A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be heart. Then the probability of the missing card to be a heart is:

If \(\vec{a}=2 \hat{i}-3 \hat{j}+\hat{k}\) and \(\vec{b}=2 \hat{i}+\hat{j}-\hat{k}\), then which of the following statements is/are correct?
(A) \(\vec{a}\) and \(\vec{b}\) are collinear
(B) \(\vec{a}\) and \(\vec{b}\) are perpendicular
(C) Angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{4}\)
(D) \(|\vec{a}+\vec{b}|=2 \sqrt{5}\)
Choose the correct answer from the options given below:

Rakshita plans to buy a house for Rs. 1,00,00,000 with down payment of 20% of the value of the house paid by her mother. Rest of the amount she wishes to pay in 25 years by equal monthly instalments at an interest of 9% per annum compounded monthly Then the EMI paid by her is \(\left(\right.\) Given \(\left.(1.0075)^{300}=9\right)\) :

Let \(f\) be a function defined by \(f(x)=2 x^3-3 x^2-36 x+2\), then which of the following are correct ?
(A) The critical points of \(f ( x )\) are \(- 2\) and 3 .
(B) The function \(f(x)\) increases in the interval \((3, \infty)\)
(C) The function \(f(x)\) decreases in the interval \((-2,3)\)
(D) The function \(f(x)\) increases in the interval \((-2,3)\)
Choose the correct answer from the options given below :

The area of the region bounded by the curve \(y=\sin x\) and the \(x\)-axis between \(x=\pi / 2\) and \(x=3 \pi / 2\) 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 x = a sin t and y = a \(\left(\cos t+\log \tan \frac{t}{2}\right)\) then \(\frac{d^2 y}{d x^2}\) is:

If the magnetic flux through a coil of 1200 turns increases from zero to \(5.4 \times 10^{-5} \text{ Wb}\) in \(2.7 \text{ ms}\), what is the magnitude of the induced emf during this time?

Out of the given statement, choose the correct statement.
(A) The direction ratios of the vector \(\vec{a}=3 \hat{i}-\hat{j}+4 \hat{k}\) is \(3,-1,4\).
(B) If \(\theta\) is the angle between two vectors \(\vec{a}\) and \(\vec{b}\), then their cross product is given as \(\vec{a} \times \vec{b}=|\vec{a}||\vec{b}| \cos \theta\).
(C) The unit vector in the direction of vector \(\vec{a}=\hat{i}+2 \hat{j}-2 \hat{k}\) is \(\hat{a}=\frac{1}{3}(\hat{i}+2 \hat{j}-2 \hat{k})\).
(D) If \(\vec{a}=3 \hat{i}\) and \(\vec{b}=4 \hat{j}\) then \(\vec{a} \cdot \vec{b}=12\).
(E) If \(\vec{a}\) and \(\vec{b}\) represent the adjacent sides of a triangle then its area is given of \(\frac{1}{2}|\vec{a} \times \vec{b}|\).
Choose the correct answer from the options given below :

Choose the correct relation between focal length f and radius of curvature R of a convex mirror from the following:

Transgenic animals that produce useful biological products can be created by the introduction of the portion of DNA (or genes) which codes for a particular product such as human protein (α-1-antitrypsin). This protein is used to treat:

A random variable X has the following probability distribution :

X	0	1	2	3	4	5	6	7	8
P(X)	a	3a	5a	7a	9a	11a	13a	15a	17a

Then the values of ' \(a\) ' and \(P(0<X<5)\) respectively are :

The process of making multiple identical copies of any template DNA is called