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Let the centroid of an equilateral triangle \( ABC \) be at the origin. Let one of the sides of the equilateral triangle be along the straight line \( x + y = 3 \). If \( R \) and \( r \) be the radius of circumcircle and incircle respectively of \( \triangle ABC \), then \( (R + r) \) is equal to:

Let in a series of \( 2n \) observations, half of them are equal to \( a \) and remaining half are equal to \( -a \). Also by adding a constant \( b \) in each of these observations, the mean and standard deviation of new set becomes \( 5 \) and \( 20 \), respectively. Then the value of \( a^2 + b^2 \) is equal to:

Let in a Binomial distribution, consisting of \( 5 \) independent trials, probabilities of exactly \( 1 \) and \( 2 \) success be \( 0.4096 \) and \( 0.2048 \) respectively. Then the probability of getting exactly \( 3 \) successes is equal to:

Let \( f:\mathbb{R}\to\mathbb{R} \) be a function defined as .
If \( f \) is continuous at \( x=0 \), then the value of \( a+b \) is equal to:

If \( P \) and \( Q \) are two statements, then which of the following compound statement is a tautology?

A pole stands vertically inside a triangular park ABC. Let the angle of elevation of the top of the pole from each corner of the park be \( \dfrac{\pi}{3} \). If the radius of the circumcircle of \( \triangle ABC \) is \( 2 \), then the height of the pole is equal to:

Let \( y = y(x) \) be the solution of the differential equation \( \dfrac{dy}{dx} = (y+1)\left((y+1)e^{x^{2}/2} - x\right) \), \( 0 < x < 2.1 \), with \( y(2) = 0 \). Then the value of \( \dfrac{dy}{dx} \) at \( x = 1 \) is equal to:

Let a tangent be drawn to the ellipse \( \dfrac{x^{2}}{27} + y^{2} = 1 \) at \( \left(3\sqrt{3}\cos\theta,\ \sin\theta\right) \) where \( \theta \in \left(0,\ \dfrac{\pi}{2}\right) \). Then the value of \( \theta \) such that the sum of intercepts on axes made by this tangent is minimum is equal to:

Let \( S_{1} \) be the sum of first \( 2n \) terms of an arithmetic progression. Let \( S_{2} \) be the sum of first \( 4n \) terms of the same arithmetic progression. If \( (S_{2} - S_{1} )\) is \( 1000 \), then the sum of the first \( 6n \) terms of the arithmetic progression is equal to:

Let \( g(x) = \displaystyle\int\limits_{0}^{x} f(t)\,dt \), where \( f \) is continuous function in \( [0,3] \) such that \( \dfrac{1}{3} \le f(t) \le 1 \) for all \( t \in [0,1] \) and \( 0 \le f(t) \le \dfrac{1}{2} \) for all \( t \in [1,3] \). The largest possible interval in which \( g(3) \) lies is:

Let a complex number be \( w = 1 - \sqrt{3} i \). Let another complex \( z \) be such that \( |zw| = 1 \) and \( \arg(z) - \arg w = \dfrac{\pi}{2} \). Then the area of the triangle with vertices origin, \( z \) and \( w \) is equal to:

Consider a hyperbola \( H : {x^{2}} - {2y^{2}} = 4 \). Let the tangent at a point \( P(4,\sqrt{6}) \) meet the x-axis at \( Q \) and latus rectum at \( R(x_{1},y_{1}), x_{1} > 0 \). If \( F \) is a focus of \( H \) which is nearer to the point \( P \), then the area of \( \triangle QFR \) is equal to:

In a triangle \( ABC \), if \( | \overrightarrow{BC} | = 8, | \overrightarrow{CA} | = 7, | \overrightarrow{AB} | = 10 \), then the projection of the vector \( \overrightarrow{AB} \) on \( \overrightarrow{AC} \) is equal to:

Define a relation \( R \) over a class of \( n \times n \) real matrices \( A \) and \( B \) as “\( ARB \) iff there exists a non-singular matrix \( P \) such that \( P A P^{-1} = B \)”. Then which of the following is true?

Let \( \vec{a} \) and \( \vec{b} \) be two non-zero vectors perpendicular to each other and \( |\vec{a}| = |\vec{b}| \). If \( |\vec{a} \times \vec{b}| = |\vec{a}| \), then the angle between the vectors \( (\vec{a} + \vec{b} +( \vec{a} \times \vec{b})) \) and \( \vec{a} \) is equal to:

Let \( S_{1} : x^{2} + y^{2} = 9 \) and \( S_{2} : (x - 2)^{2} + y^{2} = 1 \). Then the locus of center of a variable circle \( S \) which touches \( S_{1} \) internally and \( S_{2} \) externally always passes through the points:

The area bounded by the curve \( 4y^2 = x^2 (4 - x)(x - 2) \) is equal to:

If \( 15\sin^4\alpha + 10\cos^4\alpha = 6 \), for some \( \alpha \in \mathbb{R} \), then the value of \( 27\sec^6\alpha + 8\,\text{cosec}^6\alpha \) is equal to:

Let \( f : \mathbb{R} - \{3\} \to \mathbb{R} - \{1\} \) be defined by \( f(x) = \dfrac{x - 2}{x - 3} \). Let \( g : \mathbb{R} \to \mathbb{R} \) be given as \( g(x) = 2x - 3 \). Then, the sum of all the values of \( x \) for which \( f^{-1}(x) + g^{-1}(x) = \dfrac{13}{2} \) is equal to:

Let the mirror image of the point \( (1, 3, a) \) with respect to the plane \( \vec{r}\cdot(2\hat{i} - \hat{j} + \hat{k}) - b = 0 \) be \( (-3, 5, 2) \). Then, the value of \( |\, a + b \,| \) is equal to:

Let \( I \) be an identity matrix of order \( 2 \times 2 \) and \(\displaystyle P = \begin{bmatrix} 2 & -1 \\ 5 & -3 \end{bmatrix}\). Then the value of \( n \in \mathbb{N} \) for which \( P^n = 5I - 8P \) is equal to:

Let \( P(x) \) be a real polynomial of degree \( 3 \) which vanishes at \( x = -3 \). Let \( P(x) \) have local minima at \( x = 1 \), local maxima at \( x = -1 \) and \( \displaystyle\int\limits_{-1}^{1} P(x)\,dx = 18 \), then the sum of all the coefficients of the polynomials \( P(x) \) is equal to:

Let \( P \) be a plane containing the line \(\displaystyle \frac{x-1}{3}=\frac{y+6}{4}=\frac{z+5}{2} \) and parallel to the line \(\displaystyle \frac{x-3}{4}=\frac{y-2}{-3}=\frac{z+5}{7} \). If the point \(( 1,-1,\alpha) \) lies on the plane \( P \), then the value of \( |5\alpha| \) is equal to:

Let \( {}^{n}C_{r} \) denote the binomial coefficient of \( x^{r} \) in the expansion of \( (1+x)^{n} \). If \( \displaystyle\sum_{k=0}^{10} (2^{2}+3k)\,{}^{n}C_{k}=\alpha.\,3^{10}+\beta.\,2^{10},\ \alpha,\beta\in \mathbb{R} \), then \( \alpha+\beta \) is equal to:

Let \( f:\mathbb{R}\to\mathbb{R} \) satisfy the equation \( f(x+y)=f(x)\cdot f(y) \) for all \( x,y\in\mathbb{R} \) and \( f(x)\neq 0 \) for any \( x\in\mathbb{R} \). If the function \( f \) is differentiable at \( x=0 \) and \( f'(0)=3 \), then \( \displaystyle\lim_{h\to 0}\frac{1}{h}\big(f(h)-1\big) \) is equal to:

If \( f(x) \) and \( g(x) \) are two polynomials such that the polynomial \( P(x)=f(x^{3})+x\,g(x^{3}) \) is divisible by \( x^{2}+x+1 \), then \( P(1) \) is equal to:

If \( \displaystyle \sum_{r=1}^{10} r!\,\big(r^{3}+6r^{2}+2r+5\big)=\alpha\,(11!) \), then the value of \( \alpha \) is equal to:

The term independent of \( x \) in the expansion of \(\displaystyle \left[ \frac{x + 1}{x^{2/3} - x^{1/3} + 1} - \frac{x - 1}{x - x^{1/2}} \right]^{10}, x \neq 1 \), is equal to:

Let \( y = y(x) \) be the solution of the differential equation \( x\,dy - y\,dx = \sqrt{(x^2 - y^2)}\,dx, \; x \geq 1 \), with \( y(1) = 0 \).
If the area bounded by the line \( x = 1 \), \( x = e^{\pi} \), \( y = 0 \) and \( y = y(x) \) is \( \alpha e^{2\pi} + \beta \), then the value of \( 10(\alpha + \beta) \) is equal to: