If \(f(a+b+1-x)=f(x),\) for all \(x,\) where \(a\) and \(b\) are fixed positive real numbers, then \(\frac{1}{a+b} \int\limits_{a}^{b} x(f(x)+f(x+1)) d x\) is equal to 

A card from a pack of 52 cards is lost. From the remaining 51 cards, \(n\) cards are drawn and are found to be spades. If the probability of the lost card to be a spade is \(\frac{11}{50}\), the n is equal to

Let \(O\) be the origin, \(\vec{OP} = \vec{a}\) and \(\vec{OQ} = \vec{b}\). If \(R\) is the point on \(\vec{OP}\) such that \(\vec{OP} = 5\vec{OR}\), and \(M\) is the point such that \(\vec{OQ} = 5\vec{RM}\), then \(\vec{PM}\) is equal to :

If the projection of the vector \(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) on the sum of the two vectors \(2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-5 \hat{\mathrm{k}}\) and \(-\lambda \hat{i}+2 \hat{j}+3 \hat{k}\) is \(1,\) then \(\lambda\) is equal to ..... .

The plane containing the line \(\frac{{x - 1}}{1} = \frac{{y - 2}}{2} = \frac{{z - 3}}{3}\) and parallel to the line \(\frac{x}{1} = \frac{y}{1} = \frac{z}{4}\) passes through the point

If \(^{20}{C_1} + \left( {{2^2}} \right){\,^{20}}{C_3} + \left( {{3^2}} \right){\,^{20}}{C_3} + \left( {{2^2}} \right) + ..... + \left( {{{20}^2}} \right){\,^{20}}{C_{20}} = A\left( {{2^\beta }} \right)\), then the ordered pair \((A, \beta )\) is equal to

The number of solutions of the equation \(32^{\tan ^{2} x}+32^{\sec ^{2} x}=81,0 \leq x \leq \frac{\pi}{4}\) is :

Let \(f(x)=\int_0^x\left(t+\sin \left(1-e^t\right)\right) d t, x \in \mathbb{R}\). Then \(\lim _{x \rightarrow 0} \frac{f(x)}{x^3}\) is equal to

The distance of the point \(Q(0,2,-2)\) form the line passing through the point \(\mathrm{P}(5,-4,3)\) and perpendicular to the lines \(\overrightarrow{\mathrm{r}}=(-3 \hat{\mathrm{i}}+2 \hat{\mathrm{k}})\) \(\lambda(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}), \lambda \in \mathbb{R}\) and \( \overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}})+\) \(\mu(-\hat{i}+3 \hat{J}+2 \hat{K}), \mu \in \mathbb{R}\) is

The  real number \(k\) for which  the equation, \(2{x^2} + 3x + k = 0\) has two distinct real roots in \([0, 1]\) 

The centre of the circle passing through the point \((0,1)\) and touching the parabola \(y=x^{2}\) at the point \((2,4)\) is

A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate going by bus, scooter and car, respectively, are \(\dfrac{2}{5}\), \(\dfrac{1}{5}\) and \(\dfrac{2}{5}\). The probabilities that the candidate reaches late at the examination centre are \(\dfrac{1}{5}\), \(\dfrac{1}{3}\) and \(\dfrac{1}{4}\) if the candidate uses bus, scooter and car, respectively. Given that the candidate reached late at the examination centre, the probability that the candidate travelled by bus is:

Let \(H: \dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\) be a hyperbola such that the distance between its foci is \(6\) and the distance between its directrices is \(\dfrac{8}{3}\). If the line \(x=\alpha\) intersects the hyperbola \(H\) at the points \(A\) and \(B\) such that the area of the triangle \(AOB\) is \(4\sqrt{15}\), where \(O\) is the origin, then \(\alpha^2\) equals