Let $\vec{a} = \alpha \hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} - \alpha \hat{k}$, $\alpha > 0$. If the projection of $\vec{a} \times \vec{b}$ on the vector $-\hat{i} + 2\hat{j} - 2\hat{k}$ is $30$, then $\alpha$ is equal to:

Let $E_1, E_2, E_3$ be three mutually exclusive events such that $P(E_1)=\dfrac{2+3p}{6}$, $P(E_2)=\dfrac{2-p}{8}$ and $P(E_3)=\dfrac{1-p}{2}$. If the maximum and minimum values of $p$ are $p_1$ and $p_2$, then $(p_1+p_2)$ is equal to:

$\tan\!\left(2\tan^{-1}\!\tfrac{1}{5}+\sec^{-1}\!\tfrac{\sqrt{5}}{2}+2\tan^{-1}\!\tfrac{1}{8}\right)$ is equal to:

The minimum value of the sum of the squares of the roots of $x^{2}+(3-a)x+1=2a$ is:

If $z = x+iy$ satisfies $|z|-2=0$ and $|z-i|-|z+5i|=0$, then :

Let $A=\begin{bmatrix}1\\1\\1\end{bmatrix}$ and $B=\begin{bmatrix} 9^{2} & -10^{2} & 11^{2}\\ 12^{2} & 13^{2} & -14^{2}\\ -15^{2} & 16^{2} & 17^{2} \end{bmatrix}$, then the value of $A'BA$ is:

Let $P$ and $Q$ be any points on the curves $(x-1)^{2}+(y+1)^{2}=1$ and $y=x^{2}$, respectively. The distance between $P$ and $Q$ is minimum for some value of the abscissa of $P$ in the interval:

If the maximum value of $a$, for which the function $f_a(x)=\tan^{-1}(2x)-3ax+7$ is non-decreasing in $\left(-\tfrac{\pi}{6},\,\tfrac{\pi}{6}\right)$, is $\bar a$, then $f_{\bar a}\!\left(\tfrac{\pi}{8}\right)$ is equal to :

Let $\beta=\lim_{x\to 0}\dfrac{\alpha x-(e^{3x}-1)}{\alpha x(e^{3x}-1)}$ for some $\alpha\in\mathbb{R}$. Then the value of $\alpha+\beta$ is:

The value of $\log_{e}2 \, \dfrac{d}{dx}\!\big(\log_{\cos x} \csc x\big)$ at $x=\tfrac{\pi}{4}$ is:

$\int_{0}^{20\pi} (|\sin x| + |\cos x|)^{2} \, dx$ is equal to:

Let the solution curve $y=f(x)$ of the differential equation $\dfrac{dy}{dx}+\dfrac{xy}{x^{2}-1}=\dfrac{x^{4}+2x}{\sqrt{1-x^{2}}}$, $x\in(-1,1)$, pass through the origin. Then $\displaystyle \int_{-\sqrt{3}/2}^{\sqrt{3}/2} f(x)\,dx$ is equal to:

Let the abscissae of the two points $P$ and $Q$ on a circle be the roots of $x^{2}-4x-6=0$ and the ordinates of $P$ and $Q$ be the roots of $y^{2}+2y-7=0$. If $PQ$ is a diameter of the circle $x^{2}+y^{2}+2ax+2by+c=0$, then the value of $(a+b-c)$ is _________. (A)

If the line $x-1=0$ is a directrix of the hyperbola $kx^{2}-y^{2}=6$, then the hyperbola passes through the point:

$ \text{If } 0 < x < \tfrac{1}{\sqrt{2}} \text{ and } \tfrac{\sin^{-1}x}{\alpha} = \tfrac{\cos^{-1}x}{\beta}, \text{ then the value of } \sin!\left(\tfrac{2\pi\alpha}{\alpha+\beta}\right) \text{ is :}$

$ \text{The integral } \int \dfrac{\left(1-\tfrac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{1+\tfrac{2}{\sqrt{3}}\sin 2x},dx \text{ is equal to :}$

$ \text{The area bounded by the curves } y=\lvert x^{2}-1\rvert \text{ and } y=1 \text{ is :}$

$ \text{Let } R_1 \text{ and } R_2 \text{ be two relations defined on } \mathbb{R} \text{ by } a R_1 b \Leftrightarrow ab \ge 0 \text{ and } aR_2b \Leftrightarrow a \ge b. \text{ Then,}$

$ \text{Let } f,g : \mathbb{N} - \{1\} \to \mathbb{N} \text{ be functions defined by } f(a) = \alpha, \text{ where } \alpha \text{ is the maximum of the powers of those primes } p \text{ such that } p^\alpha \text{ divides } a, \text{ and } g(a) = a+1, \text{ for all } a \in \mathbb{N} - \{1\}. \text{ Then, the function } f+g \text{ is} $

$ \text{Let the minimum value } v_{0} \text{ of } v=\lvert z\rvert^{2}+\lvert z-3\rvert^{2}+\lvert z-6i\rvert^{2},\ z\in\mathbb{C} \text{ be attained at } z=z_{0}. \text{ Then } \lvert 2z_{0}^{2}-\overline{z_{0}}^{\,3}+3\rvert^{2}+v_{0}^{2} \text{ is equal to:} $

Let $A = \begin{pmatrix} 1 & 2 \\ -2 & -5 \end{pmatrix}$. Let $\alpha, \beta \in \mathbb{R}$ be such that $\alpha A^2 + \beta A = 2I$. Then $\alpha + \beta$ is equal to :

$ \text{Suppose } a_1, a_2, \ldots, a_n, \ldots \text{ be an arithmetic progression of natural numbers. If } \dfrac{S_5}{S_9} = \dfrac{5}{17} \text{ and } 110 < a_{15} < 120, \text{ then the sum of the first ten terms of the progression is equal to:} $

$ \text{Let } f:\mathbb{R}\to\mathbb{R} \text{ be a function defined as } f(x)=a\sin\!\left(\frac{\pi\lfloor x\rfloor}{2}\right)+\lfloor 2-x\rfloor,\ a\in\mathbb{R}, \text{ where } \lfloor t\rfloor \text{ is the greatest integer } \le t. \text{ If } \lim_{x\to -1} f(x) \text{ exists, then the value of } \int_{0}^{4} f(x)\,dx \text{ is equal to:}$

$ I = \int_{\pi/4}^{\pi/3} \left( \frac{8 \sin x - \sin 2x}{x} \right) dx. \ \text{ Then} $

The area of the smaller region enclosed by the curves $y^2 = 8x + 4$ and $x^2 + y^2 + 4\sqrt{3}x - 4 = 0$ is equal to

Let $y = y_{1}(x)$ and $y = y_{2}(x)$ be two distinct solutions of the differential equation $\dfrac{dy}{dx} = x + y$, with $y_{1}(0) = 0$ and $y_{2}(0) = 1$ respectively. Then, the number of points of intersection of $y = y_{1}(x)$ and $y = y_{2}(x)$ is

Let $\vec{a} = \alpha \hat{i} + \hat{j} + \beta \hat{k}$ and $\vec{b} = 3\hat{i} - 5\hat{j} + 4\hat{k}$ be two vectors, such that $\vec{a} \times \vec{b} = -\hat{i} + 9\hat{j} + 12\hat{k}$. Then the projection of $\vec{b} - 2\vec{a}$ on $\vec{b} + \vec{a}$ is equal to:

Let $S$ be the sample space of all five digit numbers. It $p$ is the probability that a randomly selected number from $S$, is a multiple of $7$ but not divisible by $5$, then $9p$ is equal to :

If the circle $x^{2} + y^{2} - 2gx + 6y - 19c = 0,; g,c \in \mathbb{R}$ passes through the point $(6,1)$ and its centre lies on the line $x - 2cy = 8$, then the length of intercept made by the circle on $x$-axis is :

$2\sqrt{23}$

Let $A(1,1)$, $B(-4,3)$, $C(-2,-5)$ be vertices of a triangle $ABC$, $P$ be a point on side $BC$, and $\Delta_1$ and $\Delta_2$ be the areas of triangles $APB$ and $ABC$, respectively. If $\Delta_1:\Delta_2=4:7$, then the area enclosed by the lines $AP$, $AC$ and the $x$-axis is:

Let a function $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as :

$f(x)= \begin{cases}\int\limits_{0}^{x}(5-|t-3|) d t, & x>4 \\ x^{2}+b x & , x \leq 4\end{cases}$

where $\mathrm{b} \in \mathbb{R}$. If $f$ is continuous at $x=4$, then which of the following statements is NOT true?

The domain of the function $f(x)=\sin^{-1}!\big([,2x^{2}-3,]\big)+\log_{2}!\left(\log_{1/2}(x^{2}-5x+5)\right)$, where $[,\cdot,]$ is the greatest integer function, is:

If $\alpha, \beta$ are the roots of the equation $x^{2} - \left(5 + 3\sqrt{\log_{3}5} - 5\sqrt{\log_{5}3}\right)x + 3\left(3^{\tfrac{1}{3}\log_{3}5} - 5^{\tfrac{2}{3}\log_{5}3} - 1\right) = 0$, then the equation, whose roots are $\alpha + \tfrac{1}{\beta}$ and $\beta + \tfrac{1}{\alpha}$, is:

If for $p\ne q\ne 0$, the function $f(x)=\dfrac{\sqrt[7]{p(729+x)}-3}{\sqrt[3]{729+qx}-9}$ is continuous at $x=0$, then:

Let $A = \{ z \in C:1 \le |z - (1 + i)| \le 2\} $

and $B = \{ z \in A:|z - (1 - i)| = 1\} $. Then, B :

Let $f(x)=2+|x|-|x-1|+|x+1|,;x\in\mathbb{R}$. Consider (S1): $f'!\left(-\tfrac{3}{2}\right)+f'!\left(-\tfrac{1}{2}\right)+f'!\left(\tfrac{1}{2}\right)+f'!\left(\tfrac{3}{2}\right)=2$ (S2): $\displaystyle \int_{-2}^{2} f(x),dx = 12$

The remainder when 3²⁰²² is divided by 5 is :

Let the sum of an infinite G.P., whose first term is $a$ and the common ratio is $r$, be $5$. Let the sum of its first five terms be $\dfrac{98}{25}$. Then the sum of the first $21$ terms of an A.P., whose first term is $10ar$, $n^{\text{th}}$ term is $a_n$ and the common difference is $10ar^{2}$, is equal to:

The area of the region enclosed by $y\le 4x^{2}$, $x^{2}\le 9y$ and $y\le 4$ is equal to:

Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 black, 2 red and n white balls. One bag is chosen at random and 2 balls drawn from it at random, are found to be 1 red and 1 black. If the probability that both balls come from Bag A is ${6 \over {11}}$, then n is equal to __________.

The value of $\displaystyle \int_{0}^{2}!\left(,|2x^{2}-3x|+\big[x-\tfrac{1}{2}\big]\right),dx$, where $[\cdot]$ is the greatest integer function, is equal to:

The number of values of $\alpha$ for which the system of equations :

x + y + z = $\alpha$

$\alpha$x + 2$\alpha$y + 3z = $-$1

x + 3$\alpha$y + 5z = 4

is inconsistent, is

Consider a curve $y=y(x)$ in the first quadrant as shown in the figure. Let the area $A_{1}$ be twice the area $A_{2}$. Then the normal to the curve perpendicular to the line $2x-12y=15$ does NOT pass through the point:

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($\alpha$³ + $\beta$³)² is equal to :

The equations of the sides $AB$, $BC$ and $CA$ of a triangle $ABC$ are $2x+y=0$, $x+py=39$ and $x-y=3$ respectively and $P(2,3)$ is its circumcentre. Then which of the following is NOT true?

The set of all values of k for which ${({\tan ^{ - 1}}x)^3} + {({\cot ^{ - 1}}x)^3} = k{\pi ^3},\,x \in R$, is the interval :

If the length of the perpendicular drawn from the point $P(a,4,2),;a>0$ on the line $\dfrac{x+1}{2}=\dfrac{y-3}{3}=\dfrac{z-1}{-1}$ is $2\sqrt{6}$ units and $Q(\alpha_{1},\alpha_{2},\alpha_{3})$ is the image of the point $P$ in this line, then $a+\sum_{i=1}^{3}\alpha_{i}$ is equal to:

For the function $f(x) = 4{\log _e}(x - 1) - 2{x^2} + 4x + 5,\,x > 1$, which one of the following is NOT correct?

A six faced die is biased such that

$3 \times P(\text{a prime number}) = 6 \times P(\text{a composite number}) = 2 \times P(1).$

Let $X$ be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of $X$ is:

The sum of absolute maximum and absolute minimum values of the function $f(x) = |2{x^2} + 3x - 2| + \sin x\cos x$ in the interval [0, 1] is :

Let $C$ be the centre of the circle $x^{2}+y^{2}-x+2y=\dfrac{11}{4}$ and $P$ be a point on the circle. A line passes through the point $C$, makes an angle of $\dfrac{\pi}{4}$ with the line $CP$ and intersects the circle at the points $Q$ and $R$. Then the area of the triangle $PQR$ (in unit$^{2}$) is :

If $\{ {a_i}\} _{i = 1}^n$, where n is an even integer, is an arithmetic progression with common difference 1, and $\sum\limits_{i = 1}^n {{a_i} = 192} ,\,\sum\limits_{i = 1}^{n/2} {{a_{2i}} = 120} $, then n is equal to :

Let the solution curve of the differential equation $x,dy = \left(\sqrt{x^{2}+y^{2}}+y\right)dx,; x>0,$ intersect the line $x=1$ at $y=0$ and the line $x=2$ at $y=\alpha$. Then the value of $\alpha$ is:

If x = x(y) is the solution of the differential equation $y{{dx} \over {dy}} = 2x + {y^3}(y + 1){e^y},\,x(1) = 0$; then x(e) is equal to :

Considering only the principal values of the inverse trigonometric functions, the domain of the function $f(x)=\cos^{-1}!\left(\dfrac{x^{2}-4x+2}{x^{2}+3}\right)$ is:

Let $\widehat a$, $\widehat b$ be unit vectors. If $\overrightarrow c $ be a vector such that the angle between $\widehat a$ and $\overrightarrow c $ is ${\pi \over {12}}$, and $\widehat b = \overrightarrow c + 2\left( {\overrightarrow c \times \widehat a} \right)$, then ${\left| {6\overrightarrow c } \right|^2}$ is equal to :

Let the vectors $\vec a=(1+t)\hat i+(1-t)\hat j+\hat k$, $\vec b=(1-t)\hat i+(1+t)\hat j+2\hat k$ and $\vec c=t\hat i-t\hat j+\hat k$, $t\in\mathbb R$ be such that for $\alpha,\beta,\gamma\in\mathbb R$, $\alpha\vec a+\beta\vec b+\gamma\vec c=\vec 0\Rightarrow \alpha=\beta=\gamma=0$. Then, the set of all values of $t$ is:

The domain of the function $f(x) = {{{{\cos }^{ - 1}}\left( {{{{x^2} - 5x + 6} \over {{x^2} - 9}}} \right)} \over {{{\log }_e}({x^2} - 3x + 2)}}$ is :

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation $\cos^{-1}(x)-2\sin^{-1}(x)=\cos^{-1}(2x)$ is equal to:

Let $x * y = {x^2} + {y^3}$ and $(x * 1) * 1 = x * (1 * 1)$.

Then a value of $2{\sin ^{ - 1}}\left( {{{{x^4} + {x^2} - 2} \over {{x^4} + {x^2} + 2}}} \right)$ is :

Let a vector $\vec{a}$ has magnitude $9$. Let a vector $\vec{b}$ be such that for every $(x,y)\in\mathbb{R}\times\mathbb{R}-{(0,0)}$, the vector $(x\vec{a}+y\vec{b})$ is perpendicular to the vector $(6y\vec{a}-18x\vec{b})$. Then the value of $|\vec{a}\times\vec{b}|$ is equal to:

The sum of all the real roots of the equation $({e^{2x}} - 4)(6{e^{2x}} - 5{e^x} + 1) = 0$ is

For $t\in(0,2\pi)$, if $\triangle ABC$ is an equilateral triangle with vertices $A(\sin t,-\cos t)$, $B(\cos t,\sin t)$ and $C(a,b)$ such that its orthocentre lies on a circle with centre $\left(1,\tfrac{1}{3}\right)$, then $(a^{2}-b^{2})$ is equal to:

Let the system of linear equations $x + y + \alpha z = 2$, $3x + y + z = 4$, $x + 2z = 1$ have a unique solution $(x^*, y^*, z^*)$. If $(\alpha, x^*)$, $(y^*, \alpha)$ and $(x^*, -y^*)$ are collinear points, then the sum of absolute values of all possible values of $\alpha$ is ?

For $\alpha \in \mathbb{N}$, consider a relation $R$ on $\mathbb{N}$ given by $R={(x,y):3x+\alpha y \text{ is a multiple of } 7}$. The relation $R$ is an equivalence relation if and only if:

Let x, y > 0. If x³y² = 2¹⁵, then the least value of 3x + 2y is

Out of $60%$ female and $40%$ male candidates appearing in an exam, $60%$ candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability that the chosen candidate is a female, is:

Let $$f(x) = \left\{ {\matrix{ {{{\sin (x - [x])} \over {x - [x]}}} & {,\,x \in ( - 2, - 1)} \cr {\max \{ 2x,3[|x|]\} } & {,\,|x| < 1} \cr 1 & {,\,otherwise} \cr } } \right.$$

where [t] denotes greatest integer $$\le$$ t. If m is the number of points where $$f$$ is not continuous and n is the number of points where $$f$$ is not differentiable, then the ordered pair (m, n) is :

If $y=y(x),\; x\in(0,\pi/2)$ be the solution curve of the differential equation $$(\sin^{2}2x)\dfrac{dy}{dx}+(8\sin^{2}2x+2\sin 4x)y=2e^{-4x}(2\sin 2x+\cos 2x),$$ with $y(\pi/4)=e^{-\pi}$, then $y(\pi/6)$ is equal to :

The value of the integral $\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {(1 + {e^x})({{\sin }^6}x + {{\cos }^6}x)}}} $ is equal to

A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with :

Let $S_{1} = \{ z_{1} \in \mathbb{C} : |z_{1} - 3| = \tfrac{1}{2} \}$ and $S_{2} = \{ z_{2} \in \mathbb{C} : |z_{2} - |z_{2} + 1|| = |z_{2} + |z_{2} - 1|| \}$. Then, for $z_{1} \in S_{1}$ and $z_{2} \in S_{2}$, the least value of $|z_{2} - z_{1}|$ is :

Let the maximum area of the triangle that can be inscribed in the ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over 4} = 1,\,a > 2$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the y-axis, be $6\sqrt 3 $. Then the eccentricity of the ellipse is :

If the minimum value of $f(x)=\dfrac{5x^{2}}{2}+\dfrac{\alpha}{x^{5}},; x>0,$ is $14$, then the value of $\alpha$ is equal to:

Let the area of the triangle with vertices $A(1,\alpha)$, $B(\alpha,0)$ and $C(0,\alpha)$ be $4$ sq. units. If the points $(\alpha,-\alpha)$, $(-\alpha,\alpha)$ and $(\alpha^2,\beta)$ are collinear, then $\beta$ is equal to:

$ \text{Let } \alpha, \beta \text{ and } \gamma \text{ be three positive real numbers. Let } f(x) = \alpha x^{5} + \beta x^{3} + \gamma x,; x \in \mathbb{R} \text{ and } g : \mathbb{R} \to \mathbb{R} \text{ be such that } g(f(x)) = x \text{ for all } x \in \mathbb{R}. \text{ If } a_{1}, a_{2}, a_{3}, \ldots, a_{n} \text{ be in arithmetic progression with mean zero, then the value of } f!\left(g!\left(\frac{1}{n}\sum_{i=1}^{n} f(a_{i})\right)\right) \text{ is equal to:}$

The number of distinct real roots of the equation

x⁷ $-$ 7x $-$ 2 = 0 is

The minimum value of the twice differentiable function $f(x)=\int_{0}^{x} e^{,x-t},f'(t),dt-(x^{2}-x+1)e^{x},; x\in\mathbb{R}$, is:

$ \text{Let } \alpha, \beta \text{ be the roots of the equation } x^{2} - \sqrt{2}x + \sqrt{6} = 0 \text{ and } \dfrac{1}{\alpha^{2}} + 1, ; \dfrac{1}{\beta^{2}} + 1 \text{ be the roots of the equation } x^{2} + ax + b = 0. $ $\text{Then the roots of the equation } x^{2} - (a+b-2)x + (a+b+2) = 0 \text{ are :}$

$S = \{\, x \in [-6,3] \setminus \{-2,2\} \;:\; \dfrac{|x+3|-1}{|x|-2} \geq 0 \,\}$ $T = \{\, x \in \mathbb{Z} \;:\; x^{2} - 7|x| + 9 \leq 0 \,\}$ Then the number of elements in $S \cap T$ is :

$ \text{Let } A \text{ and } B \text{ be any two } 3\times 3 \text{ symmetric and skew-symmetric matrices respectively. Then which of the following is NOT true?} $

$ \text{Let } f(x)=ax^{2}+bx+c \text{ be such that } f(1)=3,\ f(-2)=\lambda \text{ and } f(3)=4. $ $ \text{If } f(0)+f(1)+f(-2)+f(3)=14,\ \text{then } \lambda \text{ is equal to:} $

The function $f : \mathbb{R} \to \mathbb{R}$ defined by $$ f(x) = \lim_{n \to \infty} \frac{\cos(2 \pi x) - x^{2n} \sin(x-1)}{1 + x^{2n+1} - x^{2n}} $$ is continuous for all $x$ in :

The value of $\int\limits_0^\pi {{{{e^{\cos x}}\sin x} \over {(1 + {{\cos }^2}x)({e^{\cos x}} + {e^{ - \cos x}})}}dx} $ is equal to:

The function $f(x) = x e^{\,x(1-x)}, \, x \in \mathbb{R}$ is :

Let f : N $\to$ R be a function such that $f(x + y) = 2f(x)f(y)$ for natural numbers x and y. If f(1) = 2, then the value of $\alpha$ for which

$\sum\limits_{k = 1}^{10} {f(\alpha + k) = {{512} \over 3}({2^{20}} - 1)} $

holds, is :

The sum of the absolute maximum and absolute minimum values of the function $$f(x) = \tan^{-1}(\sin x - \cos x)$$ in the interval $[0,\pi]$ is :

Let $A$ be a $3 \times 3$ real matrix such that $A\begin{pmatrix}1 \\ 1 \\ 0\end{pmatrix} = \begin{pmatrix}1 \\ 1 \\ 0\end{pmatrix}, \quad A\begin{pmatrix}1 \\ 0 \\ 1\end{pmatrix} = \begin{pmatrix}-1 \\ 0 \\ 1\end{pmatrix}, \quad A\begin{pmatrix}0 \\ 0 \\ 1\end{pmatrix} = \begin{pmatrix}1 \\ 1 \\ 2\end{pmatrix}.$ If $X = (x_1, x_2, x_3)^T$ and $I$ is an identity matrix of order $3$, then the system $(A - 2I)X = \begin{pmatrix}4 \\ 1 \\ 1\end{pmatrix}$ has:

Let $x(t) = 2\sqrt{2}\cos t \sqrt{\sin 2t}$ and $y(t) = 2\sqrt{2}\sin t \sqrt{\sin 2t}, \; t \in (0,\tfrac{\pi}{2}).$ Then $\dfrac{1+\left(\tfrac{dy}{dx}\right)^2}{\tfrac{d^2y}{dx^2}}$ at $t=\tfrac{\pi}{4}$ is equal to :

Let f : R $\to$ R be defined as $f(x) = {x^3} + x - 5$. If g(x) is a function such that $f(g(x)) = x,\forall 'x' \in R$, then g'(63) is equal to ________________.

Let $I_n(x) = \int_0^x \dfrac{1}{(t^2+5)^n} \, dt, \; n = 1, 2, 3, \dots$ Then :

If ${1 \over {2\,.\,{3^{10}}}} + {1 \over {{2^2}\,.\,{3^9}}} + \,\,.....\,\, + \,\,{1 \over {{2^{10}}\,.\,3}} = {K \over {{2^{10}}\,.\,{3^{10}}}}$, then the remainder when K is divided by 6 is :

$ \text{The area enclosed by the curves } y=\log_{e}(x+e^{2}),; x=\log_{e}!\left(\dfrac{2}{y}\right) \text{ and } x=\log_{e}2,\ \text{above the line } y=1,\ \text{is:} $

Let f(x) be a polynomial function such that $f(x) + f'(x) + f''(x) = {x^5} + 64$. Then, the value of $\mathop {\lim }\limits_{x \to 1} {{f(x)} \over {x - 1}}$ is equal to:

$ \text{Let } y=y(x) \text{ be the solution curve of the differential equation } \dfrac{dy}{dx}+\dfrac{1}{x^{2}-1},y=\left(\dfrac{x-1}{x+1}\right)^{1/2},; x>1,\ \text{passing through the point } \left(2,\sqrt{\tfrac{1}{3}}\right). \text{ Then } \sqrt{7},y(8) \text{ is equal to:} $

Let E₁ and E₂ be two events such that the conditional probabilities $P({E_1}|{E_2}) = {1 \over 2}$, $P({E_2}|{E_1}) = {3 \over 4}$ and $P({E_1} \cap {E_2}) = {1 \over 8}$. Then :

Let the hyperbola $H : \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ pass through the point $(2\sqrt{2}, -2\sqrt{2})$. A parabola is drawn whose focus is same as the focus of $H$ with positive abscissa and the directrix of the parabola passes through the other focus of $H$. If the length of the latus rectum of the parabola is $e$ times the length of the latus rectum of $H$, where $e$ is the eccentricity of $H$, then which of the following points lies on the parabola?

Let $A = \left[ {\matrix{ 0 & { - 2} \cr 2 & 0 \cr } } \right]$. If M and N are two matrices given by $M = \sum\limits_{k = 1}^{10} {{A^{2k}}} $ and $N = \sum\limits_{k = 1}^{10} {{A^{2k - 1}}} $ then MN² is :

$ \text{Let } S \text{ be the set of all } a \in \mathbb{R} \text{ for which the angle between the vectors } \vec{u}=a(\log_{e} b),\hat{i}-6\hat{j}+3\hat{k} \text{ and } \vec{v}=(\log_{e} b),\hat{i}+2\hat{j}+2a(\log_{e} b),\hat{k},\ (b>1), \text{ is acute. Then } S \text{ is equal to:} $

Let $g:(0,\infty ) \to R$ be a differentiable function such that $\int {\left( {{{x(\cos x - \sin x)} \over {{e^x} + 1}} + {{g(x)\left( {{e^x} + 1 - x{e^x}} \right)} \over {{{({e^x} + 1)}^2}}}} \right)dx = {{x\,g(x)} \over {{e^x} + 1}} + c} $, for all x > 0, where c is an arbitrary constant. Then :

$ \text{Let A and B be two events such that } P(B|A)=\frac{2}{5}, P(A|B)=\frac{1}{7},; \text{and } P(A\cap B)=\frac{1}{9}. $ Consider:(S1) $P(A' \cup B)=\frac{5}{6}$ (S2) $P(A' \cap B')=\frac{1}{18}$

Let $f:R \to R$ and $g:R \to R$ be two functions defined by $f(x) = {\log _e}({x^2} + 1) - {e^{ - x}} + 1$ and $g(x) = {{1 - 2{e^{2x}}} \over {{e^x}}}$. Then, for which of the following range of $\alpha$, the inequality $f\left( {g\left( {{{{{(\alpha - 1)}^2}} \over 3}} \right)} \right) > f\left( {g\left( {\alpha -{5 \over 3}} \right)} \right)$ holds ?

Let $R$ be a relation from the set ${1,2,3,\dots,60}$ to itself such that R={(a,b):b=pq,    where p,q≥3 are prime numbers}.R = \{(a,b) : b = pq, \;\; \text{where $p,q \geq 3$ are prime numbers} \}.R={(a,b):b=pq,where p,q≥3 are prime numbers}. Then, the number of elements in $R$ is :

Let $\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k$ ${a_i} > 0$, $i = 1,2,3$ be a vector which makes equal angles with the coordinate axes OX, OY and OZ. Also, let the projection of $\overrightarrow a $ on the vector $3\widehat i + 4\widehat j$ be 7. Let $\overrightarrow b $ be a vector obtained by rotating $\overrightarrow a $ with 90$^\circ$. If $\overrightarrow a $, $\overrightarrow b $ and x-axis are coplanar, then projection of a vector $\overrightarrow b $ on $3\widehat i + 4\widehat j$ is equal to:

If $z = 2 + 3i$, then $z^5 + (\bar{z})^5$ is equal to:

Let $y = y(x)$ be the solution of the differential equation $(x + 1)y' - y = {e^{3x}}{(x + 1)^2}$, with $y(0) = {1 \over 3}$. Then, the point $x = - {4 \over 3}$ for the curve $y = y(x)$ is :

Let $A$ and $B$ be two $3 \times 3$ non-zero real matrices such that $AB$ is a zero matrix. Then

If the solution curve $y = y(x)$ of the differential equation ${y^2}dx + ({x^2} - xy + {y^2})dy = 0$, which passes through the point (1, 1) and intersects the line $y = \sqrt 3 x$ at the point $(\alpha ,\sqrt 3 \alpha )$, then value of ${\log _e}(\sqrt 3 \alpha )$ is equal to :

If $\dfrac{1}{(20-a)(40-a)} + \dfrac{1}{(40-a)(60-a)} + \cdots + \dfrac{1}{(180-a)(200-a)} = \dfrac{1}{256}$, then the maximum value of $a$ is :

Let $x = 2t$, $y = {{{t^2}} \over 3}$ be a conic. Let S be the focus and B be the point on the axis of the conic such that $SA \bot BA$, where A is any point on the conic. If k is the ordinate of the centroid of the $\Delta$SAB, then $\mathop {\lim }\limits_{t \to 1} k$ is equal to :

If $\lim_{x \to 0} \dfrac{\alpha e^{x^2} + \beta e^{-x} + \gamma \sin x}{x \sin^2 x} = \dfrac{2}{3}$, where $\alpha, \beta, \gamma \in \mathbb{R}$, then which of the following is NOT correct?

Let a circle C in complex plane pass through the points ${z_1} = 3 + 4i$, ${z_2} = 4 + 3i$ and ${z_3} = 5i$. If $z( \ne {z_1})$ is a point on C such that the line through z and z₁ is perpendicular to the line through z₂ and z₃, then $arg(z)$ is equal to :

The integral $\int_{0}^{\tfrac{\pi}{2}} \dfrac{1}{3 + 2 \sin x + \cos x} , dx$ is equal to :

Let $A = \{ x \in R:|x + 1| < 2\} $ and $B = \{ x \in R:|x - 1| \ge 2\} $. Then which one of the following statements is NOT true?

Let the solution curve $y = y(x)$ of the differential equation $\left(1 + e^{2x}\right)\left(\dfrac{dy}{dx} + y\right) = 1$ pass through the point $\left(0, \dfrac{\pi}{2}\right)$. Then, $\lim_{x \to \infty} e^x y(x)$ is equal to :

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 the equation ${x^2} - 2bx + 21 = 0$, then ${\alpha ^2} + {\beta ^2}$ is equal to :

$ \text{Let the solution curve } y=y(x) \text{ of the differential equation } (1+e^{2x})!\left(\dfrac{dy}{dx}+y\right)=1 \text{ pass through the point } \left(0,\dfrac{\pi}{2}\right). $ $ \text{Then } \lim_{x\to\infty} e^{x}y(x) \text{ is equal to:} $

Let z₁ and z₂ be two complex numbers such that ${\overline z _1} = i{\overline z _2}$ and $\arg \left( {{{{z_1}} \over {{{\overline z }_2}}}} \right) = \pi $. Then :

Let a line L pass through the point of intersection of the lines $b x+10 y-8=0$ and $2 x-3 y=0, \mathrm{~b} \in \mathbf{R}-\left\{\frac{4}{3}\right\}$. If the line $\mathrm{L}$ also passes through the point $(1,1)$ and touches the circle $17\left(x^{2}+y^{2}\right)=16$, then the eccentricity of the ellipse $\frac{x^{2}}{5}+\frac{y^{2}}{\mathrm{~b}^{2}}=1$ is :

The system of equations

$ - kx + 3y - 14z = 25$

$ - 15x + 4y - kz = 3$

$ - 4x + y + 3z = 4$

is consistent for all k in the set

Let the circumcentre of a triangle with vertices A(a, 3), B(b, 5) and C(a, b), ab > 0 be P(1,1). If the line AP intersects the line BC at the point Q$\left(k_{1}, k_{2}\right)$, then $k_{1}+k_{2}$ is equal to :

$\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{{\tan }^2}x\left( {{{(2{{\sin }^2}x + 3\sin x + 4)}^{{1 \over 2}}} - {{({{\sin }^2}x + 6\sin x + 2)}^{{1 \over 2}}}} \right)} \right)$ is equal to

Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{4}$. If $\theta$ is the angle between the vectors $(\hat{a}+\hat{b})$ and $(\hat{a}+2 \hat{b}+2(\hat{a} \times \hat{b}))$, then the value of $164 \,\cos ^{2} \theta$ is equal to :

The area of the region enclosed between the parabolas y² = 2x $-$ 1 and y² = 4x $-$ 3 is

If $f(\alpha)=\int\limits_{1}^{\alpha} \frac{\log _{10} \mathrm{t}}{1+\mathrm{t}} \mathrm{dt}, \alpha>0$ then $f\left(\mathrm{e}^{3}\right)+f\left(\mathrm{e}^{-3}\right)$ is equal to :

The coefficient of x¹⁰¹ in the expression ${(5 + x)^{500}} + x{(5 + x)^{499}} + {x^2}{(5 + x)^{498}} + \,\,.....\,\, + \,\,{x^{500}}$, x > 0, is

The area of the region

$\left\{(x, y):|x-1| \leq y \leq \sqrt{5-x^{2}}\right\}$ is equal to :

Water is being filled at the rate of 1 cm³ / sec in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the height of the water level is 10 cm, the rate (in cm² / sec) at which the wet conical surface area of the vessel increases is

If ${b_n} = \int_0^{{\pi \over 2}} {{{{{\cos }^2}nx} \over {\sin x}}dx,\,n \in N} $, then

$ \text{Let the focal chord of the parabola } P: y^{2}=4x \text{ along the line } L: y=mx+c,; m>0 \text{ meet the parabola at the points } M \text{ and } N. \text{ Let the line } L \text{ be a tangent to the hyperbola } H: x^{2}-y^{2}=4. \text{ If } O \text{ is the vertex of } P \text{ and } F \text{ is the focus of } H \text{ on the positive } x\text{-axis, then the area of the quadrilateral } OMFN \text{ is:} $

If $y = y(x)$ is the solution of the differential equation $2{x^2}{{dy} \over {dx}} - 2xy + 3{y^2} = 0$ such that $y(e) = {e \over 3}$, then y(1) is equal to :

$ \text{The number of points where the function } f:\mathbb{R}\to\mathbb{R},\quad f(x)=|x-1|\cos|x-2|\sin|x-1|+(x-3),|x^{2}-5x+4|,\ \text{is NOT differentiable, is:} $

The value of 2sin (12$^\circ$) $-$ sin (72$^\circ$) is :

A biased die is marked with numbers 2, 4, 8, 16, 32, 32 on its faces and the probability of getting a face with mark n is ${1 \over n}$. If the die is thrown thrice, then the probability, that the sum of the numbers obtained is 48, is :

$ \text{Let } f(x)=3^{(x^{2}-2)^{3}+4},; x\in\mathbb{R}. \text{ Then which of the following statements are true?} $ $P: x=0 \text{ is a point of local minima of } f$ $Q: x=\sqrt{2} \text{ is a point of inflection of } f$ $R: f' \text{ is increasing for } x>\sqrt{2}$

The value of ${\tan ^{ - 1}}\left( {{{\cos \left( {{{15\pi } \over 4}} \right) - 1} \over {\sin \left( {{\pi \over 4}} \right)}}} \right)$ is equal to

If $z \neq 0$ be a complex number such that $\left|z - \frac{1}{z}\right| = 2$, then the maximum value of $|z|$ is :

The line y = x + 1 meets the ellipse ${{{x^2}} \over 4} + {{{y^2}} \over 2} = 1$ at two points P and Q. If r is the radius of the circle with PQ as diameter then (3r)² is equal to :

Let the function $ f(x) = \begin{cases} \dfrac{\log_e(1+5x) - \log_e(1+\alpha x)}{x}, & x \neq 0 \\ 10, & x = 0 \end{cases} $ be continuous at $x=0$. Then $\alpha$ is equal to:

Let $f(x) = {{x - 1} \over {x + 1}},\,x \in R - \{ 0, - 1,1\} $. If ${f^{n + 1}}(x) = f({f^n}(x))$ for all n $\in$ N, then ${f^6}(6) + {f^7}(7)$ is equal to

Which of the following matrices can NOT be obtained from the matrix $\begin{bmatrix}-1 & 2 \\ 1 & -1\end{bmatrix}$ by a single elementary row operation?

Let $A = \left\{ {z \in C:\left| {{{z + 1} \over {z - 1}}} \right| < 1} \right\}$ and $B = \left\{ {z \in C:\arg \left( {{{z - 1} \over {z + 1}}} \right) = {{2\pi } \over 3}} \right\}$. Then A $\cap$ B is :

If $[t]$ denotes the greatest integer $\leq t$, then the value of $ \int_{0}^{1} \left[ 2x - |3x^{2} - 5x + 2| + 1 \right] \, dx $ is :

The ordered pair (a, b), for which the system of linear equations

3x $-$ 2y + z = b

5x $-$ 8y + 9z = 3

2x + y + az = $-$1

has no solution, is :

For $ I(x) = \int \frac{\sec^{2}x - 2022}{\sin^{2022}x} \, dx, $ if $ I\!\left(\frac{\pi}{4}\right) = 2^{1011}, $ then

The remainder when (2021)²⁰²³ is divided by 7 is :

If the solution curve of the differential equation $ \dfrac{dy}{dx}=\dfrac{x+y-2}{x-y} $ passes through the points $(2,1)$ and $(k+1,2)$, $k>0$, then

$\mathop {\lim }\limits_{x \to {1 \over {\sqrt 2 }}} {{\sin ({{\cos }^{ - 1}}x) - x} \over {1 - \tan ({{\cos }^{ - 1}}x)}}$ is equal to :

Let $y=y(x)$ be the solution curve of the differential equation $ \frac{dy}{dx}+\left(\frac{2x^{2}+11x+13}{x^{3}+6x^{2}+11x+6}\right)y=\frac{x+3}{x+1},\quad x>-1, $ which passes through the point $(0,1)$. Then $y(1)$ is equal to:

Let f, g : R $\to$ R be two real valued functions defined as $f(x) = \left\{ {\matrix{ { - |x + 3|} & , & {x < 0} \cr {{e^x}} & , & {x \ge 0} \cr } } \right.$ and $g(x) = \left\{ {\matrix{ {{x^2} + {k_1}x} & , & {x < 0} \cr {4x + {k_2}} & , & {x \ge 0} \cr } } \right.$, where k₁ and k₂ are real constants. If (gof) is differentiable at x = 0, then (gof) ($-$ 4) + (gof) (4) is equal to :

Let $m_1, m_2$ be the slopes of two adjacent sides of a square of side $a$ such that $a^{2}+11a+3\left(m_{1}^{2}+m_{2}^{2}\right)=220.$ If one vertex of the square is $\big(10(\cos\alpha-\sin\alpha),\,10(\sin\alpha+\cos\alpha)\big)$, where $\alpha\in(0,\tfrac{\pi}{2})$, and the equation of one diagonal is $(\cos\alpha-\sin\alpha)x+(\sin\alpha+\cos\alpha)y=10$, then $ 72\left(\sin^{4}\alpha+\cos^{4}\alpha\right)+a^{2}-3a+13 $ is equal to:

The sum of the absolute minimum and the absolute maximum values of the function f(x) = |3x $-$ x² + 2| $-$ x in the interval [$-$1, 2] is :

Let $A(\alpha,-2)$, $B(\alpha,6)$ and $C\!\left(\dfrac{\alpha}{4},-2\right)$ be vertices of $\triangle ABC$. If $\left(5,\dfrac{\alpha}{4}\right)$ is the circumcentre of $\triangle ABC$, then which of the following is NOT correct about $\triangle ABC$?

The area bounded by the curve y = |x² $-$ 9| and the line y = 3 is :

Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be black in colour. Then the probability that the transferred ball is red is:

Let R be the point (3, 7) and let P and Q be two points on the line x + y = 5 such that PQR is an equilateral triangle. Then the area of $\Delta$PQR is :

Let $S=\{\,z=x+iy:\ |z-1+i|\ge |z|,\ |z|<2,\ |z+i|=|z-1|\,\}$. Then the set of all values of $x$, for which $w=2x+iy\in S$ for some $y\in\mathbb{R}$, is:

If the two lines ${l_1}:{{x - 2} \over 3} = {{y + 1} \over {-2}},\,z = 2$ and ${l_2}:{{x - 1} \over 1} = {{2y + 3} \over \alpha } = {{z + 5} \over 2}$ are perpendicular, then an angle between the lines l₂ and ${l_3}:{{1 - x} \over 3} = {{2y - 1} \over { - 4}} = {z \over 4}$ is :

Let $\vec{a},\vec{b},\vec{c}$ be three coplanar concurrent vectors such that the angles between any two of them are the same. If the product of their magnitudes is $14$ and $ (\vec{a}\times\vec{b})\cdot(\vec{b}\times\vec{c}) +(\vec{b}\times\vec{c})\cdot(\vec{c}\times\vec{a}) +(\vec{c}\times\vec{a})\cdot(\vec{a}\times\vec{b})=168, $ then $|\vec{a}|+|\vec{b}|+|\vec{c}|$ is equal to:

The domain of the function $ f(x)=\sin^{-1}\!\left(\frac{x^{2}-3x+2}{x^{2}+2x+7}\right) $ is:

Let $f(x) = 2{\cos ^{ - 1}}x + 4{\cot ^{ - 1}}x - 3{x^2} - 2x + 10$, $x \in [ - 1,1]$. If [a, b] is the range of the function f, then 4a $-$ b is equal to :

If the system of equations $x+y+z=6$ $2x+5y+\alpha z=\beta$ $x+2y+3z=14$ has infinitely many solutions, then $\alpha+\beta$ is equal to:

Let f : R $\to$ R be defined as f (x) = x $-$ 1 and g : R $-$ {1, $-$1} $\to$ R be defined as $g(x) = {{{x^2}} \over {{x^2} - 1}}$.

Then the function fog is :

If the system of equations

$\alpha$x + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = $\beta$

has infinitely many solutions, then the ordered pair ($\alpha$, $\beta$) is equal to :

$\mathop {\lim }\limits_{x \to 0} {{\cos (\sin x) - \cos x} \over {{x^4}}}$ is equal to :

Let f(x) = min {1, 1 + x sin x}, 0 $\le$ x $\le$ 2$\pi $. If m is the number of points, where f is not differentiable and n is the number of points, where f is not continuous, then the ordered pair (m, n) is equal to

The area of the region bounded by y² = 8x and y² = 16(3 $-$ x) is equal to:

If $\int {{1 \over x}\sqrt {{{1 - x} \over {1 + x}}} dx = g(x) + c} $, $g(1) = 0$, then $g\left( {{1 \over 2}} \right)$ is equal to :

If $y = y(x)$ is the solution of the differential equation $x{{dy} \over {dx}} + 2y = x\,{e^x}$, $y(1) = 0$ then the local maximum value of the function $z(x) = {x^2}y(x) - {e^x},\,x \in R$ is :

f the solution of the differential equation ${{dy} \over {dx}} + {e^x}\left( {{x^2} - 2} \right)y = \left( {{x^2} - 2x} \right)\left( {{x^2} - 2} \right){e^{2x}}$ satisfies $y(0) = 0$, then the value of y(2) is _______________.

The locus of the mid point of the line segment joining the point (4, 3) and the points on the ellipse ${x^2} + 2{y^2} = 4$ is an ellipse with eccentricity :

Let $\overrightarrow a = \widehat i + \widehat j + 2\widehat k$, $\overrightarrow b = 2\widehat i - 3\widehat j + \widehat k$ and $\overrightarrow c = \widehat i - \widehat j + \widehat k$ be three given vectors. Let $\overrightarrow v $ be a vector in the plane of $\overrightarrow a $ and $\overrightarrow b $ whose projection on $\overrightarrow c $ is ${2 \over {\sqrt 3 }}$. If $\overrightarrow v \,.\,\widehat j = 7$, then $\overrightarrow v \,.\,\left( {\widehat i + \widehat k} \right)$ is equal to :

$16\sin (20^\circ )\sin (40^\circ )\sin (80^\circ )$ is equal to :

If the inverse trigonometric functions take principal values then ${\cos ^{ - 1}}\left( {{3 \over {10}}\cos \left( {{{\tan }^{ - 1}}\left( {{4 \over 3}} \right)} \right) + {2 \over 5}\sin \left( {{{\tan }^{ - 1}}\left( {{4 \over 3}} \right)} \right)} \right)$ is equal to :

The area of the polygon, whose vertices are the non-real roots of the equation $\overline z = i{z^2}$ is :

Let the system of linear equations $x + 2y + z = 2$, $\alpha x + 3y - z = \alpha $, $ - \alpha x + y + 2z = - \alpha $ be inconsistent. Then $\alpha$ is equal to :

$x = \sum\limits_{n = 0}^\infty {{a^n},y = \sum\limits_{n = 0}^\infty {{b^n},z = \sum\limits_{n = 0}^\infty {{c^n}} } } $, where a, b, c are in A.P. and |a| < 1, |b| < 1, |c| < 1, abc $\ne$ 0, then :

Let a be an integer such that $\mathop {\lim }\limits_{x \to 7} {{18 - [1 - x]} \over {[x - 3a]}}$ exists, where [t] is greatest integer $\le$ t. Then a is equal to :

If ${\cos ^{ - 1}}\left( {{y \over 2}} \right) = {\log _e}{\left( {{x \over 5}} \right)^5},\,|y| < 2$, then :

If $\int {{{({x^2} + 1){e^x}} \over {{{(x + 1)}^2}}}dx = f(x){e^x} + C} $, where C is a constant, then ${{{d^3}f} \over {d{x^3}}}$ at x = 1 is equal to :

The value of the integral $\int\limits_{ - 2}^2 {{{|{x^3} + x|} \over {({e^{x|x|}} + 1)}}dx} $ is equal to :

If ${{dy} \over {dx}} + {{{2^{x - y}}({2^y} - 1)} \over {{2^x} - 1}} = 0$, x, y > 0, y(1) = 1, then y(2) is equal to :

In an isosceles triangle ABC, the vertex A is (6, 1) and the equation of the base BC is 2x + y = 4. Let the point B lie on the line x + 3y = 7. If ($\alpha$, $\beta$) is the centroid of $\Delta$ABC, then 15($\alpha$ + $\beta$) is equal to :

Let the eccentricity of an ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$, $a > b$, be ${1 \over 4}$. If this ellipse passes through the point $\left( { - 4\sqrt {{2 \over 5}} ,3} \right)$, then ${a^2} + {b^2}$ is equal to :

If two straight lines whose direction cosines are given by the relations $l + m - n = 0$, $3{l^2} + {m^2} + cnl = 0$ are parallel, then the positive value of c is :

Let $\overrightarrow a = \widehat i + \widehat j - \widehat k$ and $\overrightarrow c = 2\widehat i - 3\widehat j + 2\widehat k$. Then the number of vectors $\overrightarrow b $ such that $\overrightarrow b \times \overrightarrow c = \overrightarrow a $ and $|\overrightarrow b | \in $ {1, 2, ........, 10} is :

Five numbers ${x_1},{x_2},{x_3},{x_4},{x_5}$ are randomly selected from the numbers 1, 2, 3, ......., 18 and are arranged in the increasing order $({x_1} < {x_2} < {x_3} < {x_4} < {x_5})$. The probability that ${x_2} = 7$ and ${x_4} = 11$ is :

The value of $\cos \left( {{{2\pi } \over 7}} \right) + \cos \left( {{{4\pi } \over 7}} \right) + \cos \left( {{{6\pi } \over 7}} \right)$ is equal to :

$\sin^{-1}\left(\sin \frac{2\pi}{3}\right) + \cos^{-1}\left(\cos \frac{7\pi}{6}\right) + \tan^{-1}\left(\tan \frac{3\pi}{4}\right)$ is equal to:

The number of points of intersection of $|z - (4 + 3i)| = 2$ and $|z| + |z - 4| = 6$, z $\in$ C, is :

If a₁, a₂, a₃ ...... and b₁, b₂, b₃ ....... are A.P., and a₁ = 2, a₁₀ = 3, a₁b₁ = 1 = a₁₀b₁₀, then a₄ b₄ is equal to

If m and n respectively are the number of local maximum and local minimum points of the function $f(x) = \int\limits_0^{{x^2}} {{{{t^2} - 5t + 4} \over {2 + {e^t}}}dt} $, then the ordered pair (m, n) is equal to

Let f be a differentiable function in $\left( {0,{\pi \over 2}} \right)$. If $\int\limits_{\cos x}^1 {{t^2}\,f(t)dt = {{\sin }^3}x + \cos x} $, then ${1 \over {\sqrt 3 }}f'\left( {{1 \over {\sqrt 3 }}} \right)$ is equal to

The integral $\int\limits_0^1 {{1 \over {{7^{\left[ {{1 \over x}} \right]}}}}dx} $, where [ . ] denotes the greatest integer function, is equal to

If the solution curve of the differential equation $(({\tan ^{ - 1}}y) - x)dy = (1 + {y^2})dx$ passes through the point (1, 0), then the abscissa of the point on the curve whose ordinate is tan(1), is

If the equation of the parabola, whose vertex is at (5, 4) and the directrix is $3x + y - 29 = 0$, is ${x^2} + a{y^2} + bxy + cx + dy + k = 0$, then $a + b + c + d + k$ is equal to :

The set of values of k, for which the circle $C:4{x^2} + 4{y^2} - 12x + 8y + k = 0$ lies inside the fourth quadrant and the point $\left( {1, - {1 \over 3}} \right)$ lies on or inside the circle C, is :

The shortest distance between the lines ${{x - 3} \over 2} = {{y - 2} \over 3} = {{z - 1} \over { - 1}}$ and ${{x + 3} \over 2} = {{y - 6} \over 1} = {{z - 5} \over 3}$, is :

Let $\overrightarrow a $ and $\overrightarrow b $ be the vectors along the diagonals of a parallelogram having area $2\sqrt 2 $. Let the angle between $\overrightarrow a $ and $\overrightarrow b $ be acute, $|\overrightarrow a | = 1$, and $|\overrightarrow a \,.\,\overrightarrow b | = |\overrightarrow a \times \overrightarrow b |$. If $\overrightarrow c = 2\sqrt 2 \left( {\overrightarrow a \times \overrightarrow b } \right) - 2\overrightarrow b $, then an angle between $\overrightarrow b $ and $\overrightarrow c $ is :

The mean and variance of the data 4, 5, 6, 6, 7, 8, x, y, where x < y, are 6 and ${9 \over 4}$ respectively. Then ${x^4} + {y^2}$ is equal to :

If a point A(x, y) lies in the region bounded by the y-axis, straight lines 2y + x = 6 and 5x $-$ 6y = 30, then the probability that y < 1 is :

The value of $\cot \left( {\sum\limits_{n = 1}^{50} {{{\tan }^{ - 1}}\left( {{1 \over {1 + n + {n^2}}}} \right)} } \right)$ is :

$\alpha = \sin 36^\circ $ is a root of which of the following equation?

Let a function f : N $\to$ N be defined by

$f(n) = \left[ {\matrix{ {2n,} & {n = 2,4,6,8,......} \cr {n - 1,} & {n = 3,7,11,15,......} \cr {{{n + 1} \over 2},} & {n = 1,5,9,13,......} \cr } } \right.$ then, f is

If the system of linear equations

$2x + 3y - z = - 2$

$x + y + z = 4$

$x - y + |\lambda |z = 4\lambda - 4$

where, $\lambda$ $\in$ R, has no solution, then

Let A₁, A₂, A₃, ....... be an increasing geometric progression of positive real numbers. If A₁A₃A₅A₇ = ${1 \over {1296}}$ and A₂ + A₄ = ${7 \over {36}}$, then the value of A₆ + A₈ + A₁₀ is equal to

Let [t] denote the greatest integer less than or equal to t. Then, the value of the integral $\int\limits_0^1 {[ - 8{x^2} + 6x - 1]dx} $ is equal to :

Let f : R $\to$ R be defined as

$f(x) = \left[ {\matrix{ {[{e^x}],} & {x < 0} \cr {a{e^x} + [x - 1],} & {0 \le x < 1} \cr {b + [\sin (\pi x)],} & {1 \le x < 2} \cr {[{e^{ - x}}] - c,} & {x \ge 2} \cr } } \right.$

where a, b, c $\in$ R and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true?

The area of the region S = {(x, y) : y² $\le$ 8x, y $\ge$ $\sqrt2$x, x $\ge$ 1} is

Let the solution curve $y = y(x)$ of the differential equation

$\left[ {{x \over {\sqrt {{x^2} - {y^2}} }} + {e^{{y \over x}}}} \right]x{{dy} \over {dx}} = x + \left[ {{x \over {\sqrt {{x^2} - {y^2}} }} + {e^{{y \over x}}}} \right]y$

pass through the points (1, 0) and (2$\alpha$, $\alpha$), $\alpha$ > 0. Then $\alpha$ is equal to

Let y = y(x) be the solution of the differential equation $x(1 - {x^2}){{dy} \over {dx}} + (3{x^2}y - y - 4{x^3}) = 0$, $x > 1$, with $y(2) = - 2$. Then y(3) is equal to :

The number of real solutions of :- ${x^7} + 5{x^3} + 3x + 1 = 0$ is equal to ____________.

Let R₁ = {(a, b) $\in$ N $\times$ N : |a $-$ b| $\le$ 13} and

R₂ = {(a, b) $\in$ N $\times$ N : |a $-$ b| $\ne$ 13}. Then on N :

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 equal to :

The term independent of x in the expansion of $(1 - {x^2} + 3{x^3}){\left( {{5 \over 2}{x^3} - {1 \over {5{x^2}}}} \right)^{11}},\,x \ne 0$ is :