The area of the region   \(A=\left\{(x, y):|\cos x-\sin x| \leq y \leq \sin x, 0 \leq x \leq \frac{\pi}{2}\right\}\)

Let \(y = y(x)\) be the solution of the differential equation \(x\sin\left(\dfrac{y}{x}\right)dy = \left(y\sin\left(\dfrac{y}{x}\right) - x\right)dx\), \(y(1) = \dfrac{\pi}{2}\) and let \(\alpha = \cos\left(\dfrac{y(e^{12})}{e^{12}}\right)\). Then the number of integral values of \(p\), for which the equation \(x^2 + y^2 - 2px + 2py + \alpha + 2 = 0\) represents a circle of radius \(r \leq 6\), is __________.

The maximum value of \(z\) in the following equation \(z=6 x y+y^{2},\) where \(3 x+4 y \leq 100\) and \(4 x+3 y \leq 75\) for \(x \geq 0\) and \(y \geq 0\) is \(......\)

If in a regular polygon the number of diagonals is \(54\), then the number of sides of this polygon is

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

Let \(d\) be the distance of the point of intersection of the lines \(\frac{x+6}{3}=\frac{y}{2}=\frac{z+1}{1} \quad\) and \(\frac{x-7}{4}=\frac{y-9}{3}=\frac{z-4}{2}\) from the point \((7,8,9)\). Then \(\mathrm{d}^2+6\) is equal to :

If \(3^{2 \sin 2 \alpha-1},14\) and \(3^{4-2 \sin 2 \alpha}\) are the first three terms of an \(A.P.\) for some \(\alpha\), then the sixth term of this \(A.P.\) is 

Suppose the vectors \(x_{1}, x_{2}\) and \(x_{3}\) are the solutions of the system of linear equations, \(Ax = b\) when the vector \(b\) on the right side is equal to \(b _{1}, b _{2}\) and \(b _{3}\) respectively. If \(x =\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], x _{2}=\left[\begin{array}{l}0 \\ 2 \\ 1\end{array}\right], x _{3}=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], b _{1}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\) \(b _{2}=\left[\begin{array}{l}0 \\ 2 \\ 0\end{array}\right]\) and \(b _{3}=\left[\begin{array}{l}0 \\ 0 \\ 2\end{array}\right],\) then the determinant of \(A\) is equal to

If the system of equations  \(x+2 y+3 z=3\)  ; \(4 x+3 y-4 z=4\) ; \(8 x+4 y-\lambda z=9+\mu\) has infinitely many solutions, then the ordered pair \((\lambda, \mu)\) is equal to

Let \(y=y(x)\) satisfies the equation \(\frac{d y}{d x}-|A|=0\), for all \(x>0\), where \(A=\left[\begin{array}{ccc}y & \sin x & 1 \\ 0 & -1 & 1 \\ 2 & 0 & \frac{1}{x}\end{array}\right] .\) If \(y(\pi)=\pi+2\), then the value of \(y\left(\frac{\pi}{2}\right)\) is:

If \(y^{1 / 4}+y^{-1 / 4}=2 x\), and \(\left(x^{2}-1\right) \frac{d^{2} y}{d x^{2}}+\alpha x \frac{d y}{d x}+\beta y=0\) then \(|\alpha-\beta|\) is equal to ...... .

Two integers are selected at random from the set \(\{1, 2, …, 11\}.\) Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is

If the points \(P\) and \(Q\) are respectively the circumcentre and the orthocentre of a \(\triangle ABC\), then \(\overrightarrow{ PA }+\overrightarrow{ PB }+\overrightarrow{ PC }\) is equal to