If \(A^{-1}\) exists for the matrix \(A=\left[\begin{array}{ccc}1 & \lambda & -1 \\ -1 & 1 & 0 \\ \lambda & 1 & 1\end{array}\right]\) then

Match List I with List II

LIst - I	List - II
(A) The present value of an immediate perpetuity of ₹R payable at end of any year forever at rate of i per period is:	(I) \(\frac{R}{i}(1+i)^m\)
(B) Present value of deferred perpetuity is	(II) \(\frac{A i}{(1+i)^n-1}\)
(C) Periodic payment R in a sinking fund is given by formula (for Amount A)	(III) \(A(1+i)^{-n}\)
(D) The present value of redemption price A of a bond is given by	(IV) \(\frac{R}{i}\)

Integrating factor of \(\frac{dy}{dx} + y \cot x = 2x + x^2 \cot x\) (\(x \neq 0\)) is:

If \(\left|\begin{array}{cc}3 x & 7 \\ 4 & x\end{array}\right|=\left|\begin{array}{cc}6 & -2 \\ 4 & 2\end{array}\right|\), then the value of \(x\) is :

In a game, a man wins ₹ \( 8\) for getting a number greater than 3 and loses ₹ 3 otherwise, when a fair die is thrown.
The man decided to throw a die 4 times but to quit as and when he gets a number greater than 3.
If \(X\) denotes the amount which the man wins or loses, then which of the following are correct?
(A) All the possible values of \(X\) are \(8,5,2\) and -1 .
(B) The probability distribution of \(X\) is :

X	8	5	2	-1	-12
P(X)	1/2	1/4	1/8	1/16	1/16

(C) The mean value of \(X\) is \(\frac{75}{16}\).
(D) The variance of \(X\) is \(\frac{6615}{256}\).
Choose the correct answer from the options given below :

The derivative of \(x^3\) with respect to \(x^2\) is

Which of the following are examples of irregular trends in a time series?
(A) Decrease in production due to a sudden strike.
(B) The rise in prices before festivals.
(C) Unusual rise in income of the printing press due to the announcement of an election.
(D) Fall in crop yield due to floods.
Choose the correct answer from the options given below :

\(\int_{-\pi}^\pi \frac{e^{\sin x}}{e^{\sin x}+e^{-\sin x}} d x\) is equal to:

Read the passage carefully and answer the Questions
The term vitamin was coined from the word vital +amine since the earlier identified compounds had amino groups. Later work showed that most of them didn't contain amino groups. They are very essential for the body. They has to be taken various diet or in the form of medicines also.
Vitamins are classified into two groups depending upon their solubility in water or fat.
Fat soluble vitamins are soluble in fat and oils but insoluble in water. They are stored in liver and adipose (fat storing) tissues Water-soluble vitamins are soluble in water. These must be supplied regularly in diet because they are readily excreted in urine and cannot be stored in our body.
Which vitamin is known as Riboflavin?

Identify the correct IUPAC name of the given compound:

Out of students in a school, it is known that \(65 \%\) reside in hostel and \(35 \%\) are day scholars. Previous year results report that \(40 \%\) of the students who reside in hostel attain A grade and \(20 \%\) of day scholars attain A grade in their annual examination.
At the end of the year, one student is chosen at random from the class.
Probability of the selected student attaining grade \(A\) is:

If \(\left|\begin{array}{lll}a+x & a-x & a-x \\ a-x & a+x & a-x \\ a-x & a-x & a+x\end{array}\right|=0\), then the values of \(x\) are:

The technique in which the semen collected either from the husband or the donor is artificially introduced into the uterus is known as-