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## Numerical

JEE Advanced 2022 Paper 2 Online
The product of all positive real values of $x$ satisfying the equation $$x^{\left(16\left(\log _{5} x\right)^{3}-68 \log _{5} x\right)}=5^{-16}$$ ...
JEE Advanced 2021 Paper 1 Online
For x $$\in$$ R, the number of real roots of the equation $$3{x^2} - 4\left| {{x^2} - 1} \right| + x - 1 = 0$$ is ________.
JEE Advanced 2018 Paper 1 Offline
Let a, b, c three non-zero real numbers such that the equation $$\sqrt 3 a\cos x + 2b\sin x = c,x \in \left[ { - {\pi \over 2},{\pi \over 2}} \right... IIT-JEE 2012 Paper 1 Offline The value of$$6 + {\log _{3/2}}\left( {{1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sq...
IIT-JEE 2011 Paper 1 Offline
The minimum value of the sum of real numbers $${a^{ - 5}},\,{a^{ - 4}},\,3{a^{ - 3}},\,1,\,{a^8}$$ and $${a^{10}}$$ where $$a > 0$$ is
IIT-JEE 2011 Paper 2 Offline
The number of distinct real roots of $${x^4} - 4{x^3} + 12{x^2} + x - 1 = 0$$
IIT-JEE 2009 Paper 2 Offline
The smallest value of $$k$$, for which both the roots of the equation $${x^2} - 8kx + 16\left( {{k^2} - k + 1} \right) = 0$$$are real, distinct and... ## MCQ (Single Correct Answer) JEE Advanced 2020 Paper 1 Offline Suppose a, b denote the distinct real roots of the quadratic polynomial x2 + 20x $$-$$ 2020 and suppose c, d denote the distinct complex roots of the ... JEE Advanced 2017 Paper 2 Offline a12 = ? JEE Advanced 2017 Paper 2 Offline If a4 = 28, then p + 2q = JEE Advanced 2016 Paper 1 Offline Let $$- {\pi \over 6} < \theta < - {\pi \over {12}}.$$ Suppose $${\alpha _1}$$ and $${\beta_1}$$ are the roots of the equation $${x^2} - 2... JEE Advanced 2014 Paper 2 Offline The quadratic equation$$p(x) = 0$$with real coefficients has purely imaginary roots. Then the equation$$p(p(x))=0$$has IIT-JEE 2012 Paper 2 Offline Let$$\alpha$$(a) and$$\beta$$(a) be the roots of the equation$$(\root 3 \of {1 + a} - 1){x^2} + (\sqrt {1 + a} - 1)x + (\root 6 \of {1 + a} - 1)... IIT-JEE 2011 Paper 1 Offline Let $$\alpha$$ and $$\beta$$ be the roots of $${x^2} - 6x - 2 = 0,$$ with $$\alpha > \beta .$$ If $${a_n} = {\alpha ^n} - {\beta ^n}$$ for $$... IIT-JEE 2011 Paper 1 Offline Let$$\left( {{x_0},{y_0}} \right)$$be the solution of the following equations$$\matrix{ {{{\left( {2x} \right)}^{\ell n2}}\, = {{\left( {3y} \... IIT-JEE 2011 Paper 2 Offline A value of $$b$$ for which the equations $$\matrix{ {{x^2} + bx - 1 = 0} \cr {{x^2} + x + b = 0} \cr }$$$ have one root in common is ...
IIT-JEE 2010 Paper 1 Offline
Let $$p$$ and $$q$$ be real numbers such that $$p \ne 0,\,{p^3} \ne q$$ and $${p^3} \ne - q.$$ If $${p^3} \ne - q.$$ and $$\,\beta$$ are nonzero c...
IIT-JEE 2008 Paper 2 Offline
Let $$a,\,b,c$$, $$p,q$$ be real numbers. Suppose $$\alpha ,\,\beta$$ are the roots of the equation $${x^2} + 2px + q = 0$$ and $$\alpha ,{1 \over \b... IIT-JEE 2007 Let$$\alpha ,\,\beta $$be the roots of the equation$${x^2} - px + r = 0$$and$${\alpha \over 2},\,2\beta $$be the roots of the equation$${x^2} ...
IIT-JEE 2006
Let $$a,\,b,\,c$$ be the sides of triangle where $$a \ne b \ne c$$ and $$\lambda \in R$$. If the roots of the equation $${x^2} + 2\left( {a + b + c}... IIT-JEE 2004 Screening If one root is square of the other root of the equation$${x^2} + px + q = 0$$, then the realation between$$p$$and$$q$$is IIT-JEE 2004 Screening For all$$'x',{x^2} + 2ax + 10 - 3a > 0,$$then the interval in which '$$a$$' lies is IIT-JEE 2003 Screening If$$\,\alpha \in \left( {0,{\pi \over 2}} \right)\,\,then\,\,\sqrt {{x^2} + x} + {{{{\tan }^2}\alpha } \over {\sqrt {{x^2} + x} }}$$is always gre... IIT-JEE 2002 Screening If$${a_1},{a_2}.......,{a_n}$$are positive real numbers whose product is a fixed number c, then the minimum value of$${a_1} + {a_2} + ..... + {a_{n...
IIT-JEE 2002 Screening
The set of all real numbers x for which $${x^2} - \left| {x + 2} \right| + x > 0$$, is
IIT-JEE 2000 Screening
If b > a, then the equation (x - a) (x - b) - 1 = 0 has
IIT-JEE 2000 Screening
If a, b, c, d are positive real numbers such that a + b + c + d = 2, then M = (a + b) (c + d) satisfies the relation
IIT-JEE 2000 Screening
For the equation $$3{x^2} + px + 3 = 0$$. p > 0, if one of the root is square of the other, then p is equal to
IIT-JEE 2000 Screening
If $$\alpha \,and\,\beta$$ $$(\alpha \, < \,\beta )$$ are the roots of the equation $${x^2} + bx + c = 0\,$$, where $$c < 0 < b$$, then
IIT-JEE 1999
If the roots of the equation $${x^2} - 2ax + {a^2} + a - 3 = 0$$ are real and less than 3, then
IIT-JEE 1998
Number of divisor of the form 4$$n$$$$+ 2\left( {n \ge 0} \right)$$ of the integer 240 is
IIT-JEE 1994
The number of points of intersection of two curves y = 2 sin x and y $$= 5{x^2} + 2x + 3$$ is
IIT-JEE 1994
If p, q, r are + ve and are on A.P., the roots of quadratic equation $$p{x^2} + qx + r = 0$$ are all real for
IIT-JEE 1994
Let $$p,q \in \left\{ {1,2,3,4} \right\}\,$$. The number of equations of the form $$p{x^2} + qx + 1 = 0$$ having real roots is
IIT-JEE 1992
Let $$\alpha \,,\,\beta$$ be the roots of the equation (x - a) (x - b) = c, $$c \ne 0$$. Then the roots of the equation $$(x - \alpha \,)\,(x - \beta... IIT-JEE 1991 The product of$$n$$positive numbers is unity. Then their sum is IIT-JEE 1990 The number of solutions of the equation sin$${(e)^x} = {5^x} + {5^{ - x}}$$is IIT-JEE 1987 If$$a,\,b,\,c,\,d$$and p are distinct real numbers such that$$$\left( {{a^2} + {b^2} + {c^2}} \right){p^2} - 2\left( {ab + bc + cd} \right)p + \le... IIT-JEE 1986 If $$a,\,b$$ and $$c$$ are distinct positive numbers, then the expression $$\left( {b + c - a} \right)\left( {c + a - b} \right)\left( {a + b - c} \r... IIT-JEE 1985 If$${\log _{0.3}}\,(x\, - \,1) < {\log _{0.09}}(x - 1)$$, then x lies in the interval- IIT-JEE 1984 The equation$$x - {2 \over {x - 1}} = 1 - {2 \over {x - 1}}$$has IIT-JEE 1984 If$$\,{a^2} + {b^2} + {c^2} = 1$$, then ab + bc + ca lies in the interval IIT-JEE 1982 The number of real solutions of the equation$${\left| x \right|^2} - 3\left| x \right| + 2 = 0$$is IIT-JEE 1982 Two towns A and B are 60 km apart. A school is to be built to serve 150 students in town A and 50 students in town B. If the total distance to be trav... IIT-JEE 1982 If p, q, r are any real numbers, then IIT-JEE 1982 The largest interval for which$${x^{12}} - {x^9} + {x^4} - x + 1 > 0$$is IIT-JEE 1980 Both the roots of the equation (x - b) (x - c) + (x - a) (x - c) + (x - a) (x - b) = 0 are always IIT-JEE 1980 The least value of the expression$$2\,\,{\log _{10}}\,x\, - \,{\log _x}(0.01)$$for x > 1, is IIT-JEE 1980 If$$\,({x^2} + px + 1)\,$$is a factor of$$(a{x^3} + bx + c)$$, then IIT-JEE 1979 The equation x + 2y + 2z = 1 and 2x + 4y + 4z = 9 have IIT-JEE 1979 If x, y and z are real and different and$$\,u = {x^2} + 4{y^2} + 9{z^2} - 6yz - 3zx - 2xy$$, then u is always. IIT-JEE 1979 Let a > 0, b > 0 and c > 0. Then the roots of the equation$$a{x^2} + bx + c = 0$$IIT-JEE 1979 If$$\ell $$, m, n are real,$$\ell \ne m$$, then the roots by the equation :$$(\ell - m)\,{x^2} - 5\,(\ell + m)\,x - 2\,(\ell - m) = 0$$are... ## MCQ (More than One Correct Answer) JEE Advanced 2019 Paper 1 Offline Let$$\alpha $$and$$\beta $$be the roots of$${x^2} - x - 1 = 0$$, with$$\alpha $$>$$\beta $$. For all positive integers n, define$${a_n} = {{... JEE Advanced 2015 Paper 2 Offline Let $$S$$ be the set of all non-zero real numbers $$\alpha$$ such that the quadratic equation $$\alpha {x^2} - x + \alpha = 0$$ has two distinct rea... JEE Advanced 2013 Paper 2 Offline If $${3^x}\, = \,{4^{x - 1}},$$ then $$x\, =$$ IIT-JEE 1989 The equation $${x^{3/4{{\left( {{{\log }_2}\,\,x} \right)}^2} + {{\log }_2}\,\,x - 5/4}} = \sqrt 2$$ has IIT-JEE 1989 If $$\alpha$$ and $$\beta$$ are the roots of $${x^2}$$+ px + q = 0 and $${\alpha ^4},{\beta ^4}$$ are the roots of $$\,{x^2} - rx + s = 0$$, then th... IIT-JEE 1989 Let a, b, c be real numbers, $$a \ne 0$$. If $$\alpha \,$$ is a root of $${a^2}{x^2} + bx + c = 0$$. $$\beta \,$$ is the root of $${a^2}{x^2} - bx - c... IIT-JEE 1986 If$$S$$is the set of all real$$x$$such that$${{2x - 1} \over {2{x^3} + 3{x^2} + x}}$$is positive, then$$S$$contains IIT-JEE 1984 For real$$x$$, the function$$\,{{\left( {x - a} \right)\left( {x - b} \right)} \over {x - c}}$$will assume all real values provided ## Subjective IIT-JEE 2006 Let$$a$$and$$b$$be the roots of the equation$${x^2} - 10cx - 11d = 0$$and those$${x^2} - 10ax - 11b = 0$$are$$c$$,$$d$$then the value of$$... IIT-JEE 2004 If $$a,\,b,c$$ are positive real numbers. Then prove that $${\left( {a + 1} \right)^7}{\left( {b + 1} \right)^7}{\left( {c + 1} \right)^7} > {7^7... IIT-JEE 2003 If$${x^2} + \left( {a - b} \right)x + \left( {1 - a - b} \right) = 0$$where$$a,\,b\, \in \,R$$then find the values of a for which equation has une... IIT-JEE 2001 Let$$a,\,b,\,c$$be real numbers with$$a \ne 0$$and let$$\alpha ,\,\beta $$be the roots of the equation$$a{x^2} + bx + c = 0$$. Express the root... IIT-JEE 2000 If$$\alpha ,\,\beta $$are the roots of$$a{x^2} + bx + c = 0$$,$$\,\left( {a \ne 0} \right)$$and$$\alpha + \delta ,\,\,\beta + \delta $$are t... IIT-JEE 1997 Let$$S$$be a square of unit area. Consider any quadrilateral which has one vertex on each side of$$S$$. If$$a,\,b,\,c$$and$$d$$denote the lengt... IIT-JEE 1995 Let$$a,\,b,\,c$$be real. If$$a{x^2} + bx + c = 0$$has two real roots$$\alpha $$and$$\beta ,$$where$$\alpha < - 1$$and$$\beta > 1,$...
IIT-JEE 1988
Solve $$\left| {{x^2} + 4x + 3} \right| + 2x + 5 = 0$$
IIT-JEE 1987
Find the set of all $$x$$ for which $${{2x} \over {\left( {2{x^2} + 5x + 2} \right)}}\, > \,{1 \over {\left( {x + 1} \right)}}$$
IIT-JEE 1986
For $$a \le 0,$$ determine all real roots of the equation $${x^2} - 2a\left| {x - a} \right| - 3{a^2} = 0$$$IIT-JEE 1985 Solve for $$x$$ ; $${\left( {5 + 2\sqrt 6 } \right)^{{x^2} - 3}} + {\left( {5 - 2\sqrt 6 } \right)^{{x^2} - 3}} = 10$$ IIT-JEE 1983 If one root of the quadratic equation $$a{x^2} + bx + c = 0$$ is equal to the $$n$$-th power of the other, then show that $${\left( {a{c^n}} \right)... IIT-JEE 1983 Find all real values of$$x$$which satisfy$${x^2} - 3x + 2 > 0$$and$${x^2} - 2x - 4 \le 0$$IIT-JEE 1982 Show that the equation$${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$$has no real solution. IIT-JEE 1982$$mn$$squares of equal size are arranged to from a rectangle of dimension$$m$$by$$n$$, where$$m$$and$$n$$are natural numbers. Two squares will... IIT-JEE 1980 Let$$y = \sqrt {{{\left( {x + 1} \right)\left( {x - 3} \right)} \over {\left( {x - 2} \right)}}} $$Find all the real values of$$x,$$for which$$y... IIT-JEE 1980 Given $${n^4} < {10^n}$$ for a fixed positive integer $$n \ge 2,$$ prove that $${\left( {n + 1} \right)^4} < {10^{n + 1}}.$$ IIT-JEE 1980 For what values of $$m,$$ does the system of equations $$\matrix{ {3x + my = m} \cr {2x - 5y = 20} \cr }$$$ has solution satisfying the...
IIT-JEE 1980
Find the solution set of the system $$\matrix{ {x + 2y + z = 1;} \cr {2x - 3y - w = 2;} \cr {x \ge 0;\,y \ge 0;\,z \ge 0;\,w \ge 0.} \... IIT-JEE 1979 If$$\alpha ,\,\beta $$are the roots of$${x^2} + px + q = 0$$and$$\gamma ,\,\delta $$are the roots of$${x^2} + rx + s = 0,$$evaluate$$\left( {...
IIT-JEE 1978
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