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## Numerical

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Let $$\alpha, \beta(\alpha>\beta)$$ be the roots of the quadratic equation $$x^{2}-x-4=0 .$$ If $$P_{n}=\alpha^{n}-\beta... JEE Main 2022 (Online) 29th July Evening Shift The sum of all real values of$$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...
JEE Main 2022 (Online) 28th July Morning Shift
The number of distinct real roots of the equation $$x^{5}\left(x^{3}-x^{2}-x+1\right)+x\left(3 x^{3}-4 x^{2}-2 x+4\right... JEE Main 2022 (Online) 26th July Morning Shift If for some$$\mathrm{p}, \mathrm{q}, \mathrm{r} \in \mathbf{R}$$, not all have same sign, one of the roots of the equat... JEE Main 2022 (Online) 26th July Morning Shift Let for$$f(x) = {a_0}{x^2} + {a_1}x + {a_2},\,f'(0) = 1$$and$$f'(1) = 0$$. If a0, a1, a2 are in an arithmatico-geomet... JEE Main 2022 (Online) 30th June Morning Shift Let p and q be two real numbers such that p + q = 3 and p4 + q4 = 369. Then$${\left( {{1 \over p} + {1 \over q}} \right...
JEE Main 2022 (Online) 26th June Evening 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 2022 (Online) 26th 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 $${\log _e}p$$...
JEE Main 2022 (Online) 27th June Morning Shift
Let $$\alpha$$, $$\beta$$ be the roots of the equation $${x^2} - 4\lambda x + 5 = 0$$ and $$\alpha$$, $$\gamma$$ be the ...
JEE Main 2022 (Online) 27th June Evening 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) 28th June Morning 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 $$-$$ ...
JEE Main 2021 (Online) 1st September Evening Shift
Let $$\lambda$$ $$\ne$$ 0 be in R. If $$\alpha$$ and $$\beta$$ are the roots of the equation x2 $$-$$ x + 2$$\lambda$$ =...
JEE Main 2021 (Online) 26th August Evening Shift
The sum of all integral values of k (k $$\ne$$ 0) for which the equation $${2 \over {x - 1}} - {1 \over {x - 2}} = {2 \o... JEE Main 2021 (Online) 26th August Morning Shift If a + b + c = 1, ab + bc + ca = 2 and abc = 3, then the value of a4 + b4 + c4 is equal to ______________.... JEE Main 2021 (Online) 25th July Evening Shift The number of real roots of the equation e4x$$-$$e3x$$-$$4e2x$$-$$ex + 1 = 0 is equal to ______________.... JEE Main 2021 (Online) 27th July Evening Shift If$$\alpha$$,$$\beta$$are roots of the equation$${x^2} + 5(\sqrt 2 )x + 10 = 0$$,$$\alpha$$&gt;$$\beta$$and$${P...
JEE Main 2021 (Online) 25th July Morning Shift
The number of solutions of the equation $${\log _{(x + 1)}}(2{x^2} + 7x + 5) + {\log _{(2x + 5)}}{(x + 1)^2} - 4 = 0$$, ...
JEE Main 2021 (Online) 20th July Evening Shift
Let $$\alpha$$ and $$\beta$$ be two real numbers such that $$\alpha$$ + $$\beta$$ = 1 and $$\alpha$$$$\beta$$ = $$-$$1. ...
JEE Main 2021 (Online) 26th February Evening Shift
The sum of 162th power of the roots of the equation x3 $$-$$ 2x2 + 2x $$-$$ 1 = 0 is ________....
JEE Main 2021 (Online) 26th February Morning Shift
The number of solutions of the equation log4(x $$-$$ 1) = log2(x $$-$$ 3) is _________.
JEE Main 2021 (Online) 26th February Morning Shift
The number of the real roots of the equation $${(x + 1)^2} + |x - 5| = {{27} \over 4}$$ is ________.
JEE Main 2021 (Online) 24th February Evening Slot
The least positive value of 'a' for which the equation 2x2 + (a – 10)x + $${{33} \over 2}$$ = 2a has real roots is...
JEE Main 2020 (Online) 8th January Morning Slot

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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 val...
JEE Main 2022 (Online) 29th July Morning Shift
Let $$\alpha$$, $$\beta$$ be the roots of the equation $$x^{2}-\sqrt{2} x+\sqrt{6}=0$$ and $$\frac{1}{\alpha^{2}}+1, \fr... JEE Main 2022 (Online) 28th July Evening Shift If$$\alpha, \beta$$are the roots of the equation$$ x^{2}-\left(5+3^{\sqrt{\log _{3} 5}}-5^{\sqrt{\log _{5} 3}}\right)...
JEE Main 2022 (Online) 27th 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) 26th July Evening 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 eq...
JEE Main 2022 (Online) 30th June Morning Shift
If the sum of the squares of the reciprocals of the roots $$\alpha$$ and $$\beta$$ of the equation 3x2 + $$\lambda$$x $$... JEE Main 2022 (Online) 24th June Morning Shift The number of distinct real roots of the equation x7$$-$$7x$$-$$2 = 0 is JEE Main 2022 (Online) 24th June Evening Shift The sum of all the real roots of the equation$$({e^{2x}} - 4)(6{e^{2x}} - 5{e^x} + 1) = 0$$is JEE Main 2022 (Online) 24th June Evening Shift Let a, b$$\in$$R be such that the equation$$a{x^2} - 2bx + 15 = 0$$has a repeated root$$\alpha$$. If$$\alpha$$and... JEE Main 2022 (Online) 25th June Evening Shift The number of distinct real roots of x4$$-$$4x + 1 = 0 is : JEE Main 2022 (Online) 27th June Morning Shift The number of real solutions of$${x^7} + 5{x^3} + 3x + 1 = 0$$is equal to ____________. JEE Main 2022 (Online) 28th June Morning Shift Let f(x) be a quadratic polynomial such that f($$-$$2) + f(3) = 0. If one of the roots of f(x) = 0 is$$-$$1, then the s... JEE Main 2022 (Online) 28th June Evening Shift Let$$\alpha$$be a root of the equation 1 + x2 + x4 = 0. Then, the value of$$\alpha$$1011 +$$\alpha$$2022$$-\al...
JEE Main 2022 (Online) 29th June 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, $$... JEE Main 2021 (Online) 1st September Evening Shift The sum of the roots of the equation$$x + 1 - 2{\log _2}(3 + {2^x}) + 2{\log _4}(10 - {2^{ - x}}) = 0$$, is : JEE Main 2021 (Online) 31st August Evening Shift cosec18$$^\circ$$is a root of the equation : JEE Main 2021 (Online) 31st August Morning Shift The set of all values of K &gt;$$-$$1, for which the equation$${(3{x^2} + 4x + 3)^2} - (k + 1)(3{x^2} + 4x + 3)(3{x^2}...
JEE Main 2021 (Online) 27th August Evening Shift
Let $$\alpha$$, $$\beta$$ be two roots of the equation x2 + (20)1/4x + (5)1/2 = 0. Then $$\alpha$$8 + $$\beta$$8 is equa...
JEE Main 2021 (Online) 27th July Morning Shift
The number of real solutions of the equation, x2 $$-$$ |x| $$-$$ 12 = 0 is :
JEE Main 2021 (Online) 25th July Evening Shift
Let $$\alpha = \mathop {\max }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\}$$ and $$\beta = \mathop {\min }\li... JEE Main 2021 (Online) 27th July Evening Shift The number of real roots of the equation$${e^{6x}} - {e^{4x}} - 2{e^{3x}} - 12{e^{2x}} + {e^x} + 1 = 0$$is : JEE Main 2021 (Online) 25th July Morning Shift If$$\alpha$$and$$\beta$$are the distinct roots of the equation$${x^2} + {(3)^{1/4}}x + {3^{1/2}} = 0$$, then the va... JEE Main 2021 (Online) 20th July Morning Shift The value of$$3 + {1 \over {4 + {1 \over {3 + {1 \over {4 + {1 \over {3 + ....\infty }}}}}}}}$$is equal to JEE Main 2021 (Online) 18th March Morning Shift Let$$\alpha$$,$$\beta$$,$$\gamma$$be the real roots of the equation, x3 + ax2 + bx + c = 0, (a, b, c$$\in$$R and a... JEE Main 2021 (Online) 18th March Morning Shift The value of$$4 + {1 \over {5 + {1 \over {4 + {1 \over {5 + {1 \over {4 + ......\infty }}}}}}}}$$is : JEE Main 2021 (Online) 17th March Morning Shift Let$$\alpha$$and$$\beta$$be the roots of x2$$-$$6x$$-$$2 = 0. If an =$$\alpha$$n$$-\beta$$n for n$$ \ge ...
JEE Main 2021 (Online) 25th February Evening Shift
The integer 'k', for which the inequality x2 $$-$$ 2(3k $$-$$ 1)x + 8k2 $$-$$ 7 &gt; 0 is valid for every x in R, is :...
JEE Main 2021 (Online) 25th February Morning Slot
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 2021 (Online) 24th February Morning 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 Evening Slot
If $$\alpha$$ and $$\beta$$ be two roots of the equation x2 – 64x + 256 = 0. Then the value of $${\left( {{{{\alpha ^3... JEE Main 2020 (Online) 6th September Morning Slot If$$\alpha $$and$$\beta $$are the roots of the equation, 7x2 – 3x – 2 = 0, then the value of$${\alpha \over {1 - {...
JEE Main 2020 (Online) 5th September Evening Slot
The product of the roots of the equation 9x2 - 18|x| + 5 = 0 is :
JEE Main 2020 (Online) 5th September Morning Slot
Let $$\lambda \ne 0$$ be in R. If $$\alpha$$ and $$\beta$$ are the roots of the equation, x2 - x + 2$$\lambda$$ = 0...
JEE Main 2020 (Online) 4th September Evening Slot
Let $$\alpha$$ and $$\beta$$ be the roots of x2 - 3x + p=0 and $$\gamma$$ and $$\delta$$ be the roots of x2 - 6x + q...
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
The set of all real values of $$\lambda$$ for which the quadratic equations, ($$\lambda$$2 + 1)x2 – 4$$\lambda$$x +...
JEE Main 2020 (Online) 3rd September Evening Slot
If $$\alpha$$ and $$\beta$$ are the roots of the equation x2 + px + 2 = 0 and $${1 \over \alpha }$$ and $${1 \over \b... JEE Main 2020 (Online) 3rd September Morning Slot Let f(x) be a quadratic polynomial such that f(–1) + f(2) = 0. If one of the roots of f(x) = 0 is 3, then its other root... JEE Main 2020 (Online) 2nd September Evening Slot Let$$\alpha $$and$$\beta $$be the roots of the equation 5x2 + 6x – 2 = 0. If Sn =$$\alpha $$n +$$\beta $$n, n = 1... JEE Main 2020 (Online) 2nd September Morning Slot Let a, b$$ \in $$R, a$$ \ne $$0 be such that the equation, ax2 – 2bx + 5 = 0 has a repeated root$$\alpha $$, which ... JEE Main 2020 (Online) 9th January Evening Slot The number of real roots of the equation, e4x + e3x – 4e2x + ex + 1 = 0 is :... JEE Main 2020 (Online) 9th January Morning Slot Let$$\alpha = {{ - 1 + i\sqrt 3 } \over 2}$$. If$$a = \left( {1 + \alpha } \right)\sum\limits_{k = 0}^{100} {{\alpha ...
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 $$\alpha$$ and $$\beta$$ be the roots of the equation x2 - x - 1 = 0. If pk = $${\left( \alpha \right)^k} + {\l... JEE Main 2020 (Online) 7th January Evening Slot Let$$\alpha $$and$$\beta $$be two real roots of the equation (k + 1)tan2x -$$\sqrt 2 $$.$$\lambda $$tanx = (1 - k... JEE Main 2020 (Online) 7th January Morning Slot If$$\alpha $$,$$\beta $$and$$\gamma $$are three consecutive terms of a non-constant G.P. such that the equations$$...
JEE Main 2019 (Online) 12th April Evening Slot
The number of real roots of the equation 5 + |2x – 1| = 2x (2x – 2) is
JEE Main 2019 (Online) 10th April Evening Slot
If $$\alpha$$ and $$\beta$$ are the roots of the quadratic equation, x2 + x sin $$\theta$$ - 2 sin $$\theta$$ = 0, ...
JEE Main 2019 (Online) 10th April Morning Slot
All the pairs (x, y) that satisfy the inequality $${2^{\sqrt {{{\sin }^2}x - 2\sin x + 5} }}.{1 \over {{4^{{{\sin }^2}y}... JEE Main 2019 (Online) 10th April Morning Slot If m is chosen in the quadratic equation (m2 + 1) x2 – 3x + (m2 + 1)2 = 0 such that the sum of its roots is greatest, ... JEE Main 2019 (Online) 9th April Evening Slot Let p, q$$ \in $$R. If 2 -$$\sqrt 3$$is a root of the quadratic equation, x2 + px + q = 0, then : JEE Main 2019 (Online) 9th April Morning Slot The number of integral values of m for which the equation (1 + m2 )x2 – 2(1 + 3m)x + (1 + 8m) = 0 has no real root is :... JEE Main 2019 (Online) 8th April Evening Slot The sum of the solutions of the equation$$\left| {\sqrt x - 2} \right| + \sqrt x \left( {\sqrt x - 4} \right) + 2 = ...
JEE Main 2019 (Online) 8th April Morning 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...
JEE Main 2019 (Online) 12th January Evening 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 valu...
JEE Main 2019 (Online) 12th January Morning Slot
Let $$\alpha$$ and $$\beta$$ be the roots of the quadratic equation x2 sin $$\theta$$ – x(sin $$\theta$$ cos $$\the... JEE Main 2019 (Online) 11th January Evening Slot If one real root of the quadratic equation 81x2 + kx + 256 = 0 is cube of the other root, then a value of k is ... JEE Main 2019 (Online) 11th January Morning Slot The value of$$\lambda $$such that sum of the squares of the roots of the quadratic equation, x2 + (3 –$$\lambda $$)x ... JEE Main 2019 (Online) 10th January Evening Slot Consider the quadratic equation (c – 5)x2 – 2cx + (c – 4) = 0, c$$ \ne $$5. Let S be the set of all integral values of... JEE Main 2019 (Online) 10th January Morning Slot The number of all possible positive integral values of$$\alpha $$&nbsp; for which the roots of the quadratic equation,... JEE Main 2019 (Online) 9th January Evening Slot If both the roots of the quadratic equation x2$$-$$mx + 4 = 0 are real and distinct and they lie in the interval [1, 5... JEE Main 2019 (Online) 9th January Evening Slot If an angle A of a$$\Delta $$ABC satiesfies 5 cosA + 3 = 0, then the roots of the quadratic equation, 9x2 + 27x + 20 =... JEE Main 2018 (Online) 16th April Morning Slot Let p, q and r be real numbers (p$$ \ne $$q, r$$ \ne $$0), such that the roots of the equation$${1 \over {x + p}} ...
JEE Main 2018 (Online) 16th April Morning 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 o...
JEE Main 2018 (Online) 15th April Evening Slot
If tanA and tanB are the roots of the quadratic equation, 3x2 $$-$$ 10x $$-$$ 25 = 0, then the value of 3 sin2(A + B) $$... JEE Main 2018 (Online) 15th April Morning Slot If$$\lambda  \in $$R is such that the sum of the cubes of the roots of the equation, x2 + (2$$-\lambda $$)... JEE Main 2018 (Online) 15th April Morning Slot Let S = {$$x \in $$R :$$x \ge $$0 and$$2\left| {\sqrt x - 3} \right| + \sqrt x \left( {\sqrt x - 6} \ri...
JEE Main 2018 (Offline)
The sum of all the real values of x satisfying the equation 2(x$$-$$1)(x2 + 5x $$-$$ 50) = 1 is : ...
JEE Main 2017 (Online) 9th 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 r...
JEE Main 2017 (Online) 8th April Morning Slot
If for a positive integer n, the quadratic equation $$x\left( {x + 1} \right) + \left( {x + 1} \right)\left( {x + 2} \ri... JEE Main 2017 (Offline) If x is a solution of the equation,$$\sqrt {2x + 1}  - \sqrt {2x - 1} = 1,\,\,\left( {x \ge {1 \over 2}} \ri...
JEE Main 2016 (Online) 10th 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 equ...
JEE Main 2016 (Online) 9th April Morning Slot
The sum of all real values of $$x$$ satisfying the equation $${\left( {{x^2} - 5x + 5} \right)^{{x^2} + 4x - 60}}\, = 1... JEE Main 2016 (Offline) Let$$\alpha $$and$$\beta $$be the roots of equation$${x^2} - 6x - 2 = 0$$. If$${a_n} = {\alpha ^n} - {\beta ^n},$$... JEE Main 2015 (Offline) Let$$\alpha $$and$$\beta $$be the roots of equation$$p{x^2} + qx + r = 0,p \ne 0.$$If$$p,\,q,\,r$$in A.P. a... JEE Main 2014 (Offline) If$$a \in R$$and the equation$$ - 3{\left( {x - \left[ x \right]} \right)^2} + 2\left( {x - \left[ x \right]} \right)...
JEE Main 2014 (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 ...
JEE Main 2013 (Offline)
The number of values of $$k$$, for which the system of equations : $$\matrix{ {\left( {k + 1} \right)x + 8y = 4k} \... JEE Main 2013 (Offline) The real number$$k$$for which the equation,$$2{x^3} + 3x + k = 0$$has two distinct real roots in$$\left[ {0,\,1} \r...
JEE Main 2013 (Offline)
The equation $${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$$ has:
AIEEE 2012
If $$\alpha$$ and $$\beta$$ are the roots of the equation $${x^2} - x + 1 = 0,$$ then $${\alpha ^{2009}} + {\beta ^{2... AIEEE 2010 If the roots of the equation$$b{x^2} + cx + a = 0$$imaginary, then for all real values of$$x$$, the expression$$3{b^...
AIEEE 2009
If $$\,\left| {z - {4 \over z}} \right| = 2,$$ then the maximum value of $$\,\left| z \right|$$ is equal to :
AIEEE 2009
The quadratic equations $${x^2} - 6x + a = 0$$ and $${x^2} - cx + 6 = 0$$ have one root in common. The other roots of th...
AIEEE 2009
STATEMENT - 1 : For every natural number $$n \ge 2,$$ $${1 \over {\sqrt 1 }} + {1 \over {\sqrt 2 }} + ........ + {1 \o... AIEEE 2008 If the difference between the roots of the equation$${x^2} + ax + 1 = 0$$is less than$$\sqrt 5 ,$$then the set of po... AIEEE 2007 If$$x$$is real, the maximum value of$${{3{x^2} + 9x + 17} \over {3{x^2} + 9x + 7}}$$is AIEEE 2006 All the values of$$m$$for which both roots of the equation$${x^2} - 2mx + {m^2} - 1 = 0$$are greater than$$ - 2$$b... AIEEE 2006 If the roots of the quadratic equation$${x^2} + px + q = 0$$are$$\tan {30^ \circ }$$and$$\tan {15^ \circ }$$, respe... AIEEE 2006 If the roots of the equation$${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 -...
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...
AIEEE 2005
In a triangle $$PQR,\;\;\angle R = {\pi \over 2}.\,\,If\,\,\tan \,\left( {{P \over 2}} \right)$$ and $$\tan \left( {{Q... AIEEE 2005 If one root of the equation$${x^2} + px + 12 = 0$$is 4, while the equation$${x^2} + px + q = 0$$has equal roots, t... AIEEE 2004 If$$\left( {1 - p} \right)$$is a root of quadratic equation$${x^2} + px + \left( {1 - p} \right) = 0$$then its ro... AIEEE 2004 Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation AIEEE 2004 The real number$$x$$when added to its inverse gives the minimum value of the sum at$$x$$equal to AIEEE 2003 The number of real solutions of the equation$${x^2} - 3\left| x \right| + 2 = 0$$is AIEEE 2003 The value of '$$a$$' for which one root of the quadratic equation$$$\left( {{a^2} - 5a + 3} \right){x^2} + \left( {3a ... 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 re... AIEEE 2003 If $$a,\,b,\,c$$ are distinct $$+ ve$$ real numbers and $${a^2} + {b^2} + {c^2} = 1$$ then $$ab + bc + ca$$ is AIEEE 2002 If $$p$$ and $$q$$ are the roots of the equation $${x^2} + px + q = 0,$$ then AIEEE 2002 Difference between the corresponding roots of $${x^2} + ax + b = 0$$ and $${x^2} + bx + a = 0$$ is same and $$a \ne b,... AIEEE 2002 Product of real roots of equation$${t^2}{x^2} + \left| x \right| + 9 = 0$$AIEEE 2002 If$$\alpha \ne \beta $$but$${\alpha ^2} = 5\alpha - 3$$and$${\beta ^2} = 5\beta - 3$$then the equation having$...
AIEEE 2002

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