JEE Main
Mathematics
Previous Years Questions

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...
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...
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 :
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 :
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 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
The equation $${x^2} - 4x + [x] + 3 = x[x]$$, where $$[x]$$ denotes the greatest integer function, has :
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 :...
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...
If $$\alpha, \beta$$ are the roots of the equation $$x^{2}-\left(5+3^{\sqrt{\log _{3} 5}}-5^{\sqrt{\log _{5} 3}}\right)x+3\left(3^{\left(\log _{3} 5\... The minimum value of the sum of the squares of the roots of$$x^{2}+(3-a) x+1=2 a$$is: Let S be the set of all integral values of$$\alpha$$for which the sum of squares of two real roots of the quadratic equation$$3{x^2} + (\alpha - 6...
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 :...
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...
The number of real solutions of $${x^7} + 5{x^3} + 3x + 1 = 0$$ is equal to ____________.
The number of distinct real roots of x4 $$-$$ 4x + 1 = 0 is :
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...
The sum of all the real roots of the equation $$({e^{2x}} - 4)(6{e^{2x}} - 5{e^x} + 1) = 0$$ is
The number of distinct real roots of the equation x7 $$-$$ 7x $$-$$ 2 = 0 is
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($$\alp... The numbers of pairs (a, b) of real numbers, such that whenever$$\alpha$$is a root of the equation x2 + ax + b = 0,$$\alpha$$2$$-$$2 is also a ro... The sum of the roots of the equation$$x + 1 - 2{\log _2}(3 + {2^x}) + 2{\log _4}(10 - {2^{ - x}}) = 0$$, is : cosec18$$^\circ$$is a root of the equation : The set of all values of K >$$-$$1, for which the equation$${(3{x^2} + 4x + 3)^2} - (k + 1)(3{x^2} + 4x + 3)(3{x^2} + 4x + 2) + k{(3{x^2} + 4x + ...
Let $$\alpha = \mathop {\max }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\}$$ and $$\beta = \mathop {\min }\limits_{x \in R} \{ {8^{2\sin 3x... Let$$\alpha$$,$$\beta$$be two roots of the equation x2 + (20)1/4x + (5)1/2 = 0. Then$$\alpha$$8 +$$\beta$$8 is equal to ... The number of real solutions of the equation, x2$$-$$|x|$$-$$12 = 0 is : The number of real roots of the equation$${e^{6x}} - {e^{4x}} - 2{e^{3x}} - 12{e^{2x}} + {e^x} + 1 = 0$$is : If$$\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...
Let $$\alpha$$, $$\beta$$, $$\gamma$$ be the real roots of the equation, x3 + ax2 + bx + c = 0, (a, b, c $$\in$$ R and a, b $$\ne$$ 0). If the system ...
The value of $$3 + {1 \over {4 + {1 \over {3 + {1 \over {4 + {1 \over {3 + ....\infty }}}}}}}}$$ is equal to
The value of $$4 + {1 \over {5 + {1 \over {4 + {1 \over {5 + {1 \over {4 + ......\infty }}}}}}}}$$ is :
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 $${{{a... The integer 'k', for which the inequality x2$$-$$2(3k$$-$$1)x + 8k2$$-$$7 > 0 is valid for every x in R, is : Let p and q be two positive numbers such that p + q = 2 and p4+q4 = 272. Then p and q are roots of the equation : If$$\alpha $$and$$\beta $$are the roots of the equation 2x(2x + 1) = 1, then$$\beta $$is equal to : If$$\alpha $$and$$\beta $$be two roots of the equation x2 – 64x + 256 = 0. Then the value of$${\left( {{{{\alpha ^3}} \over {{\beta ^5}}}} \right...
If $$\alpha$$ and $$\beta$$ are the roots of the equation, 7x2 – 3x – 2 = 0, then the value of $${\alpha \over {1 - {\alpha ^2}}} + {\beta \over {... The product of the roots of the equation 9x2 - 18|x| + 5 = 0 is : Let$$\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 ...
Let [t] denote the greatest integer $$\le$$ t. Then the equation in x, [x]2 + 2[x+2] - 7 = 0 has :
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 $$\alpha, \beta, \gam... The set of all real values of$$\lambda $$for which the quadratic equations, ($$\lambda $$2 + 1)x2 – 4$$\lambda $$x + 2 = 0 always have exactly one... If$$\alpha $$and$$\beta $$are the roots of the equation x2 + px + 2 = 0 and$${1 \over \alpha }$$and$${1 \over \beta }$$are the roots of the e... Let f(x) be a quadratic polynomial such that f(–1) + f(2) = 0. If one of the roots of f(x) = 0 is 3, then its other root lies in : Let$$\alpha $$and$$\beta $$be the roots of the equation 5x2 + 6x – 2 = 0. If Sn =$$\alpha $$n +$$\beta $$n, n = 1, 2, 3...., then :... Let a, b$$ \in $$R, a$$ \ne $$0 be such that the equation, ax2 – 2bx + 5 = 0 has a repeated root$$\alpha $$, which is also a root of the equation... The number of real roots of the equation, e4x + e3x – 4e2x + ex + 1 = 0 is : Let S be the set of all real roots of the equation, 3x(3x – 1) + 2 = |3x – 1| + |3x – 2|. Then S : Let$$\alpha = {{ - 1 + i\sqrt 3 } \over 2}$$. If$$a = \left( {1 + \alpha } \right)\sum\limits_{k = 0}^{100} {{\alpha ^{2k}}} $$and$$b = \sum\limi...
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 $... 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 ... 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 +... The number of real roots of the equation 5 + |2x – 1| = 2x (2x – 2) is 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... If $$\alpha$$ and $$\beta$$ are the roots of the quadratic equation, x2 + x sin $$\theta$$ - 2 sin $$\theta$$ = 0, $$\theta \in \left( {0,{\pi ... If m is chosen in the quadratic equation (m2 + 1) x2 – 3x + (m2 + 1)2 = 0 such that the sum of its roots is greatest, then the absolute difference o... Let p, q$$ \in $$R. If 2 -$$\sqrt 3$$is a root of the quadratic equation, x2 + px + q = 0, then : The number of integral values of m for which the equation (1 + m2 )x2 – 2(1 + 3m)x + (1 + 8m) = 0 has no real root is : The sum of the solutions of the equation$$\left| {\sqrt x - 2} \right| + \sqrt x \left( {\sqrt x - 4} \right) + 2 = 0$$(x > 0) is equal to:... The number of integral values of m for which the quadratic expression, (1 + 2m)x2 – 2(1 + 3m)x + 4(1 + m), x$$ \in $$R, is always positive, is :... If$$\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 + ... Let $$\alpha$$ and $$\beta$$ be the roots of the quadratic equation x2 sin $$\theta$$ – x(sin $$\theta$$ cos $$\theta$$ + 1) + cos $$\theta$$ =... If one real root of the quadratic equation 81x2 + kx + 256 = 0 is cube of the other root, then a value of k is 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... 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... 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... The number of all possible positive integral values of $$\alpha$$ for which the roots of the quadratic equation, 6x2 $$-$$ 11x + $$\alpha$$ =... Let p, q and r be real numbers (p $$\ne$$ q, r $$\ne$$ 0), such that the roots of the equation $${1 \over {x + p}} + {1 \over {x + q}} = {1 \over... If an angle A of a$$\Delta $$ABC satiesfies 5 cosA + 3 = 0, then the roots of the quadratic equation, 9x2 + 27x + 20 = 0 are : Let S = {$$x \in $$R :$$x \ge $$0 and$$2\left| {\sqrt x - 3} \right| + \sqrt x \left( {\sqrt x - 6} \right) + 6 = 0$$}. Then S... If f(x) is a quadratic expression such that f (1) + f (2) = 0, and$$-$$1 is a root of f (x) = 0, then the other root of f(x) = 0 is : If$$\lambda  \in $$R is such that the sum of the cubes of the roots of the equation, x2 + (2$$-\lambda $$) x + (10$$-\lambda $$) =... If tanA and tanB are the roots of the quadratic equation, 3x2$$-$$10x$$-$$25 = 0, then the value of 3 sin2(A + B)$$-$$10 sin(A + B).cos(A + B) ... The sum of all the real values of x satisfying the equation 2(x$$-$$1)(x2 + 5x$$-$$50) = 1 is : Let p(x) be a quadratic polynomial such that p(0)=1. If p(x) leaves remainder 4 when divided by x$$-$$1 and it leaves remainder 6 when divided by x +... If for a positive integer n, the quadratic equation$$x\left( {x + 1} \right) + \left( {x + 1} \right)\left( {x + 2} \right) + .... + \left( {x + ... If x is a solution of the equation, $$\sqrt {2x + 1}$$ $$- \sqrt {2x - 1} = 1,$$ $$\,\,\left( {x \ge {1 \over 2}} \right),$$ then $$\sqrt {4{x^2} -... If the equations x2 + bx−1 = 0 and x2 + x + b = 0 have a common root different from −1, then$$\left| b \right|$$is equal to : The sum of all real values of$$x$$satisfying the equation$${\left( {{x^2} - 5x + 5} \right)^{{x^2} + 4x - 60}}\, = 1$$is : Let$$\alpha $$and$$\beta $$be the roots of equation$${x^2} - 6x - 2 = 0$$. If$${a_n} = {\alpha ^n} - {\beta ^n},$$for$$n \ge 1,$$then the va... If$$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$$] ... Let$$\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... The number of values of $$k$$, for which the system of equations : $$\matrix{ {\left( {k + 1} \right)x + 8y = 4k} \cr {kx + \left( {k + 3} \r... The real number$$k$$for which the equation,$$2{x^3} + 3x + k = 0$$has two distinct real roots in$$\left[ {0,\,1} \right]$$If the equations$${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\,$$is The equation$${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$$has: If$$\alpha $$and$$\beta $$are the roots of the equation$${x^2} - x + 1 = 0,$$then$${\alpha ^{2009}} + {\beta ^{2009}} = $$The quadratic equations$${x^2} - 6x + a = 0$$and$${x^2} - cx + 6 = 0$$have one root in common. The other roots of the first and second equations a... If$$\,\left| {z - {4 \over z}} \right| = 2,$$then the maximum value of$$\,\left| z \right|$$is equal to : If the roots of the equation$$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}$$is : STATEMENT - 1 : For every natural number$$n \ge 2,${1 \over {\sqrt 1 }} + {1 \over {\sqrt 2 }} + ........ + {1 \over {\sqrt n }} > \sqrt n ....
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
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 $$2... All the values of$$m$$for which both roots of the equation$${x^2} - 2mx + {m^2} - 1 = 0$$are greater than$$ - 2$$but less then 4, lie in the int... If$$x$$is real, the maximum value of$${{3{x^2} + 9x + 17} \over {3{x^2} + 9x + 7}}$$is In a triangle$$PQR,\;\;\angle R = {\pi \over 2}.\,\,If\,\,\tan \,\left( {{P \over 2}} \right)$$and$$ \tan \left( {{Q \over 2}} \right)$$are the r... If both the roots of the quadratic equation$${x^2} - 2kx + {k^2} + k - 5 = 0$$are less than 5, then$$k$$lies in the interval The value of$$a$$for which the sum of the squares of the roots of the equation$${x^2} - \left( {a - 2} \right)x - a - 1 = 0$$assume the least valu... If the roots of the equation$${x^2} - bx + c = 0$$be two consecutive integers, then$${b^2} - 4c$$equals Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation If$$\left( {1 - p} \right)$$is a root of quadratic equation$${x^2} + px + \left( {1 - p} \right) = 0$$then its root are If one root of the equation$${x^2} + px + 12 = 0$$is 4, while the equation$${x^2} + px + q = 0$$has equal roots, then the value of$$'q'$$is ... If the sum of the roots of the quadratic equation$$a{x^2} + bx + c = 0$$is equal to the sum of the squares of their reciprocals, then$${a \over c},...
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... The number of real solutions of the equation $${x^2} - 3\left| x \right| + 2 = 0$$ is The real number $$x$$ when added to its inverse gives the minimum value of the sum at $$x$$ equal to If $$\alpha \ne \beta$$ but $${\alpha ^2} = 5\alpha - 3$$ and $${\beta ^2} = 5\beta - 3$$ then the equation having $$\alpha /\beta$$ and $$\beta ... Product of real roots of equation$${t^2}{x^2} + \left| x \right| + 9 = 0$$Difference between the corresponding roots of$${x^2} + ax + b = 0$$and$${x^2} + bx + a = 0$$is same and$$a \ne b,$$then If$$p$$and$$q$$are the roots of the equation$${x^2} + px + q = 0,$$then If$$a,\,b,\,c$$are distinct$$ + ve$$real numbers and$${a^2} + {b^2} + {c^2} = 1$$then$$ab + bc + ca$$is ## Numerical If the value of real number a>0 for which x^2-5 a x+1=0 and x^2-a x-5=0 have a common real root is \frac{3}{\sqrt{2 \beta}} then \beta is eq... Let$$\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 ... 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 $${\alpha ^4} + {\beta ... Let$$S = \left\{ {\alpha :{{\log }_2}({9^{2\alpha - 4}} + 13) - {{\log }_2}\left( {{5 \over 2}.\,{3^{2\alpha - 4}} + 1} \right) = 2} \right\}$$. Th... Let$$\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... Let $$\alpha, \beta(\alpha>\beta)$$ be the roots of the quadratic equation $$x^{2}-x-4=0 .$$ If $$P_{n}=\alpha^{n}-\beta^{n}$$, $$n \in \mathrm{N}$$, ... 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 __________.... If for some $$\mathrm{p}, \mathrm{q}, \mathrm{r} \in \mathbf{R}$$, not all have same sign, one of the roots of the equation $$\left(\mathrm{p}^{2}+\ma... The number of distinct real roots of the equation$$x^{5}\left(x^{3}-x^{2}-x+1\right)+x\left(3 x^{3}-4 x^{2}-2 x+4\right)-1=0$$is ______________.... Let for$$f(x) = {a_0}{x^2} + {a_1}x + {a_2},\,f'(0) = 1$$and$$f'(1) = 0$$. If a0, a1, a2 are in an arithmatico-geometric progression, whose corresp... The number of real solutions of the equation$${e^{4x}} + 4{e^{3x}} - 58{e^{2x}} + 4{e^x} + 1 = 0$$is ___________. Let$$\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} ... 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 _________... 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 ______... The sum of the cubes of all the roots of the equation $${x^4} - 3{x^3} - 2{x^2} + 3x + 1 = 0$$ is _________. 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 : 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 $$\gamm... The sum of all integral values of k (k$$\ne$$0) for which the equation$${2 \over {x - 1}} - {1 \over {x - 2}} = {2 \over k}$$in x has no real root... The number of real roots of the equation e4x$$-$$e3x$$-$$4e2x$$-$$ex + 1 = 0 is equal to ______________.... If a + b + c = 1, ab + bc + ca = 2 and abc = 3, then the value of a4 + b4 + c4 is equal to ______________. If$$\alpha$$,$$\beta$$are roots of the equation$${x^2} + 5(\sqrt 2 )x + 10 = 0$$,$$\alpha$$>$$\beta$$and$${P_n} = {\alpha ^n} - {\beta ^n}... The number of solutions of the equation $${\log _{(x + 1)}}(2{x^2} + 7x + 5) + {\log _{(2x + 5)}}{(x + 1)^2} - 4 = 0$$, x > 0, is Let $$\alpha$$ and $$\beta$$ be two real numbers such that $$\alpha$$ + $$\beta$$ = 1 and $$\alpha$$$$\beta$$ = $$-$$1. Let pn = ($$\alpha$$)n + ($$\b... The number of solutions of the equation log4(x$$-$$1) = log2(x$$-$$3) is _________. The sum of 162th power of the roots of the equation x3$$-$$2x2 + 2x$$-$$1 = 0 is ________. The number of the real roots of the equation$${(x + 1)^2} + |x - 5| = {{27} \over 4}$$is ________. The least positive value of 'a' for which the equation 2x2 + (a – 10)x +$${{33} \over 2}$\$ = 2a has real roots is
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