# Quadratic Equation and Inequalities · Mathematics · JEE Main

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## MCQ (Single Correct Answer)

JEE Main 2024 (Online) 9th April Evening Shift
Let $$\alpha, \beta ; \alpha>\beta$$, be the roots of the equation $$x^2-\sqrt{2} x-\sqrt{3}=0$$. Let $$\mathrm{P}_n=\alpha^n-\beta^n, n \in \mathrm{N... JEE Main 2024 (Online) 9th April Morning Shift Let$$\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{...
JEE Main 2024 (Online) 4th April Morning Shift
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 $$\frac{1}{6 a+b}$$, ...
JEE Main 2024 (Online) 1st February Evening Shift
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 ...
JEE Main 2024 (Online) 1st February Morning Shift
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 :
JEE Main 2024 (Online) 31st January Morning Shift
Let $$\mathrm{S}$$ be the set of positive integral values of $$a$$ for which $$\frac{a x^2+2(a+1) x+9 a+4}{x^2-8 x+32} ... JEE Main 2024 (Online) 27th January Evening Shift If$$\alpha, \beta$$are the roots of the equation,$$x^2-x-1=0$$and$$S_n=2023 \alpha^n+2024 \beta^n$$, then : JEE Main 2023 (Online) 15th April Morning Shift The number of real roots of the equation x|x|-5|x+2|+6=0, is : JEE Main 2023 (Online) 13th April Evening Shift Let$$\alpha, \beta$$be the roots of the equation$$x^{2}-\sqrt{2} x+2=0$$. Then$$\alpha^{14}+\beta^{14}$$is equal to JEE Main 2023 (Online) 13th April Morning Shift The set of all$$a \in \mathbb{R}$$for which the equation$$x|x-1|+|x+2|+a=0$$has exactly one real root, is : JEE Main 2023 (Online) 12th April Morning Shift Let$$\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}}{\al...
JEE Main 2023 (Online) 8th April Morning Shift
Let $$\alpha, \beta, \gamma$$ be the three roots of the equation $$x^{3}+b x+c=0$$. If $$\beta \gamma=1=-\alpha$$, then $$b^{3}+2 c^{3}-3 \alpha^{3}-6... JEE Main 2023 (Online) 6th April Morning Shift Let$$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)}...
JEE Main 2023 (Online) 6th April Morning Shift
The sum of all the roots of the equation $$\left|x^{2}-8 x+15\right|-2 x+7=0$$ is :
JEE Main 2023 (Online) 1st February Evening Shift
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 inte...
JEE Main 2023 (Online) 1st February Morning Shift
Let $$S = \left\{ {x:x \in \mathbb{R}\,\mathrm{and}\,{{(\sqrt 3 + \sqrt 2 )}^{{x^2} - 4}} + {{(\sqrt 3 - \sqrt 2 )}^{{x^2} - 4}} = 10} \right\}$$. T...
JEE Main 2023 (Online) 31st January Evening Shift
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 :
JEE Main 2023 (Online) 31st January Morning Shift
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 :
JEE Main 2023 (Online) 29th January Morning Shift
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 ...
JEE Main 2023 (Online) 24th January Evening Shift
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
JEE Main 2023 (Online) 24th January Morning Shift
The equation $${x^2} - 4x + [x] + 3 = x[x]$$, where $$[x]$$ denotes the greatest integer function, has :
JEE Main 2022 (Online) 29th July Morning Shift
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 :...
JEE Main 2022 (Online) 28th July Evening Shift
$$\text { Let } S=\left\{x \in[-6,3]-\{-2,2\}: \frac{|x+3|-1}{|x|-2} \geq 0\right\} \text { and }$$$$T=\left\{x \in \mathbb{Z}: x^{2}-7|x|+9 \leq 0\... JEE Main 2022 (Online) 28th July Evening Shift Let$$\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$$be the ro... JEE Main 2022 (Online) 27th July Evening Shift If$$\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\...
JEE Main 2022 (Online) 26th July Evening Shift
The minimum value of the sum of the squares of the roots of $$x^{2}+(3-a) x+1=2 a$$ is:
JEE Main 2022 (Online) 25th July Morning Shift
If $$\alpha, \beta, \gamma, \delta$$ are the roots of the equation $$x^{4}+x^{3}+x^{2}+x+1=0$$, then $$\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta... JEE Main 2022 (Online) 30th June Morning Shift Let$${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^{...
JEE Main 2022 (Online) 30th June Morning Shift
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 $$3{x^2} + (\alpha - 6... JEE Main 2022 (Online) 29th June Evening Shift Let$$\alpha$$be a root of the equation 1 + x2 + x4 = 0. Then, the value of$$\alpha$$1011 +$$\alpha$$2022$$-\alpha$$3033 is equal to :... JEE Main 2022 (Online) 28th June Evening Shift Let f(x) be a quadratic polynomial such that f($$-$$2) + f(3) = 0. If one of the roots of f(x) = 0 is$$-$$1, then the sum of the roots of f(x) = 0 is... JEE Main 2022 (Online) 27th June Morning Shift The number of distinct real roots of x4$$-$$4x + 1 = 0 is : JEE Main 2022 (Online) 25th June Evening Shift Let$$A = \{ x \in R:|x + 1|
JEE Main 2022 (Online) 25th June Evening Shift
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 th...
JEE Main 2022 (Online) 24th June Evening Shift
The sum of all the real roots of the equation $$({e^{2x}} - 4)(6{e^{2x}} - 5{e^x} + 1) = 0$$ is
JEE Main 2022 (Online) 24th June Evening Shift
The number of distinct real roots of the equation x7 $$-$$ 7x $$-$$ 2 = 0 is
JEE Main 2022 (Online) 24th June Morning Shift
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($$\alp... JEE Main 2021 (Online) 1st September Evening Shift The numbers of pairs (a, b) of real numbers, such that whenever$$\alpha$$is a root of the equation x2 + ax + b = 0,$$\alpha$$2$$-$$2 is also a ro... JEE Main 2021 (Online) 31st August Evening Shift The sum of the roots of the equation$$x + 1 - 2{\log _2}(3 + {2^x}) + 2{\log _4}(10 - {2^{ - x}}) = 0$$, is : JEE Main 2021 (Online) 31st August Morning Shift cosec18$$^\circ$$is a root of the equation : JEE Main 2021 (Online) 27th August Evening Shift The set of all values of K >$$-$$1, for which the equation$${(3{x^2} + 4x + 3)^2} - (k + 1)(3{x^2} + 4x + 3)(3{x^2} + 4x + 2) + k{(3{x^2} + 4x + ...
JEE Main 2021 (Online) 27th July Evening Shift
Let $$\alpha = \mathop {\max }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\}$$ and $$\beta = \mathop {\min }\limits_{x \in R} \{ {8^{2\sin 3x... JEE Main 2021 (Online) 27th July Morning Shift Let$$\alpha$$,$$\beta$$be two roots of the equation x2 + (20)1/4x + (5)1/2 = 0. Then$$\alpha$$8 +$$\beta$$8 is equal to ... JEE Main 2021 (Online) 25th July Evening Shift If [x] be the greatest integer less than or equal to x, then$$\sum\limits_{n = 8}^{100} {\left[ {{{{{( - 1)}^n}n} \over 2}} \right]} $$is equal to :... JEE Main 2021 (Online) 25th July Evening Shift The number of real solutions of the equation, x2$$-$$|x|$$-$$12 = 0 is : JEE Main 2021 (Online) 25th July Morning Shift The number of real roots of the equation$${e^{6x}} - {e^{4x}} - 2{e^{3x}} - 12{e^{2x}} + {e^x} + 1 = 0$$is : JEE Main 2021 (Online) 22th July Evening Shift Let [x] denote the greatest integer less than or equal to x. Then, the values of x$$\in$$R satisfying the equation$${[{e^x}]^2} + [{e^x} + 1] - 3 = 0...
JEE Main 2021 (Online) 20th July Morning Shift
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 $${\alpha ^{96}}({\alph... JEE Main 2021 (Online) 18th March Morning Shift The value of$$3 + {1 \over {4 + {1 \over {3 + {1 \over {4 + {1 \over {3 + ....\infty }}}}}}}}$$is equal to JEE Main 2021 (Online) 17th March Morning Shift The value of$$4 + {1 \over {5 + {1 \over {4 + {1 \over {5 + {1 \over {4 + ......\infty }}}}}}}}$$is : JEE Main 2021 (Online) 25th February Evening Shift Let$$\alpha$$and$$\beta$$be the roots of x2$$-$$6x$$-$$2 = 0. If an =$$\alpha$$n$$-\beta$$n for n$$ \ge $$1, then the value of$${{{a...
JEE Main 2021 (Online) 25th February Morning Shift
The integer 'k', for which the inequality x2 $$-$$ 2(3k $$-$$ 1)x + 8k2 $$-$$ 7 > 0 is valid for every x in R, is :
JEE Main 2021 (Online) 24th February Morning Shift
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 :
JEE Main 2020 (Online) 6th September Evening Slot
If $$\alpha$$ and $$\beta$$ are the roots of the equation 2x(2x + 1) = 1, then $$\beta$$ is equal to :
JEE Main 2020 (Online) 6th September Morning Slot
If $$\alpha$$ and $$\beta$$ be two roots of the equation x2 – 64x + 256 = 0. Then the value of $${\left( {{{{\alpha ^3}} \over {{\beta ^5}}}} \right... JEE Main 2020 (Online) 5th September Evening Slot If$$\alpha $$and$$\beta $$are the roots of the equation, 7x2 – 3x – 2 = 0, then the value of$${\alpha \over {1 - {\alpha ^2}}} + {\beta \over {...
JEE Main 2020 (Online) 5th September Morning Slot
The product of the roots of the equation 9x2 - 18|x| + 5 = 0 is :
JEE Main 2020 (Online) 4th September Evening Slot
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 $$\gamma ... JEE Main 2020 (Online) 4th September Morning Slot Let [t] denote the greatest integer$$ \le $$t. Then the equation in x, [x]2 + 2[x+2] - 7 = 0 has : JEE Main 2020 (Online) 4th September Morning Slot Let$$\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, \gam...
JEE Main 2020 (Online) 3rd September Evening Slot
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...
JEE Main 2020 (Online) 3rd September Morning Slot
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 e...
JEE Main 2020 (Online) 2nd September Evening Slot
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 :
JEE Main 2020 (Online) 2nd September Morning Slot
Let $$\alpha$$ and $$\beta$$ be the roots of the equation 5x2 + 6x – 2 = 0. If Sn = $$\alpha$$n + $$\beta$$n, n = 1, 2, 3...., then :...
JEE Main 2020 (Online) 9th January Evening Slot
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...
JEE Main 2020 (Online) 9th January Morning Slot
The number of real roots of the equation, e4x + e3x – 4e2x + ex + 1 = 0 is :
JEE Main 2020 (Online) 8th January Evening Slot
Let S be the set of all real roots of the equation, 3x(3x – 1) + 2 = |3x – 1| + |3x – 2|. Then S :
JEE Main 2020 (Online) 8th January Evening Slot
Let $$\alpha = {{ - 1 + i\sqrt 3 } \over 2}$$. If $$a = \left( {1 + \alpha } \right)\sum\limits_{k = 0}^{100} {{\alpha ^{2k}}}$$ and $$b = \sum\limi... JEE Main 2020 (Online) 7th January Evening Slot Let$$\alpha $$and$$\beta $$be the roots of the equation x2 - x - 1 = 0. If pk =$${\left( \alpha \right)^k} + {\left( \beta \right)^k}$$, k ... JEE Main 2020 (Online) 7th January Morning Slot Let$$\alpha $$and$$\beta $$be two real roots of the equation (k + 1)tan2x -$$\sqrt 2 $$.$$\lambda $$tanx = (1 - k), where k($$ \ne $$- 1) and ... JEE Main 2019 (Online) 12th April Evening Slot If$$\alpha $$,$$\beta $$and$$\gamma $$are three consecutive terms of a non-constant G.P. such that the equations$$\alpha $$x 2 + 2$$\beta $$x +... JEE Main 2019 (Online) 10th April Evening Slot The number of real roots of the equation 5 + |2x – 1| = 2x (2x – 2) is JEE Main 2019 (Online) 10th April Morning Slot All the pairs (x, y) that satisfy the inequality$${2^{\sqrt {{{\sin }^2}x - 2\sin x + 5} }}.{1 \over {{4^{{{\sin }^2}y}}}} \le 1$$also satisfy the e... JEE Main 2019 (Online) 10th April Morning Slot If$$\alpha $$and$$\beta $$are the roots of the quadratic equation, x2 + x sin$$\theta $$- 2 sin$$\theta $$= 0,$$\theta \in \left( {0,{\pi ...
JEE Main 2019 (Online) 9th April Evening Slot
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 o...
JEE Main 2019 (Online) 9th April Morning Slot
Let p, q $$\in$$ R. If 2 - $$\sqrt 3$$ is a root of the quadratic equation, x2 + px + q = 0, then :
JEE Main 2019 (Online) 8th April Evening Slot
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 :
JEE Main 2019 (Online) 8th April Morning Slot
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:...
JEE Main 2019 (Online) 12th January Evening Slot
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 :...
JEE Main 2019 (Online) 12th January Morning Slot
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 $$\lambda + ... JEE Main 2019 (Online) 11th January Evening Slot Let$$\alpha $$and$$\beta $$be the roots of the quadratic equation x2 sin$$\theta $$– x(sin$$\theta $$cos$$\theta $$+ 1) + cos$$\theta $$=... JEE Main 2019 (Online) 11th January Morning Slot If one real root of the quadratic equation 81x2 + kx + 256 = 0 is cube of the other root, then a value of k is JEE Main 2019 (Online) 10th January Evening Slot The value of$$\lambda $$such that sum of the squares of the roots of the quadratic equation, x2 + (3 –$$\lambda $$)x + 2 =$$\lambda $$has the lea... JEE Main 2019 (Online) 10th January Morning Slot Consider the quadratic equation (c – 5)x2 – 2cx + (c – 4) = 0, c$$ \ne $$5. Let S be the set of all integral values of c for which one root of the e... JEE Main 2019 (Online) 9th January Evening Slot If both the roots of the quadratic equation x2$$-$$mx + 4 = 0 are real and distinct and they lie in the interval [1, 5], then m lies in the interval... JEE Main 2019 (Online) 9th January Evening Slot The number of all possible positive integral values of$$\alpha $$for which the roots of the quadratic equation, 6x2$$-$$11x +$$\alpha $$=... JEE Main 2018 (Online) 16th April Morning Slot Let p, q and r be real numbers (p$$ \ne $$q, r$$ \ne $$0), such that the roots of the equation$${1 \over {x + p}} + {1 \over {x + q}} = {1 \over...
JEE Main 2018 (Online) 16th April Morning Slot
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 :
JEE Main 2018 (Offline)
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...
JEE Main 2018 (Online) 15th April Evening Slot
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 :
JEE Main 2018 (Online) 15th April Morning Slot
If $$\lambda$$ $$\in$$ R is such that the sum of the cubes of the roots of the equation, x2 + (2 $$-$$ $$\lambda$$) x + (10 $$-$$ $$\lambda$$) =...
JEE Main 2018 (Online) 15th April Morning Slot
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) $... JEE Main 2017 (Online) 9th April Morning Slot The sum of all the real values of x satisfying the equation 2(x$$-$$1)(x2 + 5x $$-$$ 50) = 1 is : JEE Main 2017 (Online) 8th April Morning Slot 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 +... JEE Main 2017 (Offline) If for a positive integer n, the quadratic equation $$x\left( {x + 1} \right) + \left( {x + 1} \right)\left( {x + 2} \right)$$$$+ .... + \left( {x + ... JEE Main 2016 (Online) 10th April Morning Slot If x is a solution of the equation,$$\sqrt {2x + 1}  - \sqrt {2x - 1} = 1,\,\,\left( {x \ge {1 \over 2}} \right),$$then$$\sqrt {4{x^2} -... JEE Main 2016 (Online) 9th April Morning Slot 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 : JEE Main 2016 (Offline) The sum of all real values of $$x$$ satisfying the equation $${\left( {{x^2} - 5x + 5} \right)^{{x^2} + 4x - 60}}\, = 1$$ is : JEE Main 2015 (Offline) 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 va... JEE Main 2014 (Offline) 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 [$$x$$] ... JEE Main 2014 (Offline) 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 $${1 \over \alpha } + {1 \o... JEE Main 2013 (Offline) If the equations$${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\,$$is AIEEE 2012 The equation$${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$$has: AIEEE 2010 If$$\alpha $$and$$\beta $$are the roots of the equation$${x^2} - x + 1 = 0,$$then$${\alpha ^{2009}} + {\beta ^{2009}} = $$AIEEE 2009 If the roots of the equation$$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}$$is : AIEEE 2008 STATEMENT - 1 : For every natural number$$n \ge 2,${1 \over {\sqrt 1 }} + {1 \over {\sqrt 2 }} + ........ + {1 \over {\sqrt n }} > \sqrt n ....
AIEEE 2008
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 a...
AIEEE 2007
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
AIEEE 2006
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 $$2... AIEEE 2006 All the values of$$m$$for which both roots of the equation$${x^2} - 2mx + {m^2} - 1 = 0$$are greater than$$ - 2$$but less then 4, lie in the int... AIEEE 2006 If$$x$$is real, the maximum value of$${{3{x^2} + 9x + 17} \over {3{x^2} + 9x + 7}}$$is AIEEE 2005 In a triangle$$PQR,\;\;\angle R = {\pi \over 2}.\,\,If\,\,\tan \,\left( {{P \over 2}} \right)$$and$$ \tan \left( {{Q \over 2}} \right)$$are the r... AIEEE 2005 If both the roots of the quadratic equation$${x^2} - 2kx + {k^2} + k - 5 = 0$$are less than 5, then$$k$$lies in the interval AIEEE 2005 The value of$$a$$for which the sum of the squares of the roots of the equation$${x^2} - \left( {a - 2} \right)x - a - 1 = 0$$assume the least valu... AIEEE 2005 If the roots of the equation$${x^2} - bx + c = 0$$be two consecutive integers, then$${b^2} - 4c$$equals AIEEE 2005 The value of$$a$$for which the sum of the squares of the roots of the equation$${x^2} - \left( {a - 2} \right)x - a - 1 = 0$$assume the least v... AIEEE 2005 If the roots of the equation$${x^2} - bx + c = 0$$be two consecutive integers, then$${b^2} - 4c$$equals AIEEE 2004 Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation AIEEE 2004 If$$\left( {1 - p} \right)$$is a root of quadratic equation$${x^2} + px + \left( {1 - p} \right) = 0$$then its root are AIEEE 2004 If one root of the equation$${x^2} + px + 12 = 0$$is 4, while the equation$${x^2} + px + q = 0$$has equal roots, then the value of$$'q'$$is ... AIEEE 2003 If the sum of the roots of the quadratic equation$$a{x^2} + bx + c = 0$$is equal to the sum of the squares of their reciprocals, then$${a \over c},...
AIEEE 2003
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 tw... AIEEE 2003 The number of real solutions of the equation $${x^2} - 3\left| x \right| + 2 = 0$$ is AIEEE 2002 If $$\alpha \ne \beta$$ but $${\alpha ^2} = 5\alpha - 3$$ and $${\beta ^2} = 5\beta - 3$$ then the equation having $$\alpha /\beta$$ and $$\beta ... AIEEE 2002 Product of real roots of equation$${t^2}{x^2} + \left| x \right| + 9 = 0$$AIEEE 2002 Difference between the corresponding roots of$${x^2} + ax + b = 0$$and$${x^2} + bx + a = 0$$is same and$$a \ne b,$$then AIEEE 2002 If$$p$$and$$q$$are the roots of the equation$${x^2} + px + q = 0,$$then AIEEE 2002 If$$a,\,b,\,c$$are distinct$$ + ve$$real numbers and$${a^2} + {b^2} + {c^2} = 1$$then$$ab + bc + ca$$is ## Numerical JEE Main 2024 (Online) 31st January Evening Shift Let$$a, b, c$$be the lengths of three sides of a triangle satistying the condition$$\left(a^2+b^2\right) x^2-2 b(a+c) x+\left(b^2+c^2\right)=0$$. I... JEE Main 2024 (Online) 30th January Evening Shift The number of real solutions of the equation$$x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0$$is _________. JEE Main 2024 (Online) 30th January Morning Shift Let$$\alpha, \beta \in \mathbf{N}$$be roots of the equation$$x^2-70 x+\lambda=0$$, where$$\frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathbf{N}$$... JEE Main 2024 (Online) 29th January Evening Shift Let the set$$C=\left\{(x, y) \mid x^2-2^y=2023, x, y \in \mathbb{N}\right\}$$. Then$$\sum_\limits{(x, y) \in C}(x+y)$$is equal to _________. JEE Main 2023 (Online) 13th April Evening Shift Let$$[\alpha]$$denote the greatest integer$$\leq \alpha$$. Then$$[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\ldots+[\sqrt{120}]$$is equal to __________... JEE Main 2023 (Online) 11th April Evening Shift The number of points, where the curve$$f(x)=\mathrm{e}^{8 x}-\mathrm{e}^{6 x}-3 \mathrm{e}^{4 x}-\mathrm{e}^{2 x}+1, x \in \mathbb{R}$$cuts$$x$$-ax... JEE Main 2023 (Online) 11th April Morning Shift If$$a$$and$$b$$are the roots of the equation$$x^{2}-7 x-1=0$$, then the value of$$\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}$$is equal t... JEE Main 2023 (Online) 8th April Evening Shift Let m and$$\mathrm{n}$$be the numbers of real roots of the quadratic equations$$x^{2}-12 x+[x]+31=0$$and$$x^{2}-5|x+2|-4=0$$respectively, where... JEE Main 2023 (Online) 30th January Evening Shift If the value of real number a>0 for which x^2-5 a x+1=0 and x^2-a x-5=0 have a common real root is \frac{3}{\sqrt{2 \beta}} then \beta is eq... JEE Main 2023 (Online) 29th January Evening Shift Let$$\alpha_1,\alpha_2,....,\alpha_7$$be the roots of the equation$${x^7} + 3{x^5} - 13{x^3} - 15x = 0$$and$$|{\alpha _1}| \ge |{\alpha _2}| \ge ... JEE Main 2023 (Online) 25th January Evening Shift Let $$\alpha \in\mathbb{R}$$ and let $$\alpha,\beta$$ be the roots of the equation $${x^2} + {60^{{1 \over 4}}}x + a = 0$$. If $${\alpha ^4} + {\beta ... JEE Main 2023 (Online) 24th January Morning Shift Let$$\lambda \in \mathbb{R}$$and let the equation E be$$|x{|^2} - 2|x| + |\lambda - 3| = 0$$. Then the largest element in the set S = {$$x+\lambda... JEE Main 2022 (Online) 29th July Evening Shift Let $$\alpha, \beta(\alpha>\beta)$$ be the roots of the quadratic equation $$x^{2}-x-4=0 .$$ If $$P_{n}=\alpha^{n}-\beta^{n}$$, $$n \in \mathrm{N}$$, ... JEE Main 2022 (Online) 28th July Morning Shift The sum of all real values of $$x$$ for which $$\frac{3 x^{2}-9 x+17}{x^{2}+3 x+10}=\frac{5 x^{2}-7 x+19}{3 x^{2}+5 x+12}$$ is equal to __________.... JEE Main 2022 (Online) 26th July Morning Shift If for some $$\mathrm{p}, \mathrm{q}, \mathrm{r} \in \mathbf{R}$$, not all have same sign, one of the roots of the equation $$\left(\mathrm{p}^{2}+\ma... JEE Main 2022 (Online) 26th July Morning Shift The number of distinct real roots of the equation$$x^{5}\left(x^{3}-x^{2}-x+1\right)+x\left(3 x^{3}-4 x^{2}-2 x+4\right)-1=0$$is ______________.... JEE Main 2022 (Online) 28th June Morning Shift The number of real solutions of the equation$${e^{4x}} + 4{e^{3x}} - 58{e^{2x}} + 4{e^x} + 1 = 0$$is ___________. JEE Main 2022 (Online) 27th June Evening Shift Let$$\alpha$$,$$\beta$$be the roots of the equation$${x^2} - 4\lambda x + 5 = 0$$and$$\alpha$$,$$\gamma$$be the roots of the equation$${x^2} ... JEE Main 2022 (Online) 27th June Morning Shift If the sum of all the roots of the equation $${e^{2x}} - 11{e^x} - 45{e^{ - x}} + {{81} \over 2} = 0$$ is $${\log _e}p$$, then p is equal to _________... JEE Main 2022 (Online) 26th June Evening Shift Let p and q be two real numbers such that p + q = 3 and p4 + q4 = 369. Then $${\left( {{1 \over p} + {1 \over q}} \right)^{ - 2}}$$ is equal to ______... JEE Main 2022 (Online) 26th June Morning Shift The sum of the cubes of all the roots of the equation $${x^4} - 3{x^3} - 2{x^2} + 3x + 1 = 0$$ is _________. JEE Main 2021 (Online) 1st September Evening Shift Let f(x) be a polynomial of degree 3 such that $$f(k) = - {2 \over k}$$ for k = 2, 3, 4, 5. Then the value of 52 $$-$$ 10f(10) is equal to : JEE Main 2021 (Online) 26th August Evening Shift 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 $$\gamm... JEE Main 2021 (Online) 26th August Morning Shift The sum of all integral values of k (k$$\ne$$0) for which the equation$${2 \over {x - 1}} - {1 \over {x - 2}} = {2 \over k}$$in x has no real root... JEE Main 2021 (Online) 27th July Evening Shift The number of real roots of the equation e4x$$-$$e3x$$-$$4e2x$$-$$ex + 1 = 0 is equal to ______________.... JEE Main 2021 (Online) 25th July Evening Shift If a + b + c = 1, ab + bc + ca = 2 and abc = 3, then the value of a4 + b4 + c4 is equal to ______________. JEE Main 2021 (Online) 25th July Morning Shift If$$\alpha$$,$$\beta$$are roots of the equation$${x^2} + 5(\sqrt 2 )x + 10 = 0$$,$$\alpha$$>$$\beta$$and$${P_n} = {\alpha ^n} - {\beta ^n}... JEE Main 2021 (Online) 26th February Evening Shift Let $$\alpha$$ and $$\beta$$ be two real numbers such that $$\alpha$$ + $$\beta$$ = 1 and $$\alpha$$$$\beta$$ = $$-$$1. Let pn = ($$\alpha$$)n + ($$\b... JEE Main 2021 (Online) 26th February Morning Shift The sum of 162th power of the roots of the equation x3$$-$$2x2 + 2x$$-$$1 = 0 is ________. JEE Main 2021 (Online) 24th February Evening Shift The number of the real roots of the equation$${(x + 1)^2} + |x - 5| = {{27} \over 4}$$is ________. JEE Main 2020 (Online) 8th January Morning Slot The least positive value of 'a' for which the equation 2x2 + (a – 10)x +$${{33} \over 2}$\$ = 2a has real roots is
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