Paragraph:

Given that for each \(a \in(0,1)\),

\(\lim _{h \rightarrow 0^{+}} \int_{h}^{1-h} t^{-a}(1-t)^{a-1} d t\)

exists. Let this limit be \(g(a)\). In addition, it is given that the function \(g(a)\) is differentiable on \((0,1)\).


Question:

The value of \(g\left(\frac{1}{2}\right)\) is

Let $\alpha , \beta , \gamma , \delta$ be real numbers such that ${\alpha }^2+{\beta }^2+{\gamma }^2\ne 0$ and $\alpha +\gamma =1$. Suppose the point $\left(3,2,-1\right)$ is the mirror image of the point $\left(1,0,-1\right)$ with respect to the plane $\alpha x+\beta y+\gamma z=\delta$. Then which of the following statements is/are TRUE?

Match the following

For any $y\in \mathrm{ℝ}$, let ${\cot }^{-1}\left(y\right)\in \left(0,\pi \right)$ and ${\tan }^{-1}\left(y\right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then the sum of all the solutions of the equation ${\tan }^{-1}\left(\frac{6y}{9-y^2}\right)+{\cot }^{-1}\left(\frac{9-y^2}{6y}\right)=\frac{2\pi }{3}$ for $0<\left|y\right|<3$, is equal to

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Let \(p, q\) be integers and let \(\alpha, \beta\) be the roots of the equation, \(x^{2}-x-1=0\), where \(\alpha \neq \beta\). For \(n=0,1,2, \ldots\), let \(a_{n}=p \alpha^{n}+q \beta^{n}\)
FACT: If \(a\) and \(b\) are rational numbers and \(a+b \sqrt{5}=0\), then \(a=0=b\).

Question:
If \(a_{4}=28\), then \(p+2 q=\)

Let $S=\left(0,1\right)\cup \left(1, 2\right)\cup \left(3, 4\right)$ and $T=\left\{0, 1, 2, 3\right\}$. Then which of the following statements is(are) true?

Let \(P(3,2,6)\) be a point in space and \(Q\) be a point on the line \(\mathbf{r}=(\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})+\mu(-3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+5 \hat{\mathbf{k}})\).
Then, the value of \(\mu\) for which the vector \(\mathbf{P Q}\) is parallel to the plane \(x-4 y+3 z=1\) is

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Let \(M=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^{2}+y^{2} \leq r^{2}\right\}\),
where \(r>0 .\) Consider the geometric progression \(a_{n}=\frac{1}{2^{n-1}}, n=1,2,3, \ldots .\) Let \(S_{0}=0\) and, for \(n \geq 1\), let \(S_{n}\) denote the sum of the first \(n\) terms of this progression. For \(n \geq 1\), let \(C_{n}\) denote the circle with center \(\left(S_{n-1}, 0\right)\) and radius \(a_{n}\), and \(D_{n}\) denote the circle with center \(\left(S_{n-1}, S_{n-1}\right)\) and radius \(a_{n}\).

Question:
Consider \(M\) with \(r=\frac{1025}{513}\). Let \(k\) be the number of all those circles \(C_{n}\) that are inside \(M\). Let \(l\) be the maximum possible number of circles among these \(k\) circles such that no two circles intersect. Then

Let $f\left(x\right)=\frac{\sin \pi x}{x^2}, x>0.$
Let $x_1<x_2<x_3<\dots <x_n<\dots$ be all the points of local maximum of $f$ and $y_1<y_2<y_3<\dots <y_n<\dots$ be all the points of local minimum of $f.$
Then which of the following options is/are correct?

\(\lim _{x \rightarrow \frac{\pi}{4}} \frac{\int_2^{\sec ^2 x} f(t) d t}{x^2-\frac{\pi^2}{16}}\) equals

List-I contains various reaction sequences and List-II contains the possible products. Match each entry in List-I with the appropriate entry in List-II and choose the correct option.

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Let \(p\) be an odd prime number and \(T_p\) be the following set of \(2 \times 2\) matrices
\[
T_p=\left\{A=\left[\begin{array}{ll}
a & b \\
c & a
\end{array}\right] ; a, b, c \in\{0,1,2, \ldots, p-1\}\right\}
\]Question:
The number of \(A\) in \(T_p\) such that the trace of \(A\) is not divisible by \(p\) but det \((A)\) is divisible by \(p\) is
[Note : The trace of a matrix is the sum of its diagonal entries.]

For a double strand DNA, one strand is given below:

The amount of energy required to split the double strand DNA into two single strands is _______ kcal \(\mathrm{mol}^{-1}.\)
[Given: Average energy per \(\mathrm{H}\)-bond for A-T base pair \(=1.0 \mathrm{kcal} \mathrm{mol}^{-1}, \mathrm{G}-\mathrm{C}\) base pair \(=1.5 \mathrm{kcal}\) \(\mathrm{mol}^{-1}\), and A-U base pair \(=1.25 \mathrm{kcal} \mathrm{mol}^{-1}.\) Ignore electrostatic repulsion between the phosphate groups.]