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The sum \( 1+3+11+25+45+71+\cdots \) upto \( 20 \) terms, is equal to:

Let \( g \) be a differentiable function such that \( \int_{0}^{x} g(t)\,dt=x-\int_{0}^{x} t g(t)\,dt,\, x\ge 0 \) and let \( y=y(x) \) satisfy the differential equation \( \dfrac{dy}{dx}-y\tan x=2(x+1)\sec x\, g(x),\, x\in\left[0,\dfrac{\pi}{2}\right] \). If \( y(0)=0 \), then \( y\!\left(\dfrac{\pi}{3}\right) \) is equal to:

The sum of all rational terms in the expansion of \( (2+\sqrt{3})^8 \) is:

Let \( \alpha \) and \( \beta \) be the roots of \( x^2+\sqrt{3}x-16=0 \), and \( \gamma \) and \( \delta \) be the roots of \( x^2+3x-1=0 \). If \( P_n=\alpha^n+\beta^n \) and \( Q_n=\gamma^n+\delta^n \), then \( \dfrac{P_{25}+\sqrt{3}P_{24}}{2P_{23}}+\dfrac{Q_{25}-Q_{23}}{Q_{24}} \) is equal to:

Let a line passing through the point \( (4,1,0) \) intersect the line \( L_1:\; \dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4} \) at the point \( A(\alpha,\beta,\gamma) \) and the line \( L_2:\; x-6=y=-z+4 \) at the point \( B(a,b,c) \).

If \(  \displaystyle \sum_{r=1}^{9} \left( \frac{r+3}{2^{r}} \right) \, {}^{.9}C_{r} = \alpha \left( \frac{3}{2} \right)^{9} - \beta \), \( \alpha , \beta \in \) \(\mathbb{R}\)  , then \( (\alpha + \beta)^{2} \) is equal to:

Line \( L_{1} \) passes through the point \( (1,2,3) \) and is parallel to \( z \)-axis. Line \( L_{2} \) passes through the point \( (\lambda,5,6) \) and is parallel to \( y \)-axis. Let for \( \lambda = \lambda_{1}, \lambda_{2}, \lambda_{2} < \lambda_{1} \), the shortest distance between the two lines be \( 3 \). Then the square of the distance of the point \( (\lambda_{1}, \lambda_{2}, 7) \) from the line \( L_{1} \) is:

Let \( A = \{-3,-2,-1,0,1,2,3\} \). Let \( R \) be a relation on \( A \) defined by \( xRy \) if and only if \( 0 \le x^{2} + 2y \le 4 \). Let \( l \) be the number of elements in \( R \) and \( m \) be the minimum number of elements required to be added in \( R \) to make it a reflexive relation. Then \( l + m \) is equal to:

If the domain of the function \( f(x) = \log_{9}\!\left( \frac{2x-3}{5+4x} \right) + \sin^{-1}\!\left( \frac{4+3x}{2-x} \right) \) is \( (\alpha, \beta] \), then \( \alpha^{2} + 4\beta \) is equal to:

Let \( A \) be a matrix of order \( 3 \times 3 \) and \(|A| = 5\). If \( |2 \, \operatorname{adj}(3A \, \operatorname{adj}(2A))| = 2^{\alpha} \cdot 3^{\beta} \cdot 5^{\gamma}, \alpha,\beta,\gamma \in \) \(\mathbb{R}\)  , then \( \alpha + \beta + \gamma \) is equal to:

Let \( z \in \mathbb{C} \) be such that \( \frac{z^2 + 3i}{z - 2 + i} = 2 + 3i \). Then the sum of all possible values of \( z^2 \) is:

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines \( L_1: 2x + y + 6 = 0 \) and \( L_2: 4x + 2y - p = 0, \, p > 0 \), at the points \( A \) and \( B \), respectively. If \( AB = \frac{9}{\sqrt{2}} \) and the foot of the perpendicular from the point \( A \) on the line \( L_2 \) is \( M \), then \( \frac{AM}{BM} \) is equal to:

be continuous at \( x = 0 \). Then \( e^{a} \, b \, c \) is equal to:

The number of solutions of the equation \( 2x + 3 \tan x = \pi \), \( x \in [ -2\pi, 2\pi ] - \left\{ \pm \frac{\pi}{2}, \pm \frac{3\pi}{2} \right\} \), is:

Let the domain of the function \( f(x) = \log_{2} \log_{4} \log_{6} (3 + 4x - x^2) \) be \( (a, b) \). If \( \int_{0}^{b-a} [x^2] \, dx = p - \sqrt{q} - \sqrt{r}, \, p, q, r \in \mathbb{Z}, \, \gcd(p,q,r) = 1 \), where \( [\cdot] \) is the greatest integer function, then \( p + q + r \) is equal to:

Let \( a_1, a_2, a_3, \ldots \) be a G.P. of increasing positive numbers. If \( a_3 a_5 = 729 \) and \( a_2 + a_4 = \frac{111}{4} \), then \( 24(a_1 + a_2 + a_3) \) is equal to:

\( x \in \) \(\mathbb{R}\)  , then \( \frac{d^{2}y}{dx^{2}} + y \) is equal to:

Let \( f(x) = \int x^{3}\sqrt{3 - x^{2}} \, dx \). If \( 5 f(\sqrt{2}) = -4 \), then \( f(1) \) is equal to:

The radius of the smallest circle which touches the parabolas \( y = x^{2} + 2 \) and \( x = y^{2} + 2 \) is:

Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \ \vec{b} = 3\hat{i} + 2\hat{j} - \hat{k}, \ \vec{c} = \lambda \hat{j} + \mu \hat{k} \) and \( \hat{d} \) be a unit vector such that \( \vec{a} \times \hat{d} = \vec{b} \times \hat{d} \) and \( \vec{c} \cdot \hat{d} = 1 \). If \( \vec{c} \) is perpendicular to \( \vec{a} \), then \( |3\lambda \hat{d} + \mu \vec{c}|^{2} \) is equal to _________.

The area of the region bounded by the curve \( y = \max\{ |x|,x |x - 2| \} \), the \( x \)-axis and the lines \( x = -2 \) and \( x = 4 \) is equal to _________.

All five letter words are made using all the letters A, B, C, D, E and arranged as in an English dictionary with serial numbers. Let the word at serial number \( n \) be denoted by \( W_{n} \). Let the probability \( P(W_{n}) \) of choosing the word \( W_{n} \) satisfy \( P(W_{n}) = 2P(W_{n-1}), \ n > 1 \). If \( P(CDBEA) = \frac{2^{\alpha}}{2^{\beta} - 1}, \ \alpha, \beta \in \)\(\mathbb{N}\), then \( \alpha + \beta \) is equal to _________.

Let the product of the focal distance of the point \( P(4, 2\sqrt{3}) \) on the hyperbola \( H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) be \( 32 \). Let the length of the conjugate axis of \( H \) be \( p \) and the length of its latus rectum be \( q \). Then \( p^2 + q^2 \) is equal to ________.

If the number of seven-digit numbers, such that the sum of their digits is even, is \( m \cdot n \cdot 10^{n} \); \( m,n \in \{1,2,3,\ldots,9\} \), then \( m+n \) is equal to ________.