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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