026 MCQ

The number of the real solutions of the equation: $x|x+3|+|x-1|-2=0$ is

A

3

B

2

C

5

D

4

Question 5: Palette 5/160

JEE Mains 2026 MCQ

If the domain of the function

$ f(x)=\log _{\left(10 x^2-17 x+7\right)}\left(18 x^2-11 x+1\right) $

is $(-\infty, a) \cup(b, c) \cup(d, \infty)-\{e\}$, then

$90(a+b+c+d+e)$ equals:

A

170

B

316

C

177

D

307

Question 6: Palette 6/160

JEE Mains 2026 MCQ

If $\alpha$ and $\beta(\alpha<\beta)$ are the roots of the equation $(-2+\sqrt{3})(|\sqrt{x}-3|)+(x-6 \sqrt{x})+(9-2 \sqrt{3})=0, x \geqslant 0$, then $\sqrt{\frac{\beta}{\alpha}}+\sqrt{\alpha \beta}$ is equal to :

A

8

B

10

C

9

D

11

Question 7: Palette 7/160

JEE Mains 2026 MCQ

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :

A

36 days

B

24 days

C

30 days

D

42 days

Question 8: Palette 8/160

JEE Mains 2026 MCQ

Let $\alpha, \beta$ be the roots of the quadratic equation $12 x^2-20 x+3 \lambda=0, \lambda \in \mathbf{Z}$. If $\frac{1}{2} \leqslant|\beta-\alpha| \leqslant \frac{3}{2}$, then the sum of all possible values of $\lambda$ is :

A

3

B

6

C

1

D

4

Question 9: Palette 9/160

JEE Mains 2026 MCQ

The number of distinct real solutions of the equation $x|x+4|+3|x+2|+10=0$ is

A

2

B

3

C

0

D

1

Question 10: Palette 10/160

JEE Mains 2026 MCQ

Let $\alpha$ and $\beta$ be the roots of the equation $x^2+2 a x+(3 a+10)=0$ such that $\alpha<1<\beta$. Then the set of all possible values of $a$ is :

A

$\left(-\infty, \frac{-11}{5}\right) \cup(5, \infty)$

B

$\left(-\infty, \frac{-11}{5}\right)$

C

$(-\infty,-3)$

D

$(-\infty,-2) \cup(5, \infty)$

Question 11: Palette 11/160

JEE Mains 2026 MCQ

The sum of all the roots of the equation $(x-1)^2-5|x-1|+6=0$, is :

A

3

B

1

C

4

D

5

Question 12: Palette 12/160

JEE Mains 2026 MCQ

Let $\mathop {\lim }\limits_{x \to 2} \frac{(\tan (x-2))\left(\mathrm{r} x^2+(\mathrm{p}-2) x-2 \mathrm{p}\right)}{(x-2)^2}=5$ for some $\mathrm{r}, \mathrm{p} \in \boldsymbol{R}$. If the set of all possible values of q , such that the roots of the equation $\mathrm{r} x^2-\mathrm{p} x+\mathrm{q}=0$ lie in $(0,2)$, be the interval $(\alpha, \beta]$, then $4(\alpha+\beta)$ equals :

A

11

B

13

C

17

D

21

Question 13: Palette 13/160

JEE Mains 2026 MCQ

If the quadratic equation $(\lambda+2) x^2-3 \lambda x+4 \lambda=0, \lambda \neq-2$, has two positive roots, then the number of possible integral values of $\lambda$ is :

A

1

B

2

C

3

D

4

Question 14: Palette 14/160

JEE Mains 2026 MCQ

If the set of all solutions of $\left|x^2+x-9\right|=|x|+\left|x^2-9\right|$ is $[\alpha, \beta] \cup[\gamma, \infty)$, then ( $\alpha^2+\beta^2+\gamma^2$ ) is equal to:

A

9

B

18

C

36

D

72

Question 15: Palette 15/160

JEE Mains 2026 MCQ

Let $\alpha, \beta$ be the roots of the equation $x^2 - 3x + r = 0$, and $\frac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2 + 3x + r = 0$.

If the roots of the equation $x^2 + 6x = m$ are $2\alpha + \beta + 2r$ and $\alpha - 2\beta - \frac{r}{2}$, then $m$ is equal to :

A

-135

B

-567

C

135

D

567

Question 16: Palette 16/160

JEE Mains 2026 MCQ

Let $\alpha, \alpha+2, \alpha \in \mathbf{Z}$, be the roots of the quadratic equation $x(x+2)+(x+1)(x+3)+(x+2)(x+4)+\ldots .+(x+\mathrm{n}-1)(x+\mathrm{n}+1)=4 \mathrm{n}$ for some $\mathrm{n} \in \mathbf{N}$. Then $\mathrm{n}+\alpha$ is equal to :

A

0

B

1

C

2

D

3

Question 17: Palette 17/160

JEE Mains 2025 MCQ

The sum of the squares of the roots of $ |x-2|^2 + |x-2| - 2 = 0 $ and the squares of the roots of $ x^2 - 2|x-3| - 5 = 0 $, is

A

24

B

26

C

36

D

30

Question 18: Palette 18/160

JEE Mains 2025 MCQ

The number of real roots of the equation $x |x - 2| + 3|x - 3| + 1 = 0$ is :

A

4

B

3

C

2

D

1

Question 19: Palette 19/160

JEE Mains 2025 MCQ

Let the set of all values of $p \in \mathbb{R}$, for which both the roots of the equation $x^2-(p+2) x+(2 p+9)=0$ are negative real numbers, be the interval $(\alpha, \beta]$. Then $\beta-2 \alpha$ is equal to

A

5

B

0

C

20

D

9

Question 20: Palette 20/160

JEE Mains 2025 MCQ

Consider the equation $x^2+4 x-n=0$, where $n \in[20,100]$ is a natural number. Then the number of all distinct values of $n$, for which the given equation has integral roots, is equal to

A

6

B

5

C

8

D

7

Question 21: Palette 21/160

JEE Mains 2025 MCQ

Let the equation $x(x+2)(12-k)=2$ have equal roots. Then the distance of the point $\left(k, \frac{k}{2}\right)$ from the line $3 x+4 y+5=0$ is

A

15

B

12

C

$5 \sqrt{3}$

D

$15 \sqrt{5}$

Question 22: Palette 22/160

JEE Mains 2025 MCQ

Let $\alpha$ and $\beta$ be the roots of $x^2+\sqrt{3} x-16=0$, and $\gamma$ and $\delta$ be the roots of $x^2+3 x-1=0$. If $P_n=$ $\alpha^n+\beta^n$ and $Q_n=\gamma^n+\hat{o}^n$, then $\frac{P_{25}+\sqrt{3} P_{24}}{2 P_{23}}+\frac{Q_{25}-Q_{23}}{Q_{24}}$ is equal to

A

4

B

3

C

5

D

7

Question 23: Palette 23/160

JEE Mains 2025 MCQ

Let $\mathrm{P}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}, \mathrm{n} \in \mathrm{N}$. If $\mathrm{P}_{10}=123, \mathrm{P}_9=76, \mathrm{P}_8=47$ and $\mathrm{P}_1=1$, then the quadratic equation having roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is :

A

$x^2+x-1=0$

B

$x^2-x+1=0$

C

$x^2+x+1=0$

D

$x^2-x-1=0$

Question 24: Palette 24/160

JEE Mains 2025 MCQ

If the set of all $a \in \mathbf{R}$, for which the equation $2 x^2+(a-5) x+15=3 a$ has no real root, is the interval ( $\alpha, \beta$ ), and $X=|x \in Z ; \alpha < x < \beta|$, then $\sum\limits_{x \in X} x^2$ is equal to:

A

2139

B

2119

C

2109

D

2129

Question 25: Palette 25/160

JEE Mains 2025 MCQ

The number of solutions of the equation

$ \left( \frac{9}{x} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{x} - \frac{7}{\sqrt{x}} + 3 \right) = 0 $ is :

A

3

B

2

C

1

D

4

Question 26: Palette 26/160

JEE Mains 2025 MCQ

Let $f: \mathbf{R}-\{0\} \rightarrow(-\infty, 1)$ be a polynomial of degree 2 , satisfying $f(x) f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$. If $f(\mathrm{~K})=-2 \mathrm{~K}$, then the sum of squares of all possible values of K is :

A

9

B

1

C

6

D

7

Question 27: Palette 27/160

JEE Mains 2025 MCQ

The sum, of the squares of all the roots of the equation $x^2+|2 x-3|-4=0$, is

A

$6(2-\sqrt{2})$

B

$3(3-\sqrt{2})$

C

$3(2-\sqrt{2})$

D

$6(3-\sqrt{2})$

Question 28: Palette 28/160

JEE Mains 2025 MCQ

The number of real solution(s) of the equation $x^2+3 x+2=\min \{|x-3|,|x+2|\}$ is :

A

2

B

3

C

1

D

0

Question 29: Palette 29/160

JEE Mains 2025 MCQ

The product of all the rational roots of the equation $\left(x^2-9 x+11\right)^2-(x-4)(x-5)=3$, is equal to

A

7

B

21

C

28

D

14

Question 30: Palette 30/160

JEE Mains 2025 MCQ

Let $\alpha_\theta$ and $\beta_\theta$ be the distinct roots of $2 x^2+(\cos \theta) x-1=0, \theta \in(0,2 \pi)$. If m and M are the minimum and the maximum values of $\alpha_\theta^4+\beta_\theta^4$, then $16(M+m)$ equals :

A

27

B

17

C

25

D

24

Question 31: Palette 31/160

JEE Mains 2024 MCQ

Let $\alpha, \beta ; \alpha>\beta$, be the roots of the equation $x^2-\sqrt{2} x-\sqrt{3}=0$. Let $\mathrm{P}_n=\alpha^n-\beta^n, n \in \mathrm{N}$. Then $(11 \sqrt{3}-10 \sqrt{2}) \mathrm{P}_{10}+(11 \sqrt{2}+10) \mathrm{P}_{11}-11 \mathrm{P}_{12}$ is equal to

A

$10 \sqrt{3} \mathrm{P}_9$

B

$11 \sqrt{3} \mathrm{P}_9$

C

$11 \sqrt{2} \mathrm{P}_9$

D

$10 \sqrt{2} \mathrm{P}_9$

Question 32: Palette 32/160

JEE Mains 2024 MCQ

Let $\alpha, \beta$ be the roots of the equation $x^2+2 \sqrt{2} x-1=0$. The quadratic equation, whose roots are $\alpha^4+\beta^4$ and $\frac{1}{10}(\alpha^6+\beta^6)$, is:

A

$x^2-180 x+9506=0$

B

$x^2-195 x+9506=0$

C

$x^2-190 x+9466=0$

D

$x^2-195 x+9466=0$

Question 33: Palette 33/160

JEE Mains 2024 MCQ

The sum of all the solutions of the equation $(8)^{2 x}-16 \cdot(8)^x+48=0$ is :

A

$1+\log _8(6)$

B

$1+\log _6(8)$

C

$\log _8(6)$

D

$\log _8(4)$

Question 34: Palette 34/160

JEE Mains 2024 MCQ

Let $\alpha, \beta$ be the distinct roots of the equation $x^2-\left(t^2-5 t+6\right) x+1=0, t \in \mathbb{R}$ and $a_n=\alpha^n+\beta^n$. Then the minimum value of $\frac{a_{2023}+a_{2025}}{a_{2024}}$ is

A

$-1 / 2$

B

$-1 / 4$

C

$1 / 4$

D

$1 / 2$

Question 35: Palette 35/160

JEE Mains 2024 MCQ

If 2 and 6 are the roots of the equation $a x^2+b x+1=0$, then the quadratic equation, whose roots are $\frac{1}{2 a+b}$ and $\frac{1}{6 a+b}$, is :

A

$x^2+8 x+12=0$

B

$2 x^2+11 x+12=0$

C

$4 x^2+14 x+12=0$

D

$x^2+10 x+16=0$

Question 36: Palette 36/160

JEE Mains 2024 MCQ

Let $\alpha$ and $\beta$ be the roots of the equation $p x^2+q x-r=0$, where $p \neq 0$. If $p, q$ and $r$ be the consecutive terms of a non constant G.P. and $\frac{1}{\alpha}+\frac{1}{\beta}=\frac{3}{4}$, then the value of $(\alpha-\beta)^2$ is :

A

8

B

9

C

$\frac{20}{3}$

D

$\frac{80}{9}$

Question 37: Palette 37/160

JEE Mains 2024 MCQ

Let $\mathbf{S}=\left\{x \in \mathbf{R}:(\sqrt{3}+\sqrt{2})^x+(\sqrt{3}-\sqrt{2})^x=10\right\}$. Then the number of elements in $\mathrm{S}$ is :

A

4

B

0

C

2

D

1

Question 38: Palette 38/160

JEE Mains 2024 MCQ

Let $\mathrm{S}$ be the set of positive integral values of $a$ for which $\frac{a x^2+2(a+1) x+9 a+4}{x^2-8 x+32} < 0, \forall x \in \mathbb{R}$. Then, the number of elements in $\mathrm{S}$ is :

A

0

B

$\infty$

C

3

D

1

Question 39: Palette 39/160

JEE Mains 2024 MCQ

If $\alpha, \beta$ are the roots of the equation, $x^2-x-1=0$ and $S_n=2023 \alpha^n+2024 \beta^n$, then :

A

$2 S_{12}=S_{11}+S_{10}$

B

$S_{12}=S_{11}+S_{10}$

C

$S_{11}=S_{10}+S_{12}$

D

$2 S_{11}=S_{12}+S_{10}$

Question 40: Palette 40/160

JEE Mains 2023 MCQ

The number of real roots of the equation $x|x|-5|x+2|+6=0$, is :

A

4

B

3

C

5

D

6

Question 41: Palette 41/160

JEE Mains 2023 MCQ

Let $\alpha, \beta$ be the roots of the equation $x^{2}-\sqrt{2} x+2=0$. Then $\alpha^{14}+\beta^{14}$ is equal to

A

$-64$

B

$-64 \sqrt{2}$

C

$-128 \sqrt{2}$

D

$-128$

Question 42: Palette 42/160

JEE Mains 2023 MCQ

The set of all $a \in \mathbb{R}$ for which the equation $x|x-1|+|x+2|+a=0$ has exactly one real root, is :

A

$(-\infty, \infty)$

B

$(-6, \infty)$

C

$(-\infty,-3)$

D

$(-6,-3)$

Question 43: Palette 43/160

JEE Mains 2023 MCQ

Let $\alpha, \beta$ be the roots of the quadratic equation $x^{2}+\sqrt{6} x+3=0$. Then $\frac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}}$ is equal to :

A

72

B

9

C

729

D

81

Question 44: Palette 44/160

JEE Mains 2023 MCQ

Let $\alpha, \beta, \gamma$ be the three roots of the equation $x^{3}+b x+c=0$. If $\beta \gamma=1=-\alpha$, then $b^{3}+2 c^{3}-3 \alpha^{3}-6 \beta^{3}-8 \gamma^{3}$ is equal to :

A

21

B

19

C

$\frac{169}{8}$

D

$\frac{155}{8}$

Question 45: Palette 45/160

JEE Mains 2023 MCQ

Let $A = \{ x \in R:[x + 3] + [x + 4] \le 3\} ,$

$B = \left\{ {x \in R:{3^x}{{\left( {\sum\limits_{r = 1}^\infty {{3 \over {{{10}^r}}}} } \right)}^{x - 3}} < {3^{ - 3x}}} \right\},$ where [t] denotes greatest integer function. Then,

A

$B \subset C,A \ne B$

B

$A \subset B,A \ne B$

C

$A = B$

D

$A \cap B = \phi $

Question 46: Palette 46/160

JEE Mains 2023 MCQ

The sum of all the roots of the equation $\left|x^{2}-8 x+15\right|-2 x+7=0$ is :

A

$11+\sqrt{3}$

B

$9+\sqrt{3}$

C

$9-\sqrt{3}$

D

$11-\sqrt{3}$

Question 47: Palette 47/160

JEE Mains 2023 MCQ

The number of integral values of k, for which one root of the equation $2x^2-8x+k=0$ lies in the interval (1, 2) and its other root lies in the interval (2, 3), is :

A

2

B

0

C

1

D

3

Question 48: Palette 48/160

JEE Mains 2023 MCQ

Let $S = \left\{ {x:x \in \mathbb{R}\,\mathrm{and}\,{{(\sqrt 3 + \sqrt 2 )}^{{x^2} - 4}} + {{(\sqrt 3 - \sqrt 2 )}^{{x^2} - 4}} = 10} \right\}$. Then $n(S)$ is equal to

A

6

B

4

C

0

D

2

Question 49: Palette 49/160

JEE Mains 2023 MCQ

The equation $\mathrm{e}^{4 x}+8 \mathrm{e}^{3 x}+13 \mathrm{e}^{2 x}-8 \mathrm{e}^{x}+1=0, x \in \mathbb{R}$ has :

A

two solutions and both are negative

B

two solutions and only one of them is negative

C

four solutions two of which are negative

D

no solution

Question 50: Palette 50/160

JEE Mains 2023 MCQ

The number of real roots of the equation $\sqrt{x^{2}-4 x+3}+\sqrt{x^{2}-9}=\sqrt{4 x^{2}-14 x+6}$, is :

A

0

B

1

C

3

D

2

Question 51: Palette 51/160

JEE Mains 2023 MCQ

Let $\lambda \ne 0$ be a real number. Let $\alpha,\beta$ be the roots of the equation $14{x^2} - 31x + 3\lambda = 0$ and $\alpha,\gamma$ be the roots of the equation $35{x^2} - 53x + 4\lambda = 0$. Then ${{3\alpha } \over \beta }$ and ${{4\alpha } \over \gamma }$ are the roots of the equation

A

$7{x^2} - 245x + 250 = 0$

B

$49{x^2} - 245x + 250 = 0$

C

$49{x^2} + 245x + 250 = 0$

D

$7{x^2} + 245x - 250 = 0$

Question 52: Palette 52/160

JEE Mains 2023 MCQ

The number of real solutions of the equation $3\left( {{x^2} + {1 \over {{x^2}}}} \right) - 2\left( {x + {1 \over x}} \right) + 5 = 0$, is

A

3

B

4

C

0

D

2

Question 53: Palette 53/160

JEE Mains 2023 MCQ

The equation ${x^2} - 4x + [x] + 3 = x[x]$, where $[x]$ denotes the greatest integer function, has :

A

exactly two solutions in ($-\infty,\infty$)

B

no solution

C

a unique solution in ($-\infty,\infty$)

D

a unique solution in ($-\infty,1$)

Question 54: Palette 54/160

JEE Mains 2022 MCQ

If $\frac{1}{(20-a)(40-a)}+\frac{1}{(40-a)(60-a)}+\ldots+\frac{1}{(180-a)(200-a)}=\frac{1}{256}$, then the maximum value of $\mathrm{a}$ is :

A

198

B

202

C

212

D

218

Question 55: Palette 55/160

JEE Mains 2022 MCQ

$ \text { Let } S=\left\{x \in[-6,3]-\{-2,2\}: \frac{|x+3|-1}{|x|-2} \geq 0\right\} \text { and } $

$T=\left\{x \in \mathbb{Z}: x^{2}-7|x|+9 \leq 0\right\} \text {. } $

Then the number of elements in $\mathrm{S} \cap \mathrm{T}$ is :

A

7

B

5

C

4

D

3

Question 56: Palette 56/160

JEE Mains 2022 MCQ

Let $\alpha$, $\beta$ be the roots of the equation $x^{2}-\sqrt{2} x+\sqrt{6}=0$ and $\frac{1}{\alpha^{2}}+1, \frac{1}{\beta^{2}}+1$ be the roots of the equation $x^{2}+a x+b=0$. Then the roots of the equation $x^{2}-(a+b-2) x+(a+b+2)=0$ are :

A

non-real complex numbers

B

real and both negative

C

real and both positive

D

real and exactly one of them is positive

Question 57: Palette 57/160

JEE Mains 2022 MCQ

If $\alpha, \beta$ are the roots of the equation

$ x^{2}-\left(5+3^{\sqrt{\log _{3} 5}}-5^{\sqrt{\log _{5} 3}}\right)x+3\left(3^{\left(\log _{3} 5\right)^{\frac{1}{3}}}-5^{\left(\log _{5} 3\right)^{\frac{2}{3}}}-1\right)=0 $,

then the equation, whose roots are $\alpha+\frac{1}{\beta}$ and $\beta+\frac{1}{\alpha}$, is :

A

$3 x^{2}-20 x-12=0$

B

$3 x^{2}-10 x-4=0$

C

$3 x^{2}-10 x+2=0$

D

$3 x^{2}-20 x+16=0$

Question 58: Palette 58/160

JEE Mains 2022 MCQ

The minimum value of the sum of the squares of the roots of $x^{2}+(3-a) x+1=2 a$ is:

A

4

B

5

C

6

D

8

Question 59: Palette 59/160

JEE Mains 2022 MCQ

If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x^{4}+x^{3}+x^{2}+x+1=0$, then $\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}$ is equal to :

A

$-$4

B

$-$1

C

1

D

4

Question 60: Palette 60/160

JEE Mains 2022 MCQ

Let ${S_1} = \left\{ {x \in R - \{ 1,2\} :{{(x + 2)({x^2} + 3x + 5)} \over { - 2 + 3x - {x^2}}} \ge 0} \right\}$ and ${S_2} = \left\{ {x \in R:{3^{2x}} - {3^{x + 1}} - {3^{x + 2}} + 27 \le 0} \right\}$. Then, ${S_1} \cup {S_2}$ is equal to :

A

$( - \infty , - 2] \cup (1,2)$

B

$( - \infty , - 2] \cup [1,2]$

C

$( - 2,1] \cup [2,\infty )$

D

$( - \infty ,2]$

Question 61: Palette 61/160

JEE Mains 2022 MCQ

Let S be the set of all integral values of $\alpha$ for which the sum of squares of two real roots of the quadratic equation $3{x^2} + (\alpha - 6)x + (\alpha + 3) = 0$ is minimum. Then S :

A

is an empty set

B

is a singleton

C

contains exactly two elements

D

contains more than two elements

Question 62: Palette 62/160

JEE Mains 2022 MCQ

Let $\alpha$ be a root of the equation 1 + x² + x⁴ = 0. Then, the value of $\alpha$¹⁰¹¹ + $\alpha$²⁰²² $-$ $\alpha$³⁰³³ is equal to :

A

1

B

$\alpha$

C

1 + $\alpha$

D

1 + 2$\alpha$

Question 63: Palette 63/160

JEE Mains 2022 MCQ

Let f(x) be a quadratic polynomial such that f($-$2) + f(3) = 0. If one of the roots of f(x) = 0 is $-$1, then the sum of the roots of f(x) = 0 is equal to :

A

${{11} \over 3}$

B

${{7} \over 3}$

C

${{13} \over 3}$

D

${{14} \over 3}$

Question 64: Palette 64/160

JEE Mains 2022 MCQ

The number of distinct real roots of x⁴ $-$ 4x + 1 = 0 is :

A

4

B

2

C

1

D

0

Question 65: Palette 65/160

JEE Mains 2022 MCQ

Let $A = \{ x \in R:|x + 1| < 2\} $ and $B = \{ x \in R:|x - 1| \ge 2\} $. Then which one of the following statements is NOT true?

A

$A - B = ( - 1,1)$

B

$B - A = R - ( - 3,1)$

C

$A \cap B = ( - 3, - 1]$

D

$A \cup B = R - [1,3)$

Question 66: Palette 66/160

JEE Mains 2022 MCQ

Let a, b $\in$ R be such that the equation $a{x^2} - 2bx + 15 = 0$ has a repeated root $\alpha$. If $\alpha$ and $\beta$ are the roots of the equation ${x^2} - 2bx + 21 = 0$, then ${\alpha ^2} + {\beta ^2}$ is equal to :

A

37

B

58

C

68

D

92

Question 67: Palette 67/160

JEE Mains 2022 MCQ

The sum of all the real roots of the equation

$({e^{2x}} - 4)(6{e^{2x}} - 5{e^x} + 1) = 0$ is

A

${\log _e}3$

B

$ - {\log _e}3$

C

${\log _e}6$

D

$ - {\log _e}6$

Question 68: Palette 68/160

JEE Mains 2022 MCQ

The number of distinct real roots of the equation

x⁷ $-$ 7x $-$ 2 = 0 is

A

5

B

7

C

1

D

3

Question 69: Palette 69/160

JEE Mains 2022 MCQ

If the sum of the squares of the reciprocals of the roots $\alpha$ and $\beta$ of

the equation 3x² + $\lambda$x $-$ 1 = 0 is 15, then 6($\alpha$³ + $\beta$³)² is equal to :

A

18

B

24

C

36

D

96

Question 70: Palette 70/160

JEE Mains 2021 MCQ

The numbers of pairs (a, b) of real numbers, such that whenever $\alpha$ is a root of the equation x² + ax + b = 0, $\alpha$² $-$ 2 is also a root of this equation, is :

A

6

B

2

C

4

D

8

Question 71: Palette 71/160

JEE Mains 2021 MCQ

The sum of the roots of the equation

$x + 1 - 2{\log _2}(3 + {2^x}) + 2{\log _4}(10 - {2^{ - x}}) = 0$, is :

A

log₂ 14

B

log₂ 11

C

log₂ 12

D

log₂ 13

Question 72: Palette 72/160

JEE Mains 2021 MCQ

cosec18$^\circ$ is a root of the equation :

A

x² + 2x $-$ 4 = 0

B

4x² + 2x $-$ 1 = 0

C

x² $-$ 2x + 4 = 0

D

x² $-$ 2x $-$ 4 = 0

Question 73: Palette 73/160

JEE Mains 2021 MCQ

The set of all values of K > $-$1, for which the equation ${(3{x^2} + 4x + 3)^2} - (k + 1)(3{x^2} + 4x + 3)(3{x^2} + 4x + 2) + k{(3{x^2} + 4x + 2)^2} = 0$ has real roots, is :

A

$\left( {1,{5 \over 2}} \right]$

B

[2, 3)

C

$\left[ { - {1 \over 2},1} \right)$

D

$\left( {{1 \over 2},{3 \over 2}} \right] - \{ 1\} $

Question 74: Palette 74/160

JEE Mains 2021 MCQ

Let $\alpha = \mathop {\max }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\} $ and $\beta = \mathop {\min }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\} $. If $8{x^2} + bx + c = 0$ is a quadratic equation whose roots are $\alpha$^1/5 and $\beta$^1/5, then the value of c $-$ b is equal to :

A

42

B

47

C

43

D

50

Question 75: Palette 75/160

JEE Mains 2021 MCQ

Let $\alpha$, $\beta$ be two roots of the

equation x² + (20)^1/4x + (5)^1/2 = 0. Then $\alpha$⁸ + $\beta$⁸ is equal to

A

10

B

100

C

50

D

160

Question 76: Palette 76/160

JEE Mains 2021 MCQ

If [x] be the greatest integer less than or equal to x,

then $\sum\limits_{n = 8}^{100} {\left[ {{{{{( - 1)}^n}n} \over 2}} \right]} $ is equal to :

A

0

B

4

C

$-$2

D

2

Question 77: Palette 77/160

JEE Mains 2021 MCQ

The number of real solutions of the equation, x² $-$ |x| $-$ 12 = 0 is :

A

2

B

3

C

1

D

4

Question 78: Palette 78/160

JEE Mains 2021 MCQ

The number of real roots of the equation ${e^{6x}} - {e^{4x}} - 2{e^{3x}} - 12{e^{2x}} + {e^x} + 1 = 0$ is :

A

2

B

4

C

6

D

1

Question 79: Palette 79/160

JEE Mains 2021 MCQ

Let [x] denote the greatest integer less than or equal to x. Then, the values of x$\in$R satisfying the equation ${[{e^x}]^2} + [{e^x} + 1] - 3 = 0$ lie in the interval :

A

$\left[ {0,{1 \over e}} \right)$

B

[log_e2, log_e3)

C

[1, e)

D

[0, log_e2)

Question 80: Palette 80/160

JEE Mains 2021 MCQ

If $\alpha$ and $\beta$ are the distinct roots of the equation ${x^2} + {(3)^{1/4}}x + {3^{1/2}} = 0$, then the value of ${\alpha ^{96}}({\alpha ^{12}} - 1) + {\beta ^{96}}({\beta ^{12}} - 1)$ is equal to :

A

56 $\times$ 3²⁵

B

56 $\times$ 3²⁴

C

52 $\times$ 3²⁴

D

28 $\times$ 3²⁵

Question 81: Palette 81/160

JEE Mains 2021 MCQ

The value of $3 + {1 \over {4 + {1 \over {3 + {1 \over {4 + {1 \over {3 + ....\infty }}}}}}}}$ is equal to

A

1.5 + $\sqrt 3 $

B

2 + $\sqrt 3 $

C

3 + 2$\sqrt 3 $

D

4 + $\sqrt 3 $

Question 82: Palette 82/160

JEE Mains 2021 MCQ

The value of $4 + {1 \over {5 + {1 \over {4 + {1 \over {5 + {1 \over {4 + ......\infty }}}}}}}}$ is :

A

2 + ${2 \over 5}\sqrt {30} $

B

2 + ${4 \over {\sqrt 5 }}\sqrt {30} $

C

5 + ${2 \over 5}\sqrt {30} $

D

4 + ${4 \over {\sqrt 5 }}\sqrt {30} $

Question 83: Palette 83/160

JEE Mains 2021 MCQ

Let $\alpha$ and $\beta$ be the roots of x² $-$ 6x $-$ 2 = 0. If a_n = $\alpha$ⁿ $-$ $\beta$ⁿ for n $ \ge $ 1, then the value of ${{{a_{10}} - 2{a_8}} \over {3{a_9}}}$ is :

A

3

B

2

C

4

D

1

Question 84: Palette 84/160

JEE Mains 2021 MCQ

The integer 'k', for which the inequality x² $-$ 2(3k $-$ 1)x + 8k² $-$ 7 > 0 is valid for every x in R, is :

A

4

B

2

C

3

D

0

Question 85: Palette 85/160

JEE Mains 2021 MCQ

Let p and q be two positive numbers such that p + q = 2 and p⁴+q⁴ = 272. Then p and q are roots of the equation :

A

x² – 2x + 8 = 0

B

x² - 2x + 136=0

C

x² – 2x + 16 = 0

D

x² – 2x + 2 = 0

Question 86: Palette 86/160

JEE Mains 2020 MCQ

If $\alpha $ and $\beta $ are the roots of the equation
2x(2x + 1) = 1, then $\beta $ is equal to :

A

$ - 2\alpha \left( {\alpha + 1} \right)$

B

$ 2\alpha \left( {\alpha + 1} \right)$

C

$2{\alpha ^2}$

D

$ 2\alpha \left( {\alpha - 1} \right)$

Question 87: Palette 87/160

JEE Mains 2020 MCQ

If $\alpha $ and $\beta $ be two roots of the equation
x² – 64x + 256 = 0. Then the value of
${\left( {{{{\alpha ^3}} \over {{\beta ^5}}}} \right)^{1/8}} + {\left( {{{{\beta ^3}} \over {{\alpha ^5}}}} \right)^{1/8}}$ is :

A

1

B

3

C

2

D

4

Question 88: Palette 88/160

JEE Mains 2020 MCQ

If $\alpha $ and $\beta $ are the roots of the equation,
7x² – 3x – 2 = 0, then the value of
${\alpha \over {1 - {\alpha ^2}}} + {\beta \over {1 - {\beta ^2}}}$ is equal to :

A

${1 \over {24}}$

B

${{27} \over {32}}$

C

${{27} \over {16}}$

D

${3 \over 8}$

Question 89: Palette 89/160

JEE Mains 2020 MCQ

The product of the roots of the
equation 9x² - 18|x| + 5 = 0 is :

A

${{5} \over {9}}$

B

${{5} \over {27}}$

C

${{25} \over {81}}$

D

${{25} \over {9}}$

Question 90: Palette 90/160

JEE Mains 2020 MCQ

Let $\lambda \ne 0$ be in R. If $\alpha $ and $\beta $ are the roots of the
equation, x² - x + 2$\lambda $ = 0 and $\alpha $ and $\gamma $ are the roots of
the equation, $3{x^2} - 10x + 27\lambda = 0$, then ${{\beta \gamma } \over \lambda }$ is equal to:

A

36

B

9

C

27

D

18

Question 91: Palette 91/160

JEE Mains 2020 MCQ

Let $\alpha $ and $\beta $ be the roots of x² - 3x + p=0 and $\gamma $ and $\delta $ be the roots of x² - 6x + q = 0. If $\alpha, \beta, \gamma, \delta $ form a geometric progression.Then ratio (2q + p) : (2q - p) is:

A

9 : 7

B

5 : 3

C

3 : 1

D

33 :31

Question 92: Palette 92/160

JEE Mains 2020 MCQ

Let [t] denote the greatest integer $ \le $ t. Then the equation in x,
[x]² + 2[x+2] - 7 = 0 has :

A

no integral solution.

B

exactly two solutions.

C

exactly four integral solutions.

D

infinitely many solutions.

Question 93: Palette 93/160

JEE Mains 2020 MCQ

The set of all real values of $\lambda $ for which the quadratic equations,
($\lambda $² + 1)x² – 4$\lambda $x + 2 = 0 always have exactly one root in the interval (0, 1) is :

A

(–3, –1)

B

(2, 4]

C

(0, 2)

D

(1, 3]

Question 94: Palette 94/160

JEE Mains 2020 MCQ

If $\alpha $ and $\beta $ are the roots of the equation
x² + px + 2 = 0 and ${1 \over \alpha }$ and ${1 \over \beta }$ are the
roots of the equation 2x² + 2qx + 1 = 0, then
$\left( {\alpha - {1 \over \alpha }} \right)\left( {\beta - {1 \over \beta }} \right)\left( {\alpha + {1 \over \beta }} \right)\left( {\beta + {1 \over \alpha }} \right)$ is equal to :

A

${9 \over 4}\left( {9 - {q^2}} \right)$

B

${9 \over 4}\left( {9 + {q^2}} \right)$

C

${9 \over 4}\left( {9 - {p^2}} \right)$

D

${9 \over 4}\left( {9 + {p^2}} \right)$

Question 95: Palette 95/160

JEE Mains 2020 MCQ

Let f(x) be a quadratic polynomial such that
f(–1) + f(2) = 0. If one of the roots of f(x) = 0
is 3, then its other root lies in :

A

(–3, –1)

B

(1, 3)

C

(–1, 0)

D

(0, 1)

Question 96: Palette 96/160

JEE Mains 2020 MCQ

Let $\alpha $ and $\beta $ be the roots of the equation
5x² + 6x – 2 = 0. If S_n = $\alpha $ⁿ + $\beta $ⁿ, n = 1, 2, 3...., then :

A

5S₆ + 6S₅ = 2S₄

B

5S₆ + 6S₅ + 2S₄ = 0

C

6S₆ + 5S₅ + 2S₄ = 0

D

6S₆ + 5S₅ = 2S₄

Question 97: Palette 97/160

JEE Mains 2020 MCQ

Let a, b $ \in $ R, a $ \ne $ 0 be such that the equation, ax² – 2bx + 5 = 0 has a repeated root $\alpha $, which is also a root of the equation, x² – 2bx – 10 = 0. If $\beta $ is the other root of this equation, then $\alpha $² + $\beta $² is equal to :

A

28

B

24

C

26

D

25

Question 98: Palette 98/160

JEE Mains 2020 MCQ

The number of real roots of the equation,
e^4x + e^3x – 4e^2x + e^x + 1 = 0 is :

A

1

B

2

C

3

D

4

Question 99: Palette 99/160

JEE Mains 2020 MCQ

Let $\alpha = {{ - 1 + i\sqrt 3 } \over 2}$.
If $a = \left( {1 + \alpha } \right)\sum\limits_{k = 0}^{100} {{\alpha ^{2k}}} $ and
$b = \sum\limits_{k = 0}^{100} {{\alpha ^{3k}}} $, then a and b are the roots of the quadratic equation :

A

x² + 101x + 100 = 0

B

x² + 102x + 101 = 0

C

x² – 102x + 101 = 0

D

x² – 101x + 100 = 0

Question 100: Palette 100/160

JEE Mains 2020 MCQ

Let S be the set of all real roots of the equation,
3^x(3^x – 1) + 2 = |3^x – 1| + |3^x – 2|. Then S :

A

contains exactly two elements.

B

is an empty set.

C

is a singleton.

D

contains at least four elements.

Question 101: Palette 101/160

JEE Mains 2020 MCQ

Let $\alpha $ and $\beta $ be the roots of the equation x² - x - 1 = 0.
If p_k = ${\left( \alpha \right)^k} + {\left( \beta \right)^k}$ , k $ \ge $ 1, then which one of the following statements is not true?

A

(p₁ + p₂ + p₃ + p₄ + p₅) = 26

B

p₅ = 11

C

p₃ = p₅ – p₄

D

p₅ = p₂ · p₃

Question 102: Palette 102/160

JEE Mains 2020 MCQ

Let $\alpha $ and $\beta $ be two real roots of the equation
(k + 1)tan²x - $\sqrt 2 $ . $\lambda $tanx = (1 - k), where k($ \ne $ - 1) and $\lambda $ are real numbers. if tan² ($\alpha $ + $\beta $) = 50, then a value of $\lambda $ is:

A

5$\sqrt 2 $

B

10

C

5

D

10$\sqrt 2 $

Question 103: Palette 103/160

JEE Mains 2019 MCQ

If $\alpha $, $\beta $ and $\gamma $ are three consecutive terms of a non-constant G.P. such that the equations $\alpha $x ² + 2$\beta $x + $\gamma $ = 0 and x² + x – 1 = 0 have a common root, then $\alpha $($\beta $ + $\gamma $) is equal to :

A

$\alpha $$\gamma $

B

0

C

$\beta $$\gamma $

D

$\alpha $$\beta $

Question 104: Palette 104/160

JEE Mains 2019 MCQ

The number of real roots of the equation
5 + |2^x – 1| = 2^x (2^x – 2) is

A

2

B

1

C

3

D

4

Question 105: Palette 105/160

JEE Mains 2019 MCQ

All the pairs (x, y) that satisfy the inequality
${2^{\sqrt {{{\sin }^2}x - 2\sin x + 5} }}.{1 \over {{4^{{{\sin }^2}y}}}} \le 1$
also satisfy the equation

A

sin x = |sin y|

B

sin x = 2sin y

C

2 sin x = sin y

D

2 |sin x | = 3 sin y

Question 106: Palette 106/160

JEE Mains 2019 MCQ

If $\alpha $ and $\beta $ are the roots of the quadratic equation,
x² + x sin $\theta $ - 2 sin $\theta $ = 0, $\theta \in \left( {0,{\pi \over 2}} \right)$, then
${{{\alpha ^{12}} + {\beta ^{12}}} \over {\left( {{\alpha ^{ - 12}} + {\beta ^{ - 12}}} \right).{{\left( {\alpha - \beta } \right)}^{24}}}}$ is equal to :

A

${{{2^{12}}} \over {{{\left( {\sin \theta - 8} \right)}^6}}}$

B

${{{2^6}} \over {{{\left( {\sin \theta + 4} \right)}^{12}}}}$

C

${{{2^{12}}} \over {{{\left( {\sin \theta + 8} \right)}^{12}}}}$

D

${{{2^{12}}} \over {{{\left( {\sin \theta - 4} \right)}^{12}}}}$

Question 107: Palette 107/160

JEE Mains 2019 MCQ

If m is chosen in the quadratic equation

(m² + 1) x² – 3x + (m² + 1)² = 0

such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is :-

A

$4\sqrt 3 $

B

$8\sqrt 3 $

C

$8\sqrt 5 $

D

$10\sqrt 5 $

Question 108: Palette 108/160

JEE Mains 2019 MCQ

Let p, q $ \in $ R. If 2 - $\sqrt 3$ is a root of the quadratic equation, x² + px + q = 0, then :

A

p² – 4q – 12 = 0

B

q² – 4p – 16 = 0

C

q² + 4p + 14 = 0

D

p² – 4q + 12 = 0

Question 109: Palette 109/160

JEE Mains 2019 MCQ

The number of integral values of m for which the equation

(1 + m² )x² – 2(1 + 3m)x + (1 + 8m) = 0 has no real root is :

A

2

B

infinitely many

C

1

D

3

Question 110: Palette 110/160

JEE Mains 2019 MCQ

The sum of the solutions of the equation
$\left| {\sqrt x - 2} \right| + \sqrt x \left( {\sqrt x - 4} \right) + 2 = 0$
(x > 0) is equal to:

A

9

B

12

C

4

D

10

Question 111: Palette 111/160

JEE Mains 2019 MCQ

The number of integral values of m for which the quadratic expression, (1 + 2m)x² – 2(1 + 3m)x + 4(1 + m), x $ \in $ R, is always positive, is :

A

7

B

8

C

3

D

6

Question 112: Palette 112/160

JEE Mains 2019 MCQ

If $\lambda $ be the ratio of the roots of the quadratic equation in x, 3m²x² + m(m – 4)x + 2 = 0, then the least value of m for which $\lambda + {1 \over \lambda } = 1,$ is

A

$ - 2 + \sqrt 2 $

B

4$-$3$\sqrt 2 $

C

2 $-$ $\sqrt 3 $

D

4 $-$ 2$\sqrt 3 $

Question 113: Palette 113/160

JEE Mains 2019 MCQ

Let $\alpha $ and $\beta $ be the roots of the quadratic equation x² sin $\theta $ – x(sin $\theta $ cos $\theta $ + 1) + cos $\theta $ = 0 (0 < $\theta $ < 45^o), and $\alpha $ < $\beta $. Then $\sum\limits_{n = 0}^\infty {\left( {{\alpha ^n} + {{{{\left( { - 1} \right)}^n}} \over {{\beta ^n}}}} \right)} $ is equal to :

A

${1 \over {1 + \cos \theta }} + {1 \over {1 - \sin \theta }}$

B

${1 \over {1 - \cos \theta }} + {1 \over {1 + \sin \theta }}$

C

${1 \over {1 - \cos \theta }} - {1 \over {1 + \sin \theta }}$

D

${1 \over {1 + \cos \theta }} - {1 \over {1 - \sin \theta }}$

Question 114: Palette 114/160

JEE Mains 2019 MCQ

If one real root of the quadratic equation 81x² + kx + 256 = 0 is cube of the other root, then a value of k is

A

$-$ 81

B

$-$ 300

C

100

D

144

Question 115: Palette 115/160

JEE Mains 2019 MCQ

The value of $\lambda $ such that sum of the squares of the roots of the quadratic equation, x² + (3 – $\lambda $)x + 2 = $\lambda $ has the least value is -

A

1

B

2

C

${{15} \over 8}$

D

${4 \over 9}$

Question 116: Palette 116/160

JEE Mains 2019 MCQ

Consider the quadratic equation (c – 5)x² – 2cx + (c – 4) = 0, c $ \ne $ 5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2, 3). Then the number of elements in S is -

A

12

B

18

C

10

D

11

Question 117: Palette 117/160

JEE Mains 2019 MCQ

If both the roots of the quadratic equation x² $-$ mx + 4 = 0 are real and distinct and they lie in the interval [1, 5], then m lies in the interval :

A

($-$5, $-$4)

B

(4, 5)

C

(5, 6)

D

(3, 4)

Question 118: Palette 118/160

JEE Mains 2019 MCQ

The number of all possible positive integral values of $\alpha $  for which the roots of the quadratic equation, 6x² $-$ 11x + $\alpha $ = 0 are rational numbers is :

A

3

B

2

C

4

D

5

Question 119: Palette 119/160

JEE Mains 2018 MCQ

If an angle A of a $\Delta $ABC satiesfies 5 cosA + 3 = 0, then the roots of the quadratic equation, 9x² + 27x + 20 = 0 are :

A

secA, cotA

B

sinA, secA

C

secA, tanA

D

tanA, cosA

Question 120: Palette 120/160

JEE Mains 2018 MCQ

Let p, q and r be real numbers (p $ \ne $ q, r $ \ne $ 0), such that the roots of the equation ${1 \over {x + p}} + {1 \over {x + q}} = {1 \over r}$ are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to :

A

${{{p^2} + {q^2}} \over 2}$

B

p² + q²

C

2(p² + q²)

D

p² + q² + r²

Question 121: Palette 121/160

JEE Mains 2018 MCQ

Let S = { $x$ $ \in $ R : $x$ $ \ge $ 0 and

$2\left| {\sqrt x - 3} \right| + \sqrt x \left( {\sqrt x - 6} \right) + 6 = 0$}. Then S

A

contains exactly four elements

B

is an empty set

C

contains exactly one element

D

contains exactly two elements

Question 122: Palette 122/160

JEE Mains 2018 MCQ

If f(x) is a quadratic expression such that f (1) + f (2) = 0, and $-$ 1 is a root of f (x) = 0, then the other root of f(x) = 0 is :

A

$-$ ${5 \over 8}$

B

$-$ ${8 \over 5}$

C

${5 \over 8}$

D

${8 \over 5}$

Question 123: Palette 123/160

JEE Mains 2018 MCQ

If $\lambda $ $ \in $ R is such that the sum of the cubes of the roots of the equation,
x² + (2 $-$ $\lambda $) x + (10 $-$ $\lambda $) = 0 is minimum, then the magnitude of the difference of the roots of this equation is :

A

$4\sqrt 2 $

B

$2\sqrt 5 $

C

$2\sqrt 7 $

D

20

Question 124: Palette 124/160

JEE Mains 2018 MCQ

If tanA and tanB are the roots of the quadratic equation, 3x² $-$ 10x $-$ 25 = 0, then the value of 3 sin²(A + B) $-$ 10 sin(A + B).cos(A + B) $-$ 25 cos²(A + B) is :

A

$-$ 10

B

10

C

$-$ 25

D

25

Question 125: Palette 125/160

JEE Mains 2017 MCQ

The sum of all the real values of x satisfying the equation
2^{(x$-$1)(x2 + 5x $-$ 50)} = 1 is :

A

16

B

14

C

$-$4

D

$-$ 5

Question 126: Palette 126/160

JEE Mains 2017 MCQ

Let p(x) be a quadratic polynomial such that p(0)=1. If p(x) leaves remainder 4 when divided by x$-$ 1 and it leaves remainder 6 when divided by x + 1; then :

A

p(2) = 11

B

p(2) = 19

C

p($-$ 2) = 19

D

p($-$ 2) = 11

Question 127: Palette 127/160

JEE Mains 2017 MCQ

If for a positive integer n, the quadratic equation

$x\left( {x + 1} \right) + \left( {x + 1} \right)\left( {x + 2} \right)$$ + .... + \left( {x + \overline {n - 1} } \right)\left( {x + n} \right)$$ = 10n$

has two consecutive integral solutions, then n is equal to :

A

9

B

10

C

11

D

12

Question 128: Palette 128/160

JEE Mains 2016 MCQ

If x is a solution of the equation, $\sqrt {2x + 1} $ $ - \sqrt {2x - 1} = 1,$ $\,\,\left( {x \ge {1 \over 2}} \right),$ then $\sqrt {4{x^2} - 1} $ is equal to :

A

${3 \over 4}$

B

${1 \over 2}$

C

2

D

$2\sqrt 2 $

Question 129: Palette 129/160

JEE Mains 2016 MCQ

If the equations x² + bx−1 = 0 and x² + x + b = 0 have a common root different from −1, then $\left| b \right|$ is equal to :

A

$\sqrt 2 $

B

2

C

3

D

$\sqrt 3 $

Question 130: Palette 130/160

JEE Mains 2016 MCQ

The sum of all real values of $x$ satisfying the equation ${\left( {{x^2} - 5x + 5} \right)^{{x^2} + 4x - 60}}\, = 1$ is :

A

$6$

B

$5$

C

$3$

D

$-4$

Question 131: Palette 131/160

JEE Mains 2015 MCQ

Let $\alpha $ and $\beta $ be the roots of equation ${x^2} - 6x - 2 = 0$. If ${a_n} = {\alpha ^n} - {\beta ^n},$ for $n \ge 1,$ then the value of ${{{a_{10}} - 2{a_8}} \over {2{a_9}}}$ is equal to :

A

$3$

B

$ - 3$

C

$6$

D

$ - 6$

Question 132: Palette 132/160

JEE Mains 2014 MCQ

Let $\alpha $ and $\beta $ be the roots of equation $p{x^2} + qx + r = 0,$ $p \ne 0.$ If $p,\,q,\,r$ in A.P. and ${1 \over \alpha } + {1 \over \beta } = 4,$ then the value of $\left| {\alpha - \beta } \right|$ is :

A

${{\sqrt {34} } \over 9}$

B

${{2\sqrt 13 } \over 9}$

C

${{\sqrt {61} } \over 9}$

D

${{2\sqrt 17 } \over 9}$

Question 133: Palette 133/160

JEE Mains 2014 MCQ

If $a \in R$ and the equation $ - 3{\left( {x - \left[ x \right]} \right)^2} + 2\left( {x - \left[ x \right]} \right) + {a^2} = 0$ (where [$x$] denotes the greater integer $ \le x$) has no integral solution, then all possible values of a lie in the interval :

A

$\left( { - 2, - 1} \right)$

B

$\left( { - \infty , - 2} \right) \cup \left( {2,\infty } \right)$

C

$\left( { - 1,0} \right) \cup \left( {0,1} \right)$

D

$\left( {1,2} \right)$

Question 134: Palette 134/160

JEE Mains 2013 MCQ

If the equations ${x^2} + 2x + 3 = 0$ and $a{x^2} + bx + c = 0,$ $a,\,b,\,c\, \in \,R,$ have a common root, then $a\,:b\,:c\,$ is

A

$1:2:3$

B

$3:2:1$

C

$1:3:2$

D

$3:1:2$

Question 135: Palette 135/160

JEE Mains 2012 MCQ

The equation ${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$ has:

A

infinite number of real roots

B

no real roots

C

exactly one real root

D

exactly four real roots

Question 136: Palette 136/160

JEE Mains 2010 MCQ

If $\alpha $ and $\beta $ are the roots of the equation ${x^2} - x + 1 = 0,$ then ${\alpha ^{2009}} + {\beta ^{2009}} = $

A

$\, - 1$

B

$\, 1$

C

$\, 2$

D

$\, - 2$

Question 137: Palette 137/160

JEE Mains 2009 MCQ

If the roots of the equation $b{x^2} + cx + a = 0$ imaginary, then for all real values of $x$, the expression $3{b^2}{x^2} + 6bcx + 2{c^2}$ is :

A

less than $4ab$

B

greater than $-4ab$

C

less than $-4ab$

D

greater than $4ab$

Question 138: Palette 138/160

JEE Mains 2008 MCQ

The quadratic equations ${x^2} - 6x + a = 0$ and ${x^2} - cx + 6 = 0$ have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is

A

1

B

4

C

3

D

2

Question 139: Palette 139/160

JEE Mains 2008 MCQ

STATEMENT - 1 : For every natural number $n \ge 2,$ $${1 \over {\sqrt 1 }} + {1 \over {\sqrt 2 }} + ........ + {1 \over {\sqrt n }} > \sqrt n .$$

STATEMENT - 2 : For every natural number $n \ge 2,$, $$\sqrt {n\left( {n + 1} \right)} < n + 1.$$

A

Statement - 1 is false, Statement - 2 is true

B

Statement - 1 is true, Statement - 2 is true; Statement - 2 is a correct explanation for statement - 1

C

Statement - 1 is true, Statement - 2 is true; Statement - 2 is not a correct explanation for Statement - 1

D

Statement - 1 is true, Statement - 2 is false

Question 140: Palette 140/160

JEE Mains 2007 MCQ

If the difference between the roots of the equation ${x^2} + ax + 1 = 0$ is less than $\sqrt 5 ,$ then the set of possible values of $a$ is

A

$\left( {3,\infty } \right)$

B

$\left( { - \infty , - 3} \right)$

C

$\left( { - 3,3} \right)$

D

$\left( { - 3,\infty } \right)$

Question 141: Palette 141/160

JEE Mains 2006 MCQ

If $x$ is real, the maximum value of ${{3{x^2} + 9x + 17} \over {3{x^2} + 9x + 7}}$ is

A

${1 \over 4}$

B

$41$

C

$1$

D

${17 \over 7}$

Question 142: Palette 142/160

JEE Mains 2006 MCQ

If the roots of the quadratic equation ${x^2} + px + q = 0$ are $\tan {30^ \circ }$ and $\tan {15^ \circ }$, respectively, then the value of $2 + q - p$ is

A

2

B

3

C

0

D

1

Question 143: Palette 143/160

JEE Mains 2006 MCQ

All the values of $m$ for which both roots of the equation ${x^2} - 2mx + {m^2} - 1 = 0$ are greater than $ - 2$ but less then 4, lie in the interval

A

$ - 2 < m < 0$

B

$m > 3$

C

$ - 1 < m < 3$

D

$1 < m < 4$

Question 144: Palette 144/160

JEE Mains 2005 MCQ

In a triangle $PQR,\;\;\angle R = {\pi \over 2}.\,\,If\,\,\tan \,\left( {{P \over 2}} \right)$ and $ \tan \left( {{Q \over 2}} \right)$ are the roots of $a{x^2} + bx + c = 0,\,\,a \ne 0$ then

A

$a = b + c$

B

$c = a + b$

C

$b = c$

D

$b = a + c$

Question 145: Palette 145/160

JEE Mains 2005 MCQ

The value of $a$ for which the sum of the squares of the roots of the equation
${x^2} - \left( {a - 2} \right)x - a - 1 = 0$ assume the least value is

A

$1$

B

$0$

C

$3$

D

$2$

Question 146: Palette 146/160

JEE Mains 2005 MCQ

If the roots of the equation ${x^2} - bx + c = 0$ be two consecutive integers, then ${b^2} - 4c$ equals

A

$-2$

B

$3$

C

$2$

D

$1$

Question 147: Palette 147/160

JEE Mains 2005 MCQ

If the roots of the equation ${x^2} - bx + c = 0$ be two consecutive integers, then ${b^2} - 4c$ equals

A

$-2$

B

$3$

C

$2$

D

$1$

Question 148: Palette 148/160

JEE Mains 2005 MCQ

The value of $a$ for which the sum of the squares of the roots of the equation ${x^2} - \left( {a - 2} \right)x - a - 1 = 0$ assume the least value is :

A

$1$

B

$0$

C

$3$

D

$2$

Question 149: Palette 149/160

JEE Mains 2005 MCQ

If both the roots of the quadratic equation ${x^2} - 2kx + {k^2} + k - 5 = 0$ are less than 5, then $k$ lies in the interval

A

$\left( {5,6} \right]$

B

$\left( {6,\,\infty } \right)$

C

$\left( { - \infty ,\,4} \right)$

D

$\left[ {4,\,5} \right]$

Question 150: Palette 150/160

JEE Mains 2004 MCQ

If one root of the equation ${x^2} + px + 12 = 0$ is 4, while the equation ${x^2} + px + q = 0$ has equal roots,
then the value of $'q'$ is

A

4

B

12

C

3

D

${{49} \over 4}$

Question 151: Palette 151/160

JEE Mains 2004 MCQ

If $\left( {1 - p} \right)$ is a root of quadratic equation ${x^2} + px + \left( {1 - p} \right) = 0$ then its root are

A

$ - 1,2$

B

$ - 1,1$

C

$ 0,-1$

D

$0,1$

Question 152: Palette 152/160

JEE Mains 2004 MCQ

Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

A

${x^2} - 18x - 16 = 0$

B

${x^2} - 18x + 16 = 0$

C

${x^2} + 18x - 16 = 0$

D

${x^2} + 18x + 16 = 0$

Question 153: Palette 153/160

JEE Mains 2003 MCQ

The value of '$a$' for which one root of the quadratic equation $$\left( {{a^2} - 5a + 3} \right){x^2} + \left( {3a - 1} \right)x + 2 = 0$$
is twice as large as the other is

A

$ - {1 \over 3}$

B

$ {2 \over 3}$

C

$ - {2 \over 3}$

D

$ {1 \over 3}$

Question 154: Palette 154/160

JEE Mains 2003 MCQ

The number of real solutions of the equation ${x^2} - 3\left| x \right| + 2 = 0$ is

A

3

B

2

C

4

D

1

Question 155: Palette 155/160

JEE Mains 2003 MCQ

If the sum of the roots of the quadratic equation $a{x^2} + bx + c = 0$ is equal to the sum of the squares of their reciprocals, then ${a \over c},\,{b \over a}$ and ${c \over b}$ are in

A

Arithmetic - Geometric Progression

B

Arithmetic Progression

C

Geometric Progression

D

Harmonic Progression

Question 156: Palette 156/160

JEE Mains 2002 MCQ

Product of real roots of equation ${t^2}{x^2} + \left| x \right| + 9 = 0$

A

is always positive

B

is always negative

C

does not exist

D

none of these

Question 157: Palette 157/160

JEE Mains 2002 MCQ

If $a,\,b,\,c$ are distinct $ + ve$ real numbers and ${a^2} + {b^2} + {c^2} = 1$ then $ab + bc + ca$ is

A

less than 1

B

equal to 1

C

greater than 1

D

any real no.

Question 158: Palette 158/160

JEE Mains 2002 MCQ

If $\alpha \ne \beta $ but ${\alpha ^2} = 5\alpha - 3$ and ${\beta ^2} = 5\beta - 3$ then the equation having $\alpha /\beta $ and $\beta /\alpha \,\,$ as its roots is

A

$3{x^2} - 19x + 3 = 0$

B

$3{x^2} + 19x - 3 = 0$

C

$3{x^2} - 19x - 3 = 0$

D

${x^2} - 5x + 3 = 0$

Question 159: Palette 159/160

JEE Mains 2002 MCQ

If $p$ and $q$ are the roots of the equation ${x^2} + px + q = 0,$ then

A

$p = 1,\,\,q = - 2$

B

$p = 0,\,\,q = 1$

C

$p = - 2,\,\,q = 0$

D

$p = - 2,\,\,q = 1$

Question 160: Palette 160/160

JEE Mains 2002 MCQ

Difference between the corresponding roots of ${x^2} + ax + b = 0$ and ${x^2} + bx + a = 0$ is same and $a \ne b,$ then

A

$a + b + 4 = 0$

B

$a + b - 4 = 0$

C

$a - b - 4 = 0$

D

$a - b + 4 = 0$