In a game two players \(A\) and \(B\) take turns in throwing a pair of fair dice starting with player \(A\) and total of scores on the two dice, in each throw is noted. \(A\) wins the game if he throws a total of \(6\) before \(B\) throws a total of \(7\) and \(B\) wins the game if he throws a total of \(7\) before \(A\) throws a total of six The game stops as soon as either of the players wins. The probability of \(A\) winning the game is 

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

If \(\sum_{r=1}^{13}\left\{\frac{1}{\sin \left(\frac{\pi}{4}+(r-1) \frac{\pi}{6}\right) \sin \left(\frac{\pi}{4}+\frac{r \pi}{6}\right)}\right\}=a \sqrt{3}+b, a, b \in \mathbf{Z}\), then \(a^2+b^2\) is equal to :

The integral \(\int \frac{1}{\sqrt[4]{(x-1)^{3}(x+2)^{5}}} d x\) is equal to : (where \(\mathrm{C}\) is a constant of integration)

Let \(A\) be a symmetric matrix of order \(2\) with integer entries. If the sum of the diagonal elements of \(A ^{2}\) is \(1,\) then the possible number of such matrices is

The coefficient of \(x^{2012}\) in the expansion of \((1-x)^{2008}\left(1+x+x^2\right)^{2007}\) is equal to

Let \(\vec \alpha \, = \,3\hat i\, + \hat j\) and \(\vec \beta \, = \,2\hat i\, - \hat j + 3\hat k.\) If \(\vec \beta \, = \,{\vec \beta _1} - {\vec \beta _2},\) where \({\vec \beta _1}\) is parallel to \(\vec \alpha \) and \(\vec \beta_2 \) is perpendicular to \(\vec \alpha ,\)  then \({\vec \beta _1} \times {\vec \beta _2}\) is equal to

If \( \int_{0}^{1}4~cot^{-1}(1-2x+4x^{2})dx=a~tan^{-1}(2)-b~log_{c}(5), \) where a, b \( \in N \), then \( (2a+b) \) is equal to :

Let a line pass through two distinct points \(P(-2,-1,3)\) and \(Q\), and be parallel to the vector \(3 \hat{i}+2 \hat{j}+2 \hat{k}\). If the distance of the point Q from the point \(\mathrm{R}(1,3,3)\) is 5 , then the square of the area of \(\triangle P Q R\) is equal to :

Let \(\mathrm{A}\,(\sec \theta, 2 \tan \theta)\) and \(\mathrm{B}\,(\sec \phi, 2 \tan \phi)\), where \(\theta+\phi=\pi / 2\), be two points on the hyperbola \(2 \mathrm{x}^{2}-\mathrm{y}^{2}=2\). If \((\alpha, \beta)\) is the point of the intersection of the normals to the hyperbola at \(\mathrm{A}\) and \(\mathrm{B}\), then \((2 \beta)^{2}\) 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 :

Consider the following two statements : Statement \(I\) : For any two non-zero complex numbers \(\mathrm{z}_1, \mathrm{z}_2\) \(\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)\) and Statement \(II\) : If \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) are three distinct complex numbers and a, b, c are three positive real numbers such that \(\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}\), then \(\frac{\mathrm{a}^2}{\mathrm{y}-\mathrm{z}}+\frac{\mathrm{b}^2}{\mathrm{z}-\mathrm{x}}+\frac{\mathrm{c}^2}{\mathrm{x}-\mathrm{y}}=1\) Between the above two statements,

If the function \(f(x)=\left\{\begin{array}{ll}k_{1}(x-\pi)^{2}-1, & x \leq \pi \\ k_{2} \cos x, & x>\pi\end{array}\right.\) is twice differentiable, then the ordered pair \(\left( k _{1}, k _{2}\right)\) is equal to