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Let \(a, r, s, t\) be nonzero real numbers. Let \(P\left(a t^{2}, 2 a t\right), Q, R\left(a r^{2}, 2 a r\right)\) and \(S\left(a s^{2}, 2 a s\right)\) be distinct points on the parabola \(y^{2}=4 a x\). Suppose that \(P Q\) is the focal chord and lines \(Q R\) and \(P K\) are parallel, where \(K\) is the point \((2 a, 0)\).


Question:

The value of \(r\) is

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Consider the functions defined implicitly by the equation \(y^3-3 y+x=0\) on various intervals in the real line. If \(x \in(-\infty,-2) \cup(2, \infty)\), the equation implicitly defines a unique real valued differentiable function \(y=f(x)\). If \(x \in(-2,2)\), the equation implicitly defines a unique real valued differentiable function \(y=g(x)\), satisfying \(g(0)=0\).Question:
The area of the region bounded by the curve \(y=f(x)\), the \(X\)-axis and the lines \(x=a\) and \(x=b\), where \(-\infty < a < b < -2\), is

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Tangents are drawn from the point \(P(3,4)\) to the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) touching the ellipse at points \(A\) and \(B\).Question:
The orthocentre of the triangle \(P A B\) is

Let $f:\left[0, \infty \right)\to R$ be a continuous function such that $f\left(x\right)=1-2x+{\int }_0^x{e}^{x-t}f\left(t\right)dt$ for all $x\in \left[0,\infty \right).$ Then, which of the following statement(s) is (are) TRUE?

Let \(f(x)=x^2\) and \(g(x)=\sin x\) for all \(x \in R\). Then, the set of all \(x\) satisfying \((\) fogogof \()(x)=(\operatorname{gogof})(x)\), where \((f \circ g)(x)=f(g(x))\) is

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Let \(\psi_{1}:[0, \infty) \rightarrow \mathbb{R}, \psi_{2}:[0, \infty) \rightarrow \mathbb{R}, f:[0, \infty) \rightarrow \mathbb{R}\) and \(g:[0, \infty) \rightarrow \mathbb{R}\) be functions such that \(f(0)=g(0)=0\),

\(\psi_{1}(x)=e^{-x}+x, \quad x \geq 0\),

\(\psi_{2}(x)=x^{2}-2 x-2 e^{-x}+2, \quad x \geq 0\),

\(f(x)=\int_{-x}^{x}\left(|t|-t^{2}\right) e^{-t^{2}} d t, \quad x>0\)

and \(g(x)=\int_{0}^{x^{2}} \sqrt{t} e^{-t} d t, \quad x>0\)


Question:

Which of the following statements is TRUE ?

Let $X$ and $Y$ be two events such that $P\left(X\right)=\frac{1}{3}$, $P\left(X|Y\right)=\frac{1}{2}$ and $P\left(Y|X\right)=\frac{2}{5}.$ Then

Suppose $\det \left[\begin{array}{cc}\sum _{k=0}^{n}k& \sum _{k=0}^n{}_{\mathrm{ }}{}^n{C}_k{k}^2\\ \sum _{k=0}^n{}_{\mathrm{ }}{}^n{C}_{k}k& \sum _{k=0}^n{}_{\mathrm{ }}{}^n{C}_k{3}^k\end{array}\right]=0,$ holds for some positive integer $n.$ Then $\sum _{k=0}^n\frac{{}_{ }{}^n{C}_k}{k+1}$ equals

A student performed the experiment of determination of focal length of a concave mirror by \(u-v\) method using an optical bench of length \(1.5 \mathrm{~m}\). The focal length of the mirror used is 24 \(\mathrm{cm}\). The maximum error in the location of the image can be \(0.2 \mathrm{~cm}\).
The 5 sets of \((u, v)\) values recorded by the student (in \(\mathrm{cm})\) are : \((42,56),(48\), \(48),(60,40),(66,33),(78,39)\). The data set(s) that cannot come from experiment and is (are) incorrectly recorded, is(are)

Consider the given data with frequency distribution $\begin{array}{lllllll}{x}_i& 3& 8& 11& 10& 5& 4\\ f_i& 5& 2& 3& 2& 4& 4\end{array}$ Match each entry in List-I to the correct entries in List-II.

	List-I		List-II
$\left(\mathrm{P}\right)$	The mean of the above data is	$\left(1\right)$	$2.5$
$\left(\mathrm{Q}\right)$	The median of the above data is	$\left(2\right)$	$5$
$\left(\mathrm{R}\right)$	The mean deviation about the mean of the above data is	$\left(3\right)$	$6$
$\left(\mathrm{S}\right)$	The mean deviation about the median of the above data is	$\left(4\right)$	$2.7$
		$\left(5\right)$	$2.4$

The correct option is

The number of resonance structures for N is

Let \(\mathbb{R}^3\) denote the three-dimensional space. Take two points \(P=(1,2,3)\) and \(Q=(4,2,7)\). Let \(\operatorname{dist}(X, Y)\) denote the distance between two points \(X\) and \(Y\) in \(\mathbb{R}^3\). Let
\(\begin{gathered}S=\left\{X \in \mathbb{R}^3:(\operatorname{dist}(X, P))^2-(\operatorname{dist}(X, Q))^2=50\right\} \text { and } \\T=\left\{Y \in \mathbb{R}^3:(\operatorname{dist}(Y, Q))^2-(\operatorname{dist}(Y, P))^2=50\right\} .\end{gathered}\)
Then which of the following statements is (are) TRUE?

When 100 mL of 1.0 M HCl was mixed with 100 mL of 1.0 M NaOH in an insulated beaker at constant pressure, a temperature increase of ${5.7}^{\mathrm{o}}\mathrm{C}$ was measured for the beaker and its contents (Expt.1). Because the enthalpy of neutralization of a strong acid with a strong base is a constant $(-57.0\mathrm{ }\mathrm{k}\mathrm{J}\mathrm{ }\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1})$ , this experiment could be used to measure the calorimeter constant. In a second experiment (Expt. 2), 100 mL of 2.0 M acetic acid $({\mathrm{K}}_{\mathrm{a}}=2.0\times {10}^{-5})$ was mixed with 100 mL of 1.0 M NaOH (under identical conditions to Expt. 1) where a temperature rise of ${5.6}^{\mathrm{o}}\mathrm{C}$ was measured. (Consider heat capacity of all solutions as $4.2\mathrm{ }\mathrm{J}\mathrm{ }{\mathrm{g}}^{-1}\mathrm{ }{\mathrm{K}}^{-1}$ and density of all solutions as $1.0\mathrm{ }\mathrm{g}\mathrm{ }\mathrm{m}{\mathrm{L}}^{-1}$ )
The pH of the solution after Expt. 2 is