Paragraph:
Let \(V_r\) denotes the sum of the first \(r\) terms of an arithmetic progression \((A P)\) whose first term is \(r\) and the common difference is \((2 r-1)\). Let \(T_r=V_{r+1}-V_r-2\) and \(Q_r=T_{r+1}-T_r\) for \(r=1,2, \ldots\)
Question:
Which one of the following is a correct statement?

The total number of local maxima and local minima of the function \(f(x)=\left\{\begin{array}{lc}(2+x)^3 ; & -3 < x \leq-1 \\ x^{\frac{2}{3}} ; & -1 < x < 2\end{array}\right.\) is

Let \(p\) and \(q\) be real numbers such that \(p \neq 0, p^3 \neq q\) and \(p^3 \neq-q\). If \(\alpha\) and \(\beta\) are non-zero complex numbers satisfying \(\alpha+\beta=-p\) and \(\alpha^3+\beta^3=q\), then \(a\) quadratic equation having \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\) as its roots is

The maximum value of the expression \(\frac{1}{\sin ^2 \theta+3 \sin \theta \cos \theta+5 \cos ^2 \theta}\) is

Let $a, b$ and $\lambda$ be positive real numbers. Suppose $P$ is an end point of the latus rectum of the parabola $y^2=4\lambda x$, and suppose the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through the point $P$. If the tangents to the parabola and the ellipse at the point $P$ are perpendicular to each other, then the eccentricity of the ellipse is

Consider a triangle \(A B C\) and let \(a, b\) and \(c\) denote the lengths of the sides opposite to vertices \(A, B\) and \(C\) respectively. Suppose \(a=6, b=10\) and the area of the triangle is \(15 \sqrt{3}\). If \(\angle A C B\) is obtuse and if \(r\) denotes the radius of the incircle of the triangle, then \(r^2\) is equal to

Match the following.
 

	List – I		List – II
(A)	The number of polynomials f(x) with non – negative integer coefficients of degree $\le 2,$ satisfying f(0) =0 and $\underset{0}{\overset{1}{\int }}f\left(x\right)dx=1, is$	(P)	8
(B)	The number of point in the interval $\left[- \sqrt{13}, \sqrt{13}\right]$ at which $f\left(x\right)=\sin \left(x^2\right)+\cos \left(x^2\right)$ attains its maximum value, is	(Q)	2
(C)	$\underset{-2}{\overset{2}{\int }}\frac{3x^2}{\left(1+e^x\right)}dx$ equals	(R)	4
(D)	$\frac{\left(\underset{-\frac{1}{2}}{\overset{\frac{1}{2}}{\int }}\cos 2x .\log \left(\frac{1+x}{1-x}\right)dx\right)}{\left(\underset{0}{\overset{\frac{1}{2}}{\int }}\cos 2x .\log \left(\frac{1+x}{1-x}\right)dx\right)}$	(S)	0

For the given compound X, the total number of optically active stereoisomers is ____.

A soft plastic bottle, filled with water of density \(1 \mathrm{gm} / \mathrm{cc}\), carries an inverted glass test-tube with some air (ideal gas) trapped as shown in the figure. The test-tube has a mass of \(5 \mathrm{gm}\), and it is made of a thick glass of density \(2.5 \mathrm{gm} / \mathrm{cc}\). Initially the bottle is sealed at atmospheric pressure \(p_{0}=10^{5} \mathrm{~Pa}\) so that the volume of the trapped air is \(v_{0}=3.3 \mathrm{cc}\). When the bottle is squeezed from outside at constant temperature, the pressure inside rises and the volume of the trapped air reduces. It is found that the test tube begins to sink at pressure \(p_{0}+\Delta p\) without changing its orientation. At this pressure, the volume of the trapped air is \(v_{0}-\Delta v\).
Let \(\Delta v=X\) cc and \(\Delta p=Y \times 10^{3} \mathrm{~Pa}\).

The value of $X$is ____.

Match the transformation in Column I with appropriate options in Column II.

Column I	Column II
A. \(CO _2(s) \rightarrow CO _2(g)\)	p. Phase transition
B. \(CaCO _3(s) \rightarrow CaO (s)\) \(+~ CO _2(g)\)	q. Allotropic change
C. \(2 H \rightarrow H _2(g)\)	r. \(\Delta\text{H}\) is positive
D. \(P _{\text {(white, solid) }} \rightarrow P _{\text {(red, solid) }}\)	s. \(\Delta\text{S}\) is positive
	t. \(\Delta\text{S}\) is negative

Which of the following is (are) NOT the square of a $3\times 3$ matrix with real entries?

A long straight wire carries a current, $I=2$ ampere. A semi-circular conducting rod is placed beside it on two conducting parallel rails of negligible resistance. Both the rails are parallel to the wire. The wire, the rod and the rails lie in the same horizontal plane, as shown in the figure. Two ends of the semi-circular rod are at distances $1 \mathrm{cm}$ and $4 \mathrm{cm}$ from the wire.  At time $t=0$, the rod starts moving on the rails with a speed $v=3.0 \mathrm{m} {\mathrm{s}}^{-1}$ (see the figure). A resistor $R=1.4 \mathrm{\Omega }$ and a capacitor $C_0=5.0 \mathrm{\mu F}$ are connected in series between the rails. At time $t=0$, $C_0$ is uncharged. Which of the following statement(s) is (are) correct? [${\mu }_0=4\pi \times {10}^{-7}$ SI units. Take $\ln 2=0.7$]

Benzene and naphthalene form an ideal solution at room temperature. For this process, the true statement(s) is (are)