Mathematics
Previous Years Questions

## Numerical

The product of all positive real values of $x$ satisfying the equation $$x^{\left(16\left(\log _{5} x\right)^{3}-68 \log _{5} x\right)}=5^{-16}$$ ...
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 ________.
Let a, b, c three non-zero real numbers such that the equation $$\sqrt 3 a\cos x + 2b\sin x = c,x \in \left[ { - {\pi \over 2},{\pi \over 2}} \right... The value of$$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...
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
The number of distinct real roots of $${x^4} - 4{x^3} + 12{x^2} + x - 1 = 0$$
Let $$\left( {x,\,y,\,z} \right)$$ be points with integer coordinates satisfying the system of homogeneous equation: $$\matrix{ {3x - y - z = 0} ... The smallest value of$$k$$, for which both the roots of the equation$$${x^2} - 8kx + 16\left( {{k^2} - k + 1} \right) = 0$$are real, distinct and... ## MCQ (Single Correct Answer) Suppose a, b denote the distinct real roots of the quadratic polynomial x2 + 20x$$-$$2020 and suppose c, d denote the distinct complex roots of the ... If a4 = 28, then p + 2q = Let$$ - {\pi \over 6} < \theta < - {\pi \over {12}}.$$Suppose$${\alpha _1}$$and$${B_1}$$are the roots of the equation$${x^2} - 2x\se... The quadratic equation $$p(x)$$ $$= 0$$ with real coefficients has purely imaginary roots. Then the equation $$p(p(x))=0$$ has Let $$\alpha$$(a) and $$\beta$$(a) be the roots of the equation $$(\root 3 \of {1 + a} - 1){x^2} + (\sqrt {1 + a} - 1)x + (\root 6 \of {1 + a} - 1)... Let$$\alpha $$and$$\beta $$be the roots of$${x^2} - 6x - 2 = 0,$$with$$\alpha > \beta .$$If$${a_n} = {\alpha ^n} - {\beta ^n}$$for$$... Let $$\left( {{x_0},{y_0}} \right)$$ be the solution of the following equations $$\matrix{ {{{\left( {2x} \right)}^{\ell n2}}\, = {{\left( {3y} \... A value of$$b$$for which the equations$$$\matrix{ {{x^2} + bx - 1 = 0} \cr {{x^2} + x + b = 0} \cr } $$have one root in common is ... Let$$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 $$are nonzero c... Let$$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...
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 $${x^2} ... Let$$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}...
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
For all $$'x',{x^2} + 2ax + 10 - 3a > 0,$$ then the interval in which '$$a$$' lies is
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...
If $${a_1},{a_2}.......,{a_n}$$ are positive real numbers whose product is a fixed number c, then the minimum value of $${a_1} + {a_2} + ..... + {a_{n... The set of all real numbers x for which$${x^2} - \left| {x + 2} \right| + x > 0$$, is If b > a, then the equation (x - a) (x - b) - 1 = 0 has If a, b, c, d are positive real numbers such that a + b + c + d = 2, then M = (a + b) (c + d) satisfies the relation For the equation$$3{x^2} + px + 3 = 0$$. p > 0, if one of the root is square of the other, then p is equal to If$$\alpha \,and\,\beta (\alpha \, < \,\beta )$$are the roots of the equation$${x^2} + bx + c = 0\,$$, where$$c < 0 < b$$, then If the roots of the equation$${x^2} - 2ax + {a^2} + a - 3 = 0$$are real and less than 3, then Number of divisor of the form 4$$n + 2\left( {n \ge 0} \right)$$of the integer 240 is The number of points of intersection of two curves y = 2 sin x and y$$ = 5{x^2} + 2x + 3$$is If p, q, r are + ve and are on A.P., the roots of quadratic equation$$p{x^2} + qx + r = 0$$are all real for Let$$p,q \in \left\{ {1,2,3,4} \right\}\,$$. The number of equations of the form$$p{x^2} + qx + 1 = 0$$having real roots is Let$$\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...
The product of $$n$$ positive numbers is unity. Then their sum is
The number of solutions of the equation sin$${(e)^x} = {5^x} + {5^{ - x}}$$ is
If $$a,\,b,\,c,\,d$$ and p are distinct real numbers such that $$\left( {{a^2} + {b^2} + {c^2}} \right){p^2} - 2\left( {ab + bc + cd} \right)p + \le... If$$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...
If $${\log _{0.3}}\,(x\, - \,1) < {\log _{0.09}}(x - 1)$$, then x lies in the interval-
The equation $$x - {2 \over {x - 1}} = 1 - {2 \over {x - 1}}$$ has
If $$\,{a^2} + {b^2} + {c^2} = 1$$, then ab + bc + ca lies in the interval
The number of real solutions of the equation $${\left| x \right|^2} - 3\left| x \right| + 2 = 0$$ is
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...
If p, q, r are any real numbers, then
The largest interval for which $${x^{12}} - {x^9} + {x^4} - x + 1 > 0$$ is
Both the roots of the equation (x - b) (x - c) + (x - a) (x - c) + (x - a) (x - b) = 0 are always
The least value of the expression $$2\,\,{\log _{10}}\,x\, - \,{\log _x}(0.01)$$ for x > 1, is
If $$\,({x^2} + px + 1)\,$$ is a factor of $$(a{x^3} + bx + c)$$, then
The equation x + 2y + 2z = 1 and 2x + 4y + 4z = 9 have
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.
Let a > 0, b > 0 and c > 0. Then the roots of the equation $$a{x^2} + bx + c = 0$$
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)

Let $$\alpha$$ and $$\beta$$ be the roots of$${x^2} - x - 1 = 0$$, with $$\alpha$$ > $$\beta$$. For all positive integers n, define$${a_n} = {{... Let$$S$$be the set of all non-zero real numbers$$\alpha $$such that the quadratic equation$$\alpha {x^2} - x + \alpha = 0$$has two distinct rea... If$${3^x}\, = \,{4^{x - 1}},$$then$$x\, = $$The equation$${x^{3/4{{\left( {{{\log }_2}\,\,x} \right)}^2} + {{\log }_2}\,\,x - 5/4}} = \sqrt 2 $$has If$$\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$$, then th... Let a, b, c be real numbers,$$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...
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
For real $$x$$, the function $$\,{{\left( {x - a} \right)\left( {x - b} \right)} \over {x - c}}$$ will assume all real values provided

## Subjective

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 $$... If$$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... 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... 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... 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... 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... Let $$a,\,b,\,c$$ be real. If $$a{x^2} + bx + c = 0$$ has two real roots $$\alpha$$ and $$\beta ,$$ where $$\alpha < - 1$$ and $$\beta > 1,... Solve$$\left| {{x^2} + 4x + 3} \right| + 2x + 5 = 0$$Find the set of all$$x$$for which$${{2x} \over {\left( {2{x^2} + 5x + 2} \right)}}\, > \,{1 \over {\left( {x + 1} \right)}}$$For$$a \le 0,$$determine all real roots of the equation$$${x^2} - 2a\left| {x - a} \right| - 3{a^2} = 0$$Solve for$$x$$;$${\left( {5 + 2\sqrt 6 } \right)^{{x^2} - 3}} + {\left( {5 - 2\sqrt 6 } \right)^{{x^2} - 3}} = 10$$If one root of the quadratic equation$$a{x^2} + bx + c = 0$$is equal to the$$n$$-th power of the other, then show that$$${\left( {a{c^n}} \right)... Find all real values of $$x$$ which satisfy $${x^2} - 3x + 2 > 0$$ and $${x^2} - 2x - 4 \le 0$$ Show that the equation $${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$$ has no real solution. $$mn$$ 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... 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 $$y... Given$${n^4} < {10^n}$$for a fixed positive integer$$n \ge 2,$$prove that$${\left( {n + 1} \right)^4} < {10^{n + 1}}.$$For what values of$$m,$$does the system of equations$$$\matrix{ {3x + my = m} \cr {2x - 5y = 20} \cr } $$has solution satisfying the... Find the solution set of the system$$$\matrix{ {x + 2y + z = 1;} \cr {2x - 3y - w = 2;} \cr {x \ge 0;\,y \ge 0;\,z \ge 0;\,w \ge 0.} \... 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 $$\left( {... If$$\left( {m\,,\,n} \right) = {{\left( {1 - {x^m}} \right)\left( {1 - {x^{m - 1}}} \right).......\left( {1 - {x^{m - n + 1}}} \right)} \over {\left(... Solve for $$x:{4^x} - {3^{^{x - {1 \over 2}}}}\, = {3^{^{x + {1 \over 2}}}}\, - {2^{2x - 1}}$$ Solve for $$x:\,\sqrt {x + 1} - \sqrt {x - 1} = 1.$$ Solve the following equation for $$x:\,\,2\,{\log _x}a + {\log _{ax}}a + 3\,\,{\log _{{a^2}x}}\,a = 0,a > 0$$ Show that the square of $$\,{{\sqrt {26 - 15\sqrt 3 } } \over {5\sqrt 2 - \sqrt {38 + 5\sqrt 3 } }}$$ is a rational number. Find all integers $$x$$ for which $$\left( {5x - 1} \right) < {\left( {x + 1} \right)^2} < \left( {7x - 3} \right).$$ Sketch the solution set of the following system of inequalities: $${x^2} + {y^2} - 2x \ge 0;\,\,3x - y - 12 \le 0;\,\,y - x \le 0;\,\,y \ge 0.$$$

## Fill in the Blanks

The sum of all the real roots of the equation $${\left| {x - 2} \right|^2} + \left| {x - 2} \right| - 2 = 0$$ is ............................
Let n and k be positive such that $$n \ge {{k(k + 1)} \over 2}$$ . The number of solutions $$\,({x_1},\,{x_2},\,.....{x_k}),\,{x_1}\,\, \ge \,1,\,{x_2... If$$\,x < 0,\,\,y < 0,\,\,x + y + {x \over y} = {1 \over 2}$$and$$(x + y)\,{x \over y} = - {1 \over 2}$$, then x =..........and y =......... If the quadratic equations$${x^2} + ax + b = 0$$and$${x^2} + bx + a = 0(a \ne b)$$have a common root, then the numerical value of a + b is..... The solution of equation$${\log _7}\,{\log _5}\,\left( {\sqrt {x + 5} + \sqrt x } \right) = 0$$is ........................ If the product of the roots of the equation$$\,{x^2} - 3\,k\,x + 2\,{e^{2lnk}} - 1 = 0\,\,\,\,is\,7$$, then the roots are real for k = ................. The coeffcient of$${x^{99}}$$in the polynomial (x -1) (x - 2)...(x - 100) is .............. If$$2 + i\sqrt 3 $$is root of the equation$${x^2} + px + q = 0$$, where p and q are real, then (p, q) = (..........,....................). ## True or False If x and y are positive real numbers and m, n are any positive integers, then$${{{x^n}\,{y^m}} \over {(1 + {x^{2n}})\,(1 + {y^{2m}})}} > {1 \over ...
If $${n_1}$$, $${n_2}$$,.......$${n_p}$$ are p positive integers, whose sum is an even number, then the number of odd integers among them is odd.
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 two real roots.
If a < b < c < d, then the roots of the equation (x - a) (x - c) + 2 ( x - b) (x - d) = 0 are real and distinct.
The equation $$2{x^2} + 3x + 1 = 0$$ has an irrational root.
For every integer n > 1, the inequality $${(n!)^{1/n}} < {{n + 1} \over 2}$$ holds.
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