The area of the region bounded by \(y^2=2 x+1\) and x - y = 1 is:

If \(x=a\left(\cos t+\log \tan \frac{t}{2}\right), y=a \sin t\), then value of \(\frac{d y}{d x}\) at \(t=\pi / 4\) is

The conditions required for applicability of Rolle's theorem for a real valued function \(f(x)\) on an interval \([a, b]\) are:
(A) \(f\) is continuous on \([a, b]\)
(B) \(f\) is differentiable on \((a, b)\)
(C) \(f(a)=f(b)\)
(D) \(f^{\prime}(c)=\)
Choose the correct answer from the options given below :

The least positive integer \(x\) satisfying \(28=x(\bmod 6)\) is :

The following data are from a random sample : \(5,8,10,7,10,14\), then the point estimate of the population standard deviation is

Match List-I with List-II (Given that \(c\) is an arbitrary constant)

List-I	List-II
(A) \(\int \frac{d x}{\sqrt{a^2-x^2}}\)	(I) \(\log _e\left|x+\sqrt{x^2-a^2}\right|+c\)
(B) \(\int \sqrt{a^2-x^2} d x\)	(II) \(\sin ^{-1} \frac{x}{a}+c\)
(C) \(\int \sqrt{x^2-a^2} d x\)	(III) \(\frac{x}{2} \sqrt{a^2-x^2}+\frac{a^2}{2} \sin ^{-1} \frac{x}{a}+c\)
(D) \(\int \frac{d x}{\sqrt{x^2-a^2}}\)	(IV) \(\frac{x}{2} \sqrt{x^2-a^2}-\frac{a^2}{2} \log _e\left|x+\sqrt{x^2-a^2}\right|+c\)

Choose the correct answer from the options given below :

For the following probability distribution,

X	1	2	3	4
p(x)	\(\frac{1}{10}\)	\(\frac{1}{5}\)	\(\frac{3}{10}\)	\(\frac{2}{5}\)

The value of \(E\left(X^2\right)-[E(X)]^2\)

Identify the incorrect example of albuminous seed.

Arrange the following nuclei in the increasing order of their binding energy per nucleon: \( ^4\text{He}, ^6_3\text{Li}, ^{56}_{26}\text{Fe}, ^{238}_{92}\text{U} \)

The fruits that develop without fertilization are called as

Choose the correct statements from the following :
(A) A dielectric develops a net dipole moment in the presence of an electric field.
(B) Electrostatic potential is constant throughout the volume of the conductor.
(C) Electric field \( \vec{E} \) depends upon the distance between two thin infinite long uniformly charged plane sheets held parallel to each other.
(D) Electric field due to a uniformly charged spherical shell is zero at all points inside the shell.
(E) Electric field is in the direction in which the potential decreases steepest.

Read the passage carefully and answer the following questions
There are mainly four series of the transition metals, \(3 d\) series ( \(S c\) to Zn), \(4 d\) series (Y to Cd), 5d series (La and Hf to Hg) and 6d series which has \(A c\) and elements from Rf to Cn. The two series of the inner transition metals; \(4 f\) (Ce to Lu) and \(5 f\) (Th to Lr) are known as lanthanoids and actinoids respectively. Their chemical properties are transitional between those of \(s\) and p-block elements. The transition metals (with the exception of Zn, Cd and Hg) are very hard and have low volatility. Their melting and boiling points are high. Transition metals have incomplete d subshell either in neutral atom or in their ions. With partly filled d orbitals these elements exhibit certain characteristic properties such as display of a variety of oxidation states, formation of coloured ions and entering into complex formation with a variety of ligands. The transition metals and their compounds are known for their catalytic activity.
In series Sc to Zn, the lowest enthalpy of atomization is shown by

Let A be a non-singular square matrix of order n, then
Match List - I with List - II

List - I	List - II
(A) A (adj A)	(I) \(\frac{1}{|A|}\)
(B) |adj A|	(II) \(|A|^n\)
(C) \(\left|A^{-1}\right|\)	(III) \(|A| I\)
(D) \(|A(\operatorname{adj} A)|\)	(IV) \(|A|^{n-1}\)

Choose the correct answer from the options given below: