If the matrix \(A=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ 0 & 0 & 2 \\ 3 & -2 & 0\end{array}\right]\) is a skew symmetric matrix, then the value of \((\alpha+\beta+\gamma)^2\) is :

Let the corner points of the bounded feasible region of the linear programming problem \(( LPP ) Z=a x+b y\) be \((0,0),(2,0),\left(\frac{20}{19}, \frac{45}{19}\right)\) and \((0,3)\).
If the optimal value of \(Z\) occurs at both points \((2,0)\) and \(\left(\frac{20}{19}, \frac{45}{19}\right)\), then the relation between \(a\) and \(b\) is:

The value of \(\left|\begin{array}{lll}2^x & 1 & 6^x \\ 4^x & 1 & 3^x \\ 2^x & 1 & 6^x\end{array}\right|\), where \(x \neq 0 ;\) is:

Let \(P=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix and let \(Q=\left[b_{i j}\right]\) where \(b_{i j}=2^{i+j} a_{i j}, \forall 1 \leq i, j \leq 3\). If the determinant of \(P\) is 2 , then the determinant of \(Q\) is:

For the LPP
Maximise \(z=x+y\)
subject to \(x-y \leq-1,-x+y \leq 2, x, y \geq 0, z\) has :

A container contains 48 litres of buffalo milk. From this container 8 litres of buffalo milk was taken out and replaced with cow milk. This process was repeated further 3 more times. How much buffalo milk is there in the container now ?

The value of the integral \(I=\int e^x\left(\tan ^{-1} x+\frac{1}{1+x^2}\right) d x\) is :

To increase the current sensitivity of a moving coil galvanometer by 50%, its resistance is increased so that the new resistance becomes twice its initial resistance. By what percentage does its voltage sensitivity change?

In a half wave rectifier, the frequency of the output is :

A random variable y has the following probability distribution

y	1	2	3	4	5
P(y)	2k	3k	k	4k	5k

Match List-l with List-ll

List - I	List - II
(A) P(y > 2)	(I) \(\frac{2}{5}\)
(B) k	(II) \(\frac{2}{3}\)
(C) \(P(y \leq 3)\)	(II) \(\frac{8}{15}\)
(D) \(P(2 \leq y \leq 4)\)	(II) \(\frac{1}{15}\)

Choose the correct answer from the options given below

A coil \(C_1\) is connected to a galvanometer G. Second coil \(C_2\) is connected to a battery through a tapping key. The steady current in the coil \(C_2\) produces a steady magnetic field. As coil \(C_2\) is moved towards the coil \(C_1\), the galvanometer shows a deflection. To obtain a large deflection, one or more of the following steps can be taken:
(A) Use a rod made of soft iron inside the coil \(C_2\)
(B) Connect the coil \(C_2\) to a more powerful battery
(C) Move the arrangement of coil \(C_2\) rapidly towards the coil \(C_1\)
(D) hold the key pressed continuously
Choose the correct answer from the options given below:

For \(x \in\left(0, \frac{\pi}{2}\right), \int \frac{1}{\sin ^2 x+\sin 2 x} d x\) is equal to

The probability distribution of a discrete random variable x is given below:

\(x\)	\(2\)	\(3\)	\(4\)	\(5\)
\(p(x)\)	\(\frac{5}{k}\)	\(\frac{7}{k}\)	\(\frac{9}{k}\)	\(\frac{10}{k}\)

The value of \(k\) is