The equation of the tangent, to the curve \(y=x^2-2 x-3\) which is perpendicular to the line x + 2y + 3 = 0, is:

If \(A=\left[\begin{array}{ll}1 & 0 \\ 3 & 1\end{array}\right]\) and \(A^4=\left[\begin{array}{ll}1 & 0 \\ k & 1\end{array}\right]\), then the value of \(k\) is :

Match List - I With List - II

List - I	List - II
(A) The area of parallelogram determined by vectors \(2 \hat{i}\) and \(3 \hat{j}\)	(I) 2
(B) The value of \((\hat{i} \times \hat{j}) \cdot \hat{k}+(\hat{j} \times \hat{k}) \cdot \hat{i}\)	(II) 4
(C) The value of a for which the vectors \(2 \hat{i}-3 \hat{j}+4 \hat{k}\) and \(a \hat{i}-6 \hat{j}+8 \hat{k}\) are collinear.	(III) 0
D. The value of \(\lambda\) for which the vectors \(2 \hat{i}+\hat{j}+\hat{k}\) and \(2 \hat{i}-4 \hat{j}+\lambda \hat{k}\) are perpendicular.	(IV) 6

Choose the correct answer from the options given below.

A measurable characteristic of a population is called:

The value of \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(x^3+x \cos x+\tan ^5 x+1\right) d x\) is:

Which one of the following set of constraints represents the shaded region given below?

In a game, a man wins Rs. 5 for getting a number greater than 4 and loses Rs. 1 otherwise, when a fair die is thrown.
The man decides to throw the die thrice but quits as soon as he gets a number greater than 4 .
The expected value of the amount (in Rs.) he wins or loses is :

Answer the questions from the figure given below:

If the reaction starts with one molecule of DNA, the number of cycles required to produce one billion amplified products is:

Which one of the following diseases is non-infectious?

The correct increasing order of basic strength of amine is :
(A) \(\text{C}_6\text{H}_5\text{NH}_2 < \text{NH}_3 < \text{C}_6\text{H}_5\text{CH}_2\text{NH}_2 < \text{C}_2\text{H}_5\text{NH}_2 < (\text{C}_2\text{H}_5)_2\text{NH}\)
(B) \(\text{NH}_3 < \text{C}_6\text{H}_5\text{NH}_2 < \text{C}_6\text{H}_5\text{CH}_2\text{NH}_2 < \text{C}_2\text{H}_5\text{NH}_2 < (\text{C}_2\text{H}_5)_2\text{NH}\)
(C) \(\text{C}_6\text{H}_5\text{CH}_2\text{NH}_2 < \text{C}_6\text{H}_5\text{NH}_2 < \text{NH}_3 < \text{C}_2\text{H}_5\text{NH}_2 < (\text{C}_2\text{H}_5)_2\text{NH}\)
(D) \(\text{C}_2\text{H}_5\text{NH}_2 < (\text{C}_2\text{H}_5)_2\text{NH} < \text{C}_6\text{H}_5\text{NH}_2 < \text{NH}_3\)
Choose the correct answer from the options given below :

In Octahedral coordination entities :

Arrange the evolutionary human species in their occurrence from recent to the oldest order.
(A) Australopithecus
(B) Homo erectus
(C) Homo sapiens
(D) Neanderthal man
Choose the correct answer from the options given below:

If \(x=a \sec ^3 \theta, y=a \tan ^3 \theta\), then \(\frac{d y}{d x}\) at \(\theta=\frac{\pi}{3}\) is