If the optimal value of the objective function \(z=p x+y\) of an L.P.P occurs at two corner points \((2,11)\) and \((4,5)\) of its bounded feasible region, then its optimal value is

Match List - I with List - II. Given \(P(A)=\frac{1}{3}\) and \(P(B)=\frac{1}{5}\) where \(A\) and \(B\) are independent.

LIST I	LIST II
A. \(P(A \cap B)\)	I. \(\frac{7}{15}\)
B. \(P \left( A ^{\prime} \cap B ^{\prime}\right)\)	II. \(\frac{2}{5}\)
C. \(P\) (at least one of the two events takes place)	III. \(\frac{1}{15}\)
D. \(P\) (only one event takes place)	IV. \(\frac{8}{15}\)

Choose the correct answer from the options given below:

Let \(\left|\begin{array}{cc}x & 2 \\ 18 & x\end{array}\right|=\left|\begin{array}{ll}-4 & -2 \\ -8 &-4\end{array}\right|\). Then :
(A) \(x=-4\)
(B) \(x=-6\)
(C) \(x=4\)
(D) \(x=6\)
Choose the correct answer from the options given below :

The stationary point of \(f(x)=\sin ^4 x+\cos ^4 x, x \in\left(0, \frac{\pi}{2}\right)\) is given by:

Let \(\vec{a}=4 \hat{i}-\hat{j}+3 \hat{k}\) and \(\vec{b}=-2 \hat{i}+\hat{j}-2 \hat{k}\). Then
(A) \(\vec{a}\) is a unit vector
(B) \(\vec{a} \times \vec{b}=-\hat{i}+2 \hat{j}+2 \hat{k}\)
(C) \(\vec{a}\) and \(\vec{b}\) are parallel vectors
(D) \(\vec{a}\) and \(\vec{b}\) are neither parallel nor perpendicular vectors
Choose the \(\backslash\) textbf\{correct\} answer from the options given below :

The area of the triangle whose vertices are \((3,8),(-4,2)\) and \((5,1)\) is

Let \(\theta\) be the angle between two vectors \(\vec{a}\) and \(\vec{b}\). Then match List-I with List-II

List-I	List-II
(A) \(\sin \theta\)	(I) \(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\)
(B) \(\cos \theta\)	(II) \(|\vec{a} \times \vec{b}|\)
(C) Area of the parallelogram with adjacent sides represented by \(\vec{a}\) and \(\vec{b}\)	(III) \(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|}\)
(D) Projection of \(\vec{a}\) on \(\vec{b}\)	(IV) \(\frac{|\vec{a} \times \vec{b}|}{|\vec{a}||\vec{b}|}\)

Choose the Correct answer from the options given below :

A \( 25 \, \mu\text{F} \) capacitor, a 0.10 H inductor and a \( 25 \, \Omega \) resistor are connected in series with an ac source of emf \( \epsilon = 310 \sin 314t \). What is the frequency of AC source ?

Industrial melanism is an observation which supports evolution by:

If a random variable \(X\) has the following probability distribution :

X	0	1	2	3
P(X)	K	\(\frac{K}{2}\)	\(\frac{K}{4}\)	\(\frac{K}{8}\)

then,
Match List-I with List-II

List-I	List-II
(A) The value of \(K\) is	(I) \(2 / 15\)
(B) \(P(0<X<2)\) is	(II) \(1 / 15\)
(C) \(P(1<X<3)\) is	(III) \(8 / 15\)
(D) \(P(X>2)\) is	(IV) \(4 / 15\)

Choose the correct answer from the options given below :

Read the passage carefully and answer the Questions
Amines are classified as primary \(\left(1^{\circ}\right)\), secondary \(\left(2^{\circ}\right)\) and tertiary \(\left(3^{\circ}\right)\) depending upon the number of hydrogen atoms replaced by alkyl or aryl groups in ammonia molecules. Aliphatic amines are a stronger bases due to \(+I\) effect of an alkyl groups. Intermolecular hydrogen bonding in primary and secondary amines affects the boiling points of amines. The intermolecular association is more in primary amines than in secondary amines as there are more number of hydrogen atoms available for hydrogen bond formation. Nitro compounds are reduced to amines by passing hydrogen gas in the presence of finely divided nickel, palladium or platinum and also by reduction with metals in an acidic medium. Gabriel synthesis is used for the preparation of primary amines. Primary amines react with nitrous acid to give unstable diazonium-salts which can be used to synthesize a variety of compounds.
The reaction is an example of

If \(\begin{vmatrix} 2 & 3 & 2 \\ x & x & x \\ 4 & 9 & 1 \end{vmatrix} + 3 = 0\) then the value of x is:

A current of 2 A flows through a plane circular coil of radius 3 cm and having 50 turns. The coil has been placed in a uniform magnetic field of 0.6 T. The minimum potential energy of the coil will be