For \(x \neq-1\), if \(\int \frac{x e^x d x}{(1+x)^2}=\frac{a e^x}{(1+x)^b}+c\), where \(a, b\) are fixed numbers and \(c\) is the integration constant, then \(a+b\) is equal to

\(\int \frac{\cos 2 x-\cos 2 \alpha}{\cos x-\cos \alpha} d x\) is equal to

Two points A and B in a plane are related if OA = OB, where O is a fixed point. This relation is:

The function \(f(x)=\tan x-x\) for \(x \in\left(0, \frac{\pi}{2}\right)\) is :

The refractive index of the material of an equilateral prism is \(\sqrt {2}\). The angle of minimum deviation of that prism is

If \(A\) and \(B\) are independent events and \(P(A)=\frac{1}{2}, P(B)=\frac{1}{3}\) then
Match List-I with List-II

List-I	List-II
(A) \(P(A \cap B)\)	(I) \(\frac{1}{2}\)
(B) \(P(\bar{A}) P(B)+P(A) P(\bar{B})\)	(II) \(\frac{1}{3}\)
(C) \(P(A \mid B)+P(B \mid A)\)	(III) \(\frac{1}{6}\)
(D) \(P(\overline{A \cap B})\)	(IV) \(\frac{5}{6}\)

Choose the correct answer from the options given below :

If \(y=(x+1)\left(x^2+1\right)\left(x^4+1\right)\left(x^8+1\right)\) then \(\frac{d y}{d x}\) at \(x = -1\) is

If \(\left[\begin{array}{ll}x-y & 0 \\ x+y & 1\end{array}\right]\) is an identity matrix and \(\left[\begin{array}{ll}x & y \\ x & x\end{array}\right]\) is a singular matrix then :

If A and B are two matrices of order 2 × 2 such that A is a symmetric matrix and B is a skew-symmetric matrix, then:

Electrostatics deals with the study of forces, fields and potential arising from static charges. Which of the following statements are correct ?
(A) Electrostatic force is a conservative force
(B) Charge is quantized because only integral number of electrons can be transferred from one body to the other
(C) In a uniform electric field E , an electric dipole experiences a torque \(\tau\) and a net force F .
(D) The electric field lines start from positive charges and end at negative charges. If there is a single charge, these may start or end at infinity.
(E) If the uniform surface charge density of an infinite plane sheet is positive, the electric field is directed away from the plate.
Choose the correct answer from the options given below :

The area of the region bounded by the curve y = x + 1, x-axis and the lines x = 2 and x = 3 is

Match List - I with List - II

List - I (Vitamin)	List - II (Deficiency)
(A) Vitamin D	(I) Slow blood clotting
(B) Vitamin C	(II) Night blindness
(C) Vitamin K	(III) Rickets
(D) Vitamin A	(IV) Scurvy

Choose the correct answer from the options given below :

The lines \(\frac{x}{1}=\frac{y}{2}=\frac{z}{3}\) and \(\frac{x-3}{-3}=\frac{y-4}{-6}=\frac{z-5}{-9}\) are :