The function \(f(x)=\left\{\begin{array}{ll}\frac{\sin 2 x}{x}+\cos x & , \text { if } x \neq 0 \\ K & , \text { if } x=0\end{array}\right.\) is continuous at \(x = 0\) then the value of K is:

The objective function of an LPP is \(Z=a x+b y\). If the maximum value of the objective function is 180 , which occurs at two points \((15,15)\) and \((0,20)\), then which one of the following is true?

The probability distribution of the random variable \(X\) is given by

X	0	1	2	3
P(X)	0.2	k	2k	2k

The variance of the random variable \(X\) is given by:

If \(y=\log \left[\frac{x^2}{e^2}\right]\) then value of \(\frac{d^2 y}{d x^2}\) is :

Match List-I with List-II

List-I (Differential equation)	List-II (Order and Degree)
(A) \(\frac{d^3 y}{d x^3}+y^2+e^{d y / d x}=0\)	(I) order \(=3\), degree \(=1\)
(B) \(\left(\frac{d^2 y}{d x^2}\right)^3+\left(\frac{d y}{d x}\right)^2+\frac{d y}{d x}+1=0\)	(II) order \(=3\), degree not defined
(C) \(2 x^2 \frac{d^2 y}{d x^2}-3\left(\frac{d y}{d x}\right)^2+y=0\)	(III) order \(=2\), degree \(=3\)
(D) \(\frac{d^3 y}{d x^3}+2\left(\frac{d y}{d x}\right)^2+\frac{d y}{d x}=0\)	(IV) order \(=2\), degree \(=1\)

Choose the correct answer from the options given below :

The value of \(c \in[0, \pi]\) for which the function \(f(x)=e^x \sin x\) satisfies Rolle's theorem is:

A point charge – 5 µC is at a distance 5 cm directly above the centre of a square of side 10 cm.
The magnitude of electric flux through this square is:

Match List-l with List-ll.Find the derivatives from List-I.

List-l	List-II
(A) \(y=\sqrt{\sin x+y}\)	(I) \(\frac{-\sin x}{1+\cos y}\)
(B)\(\sin y=x \sin (a+y)\)	(II) \(\frac{\cos x}{2 y-1}\)
(C) \(y+\sin y=\cos x\)	(III) \(\frac{-1}{\sin ^2(x+y)}\)
(D) \(y=\tan (x+y)\)	(IV) \(\frac{\sin ^2(a+y)}{\sin a}\)

Choose the correct answer from the options given below:

Match List I with List II

LIST I	LIST II
A. \(\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{3 x^2}\)	I. 2
B. \(\lim _{x \rightarrow 4} \frac{x^2-16}{x-4}\)	II. 8
C. \(\lim _{x \rightarrow 0} \frac{\sin a x+4 x}{a x+\sin 4 x}\)	III. 1
D. \(\lim _{z \rightarrow 1} \frac{z^{1 / 3}-1}{z^{1 / 6}-1}\)	IV. \(\frac{2}{3}\)

A card is picked at random from a pack of 52 playing cards. Given that the picked card is a queen, the probability of this card to be a card of spade is:

"Choose the reactions given by \(\text{Cr}_2\text{O}_7^{2-}\)
(A)\(\text{Cr}_2\text{O}_7^{2-} + 14\text{H}^+ + 6\text{Fe}^{2+} \rightarrow 2\text{Cr}^{3+} + 6\text{Fe}^{3+} + 7\text{H}_2\text{O}\)
(B) \(\text{Cr}_2\text{O}_7^{2-} + 14\text{H}^+ + 3\text{H}_2\text{S} \rightarrow 2\text{Cr}^{3+} + 3\text{SO}_4^{2-} + 7\text{H}_2\text{O}\)
(C) \(\text{Cr}_2\text{O}_7^{2-} + 14\text{H}^+ + 3\text{Sn}^{2+} \rightarrow 2\text{Cr}^{3+} + 3\text{Sn}^{4+} + 7\text{H}_2\text{O}\)
(D) \(\text{Cr}_2\text{O}_7^{2-} + 14\text{H}^+ \rightarrow 3\text{Cr}^{3+} + 2\text{H}_2\text{O}\)
(E) \(\text{Cr}_2\text{O}_7^{2-} + 2\text{OH}^- \rightarrow 2\text{CrO}_4^{2-} + \text{H}_2\text{O}\)
Choose the correct answer from the options given below:"

Which of the following is not a stop codon?