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## Numerical

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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 2021 Paper 1 Online
Let a, b, c three non-zero real numbers such that the equation $$\sqrt 3 a\cos x + 2b\sin x = c,x \in \left[ { - {\pi \... JEE Advanced 2018 Paper 1 Offline The value of$$6 + {\log _{3/2}}\left( {{1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt...
IIT-JEE 2012 Paper 1 Offline
The number of distinct real roots of $${x^4} - 4{x^3} + 12{x^2} + x - 1 = 0$$
IIT-JEE 2011 Paper 2 Offline
The minimum value of the sum of real numbers $${a^{ - 5}},\,{a^{ - 4}},\,3{a^{ - 3}},\,1,\,{a^8}$$ and $${a^{10}}$$ wher...
IIT-JEE 2011 Paper 1 Offline
The smallest value of $$k$$, for which both the roots of the equation $${x^2} - 8kx + 16\left( {{k^2} - k + 1} \right)... IIT-JEE 2009 Let$$\left( {x,\,y,\,z} \right)$$be points with integer coordinates satisfying the system of homogeneous equation:$$...
IIT-JEE 2009

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Suppose a, b denote the distinct real roots of the quadratic polynomial x2 + 20x $$-$$ 2020 and suppose c, d denote the ...
JEE Advanced 2020 Paper 1 Offline
If a4 = 28, then p + 2q =
JEE Advanced 2017 Paper 2 Offline
a12 = ?
JEE Advanced 2017 Paper 2 Offline
Let $$- {\pi \over 6} &lt; \theta &lt; - {\pi \over {12}}.$$ Suppose $${\alpha _1}$$ and $${B_1}$$ are the roots o...
JEE Advanced 2016 Paper 1 Offline
The quadratic equation $$p(x)$$ $$= 0$$ with real coefficients has purely imaginary roots. Then the equation $$p(p(x))... JEE Advanced 2014 Paper 2 Offline Let$$\alpha$$(a) and$$\beta$$(a) be the roots of the equation$$(\root 3 \of {1 + a} - 1){x^2} + (\sqrt {1 + a} - 1)...
IIT-JEE 2012 Paper 2 Offline
A value of $$b$$ for which the equations $$\matrix{ {{x^2} + bx - 1 = 0} \cr {{x^2} + x + b = 0} \cr }$$$... IIT-JEE 2011 Paper 2 Offline Let $$\left( {{x_0},{y_0}} \right)$$ be the solution of the following equations $$\matrix{ {{{\left( {2x} \right)}^... IIT-JEE 2011 Paper 1 Offline Let$$\alpha $$and$$\beta $$be the roots of$${x^2} - 6x - 2 = 0,$$with$$\alpha &gt; \beta .$$If$${a_n} = {\al... IIT-JEE 2011 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 $${p^3} \ne - q.$$ ... IIT-JEE 2010 Paper 1 Offline Let $$a,\,b,c$$, $$p,q$$ be real numbers. Suppose $$\alpha ,\,\beta$$ are the roots of the equation $${x^2} + 2px + q =... IIT-JEE 2008 Let$$\alpha ,\,\beta $$be the roots of the equation$${x^2} - px + r = 0$$and$${\alpha \over 2},\,2\beta $$be the ... IIT-JEE 2007 Let$$a,\,b,\,c$$be the sides of triangle where$$a \ne b \ne c$$and$$\lambda \in R$$. If the roots of the equation... IIT-JEE 2006 For all$$'x',{x^2} + 2ax + 10 - 3a &gt; 0,$$then the interval in which '$$a$$' lies is IIT-JEE 2004 Screening If one root is square of the other root of the equation$${x^2} + px + q = 0$$, then the realation between$$p$$and$$q... IIT-JEE 2004 Screening If $$\,\alpha \in \left( {0,{\pi \over 2}} \right)\,\,then\,\,\sqrt {{x^2} + x} + {{{{\tan }^2}\alpha } \over {\sqrt ... IIT-JEE 2003 Screening The set of all real numbers x for which$${x^2} - \left| {x + 2} \right| + x &gt; 0$$, is IIT-JEE 2002 Screening If$${a_1},{a_2}.......,{a_n}$$are positive real numbers whose product is a fixed number c, then the minimum value of ... IIT-JEE 2002 Screening If$$\alpha \,and\,\beta (\alpha \, &lt; \,\beta )$$are the roots of the equation$${x^2} + bx + c = 0\,$$, where ... IIT-JEE 2000 Screening For the equation$$3{x^2} + px + 3 = 0$$. p &gt; 0, if one of the root is square of the other, then p is equal to IIT-JEE 2000 Screening If a, b, c, d are positive real numbers such that a + b + c + d = 2, then M = (a + b) (c + d) satisfies the relation IIT-JEE 2000 Screening If b &gt; a, then the equation (x - a) (x - b) - 1 = 0 has IIT-JEE 2000 Screening If the roots of the equation$${x^2} - 2ax + {a^2} + a - 3 = 0$$are real and less than 3, then IIT-JEE 1999 Number of divisor of the form 4$$n + 2\left( {n \ge 0} \right)$$of the integer 240 is IIT-JEE 1998 Let$$p,q \in \left\{ {1,2,3,4} \right\}\,$$. The number of equations of the form$$p{x^2} + qx + 1 = 0$$having real ro... IIT-JEE 1994 If p, q, r are + ve and are on A.P., the roots of quadratic equation$$p{x^2} + qx + r = 0$$are all real for IIT-JEE 1994 The number of points of intersection of two curves y = 2 sin x and y$$ = 5{x^2} + 2x + 3$$is IIT-JEE 1994 Let$$\alpha \,,\,\beta $$be the roots of the equation (x - a) (x - b) = c,$$c \ne 0$$. Then the roots of the equation... IIT-JEE 1992 The product of$$n$$positive numbers is unity. Then their sum is IIT-JEE 1991 The number of solutions of the equation sin$${(e)^x} = {5^x} + {5^{ - x}}$$is IIT-JEE 1990 If$$a,\,b,\,c,\,d$$and p are distinct real numbers such that$$$\left( {{a^2} + {b^2} + {c^2}} \right){p^2} - 2\left(...
IIT-JEE 1987
If $$a,\,b$$ and $$c$$ are distinct positive numbers, then the expression $$\left( {b + c - a} \right)\left( {c + a - b... IIT-JEE 1986 If$${\log _{0.3}}\,(x\, - \,1) &lt; {\log _{0.09}}(x - 1)$$, then x lies in the interval- IIT-JEE 1985 If$$\,{a^2} + {b^2} + {c^2} = 1$$, then ab + bc + ca lies in the interval IIT-JEE 1984 The equation$$x - {2 \over {x - 1}} = 1 - {2 \over {x - 1}}$$has IIT-JEE 1984 The largest interval for which$${x^{12}} - {x^9} + {x^4} - x + 1 &gt; 0$$is IIT-JEE 1982 If p, q, r are any real numbers, then IIT-JEE 1982 Two towns A and B are 60 km apart. A school is to be built to serve 150 students in town A and 50 students in town B. If... IIT-JEE 1982 The number of real solutions of the equation$${\left| x \right|^2} - 3\left| x \right| + 2 = 0$$is IIT-JEE 1982 If$$\,({x^2} + px + 1)\,$$is a factor of$$(a{x^3} + bx + c)$$, then IIT-JEE 1980 The least value of the expression$$2\,\,{\log _{10}}\,x\, - \,{\log _x}(0.01)$$for x &gt; 1, is IIT-JEE 1980 Both the roots of the equation (x - b) (x - c) + (x - a) (x - c) + (x - a) (x - b) = 0 are always IIT-JEE 1980 If$$\ell $$, m, n are real,$$\ell \ne m$$, then the roots by the equation :$$(\ell - m)\,{x^2} - 5\,(\ell + m)\,x...
IIT-JEE 1979
Let a &gt; 0, b &gt; 0 and c &gt; 0. Then the roots of the equation $$a{x^2} + bx + c = 0$$
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
The equation x + 2y + 2z = 1 and 2x + 4y + 4z = 9 have
IIT-JEE 1979

## MCQ (More than One Correct Answer)

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Let $$\alpha$$ and $$\beta$$ be the roots of$${x^2} - x - 1 = 0$$, with $$\alpha$$ &gt; $$\beta$$. For all positive ...
JEE Advanced 2019 Paper 1 Offline
Let $$S$$ be the set of all non-zero real numbers $$\alpha$$ such that the quadratic equation $$\alpha {x^2} - x + \alp... JEE Advanced 2015 Paper 2 Offline If$${3^x}\, = \,{4^{x - 1}},$$then$$x\, = $$JEE Advanced 2013 Paper 2 Offline Let a, b, c be real numbers,$$a \ne 0$$. If$$\alpha \,$$is a root of$${a^2}{x^2} + bx + c = 0$$.$$\beta \,$$is the... IIT-JEE 1989 If$$\alpha $$and$$\beta $$are the roots of$${x^2}$$+ px + q = 0 and$${\alpha ^4},{\beta ^4}$$are the roots of$$\...
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 $$S$$ is the set of all real $$x$$ such that $${{2x - 1} \over {2{x^3} + 3{x^2} + x}}$$ is positive, then $$S$$ conta...
IIT-JEE 1986
For real $$x$$, the function $$\,{{\left( {x - a} \right)\left( {x - b} \right)} \over {x - c}}$$ will assume all real v...
IIT-JEE 1984

## Subjective

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Let $$a$$ and $$b$$ be the roots of the equation $${x^2} - 10cx - 11d = 0$$ and those $${x^2} - 10ax - 11b = 0$$ are $$c... IIT-JEE 2006 If$$a,\,b,c$$are positive real numbers. Then prove that$$${\left( {a + 1} \right)^7}{\left( {b + 1} \right)^7}{\left... IIT-JEE 2004 If $${x^2} + \left( {a - b} \right)x + \left( {1 - a - b} \right) = 0$$ where $$a,\,b\, \in \,R$$ then find the values o... IIT-JEE 2003 Let $$a,\,b,\,c$$ be real numbers with $$a \ne 0$$ and let $$\alpha ,\,\beta$$ be the roots of the equation $$a{x^2} + ... IIT-JEE 2001 If$$\alpha ,\,\beta $$are the roots of$$a{x^2} + bx + c = 0$$,$$\,\left( {a \ne 0} \right)$$and$$\alpha + \delta... IIT-JEE 2000 Let $$S$$ be a square of unit area. Consider any quadrilateral which has one vertex on each side of $$S$$. If $$a,\,b,\,... IIT-JEE 1997 Let$$a,\,b,\,c$$be real. If$$a{x^2} + bx + c = 0$$has two real roots$$\alpha $$and$$\beta ,$$where$$\alpha &lt... IIT-JEE 1995 Solve $$\left| {{x^2} + 4x + 3} \right| + 2x + 5 = 0$$ IIT-JEE 1988 Find the set of all $$x$$ for which $${{2x} \over {\left( {2{x^2} + 5x + 2} \right)}}\, &gt; \,{1 \over {\left( {x + 1} ... IIT-JEE 1987 For$$a \le 0,$$determine all real roots of the equation$$${x^2} - 2a\left| {x - a} \right| - 3{a^2} = 0$$IIT-JEE 1986 Solve for$$x$$;$${\left( {5 + 2\sqrt 6 } \right)^{{x^2} - 3}} + {\left( {5 - 2\sqrt 6 } \right)^{{x^2} - 3}} = 10$$... IIT-JEE 1985 Find all real values of$$x$$which satisfy$${x^2} - 3x + 2 &gt; 0$$and$${x^2} - 2x - 4 \le 0$$IIT-JEE 1983 If one root of the quadratic equation$$a{x^2} + bx + c = 0$$is equal to the$$n$$-th power of the other, then show tha... IIT-JEE 1983$$mn$$squares of equal size are arranged to from a rectangle of dimension$$m$$by$$n$$, where$$m$$and$$n$$are nat... IIT-JEE 1982 Show that the equation$${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$$has no real solution. IIT-JEE 1982 Find the solution set of the system$$$\matrix{ {x + 2y + z = 1;} \cr {2x - 3y - w = 2;} \cr {x \ge 0;\,y ... IIT-JEE 1980 For what values of $$m,$$ does the system of equations $$\matrix{ {3x + my = m} \cr {2x - 5y = 20} \cr } ... IIT-JEE 1980 Given$${n^4} &lt; {10^n}$$for a fixed positive integer$$n \ge 2,$$prove that$${\left( {n + 1} \right)^4} &lt; {10^{... IIT-JEE 1980 Let $$y = \sqrt {{{\left( {x + 1} \right)\left( {x - 3} \right)} \over {\left( {x - 2} \right)}}}$$ Find all the real ... IIT-JEE 1980 If $$\alpha ,\,\beta$$ are the roots of $${x^2} + px + q = 0$$ and $$\gamma ,\,\delta$$ are the roots of $${x^2} + rx ... IIT-JEE 1979 Sketch the solution set of the following system of inequalities:$$${x^2} + {y^2} - 2x \ge 0;\,\,3x - y - 12 \le 0;\,\,...
IIT-JEE 1978
Find all integers $$x$$ for which $$\left( {5x - 1} \right) &lt; {\left( {x + 1} \right)^2} &lt; \left( {7x - 3} \right)... IIT-JEE 1978 Show that the square of$$\,{{\sqrt {26 - 15\sqrt 3 } } \over {5\sqrt 2 - \sqrt {38 + 5\sqrt 3 } }}$$is a rational num... IIT-JEE 1978 Solve the following equation for$$x:\,\,2\,{\log _x}a + {\log _{ax}}a + 3\,\,{\log _{{a^2}x}}\,a = 0,a &gt; 0$$IIT-JEE 1978 Solve for$$x:\,\sqrt {x + 1} - \sqrt {x - 1} = 1.$$IIT-JEE 1978 Solve for$$x:{4^x} - {3^{^{x - {1 \over 2}}}}\, = {3^{^{x + {1 \over 2}}}}\, - {2^{2x - 1}}$$IIT-JEE 1978 If$$\left( {m\,,\,n} \right) = {{\left( {1 - {x^m}} \right)\left( {1 - {x^{m - 1}}} \right).......\left( {1 - {x^{m - n...
IIT-JEE 1978

## Fill in the Blanks

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The sum of all the real roots of the equation $${\left| {x - 2} \right|^2} + \left| {x - 2} \right| - 2 = 0$$ is ..........
IIT-JEE 1997
Let n and k be positive such that $$n \ge {{k(k + 1)} \over 2}$$ . The number of solutions $$\,({x_1},\,{x_2},\,.....{x_... IIT-JEE 1996 If$$\,x &lt; 0,\,\,y &lt; 0,\,\,x + y + {x \over y} = {1 \over 2}$$and$$(x + y)\,{x \over y} = - {1 \over 2}$$, then... IIT-JEE 1990 The solution of equation$${\log _7}\,{\log _5}\,\left( {\sqrt {x + 5} + \sqrt x } \right) = 0$$is ...................... IIT-JEE 1986 If the quadratic equations$${x^2} + ax + b = 0$$and$${x^2} + bx + a = 0(a \ne b)$$have a common root, then the... IIT-JEE 1986 If the product of the roots of the equation$$\,{x^2} - 3\,k\,x + 2\,{e^{2lnk}} - 1 = 0\,\,\,\,is\,7$$, then the roots a... IIT-JEE 1984 If$$2 + i\sqrt 3 $$is root of the equation$${x^2} + px + q = 0$$, where p and q are real, then (p, q) = (..........,.... IIT-JEE 1982 The coeffcient of$${x^{99}}$$in the polynomial (x -1) (x - 2)...(x - 100) is .............. IIT-JEE 1982 ## True or False More If x and y are positive real numbers and m, n are any positive integers, then$${{{x^n}\,{y^m}} \over {(1 + {x^{2n}})\,(...
IIT-JEE 1989
If $$P(x) = a{x^2} + bx + c\,\,and\,\,Q(x) = - a{x^2} + dx + c$$, where $$ac \ne \,0$$, then P(x) Q(x) = 0 has at least...
IIT-JEE 1985
If $${n_1}$$, $${n_2}$$,.......$${n_p}$$ are p positive integers, whose sum is an even number, then the number of odd in...
IIT-JEE 1985
If a &lt; b &lt; c &lt; d, then the roots of the equation (x - a) (x - c) + 2 ( x - b) (x - d) = 0 are real and distinct...
IIT-JEE 1984
The equation $$2{x^2} + 3x + 1 = 0$$ has an irrational root.
IIT-JEE 1983
For every integer n &gt; 1, the inequality $${(n!)^{1/n}} &lt; {{n + 1} \over 2}$$ holds.
IIT-JEE 1981

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