The number of real solutions of the equation, x² $-$ |x| $-$ 12 = 0 is :

Consider function f : A $\to$ B and g : B $\to$ C (A, B, C $ \subseteq $ R) such that (gof)^$-$1 exists, then :

If $P = \begin{bmatrix} 1 & 0 \\ \tfrac{1}{2} & 1 \end{bmatrix}$, then $P^{50}$ is :

Let X be a random variable such that the probability function of a distribution is given by $P(X = 0) = {1 \over 2},P(X = j) = {1 \over {{3^j}}}(j = 1,2,3,...,\infty )$. Then the mean of the distribution and P(X is positive and even) respectively are :

If ${}^n{P_r} = {}^n{P_{r + 1}}$ and ${}^n{C_r} = {}^n{C_{r - 1}}$, then the value of r is equal to :

Let y = y(x) be the solution of the differential equation xdy = (y + x³ cosx)dx with y($\pi$) = 0, then $y\left( {{\pi \over 2}} \right)$ is equal to :

If the mean and variance of the following data : 6, 10, 7, 13, a, 12, b, 12 are 9 and ${{37} \over 4}$ respectively, then (a $-$ b)² is equal to :

Let $\overrightarrow a = \widehat i + \widehat j + 2\widehat k$ and $\overrightarrow b = - \widehat i + 2\widehat j + 3\widehat k$. Then the vector product $\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {\left( {\overrightarrow a \times \left( {\left( {\overrightarrow a - \overrightarrow b } \right) \times \overrightarrow b } \right)} \right) \times \overrightarrow b } \right)$ is equal to :

The value of the definite integral$ \int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{dx} \over {(1 + {e^{x\cos x}})({{\sin }^4}x + {{\cos }^4}x)}}} $ is equal to :

Let C be the set of all complex numbers. Let ${S_1} = \{ z \in C||z - 3 - 2i{|^2} = 8\} $ ${S_2} = \{ z \in C|{\mathop{\rm Re}\nolimits} (z) \ge 5\} $ and ${S_3} = \{ z \in C||z - \overline z | \ge 8\} $. Then the number of elements in ${S_1} \cap {S_2} \cap {S_3}$ is equal to :

If the area of the bounded region $R = \left\{ {(x,y):\max \{ 0,{{\log }_e}x\} \le y \le {2^x},{1 \over 2} \le x \le 2} \right\}$ is , $\alpha {({\log _e}2)^{ - 1}} + \beta ({\log _e}2) + \gamma $, then the value of ${(\alpha + \beta - 2\lambda )^2}$ is equal to :

A ray of light through (2, 1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this reflected ray is the directrix of an ellipse with eccentricity ${1 \over 3}$ and the distance of the nearer focus from this directrix is ${8 \over {\sqrt {53} }}$, then the equation of the other directrix can be :

If the coefficients of x⁷ in ${\left( {{x^2} + {1 \over {bx}}} \right)^{11}}$ and x^$-$7 in ${\left( {{x} - {1 \over {bx^2}}} \right)^{11}}$, b $\ne$ 0, are equal, then the value of b is equal to :

If $\sin \theta + \cos \theta = {1 \over 2}$, then 16(sin(2$\theta$) + cos(4$\theta$) + sin(6$\theta$)) is equal to :

Let $A = \left[ {\matrix{ 1 & 2 \cr { - 1} & 4 \cr } } \right]$. If A^$-$1 = $\alpha$I + $\beta$A, $\alpha$, $\beta$ $\in$ R, I is a 2 $\times$ 2 identity matrix then 4($\alpha$ $-$ $\beta$) is equal to :

Let $f:\left( { - {\pi \over 4},{\pi \over 4}} \right) \to R$ be defined as $f(x) = \left\{ {\matrix{ {{{(1 + |\sin x|)}^{{{3a} \over {|\sin x|}}}}} & , & { - {\pi \over 4} < x < 0} \cr b & , & {x = 0} \cr {{e^{\cot 4x/\cot 2x}}} & , & {0 < x < {\pi \over 4}} \cr } } \right.$ If f is continuous at x = 0, then the value of 6a + b² is equal to :

Let y = y(x) be solution of the differential equation ${\log _{}}\left( {{{dy} \over {dx}}} \right) = 3x + 4y$, with y(0) = 0.If $y\left( { - {2 \over 3}{{\log }_e}2} \right) = \alpha {\log _e}2$, then the value of $\alpha$ is equal to :

Let f : R $\to$ R be a function such that f(2) = 4 and f'(2) = 1. Then, the value of $\mathop {\lim }\limits_{x \to 2} {{{x^2}f(2) - 4f(x)} \over {x - 2}}$ is equal to :

The value of the integral, $\int\limits_1^3 {[{x^2} - 2x - 2]dx} $, where [x] denotes the greatest integer less than or equal to x, is :

Let a, b$ \in $R. If the mirror image of the point P(a, 6, 9) with respect to the line ${{x - 3} \over 7} = {{y - 2} \over 5} = {{z - 1} \over { - 9}}$ is (20, b, $-$a$-$9), then | a + b |, is equal to :

For the system of linear equations: $x - 2y = 1,x - y + kz = - 2,ky + 4z = 6,k \in R$, consider the following statements : 

Let f(x) be a differentiable function defined on [0, 2] such that f'(x) = f'(2 $-$ x) for all x$ \in $ (0, 2), f(0) = 1 and f(2) = e². Then the value of $\int\limits_0^2 {f(x)} dx$ is :

Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f'(x) $ \ne $ 0 for all x $ \in $ R. If $\left| {\matrix{ {f(x)} & {f'(x)} \cr {f'(x)} & {f''(x)} \cr } } \right|$ = 0, for all x$ \in $R, then the value of f(1) lies in the interval :

$f:R \to R$ be defined as$f(x) = \left\{ {\matrix{ { - 55x,} & {if\,x < - 5} \cr {2{x^3} - 3{x^2} - 120x,} & {if\, - 5 \le x \le 4} \cr {2{x^3} - 3{x^2} - 36x - 336,} & {if\,x > 4,} \cr } } \right.$ Let A = {x $ \in $ R : f is increasing}. Then A is equal to :

The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements in their intersection, is :

Let $\mathbb{N}$ be the set of natural numbers and a relation $R$ on $\mathbb{N}$ be defined by \[ R=\{(x,y)\in \mathbb{N}\times \mathbb{N} : x^{3}-3x^{2}y-xy^{2}+3y^{3}=0\}. \] Then the relation $R$ is:

A possible value of $\tan \left( {{1 \over 4}{{\sin }^{ - 1}}{{\sqrt {63} } \over 8}} \right)$ is :

Consider a circle C which touches the y-axis at (0, 6) and cuts off an intercept $6\sqrt 5 $ on the x-axis. Then the radius of the circle C is equal to :

The area of the region : $R = \{ (x,y):5{x^2} \le y \le 2{x^2} + 9\} $

The area of the region : $R = \{ (x,y):5{x^2} \le y \le 2{x^2} + 9\} $

Let y = y(x) be a solution curve of the differential equation $(y + 1){\tan ^2}x\,dx + \tan x\,dy + y\,dx = 0$, $x \in \left( {0,{\pi \over 2}} \right)$. If $\mathop {\lim }\limits_{x \to 0 + } xy(x) = 1$, then the value of $y\left( {{\pi \over 4}} \right)$ is :

Let $f(x) = 3{\sin ^4}x + 10{\sin ^3}x + 6{\sin ^2}x - 3$, $x \in \left[ { - {\pi \over 6},{\pi \over 2}} \right]$. Then, f is :

The mean and standard deviation of 20 observations were calculated as 10 and 2.5 respectively. It was found that by mistake one data value was taken as 25 instead of 35. if $\alpha$ and $\sqrt \beta $ are the mean and standard deviation respectively for correct data, then ($\alpha$, $\beta$) is :

Let S_n be the sum of the first n terms of an arithmetic progression. If S_3n = 3S_2n, then the value of ${{{S_{4n}}} \over {{S_{2n}}}}$ is :

Let A and B be independent events such that P(A) = p, P(B) = 2p. The largest value of p, for which P (exactly one of A, B occurs) = ${5 \over 9}$, is :

Let $f(x) = 3{\sin ^4}x + 10{\sin ^3}x + 6{\sin ^2}x - 3$, $x \in \left[ { - {\pi \over 6},{\pi \over 2}} \right]$. Then, f is :

Let $\theta \in \left( {0,{\pi \over 2}} \right)$. If the system of linear equations $(1 + {\cos ^2}\theta )x + {\sin ^2}\theta y + 4\sin 3\,\theta z = 0$, ${\cos ^2}\theta x + (1 + {\sin ^2}\theta )y + 4\sin 3\,\theta z = 0$, ${\cos ^2}\theta x + {\sin ^2}\theta y + (1 + 4\sin 3\,\theta )z = 0,$ has a non-trivial solution, then the value of $\theta$ is :

Let S_n be the sum of the first n terms of an arithmetic progression. If S_3n = 3S_2n, then the value of ${{{S_{4n}}} \over {{S_{2n}}}}$ is :

Let $f(x) = \cos \left( {2{{\tan }^{ - 1}}\sin \left( {{{\cot }^{ - 1}}\sqrt {{{1 - x} \over x}} } \right)} \right)$, 0 < x < 1. Then :

Let f : R $\to$ R be defined as$f(x) = \left\{ {\matrix{ {{{\lambda \left| {{x^2} - 5x + 6} \right|} \over {\mu (5x - {x^2} - 6)}},} & {x < 2} \cr {{e^{{{\tan (x - 2)} \over {x - [x]}}}},} & {x > 2} \cr {\mu ,} & {x = 2} \cr } } \right.$ where [x] is the greatest integer is than or equal to x. If f is continuous at x = 2, then $\lambda$ + $\mu$ is equal to :

The locus of the centroid of the triangle formed by any point P on the hyperbola $16{x^2} - 9{y^2} + 32x + 36y - 164 = 0$, and its foci is :

The equation $\arg \left( {{{z - 1} \over {z + 1}}} \right) = {\pi \over 4}$ represents a circle with :

The value of the definite integral $\int\limits_{\pi /24}^{5\pi /24} {{{dx} \over {1 + \root 3 \of {\tan 2x} }}} $ is :

If a line along a chord of the circle 4x² + 4y² + 120x + 675 = 0, passes through the point ($-$30, 0) and is tangent to the parabola y² = 30x, then the length of this chord is :

If b is very small as compared to the value of a, so that the cube and other higher powers of ${b \over a}$ can be neglected in the identity ${1 \over {a - b}} + {1 \over {a - 2b}} + {1 \over {a - 3b}} + ..... + {1 \over {a - nb}} = \alpha n + \beta {n^2} + \gamma {n^3}$, then the value of $\gamma$ is :

The value of $\int\limits_{{{ - 1} \over {\sqrt 2 }}}^{{1 \over {\sqrt 2 }}} {{{\left( {{{\left( {{{x + 1} \over {x - 1}}} \right)}^2} + {{\left( {{{x - 1} \over {x + 1}}} \right)}^2} - 2} \right)}^{{1 \over 2}}}dx} $ is :

Let y = y(x) be the solution of the differential equation ${{dy} \over {dx}} = 1 + x{e^{y - x}}, - \sqrt 2 < x < \sqrt 2 ,y(0) = 0$ then, the minimum value of $y(x),x \in \left( { - \sqrt 2 ,\sqrt 2 } \right)$ is equal to :

If $A = \left( {\matrix{ {{1 \over {\sqrt 5 }}} & {{2 \over {\sqrt 5 }}} \cr {{{ - 2} \over {\sqrt 5 }}} & {{1 \over {\sqrt 5 }}} \cr } } \right)$, $B = \left( {\matrix{ 1 & 0 \cr i & 1 \cr } } \right)$, $i = \sqrt { - 1} $, and Q = A^TBA, then the inverse of the matrix A Q²⁰²¹ A^T is equal to :

The area (in sq. units) of the region, given by the set $\{ (x,y) \in R \times R|x \ge 0,2{x^2} \le y \le 4 - 2x\} $

Let g : N $\to$ N be defined as g(3n + 1) = 3n + 2, g(3n + 2) = 3n + 3, g(3n + 3) = 3n + 1, for all n $\ge$ 0. Then which of the following statements is true?

Let [t] denote the greatest integer less than or equal to t. Let f(x) = x $-$ [x], g(x) = 1 $-$ x + [x], and h(x) = min{f(x), g(x)}, x $\in$ [$-$2, 2]. Then h is :

Let $f:[0,\infty ) \to [0,\infty )$ be defined as $f(x) = \int_0^x {[y]dy} $ where [x] is the greatest integer less than or equal to x. Which of the following is true?

Let $A = \left( {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 1 \cr 1 & 0 & 0 \cr } } \right)$. Then A²⁰²⁵ $-$ A²⁰²⁰ is equal to :

The values of a and b, for which the system of equations 2x + 3y + 6z = 8, x + 2y + az = 5, 3x + 5y + 9z = b, has no solution, are :

The local maximum value of the function $f(x) = {\left( {{2 \over x}} \right)^{{x^2}}}$, x > 0, is

Let 9 distinct balls be distributed among 4 boxes, B₁, B₂, B₃ and B₄. If the probability than B₃ contains exactly 3 balls is $k{\left( {{3 \over 4}} \right)^9}$ then k lies in the set :

If the value of the integral $\int\limits_0^5 {{{x + [x]} \over {{e^{x - [x]}}}}dx = \alpha {e^{ - 1}} + \beta } $, where $\alpha$, $\beta$ $\in$ R, 5$\alpha$ + 6$\beta$ = 0, and [x] denotes the greatest integer less than or equal to x; then the value of ($\alpha$ + $\beta$)² is equal to :

The number of real roots of the equation ${e^{6x}} - {e^{4x}} - 2{e^{3x}} - 12{e^{2x}} + {e^x} + 1 = 0$ is :

Let y(x) be the solution of the differential equation 2x² dy + (e^y $-$ 2x)dx = 0, x > 0. If y(e) = 1, then y(1) is equal to :

Let an ellipse $E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$, ${a^2} > {b^2}$, passes through $\left( {\sqrt {{3 \over 2}} ,1} \right)$ and has eccentricity ${1 \over {\sqrt 3 }}$. If a circle, centered at focus F($\alpha$$, 0), $\alpha$$ > 0, of E and radius ${2 \over {\sqrt 3 }}$, intersects E at two points P and Q, then PQ² is equal to :

The domain of the function ${{\mathop{\rm cosec}\nolimits} ^{ - 1}}\left( {{{1 + x} \over x}} \right)$ is :

A function f(x) is given by $f(x) = {{{5^x}} \over {{5^x} + 5}}$, then the sum of the series $f\left( {{1 \over {20}}} \right) + f\left( {{2 \over {20}}} \right) + f\left( {{3 \over {20}}} \right) + ....... + f\left( {{{39} \over {20}}} \right)$ is equal to :

A fair die is tossed until six is obtained on it. Let x be the number of required tosses, then the conditional probability P(x $\ge$ 5 | x > 2) is :

Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one functions form the set A to the set A $\times$ B. Then :

If $\sum\limits_{r = 1}^{50} {{{\tan }^{ - 1}}{1 \over {2{r^2}}} = p} $, then the value of tan p is :

The integral $\int {{{{e^{3{{\log }_e}2x}} + 5{e^{2{{\log }_e}2x}}} \over {{e^{4{{\log }_e}x}} + 5{e^{3{{\log }_e}x}} - 7{e^{2{{\log }_e}x}}}}} dx$, x > 0, is equal to : (where c is a constant of integration)

Two fair dice are thrown. The numbers on them are taken as $\lambda$ and $\mu$, and a system of linear equations, x + y + z = 5, x + 2y + 3z = $\mu$ ,x + 3y + $\lambda$z = 1, is constructed. If p is the probability that the system has a unique solution and q is the probability that the system has no solution, then :

Let A be a set of all 4-digit natural numbers whose exactly one digit is 7. Then the probability that a randomly chosen element of A leaves remainder 2 when divided by 5 is :

The locus of the mid points of the chords of the hyperbola x² $-$ y² = 4, which touch the parabola y² = 8x, is :

The shortest distance between the line x $-$ y = 1 and the curve x² = 2y is :

The value of $2\sin \left( {{\pi \over 8}} \right)\sin \left( {{{2\pi } \over 8}} \right)\sin \left( {{{3\pi } \over 8}} \right)\sin \left( {{{5\pi } \over 8}} \right)\sin \left( {{{6\pi } \over 8}} \right)\sin \left( {{{7\pi } \over 8}} \right)$ is :

Let $\alpha$ and $\beta$ be the roots of x² $-$ 6x $-$ 2 = 0. If a_n = $\alpha$$ⁿ $-$ $\beta$ⁿ for n $ \ge $ 1, then the value of ${{{a_{10}} - 2{a_8}} \over {3{a_9}}}$ is :

If ${\left( {\sqrt 3 + i} \right)^{100}} = {2^{99}}(p + iq)$, then p and q are roots of the equation :

If 0 < x, y < $\pi$ and cosx + cosy $-$ cos(x + y) = ${3 \over 2}$, then sinx + cosy is equal to :

The value of $\int\limits_{ - {\pi \over 2}}^{{\pi \over 2}} {\left( {{{1 + {{\sin }^2}x} \over {1 + {\pi ^{\sin x}}}}} \right)} \,dx$

A circle C touches the line x = 2y at the point (2, 1) and intersects the circle C₁ : x² + y² + 2y $-$ 5 = 0 at two points P and Q such that PQ is a diameter of C₁. Then the diameter of C is :

Let A be a 3 $\times$ 3 matrix with det(A) = 4. Let R_i denote the i^th row of A. If a matrix B is obtained by performing the operation R₂ $ \to $ 2R₂ + 5R₃ on 2A, then det(B) is equal to :

$\mathop {\lim }\limits_{x \to 2} \left( {\sum\limits_{n = 1}^9 {{x \over {n(n + 1){x^2} + 2(2n + 1)x + 4}}} } \right)$ is equal to :

If $\alpha$, $\beta$ $\in$ R are such that 1 $-$ 2i (here i² = $-$1) is a root of z² + $\alpha$z + $\beta$ = 0, then ($\alpha$ $-$ $\beta$) is equal to :

The minimum value of $f(x) = {a^{{a^x}}} + {a^{1 - {a^x}}}$, where a, $x \in R$ and a > 0, is equal to :

(sin−1x)2−(cos−1x)2=a,0<x<1,a;=0

then the value of

$2x^2 - 1$

is:

If ${I_n} = \int\limits_{{\pi \over 4}}^{{\pi \over 2}} {{{\cot }^n}x\,dx} $, then :

If the matrix $A = \left( {\matrix{ 0 & 2 \cr K & { - 1} \cr } } \right)$ satisfies $A({A^3} + 3I) = 2I$, then the value of K is :

In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chances of getting a particular chest disorder are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is :

$S = { z \in \mathbb{C} : \dfrac{z - i}{z + 2i} \in \mathbb R }$, then:

If for the matrix, $A = \left[ {\matrix{ 1 & { - \alpha } \cr \alpha & \beta \cr } } \right]$, $A{A^T} = {I_2}$, then the value of ${\alpha ^4} + {\beta ^4}$ is :

et y = y(x) be the solution of the differential equation ${{dy} \over {dx}} = 2(y + 2\sin x - 5)x - 2\cos x$ such that y(0) = 7. Then y($\pi$) is equal to :

If the curve x² + 2y² = 2 intersects the line x + y = 1 at two points P and Q, then the angle subtended by the line segment PQ at the origin is :

Let us consider a curve, y = f(x) passing through the point ($-$2, 2) and the slope of the tangent to the curve at any point (x, f(x)) is given by f(x) + xf'(x) = x². Then :

The following system of linear equations :- 2x + 3y + 2z = 9, 3x + 2y + 2z = 9, x $-$ y + 4z = 8

If $\alpha$, $\beta$ are the distinct roots of x² + bx + c = 0, then $\mathop {\lim }\limits_{x \to \beta } {{{e^{2({x^2} + bx + c)}} - 1 - 2({x^2} + bx + c)} \over {{{(x - \beta )}^2}}}$ is equal to :

A hyperbola passes through the foci of the ellipse ${{{x^2}} \over {25}} + {{{y^2}} \over {16}} = 1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is :

When a certain biased die is rolled, a particular face occurs with probability ${1 \over 6} - x$ and its opposite face occurs with probability ${1 \over 6} + x$. All other faces occur with probability ${1 \over 6}$. Note that opposite faces sum to 7 in any die. If 0 < x < ${1 \over 6}$, and the probability of obtaining total sum = 7, when such a die is rolled twice, is ${13 \over 96}$, then the value of x is :

The maximum value of the term independent of 't' in the expansion of ${\left( {t{x^{{1 \over 5}}} + {{{{(1 - x)}^{{1 \over {10}}}}} \over t}} \right)^{10}}$ where x$\in$(0, 1) is :

If x² + 9y² $-$ 4x + 3 = 0, x, y $\in$ R, then x and y respectively lie in the intervals :

The value of $\mathop {\lim }\limits_{h \to 0} 2\left\{ {{{\sqrt 3 \sin \left( {{\pi \over 6} + h} \right) - \cos \left( {{\pi \over 6} + h} \right)} \over {\sqrt 3 h\left( {\sqrt 3 \cosh - \sinh } \right)}}} \right\}$ is :

$\int\limits_6^{16} {{{{{\log }_e}{x^2}} \over {{{\log }_e}{x^2} + {{\log }_e}({x^2} - 44x + 484)}}dx} $$ is equal to :

Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A² is 1, then the possible number of such matrices is :

The value of $\int\limits_{ - \pi /2}^{\pi /2} {{{{{\cos }^2}x} \over {1 + {3^x}}}} dx$ is :

The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m $-$ n = 0 and mn + nl + lm = 0, is :

The number of seven digit integers with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only is :

Let $A = \left( {\matrix{ {[x + 1]} & {[x + 2]} & {[x + 3]} \cr {[x]} & {[x + 3]} & {[x + 3]} \cr {[x]} & {[x + 2]} & {[x + 4]} \cr } } \right)$, where [t] denotes the greatest integer less than or equal to t. If det(A) = 192, then the set of values of x is the interval :

Let R = {(P, Q) | P and Q are at the same distance from the origin} be a relation, then the equivalence class of (1, $-$1) is the set :

Let M and m respectively be the maximum and minimum values of the function f(x) = tan^$-$1 (sin x + cos x) in $\left[ {0,{\pi \over 2}} \right]$, then the value of tan(M $-$ m) is equal to :

The value of $\sum\limits_{n = 1}^{100} {\int\limits_{n - 1}^n {{e^{x - [x]}}dx} } $, where [ x ] is the greatest integer $ \le $ x, is :

If two tangents drawn from a point P to the parabola y² = 16(x $-$ 3) are at right angles, then the locus of point P is :

The intersection of three lines x - y = 0, x + 2y = 3 and 2x + y = 6 is a

If the solution curve of the differential equation (2x $-$ 10y³)dy + ydx = 0, passes through the points (0, 1) and (2, $\beta$), then $\beta$ is a root of the equation :

In the circle given below, let OA = 1 unit, OB = 13 unit and PQ $ \bot $ OB. Then, the area of the triangle PQB (in square units) is :

Let [$\lambda$] be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations x + y + z = 4, 3x + 2y + 5z = 3, 9x + 4y + (28 + [$\lambda$])z = [$\lambda$] has a solution is :

The value of $\left| {\matrix{ {(a + 1)(a + 2)} & {a + 2} & 1 \cr {(a + 2)(a + 3)} & {a + 3} & 1 \cr {(a + 3)(a + 4)} & {a + 4} & 1 \cr } } \right|$ is :

If ${{{{\sin }^1}x} \over a} = {{{{\cos }^{ - 1}}x} \over b} = {{{{\tan }^{ - 1}}y} \over c}$; $0 < x < 1$, then the value of $\cos \left( {{{\pi c} \over {a + b}}} \right)$ is :

The set of all values of K > $-$1, for which the equation ${(3{x^2} + 4x + 3)^2} - (k + 1)(3{x^2} + 4x + 3)(3{x^2} + 4x + 2) + k{(3{x^2} + 4x + 2)^2} = 0$ has real roots, is :

The maximum slope of the curve $y = {1 \over 2}{x^4} - 5{x^3} + 18{x^2} - 19x$ occurs at the point :

Let Z be the set of all integers,$A = \{ (x,y) \in Z \times Z:{(x - 2)^2} + {y^2} \le 4\} $, $B = \{ (x,y) \in Z \times Z:{x^2} + {y^2} \le 4\} $, $C = \{ (x,y) \in Z \times Z:{(x - 2)^2} + {(y - 2)^2} \le 4\} $, If the total number of relation from A $\cap$ B to A $\cap$ C is 2^p, then the value of p is :

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time t = 0. The number of bacteria is increased by 20% in 2 hours. If the population of bacteria is 2000 after ${k \over {{{\log }_e}\left( {{6 \over 5}} \right)}}$ hours, then ${\left( {{k \over {{{\log }_e}2}}} \right)^2}$ is equal to :

The area of the region bounded by the parabola (y $-$ 2)² = (x $-$ 1), the tangent to it at the point whose ordinate is 3 and the x-axis is :

In an increasing geometric series, the sum of the second and the sixth term is ${{25} \over 2}$ and the product of the third and fifth term is 25. Then, the sum of 4^th, 6^th and 8^th terms is equal to :

If $y(x)=\cot^{-1}\!\left(\dfrac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right),\; x\in\left(\tfrac{\pi}{2},\pi\right)$, then $\dfrac{dy}{dx}$ at $x=\tfrac{5\pi}{6}$ is:

Let f(x) be a differentiable function at x = a with f'(a) = 2 and f(a) = 4. Then $\mathop {\lim }\limits_{x \to a} {{xf(a) - af(x)} \over {x - a}}$ equals :

The value of the integral $\displaystyle \int_{0}^{1} \frac{\sqrt{x}\,dx}{(1+x)(1+3x)(3+x)}$ is:

Let A(1, 4) and B(1, $-$5) be two points. Let P be a point on the circle (x $-$ 1)² + (y $-$ 1)² = 1 such that (PA)² + (PB)² have maximum value, then the points, P, A and B lie on :

If $\mathop {\lim }\limits_{x \to \infty } \left( {\sqrt {{x^2} - x + 1} - ax} \right) = b$, then the ordered pair (a, b) is :

Let f(x) be a differentiable function at x = a with f'(a) = 2 and f(a) = 4. Then $\mathop {\lim }\limits_{x \to a} {{xf(a) - af(x)} \over {x - a}}$ equals :

If $\mathop {\lim }\limits_{x \to \infty } \left( {\sqrt {{x^2} - x + 1} - ax} \right) = b$, then the ordered pair (a, b) is :

Consider the following system of equations : x + 2y $-$ 3z = a 2x + 6y $-$ 11z = bx $-$ 2y + 7z = c, where a, b and c are real constants. Then the system of equations :

A natural number has prime factorization given by n = 2^x3^y5^z, where y and z are such that y + z = 5 and y^$-$1 + z^$-$1 = ${5 \over 6}$, y > z. Then the number of odd divisions of n, including 1, is :

If 0 < a, b < 1, and tan^$-$1a + tan^$-$1b = ${\pi \over 4}$, then the value of

$(a + b) - \left( {{{{a^2} + {b^2}} \over 2}} \right) + \left( {{{{a^3} + {b^3}} \over 3}} \right) - \left( {{{{a^4} + {b^4}} \over 4}} \right) + .....$ is :

Let $f(x) = \int\limits_0^x {{e^t}f(t)dt + {e^x}} $ be a differentiable function for all x$\in$R. Then f(x) equals :

Let $A = \{ 1,2,3,....,10\} $ and $$f:A \to A$$ be defined as $f(k) = \left\{ {\matrix{ {k + 1} & {if\,k\,is\,odd} \cr k & {if\,k\,is\,even} \cr } } \right. $ Then the number of possible functions $g:A \to A$ such that $gof = f$ is :

Let A₁ be the area of the region bounded by the curves y = sinx, y = cosx and y-axis in the first quadrant. Also, let A₂ be the area of the region bounded by the curves y = sinx, y = cosx, x-axis and x = ${\pi \over 2}$ in the first quadrant. Then,

Let $f(x) = {\sin ^{ - 1}}x$ and $g(x) = {{{x^2} - x - 2} \over {2{x^2} - x - 6}}$. If $g(2) = \mathop {\lim }\limits_{x \to 2} g(x)$, then the domain of the function fog is :

Let f : R $ \to $ R be defined as $f(x) = \left\{ \matrix{ 2\sin \left( { - {{\pi x} \over 2}} \right),if\,x < - 1 \hfill \cr |a{x^2} + x + b|,\,if - 1 \le x \le 1 \hfill \cr \sin (\pi x),\,if\,x > 1 \hfill \cr} \right.$ If f(x) is continuous on R, then a + b equals :

If the locus of the mid-point of the line segment from the point (3, 2) to a point on the circle, x² + y² = 1 is a circle of radius r, then r is equal to :

If vectors $\overrightarrow {{a_1}} = x\widehat i - \widehat j + \widehat k$ and $\overrightarrow {{a_2}} = \widehat i + y\widehat j + z\widehat k$ are collinear, then a possible unit vector parallel to the vector $x\widehat i + y\widehat j + z\widehat k$ is :

A seven digit number is formed using digits 3, 3, 4, 4, 4, 5, 5. The probability, that number so formed is divisible by 2, is :

Let a vector $\alpha \widehat i + \beta \widehat j$ be obtained by rotating the vector $\sqrt 3 \widehat i + \widehat j$ by an angle 45$^\circ$ about the origin in counterclockwise direction in the first quadrant. Then the area of triangle having vertices ($\alpha$, $\beta$), (0, $\beta$) and (0, 0) is equal to :

Let ${S_k} = \sum\limits_{r = 1}^k {{{\tan }^{ - 1}}\left( {{{{6^r}} \over {{2^{2r + 1}} + {3^{2r + 1}}}}} \right)} $. Then $\mathop {\lim }\limits_{k \to \infty } {S_k}$ is equal to :

Let the position vectors of two points P and Q be 3$\widehat i$ $-$ $\widehat j$ + 2$\widehat k$ and $\widehat i$ + 2$\widehat j$ $-$ 4$\widehat k$, respectively. Let R and S be two points such that the direction ratios of lines PR and QS are (4, $-$1, 2) and ($-$2, 1, $-$2), respectively. Let lines PR and QS intersect at T. If the vector $\overrightarrow {TA} $ is perpendicular to both $\overrightarrow {PR} $ and $\overrightarrow {QS} $ and the length of vector $\overrightarrow {TA} $ is $\sqrt 5 $ units, then the modulus of a position vector of A is :

The number of elements in the set {x $\in$ R : (|x| $-$ 3) |x + 4| = 6} is equal to :

Let a complex number z, |z| $\ne$ 1, satisfy ${\log _{{1 \over {\sqrt 2 }}}}\left( {{{|z| + 11} \over {{{(|z| - 1)}^2}}}} \right) \le 2$. Then, the largest value of |z| is equal to ____________.

Let $A = \left[ {\matrix{ i & { - i} \cr { - i} & i \cr } } \right],i = \sqrt { - 1} $. Then, the system of linear equations ${A^8}\left[ {\matrix{ x \cr y \cr } } \right] = \left[ {\matrix{ 8 \cr {64} \cr } } \right]$ has :

If n is the number of irrational terms in the expansion of ${\left( {{3^{1/4}} + {5^{1/8}}} \right)^{60}}$, then (n $-$ 1) is divisible by :

The locus of the midpoints of the chord of the circle, x² + y² = 25 which is tangent to the hyperbola, ${{{x^2}} \over 9} - {{{y^2}} \over {16}} = 1$ is :

Let the functions f : R $ \to $ R and g : R $ \to $ R be defined as :$f(x) = \left\{ {\matrix{ {x + 2,} & {x < 0} \cr {{x^2},} & {x \ge 0} \cr } } \right.$ and $g(x) = \left\{ {\matrix{ {{x^3},} & {x < 1} \cr {3x - 2,} & {x \ge 1} \cr } } \right.$ Then, the number of points in R where (fog) (x) is NOT differentiable is equal to :

Consider three observations a, b, and c such that b = a + c. If the standard deviation of a + 2, b + 2, c + 2 is d, then which of the following is true?

The range of a$\in$R for which the function f(x) = (4a $-$ 3)(x + log_e 5) + 2(a $-$ 7) cot$\left( {{x \over 2}} \right)$ sin²$\left( {{x \over 2}} \right)$, x $\ne$ 2n$\pi$, n$\in$N has critical points, is :

If for x $\in$ $\left( {0,{\pi \over 2}} \right)$, log₁₀sinx + log₁₀cosx = $-$1 and log₁₀(sinx + cosx) = ${1 \over 2}$(log₁₀ n $-$ 1), n > 0, then the value of n is equal to :

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is :

If y = y(x) is the solution of the differential equation, ${{dy} \over {dx}} + 2y\tan x = \sin x,y\left( {{\pi \over 3}} \right) = 0$, then the maximum value of the function y(x) over R is equal to:

If y = y(x) is the solution of the differential equation ${{dy} \over {dx}}$ + (tan x) y = sin x, $0 \le x \le {\pi \over 3}$, with y(0) = 0, then $y\left( {{\pi \over 4}} \right)$ equal to :

Let f be a real valued function, defined on R $-$ {$-$1, 1} and given by f(x) = 3 log_e $\left| {{{x - 1} \over {x + 1}}} \right| - {2 \over {x - 1}}$. Then in which of the following intervals, function f(x) is increasing?

Let $\overrightarrow{a} = \hat{i} + 2\hat{j} - 3\hat{k}$ and $\overrightarrow{b} = 2\hat{i} - 3\hat{j} + 5\hat{k}$. If $\overrightarrow{r} \times \overrightarrow{a} = \overrightarrow{b} \times \overrightarrow{r}$, $\overrightarrow{r} \cdot (\alpha \hat{i} + 2\hat{j} + \hat{k}) = 3$ and $\overrightarrow{r} \cdot (2\hat{i} + 5\hat{j} - \alpha \hat{k}) = -1$, $\alpha \in \mathbb{R}$, then the value of $\alpha + |\overrightarrow{r}|^2$ is equal to :

If the foot of the perpendicular from point (4, 3, 8) on the line ${L_1}:{{x - a} \over l} = {{y - 2} \over 3} = {{z - b} \over 4}$, l $\ne$ 0 is (3, 5, 7), then the shortest distance between the line L₁ and line ${L_2}:{{x - 2} \over 3} = {{y - 4} \over 4} = {{z - 5} \over 5}$ is equal to :

Consider a rectangle ABCD having 5, 7, 6, 9 points in the interior of the line segments AB, CD, BC, DA respectively. Let $\alpha$ be the number of triangles having these points from different sides as vertices and $\beta$ be the number of quadrilaterals having these points from different sides as vertices. Then ($\beta$ $-$ $\alpha$) is equal to :

Let f : S $ \to $ S where S = (0, $\infty $) be a twice differentiable function such that f(x + 1) = xf(x). If g : S $ \to $ R be defined as g(x) = log_e f(x), then the value of |g''(5) $-$ g''(1)| is equal to :

Consider the integral $I = \int_0^{10} {{{[x]{e^{[x]}}} \over {{e^{x - 1}}}}dx} $, where [x] denotes the greatest integer less than or equal to x. Then the value of I is equal to :

Let P(x) = x² + bx + c be a quadratic polynomial with real coefficients such that $\int_0^1 {P(x)dx} $ = 1 and P(x) leaves remainder 5 when it is divided by (x $-$ 2). Then the value of 9(b + c) is equal to :

Let A = {2, 3, 4, 5, ....., 30} and '$ \simeq $' be an equivalence relation on A $\times$ A, defined by (a, b) $ \simeq $ (c, d), if and only if ad = bc. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair (4, 3) is equal to :

The least value of |z| where z is complex number which satisfies the inequality $\exp \left( {{{(|z| + 3)(|z| - 1)} \over {||z| + 1|}}{{\log }_e}2} \right) \ge {\log _{\sqrt 2 }}|5\sqrt 7 + 9i|,i = \sqrt { - 1} $, is equal to :

Let $\alpha$ $\in$ R be such that the function $f(x) = \left\{ {\matrix{ {{{{{\cos }^{ - 1}}(1 - {{\{ x\} }^2}){{\sin }^{ - 1}}(1 - \{ x\} )} \over {\{ x\} - {{\{ x\} }^3}}},} & {x \ne 0} \cr {\alpha ,} & {x = 0} \cr } } \right.$ is continuous at x = 0, where {x} = x $-$ [ x ] is the greatest integer less than or equal to x. Then :

Given that the inverse trigonometric functions take principal values only. Then, the number of real values of x which satisfy ${\sin ^{ - 1}}\left( {{{3x} \over 5}} \right) + {\sin ^{ - 1}}\left( {{{4x} \over 5}} \right) = {\sin ^{ - 1}}x$ is equal to :

Let the lengths of intercepts on x-axis and y-axis made by the circle x² + y² + ax + 2ay + c = 0, (a < 0) be 2${\sqrt 2 }$ and 2${\sqrt 5 }$, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line x + 2y = 0, is equal to :

Let A($-$1, 1), B(3, 4) and C(2, 0) be given three points. A line y = mx, m > 0, intersects lines AC and BC at point P and Q respectively. Let A₁ and A₂ be the areas of $\Delta$ABC and $\Delta$PQC respectively, such that A₁ = 3A₂, then the value of m is equal to :

Let A denote the event that a 6-digit integer formed by 0, 1, 2, 3, 4, 5, 6 without repetitions, be divisible by 3. Then probability of event A is equal to :

The maximum value of $f(x) = \left| {\matrix{ {{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\cos 2x} \cr {1 + {{\sin }^2}x} & {{{\cos }^2}x} & {\cos 2x} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {\sin 2x} \cr } } \right|,x \in R$ is :

In a triangle PQR, the co-ordinates of the points P and Q are ($-$2, 4) and (4, $-$2) respectively. If the equation of the perpendicular bisector of PR is 2x $-$ y + 2 = 0, then the centre of the circumcircle of the $\Delta $PQR is :

The value of $\mathop {\lim }\limits_{x \to {0^ + }} {{{{\cos }^{ - 1}}(x - {{[x]}^2}).{{\sin }^{ - 1}}(x - {{[x]}^2})} \over {x - {x^3}}}$, where [ x ] denotes the greatest integer $ \le $ x is :

Which of the following statements is correct for the function g($\alpha$) for $\alpha$ $\in$ R such that $g(\alpha ) = \int\limits_{{\pi \over 6}}^{{\pi \over 3}} {{{{{\sin }^\alpha }x} \over {{{\cos }^\alpha }x + {{\sin }^\alpha }x}}dx} $

Which of the following is true for y(x) that satisfies the differential equation ${{dy} \over {dx}}$ = xy $-$ 1 + x $-$ y; y(0) = 0 :

The sum of possible values of x for tan^$-$1(x + 1) + cot^$-$1$\left( {{1 \over {x - 1}}} \right)$ = tan^$-$1$\left( {{8 \over {31}}} \right)$ is :

The value of $4 + {1 \over {5 + {1 \over {4 + {1 \over {5 + {1 \over {4 + ......\infty }}}}}}}}$ is :

The inverse of $y = {5^{\log x}}$ is :

If the fourth term in the expansion of ${(x + {x^{{{\log }_2}x}})^7}$ is 4480, then the value of x where x$\in$N is equal to :

The system of equations kx + y + z = 1, x + ky + z = k and x + y + zk = k² has no solution if k is equal to :

If $\cot^{-1}(\alpha) = \cot^{-1}(2) + \cot^{-1}(8) + \cot^{-1}(18) + \cot^{-1}(32) + \ldots \text{ (upto 100 terms)},$ then $\alpha$ is :

Team 'A' consists of 7 boys and n girls and Team 'B' has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then n is equal to :

The area of the triangle with vertices A(z), B(iz) and C(z + iz) is :

Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability of occurrence of 0 at even places be ${1 \over 2}$ and probability of occurrence of 0 at the odd place be ${1 \over 3}$. Then the probability that '10' is followed by '01' is equal to :

Let f : R $ \to $ R be defined as f(x) = e^$-$xsinx. If F : [0, 1] $ \to $ R is a differentiable function with that F(x) = $\int_0^x {f(t)dt} $, then the value of $\int_0^1 {(F'(x) + f(x)){e^x}dx} $ lies in the interval

If $\int {{{\cos x - \sin x} \over {\sqrt {8 - \sin 2x} }}} dx = a{\sin ^{ - 1}}\left( {{{\sin x + \cos x} \over b}} \right) + c$, where c is a constant of integration, thenthe ordered pair (a, b) is equal to :

Let $S_1, S_2$ and $S_3$ be three sets defined as $S_1 = \{z \in C : |z - 1| \le \sqrt{2}\}$ ,$S_2 = \{z \in C : \text{Re}((1 - i)z) \ge 1\}$ $S_3 = \{z \in C : \text{Im}(z) \le 1\}$ Then the set $S_1 \cap S_2 \cap S_3$

If f : R $ \to $ R is a function defined by f(x)= [x - 1]  $\cos \left( {{{2x - 1} \over 2}} \right)\pi $, where [.] denotes the greatestinteger function, then f is :

The number of solutions of the equation ${\sin ^{ - 1}}\left[ {{x^2} + {1 \over 3}} \right] + {\cos ^{ - 1}}\left[ {{x^2} - {2 \over 3}} \right] = {x^2}$, for x$\in$[$-$1, 1], and [x] denotes the greatest integer less than or equal to x, is :

The population P = P(t) at time 't' of a certain species follows the differential equation ${{dP} \over {dt}}$ = 0.5P – 450. If P(0) = 850, then the time at which population becomes zero is :

If the curve y = y(x) is the solution of the differential equation $2({x^2} + {x^{5/4}})dy - y(x + {x^{1/4}})dx = {2x^{9/4}}dx$, x > 0 which passes through the point $\left( {1,1 - {4 \over 3}{{\log }_e}2} \right)$, then the value of y(16) is equal to :

Let $f : R → R$ be defined as $f (x) = 2x – 1$ and $g : R - {1} → R$ be defined as g(x) =${{x - {1 \over 2}} \over {x - 1}}$.Then the composition function $f(g(x))$ is :

Let O be the origin. Let $\overrightarrow{OP} = x\widehat i + y\widehat j - \widehat k$ and $\overrightarrow{OQ} = -\widehat i + 2\widehat j + 3x\widehat k$, $x, y \in R, x > 0$, be such that $|\overrightarrow{PQ}| = \sqrt{20}$ and the vector $\overrightarrow{OP}$ is perpendicular $\overrightarrow{OQ}$. If $\overrightarrow{OR} = 3\widehat i + z\widehat j - 7\widehat k$, $z \in R$, is coplanar with $\overrightarrow{OP}$ and $\overrightarrow{OQ}$, then the value of $x^2 + y^2 + z^2$ is equal to :

The function $f(x) = {{4{x^3} - 3{x^2}} \over 6} - 2\sin x + \left( {2x - 1} \right)\cos x$ :

$ \text{If } \displaystyle \int_{0}^{10}\frac{[\sin 2\pi x]}{e^{,x-[x]}},dx ;=; \alpha e^{-1}+\beta e^{-1/2}+\gamma,\ \text{ where } \alpha,\beta,\gamma \text{ are integers and } [x] \text{ is the greatest integer } \le x,\ \text{then the value of } \alpha+\beta+\gamma \text{ is:} $

$\mathop {\lim }\limits_{x \to 0} {{\int\limits_0^{{x^2}} {\left( {\sin \sqrt t } \right)dt} } \over {{x^3}}}$ is equal to :

The value of the limit $\mathop {\lim }\limits_{\theta \to 0} {{\tan (\pi {{\cos }^2}\theta )} \over {\sin (2\pi {{\sin }^2}\theta )}}$ is equal to :

A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is :

A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\frac{1}{4}$ . Three stones A, B and C are placed at the points (1, 1), (2, 2) and (4, 4) respectively. Then, which of these stones is / are on the path of the man?

The value of $\mathop {\lim }\limits_{n \to \infty } {{[r] + [2r] + ... + [nr]} \over {{n^2}}}$, where r is a non-zero real number and [r] denotes the greatest integer less than or equal to r, is equal to :

Let p and q be two positive numbers such that p + q = 2 and p⁴+q⁴ = 272. Then p and q areroots of the equation :

If x, y, z are in arithmetic progression with common difference d, x $\ne$ 3d, and the determinant of the matrix $\left[ {\matrix{ 3 & {4\sqrt 2 } & x \cr 4 & {5\sqrt 2 } & y \cr 5 & k & z \cr } } \right]$ is zero, then the value of k² is :

The area (in sq. units) of the part of the circle x² + y² = 36, which is outside the parabola y² = 9x, is :

Let y = y(x) be the solution of the differential equation $\cos x(3\sin x + \cos x + 3)dy = (1 + y\sin x(3\sin x + \cos x + 3))dx,0 \le x \le {\pi \over 2},y(0) = 0$. Then, $y\left( {{\pi \over 3}} \right)$ is equal to :

If ${e^{\left( {{{\cos }^2}x + {{\cos }^4}x + {{\cos }^6}x + ...\infty } \right){{\log }_e}2}}$ satisfies the equation t² - 9t + 8 = 0, then the value of ${{2\sin x} \over {\sin x + \sqrt 3 \cos x}}\left( {0 < x < {\pi \over 2}} \right)$ is :

The solutions of the equation $\left| {\matrix{ {1 + {{\sin }^2}x} & {{{\sin }^2}x} & {{{\sin }^2}x} \cr {{{\cos }^2}x} & {1 + {{\cos }^2}x} & {{{\cos }^2}x} \cr {4\sin 2x} & {4\sin 2x} & {1 + 4\sin 2x} \cr } } \right| = 0,(0 < x < \pi )$, are
1 ${\pi \over {12}},{\pi \over 6}$
2 ${\pi \over {6}},{\pi \over 6}$
3 ${5\pi \over {12}},{7\pi \over 12}$
4 ${7\pi \over {12}},{11\pi \over 12}$
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