The volume of parallelepiped whose co-terminous edges are given by \(-3 \hat{i}+7 \hat{j}+5 \hat{k},-5 \hat{i}+7 \hat{j}-3 \hat{k}\) and \(7 \hat{i}-5 \hat{j}-3 \hat{k}\) is :

The Differential Equation representing the family of tangents to the circle \(x^2+y^2=r^2\)is:

The derivative of \(\frac{\tan ^{-1} x}{1+\tan ^{-1} x}\) with respect to \(\tan ^{-1}\) is :

If two dice are rolled 12 times and getting a total greater than 4 is considered as a success, then which of the following statements are correct?
(A) The probability of getting a total greater than 4 in a single throw of the pair of dice is \(5 / 6\).
(B) Mean = 10
(C) Variance \(=3 / 5\)
(D) The probability of getting a total less than or equal to 4 in a single throw is \(1 / 6\).
Choose the correct answer from the options given below :

The edge of a cube is increasing at a rate of \(7 cm / s\). The rate of change of area of the cube when edge of the cube is 3 cm is :

Which of the following functions is onto?

A random variable X has the following probability distribution

X	2	3	4	5
P(X)	5/k	7/k	9/k	11/k

Then the value of \(\frac{k}{4}\) is

Match List-I with List-II

List-I (Contraceptive methods)	List-II (Respective actions)
(A) Diaphragms	(I) Inhibit ovulation and implantation
(B) Contraceptive pills	(II) Increase phagocytosis of sperm within uterus
(C) Intra Uterine Devices	(III) Absence of menstrual cycle and ovulation following parturition
(D) Lactational amenorrhea	(IV) They cover the cervix blocking the entry of sperms

Choose the correct answer from the options given below:

Read the following passage carefully and answer the given questions.
Theory of Evolution
The evolution of life is a gradual process through which simple organisms are transformed into complex life forms over billions of years. It involves genetic changes, adaptation to the environment, and the emergence of diversity in species. Fossil records provide evidence of extinct life forms and transitional stages. Evolution is influenced by factors such as mutation, genetic recombination, and environmental pressure, shaping the biodiversity we see today. Charles Darwin and Alfred Russel Wallace were pioneering naturalists who developed the theory of evolution by natural selection.
Alfred Russel Wallace, a naturalist, arrived at conclusions similar to Darwin while working in:

Interstitial compounds are formed when small atoms like H, C, or N are trapped in the crystal lattice of metal. Which of the following statements are true about the interstitial compounds?
(A) Interstitial compounds have higher melting points than the pure metals.
(B) Interstitial compounds are very soft.
(C) Interstitial compounds are very reactive.
(D) Interstitial compounds retain metallic conductivity.
Choose the correct answer from the options given below:

If \(A=\left[\begin{array}{lll}1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9\end{array}\right]\), then \(|A|\) is equal to:

A 25 cm long solenoid with 100 turns carries a current of 3 A. The flux density at the center of the solenoid is

A convex lens of focal length 20 cm and two concave lenses of focal length 10 cm and 30 cm are placed in contact with each other. The focal length of the combined three lenses will be.