If \(\vec{a}, \vec{b}, \vec{c}\) are three mutually perpendicular unit vectors, then \(|\vec{a}+\vec{b}+\vec{c}|\) is equal to:

Which of the following matrices is symmetric as well as skew-symmetric?

If \(\theta\) is the angle between the vectors \(2\hat{i} - 2\hat{j} + 4\hat{k}\) and \(3\hat{i} + \hat{j} + 2\hat{k}\), then value of \(\sin \theta\) is:

If \(\left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right]\left[\begin{array}{cc}3 & 5 \\ -1 & 3\end{array}\right]=\left[\begin{array}{cc}m & 14 \\ 2 & n\end{array}\right]\), then m + n is equal to:

The probability distribution function of a normal variate with mean \(\mu\) and variance \(\sigma^2\) is given by :
\(f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2},-\infty< x< \infty,-\infty < \mu < \infty, \sigma>0\)
If \(y=f(x)\) be the normal probability curve, then which of the following is correct?
(A) The normal curve is symmetrical about the line \(x=\mu\).
(B) Mean, median and mode of the distribution coincide.
(C) \(Y\)-axis is an asymptote to the normal curve.
(D) If \(x\) increases numerically, \(f(x)\) decreases rapidly.
Choose the correct answer from the options given below :

The degree of the differential equation \(\frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^3=\left(\frac{d^3 y}{d x^3}\right)^2\) is :

Match List-I with List-II

List-I (Differential equation)	List-II (Order and degree)
(A) \(\left(y^{\prime \prime}\right)^3+\left(y^{\prime}\right)^4-6=\left(y^{\prime \prime \prime}\right)^2\)	(I) Order \(=1\), Degree = 2
(B) \(\sqrt{\left(y^{\prime}\right)^2+5}=y^{\prime \prime}\)	(II) Order \(=2\), Degree = 3
(C) \(\left(y^{\prime}\right)^2=\left(2+y^{\prime \prime}\right)^{3 / 2}\)	(III) Order \(=2\), Degree = 2
(D) \(y=x y^{\prime}+\sqrt{a^2\left(y^{\prime}\right)^2+b^2}\)	(IV) Order \(=3\), Degree = 2

Choose the correct answer from the options given below :

Which one of the following pyramids of biomass is generally inverted in the ecosystems?

Answer the question on basis of passage given below :
Adsorption is surface phenomenon and it differs from absorption which occurs throughout the body of the substance which absorbs. In physisorption the attractive forces are mainly van der Waal forces while in chemisorption actual bonding occurs between the particles of adsorbent and adsorbed more easily on the surface of a solid as compared to the gases which are liquified with difficulty. Adsorption increases with the increases in pressure and decreases as temperature is increased.
Freundlich adsorption isotherm gives a straight line on plotting.

In aldehydes, the carbonyl group is bonded to a carbon and hydrogen, while in ketones it is bonded to two carbon atoms. The carbonyl compounds in which carbonyl group is bonded to oxygen are known as carboxylic acid and their derivatives.
Which series represents compounds in increasing order of acidity?

Hoffmann bromamide degradation method is used for preparation of

\(V_2 O _6\) catalyst is used in the manufacturing of

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.
Which one of the following statements is not a connotation of the theory of special creation?