The random variable X has the following probability distribution

X	0	1	2	3
P(X)	a	a	b	b

such that E\(\left(x^2\right)\) = 2E(x), then the value of b is:

Let \(\Delta = \begin{vmatrix} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \end{vmatrix}\), then \(\Delta\) lies in the interval

If the corner points of Linear Programming Problem having objective function \(\operatorname{Max} z=2 x+y\) are \(O(0,0), A(30,0), B(20,30), C(0,50)\), then the optimal value will be at the point :

The system of equations \(x+k y=0\) \(3 x+5 y=0\) has infinitely many solutions, then \(k\) is equal to

If \(x, y, z\) are all distinct and \(\left|\begin{array}{lll}x & x^2 & 1+x^3 \\ y & y^2 & 1+y^3 \\ z & z^2 & 1+z^3\end{array}\right|=0\), then the value of \(x y z\) is :

For a random variable \(X, E(X)=3\) and \(E\left(X^2\right)=11\), then variance of \(X\) is :

Consider the LPP, with objective function optimize \(Z=2 x-y+5\) subject to constraints \(3 x+4 y \leq 60 ; x+3 y \leq 30 ; x, y \geq 0\). If the corner points of feasible region are \(A(0,10), B(12,6), C(20,0)\) and \(O(0,0)\), then match List I with List II:

LIST 1	LIST 2
Minimum value of Z	45
Maximum value of Z	50
The sum of minimum and maximum value of Z	-5
Maximum of Z - Minimum of Z	40

A Fraunhofer diffraction pattern due to a single slit of width 0.2 mm is being obtained on a screen placed at a distance of 2 m from the slit. The first minimum lies at 5 mm on either side of the central maximum on the screen. The wavelength of light is:

Answer the question on the basis of passage given below :
A colloid is a heterogeneous system in which a substance is dispersed (dispersed phase) as fine particles in another substance called dispersion medium.
Their particle size is intermediate between that of pure solutions (less than 1 nm) and suspensions (greater than 1000 nm). They are known for interesting properties such as enormous surface area, Tyndall effect, Brownian motion etc.
They are widely used in industries and day to day life.
Some prominent uses of colloids are those in smoke precipitators, purification of water, medicines, cleansers etc.
Which of the following statement is false regarding charge on sol particles ?

The area (in sq. units) of the region bounded by the curve \(y=2 x^3, x\)-axis and ordinates \(x=-1\) and \(x=1\) is.

Which of the following statements are correct?
A. Number of photoelectrons emitted per second is directly proportional to the intensity of incident radiation
B. Cut off potential is the minimum negative potential given to the plate for which photocurrent becomes maximum
C. For a given frequency of the incident radiation, the stopping potential is independent of its intensity
D. Even for the exceedingly dim incident radiation apparent time lag is \( \sim 10^{-9} \) s or less
E. Photons are electrically neutral and are not deflected by electric and magnetic fields
Choose the correct answer from the options given below:

A convex lens of glass (n = 1.5) has a focal length 18 cm in air. When it is immersed in water (n = 4/3) the focal length increases to:

The structures present in human sperm are-
(A). Nucleus
(B). Zona pellucida
(C). Acrosome
(D). Mitochondria
Choose the correct answer from the options given below: