If \(f^{\prime \prime}(x)=-f(x)\), where \(f(x)\) is a continuous double differentiable function and \(g(x)=f^{\prime}(x)\). If \(F(x)=\left(f\left(\frac{x}{2}\right)\right)^2+\left(g\left(\frac{x}{2}\right)\right)^2\) and \(F(5)=5\), then \(F(10)\) is

Consider the system of equations
where \(\quad a, b, c, d \in\{0,1\}\)
Statement 1 The probability that the system of equations has a unique solution, is \(3 / 8\).
Statement 2 The probability that the system of equations has a solution, is 1 .

Let \(P(6,3)\) be a point on the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\). If the normal at the point \(P\) intersects the \(X\)-axis at \((9,0)\), then the eccentricity of the hyperbola is

Let $f\mathbb{ }:\mathbb{R}\to \left(0, \infty \right)$ and $g\mathbb{ }:\mathbb{R}\to \mathbb{R}$ be twice differentiable functions such that $f^{″}$ and $g^{″}$ are continuous functions on $\mathbb{R}$ . Suppose $f^{\prime }\left(2\right)=g\left(2\right)=0, f^{″}\left(2\right)\ne 0$ and $g^{\prime }\left(2\right) \ne 0.\underset{x\to 2}{\lim }\frac{f\left(x\right) g\left(x\right)}{f^{\prime } \left(x\right) g^{\prime }\left(x\right)}=1,$ then

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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

A signal which can be green or red with probability \(\frac{4}{5}\) and \(\frac{1}{5}\) respectively, is received by station \(A\) and then transmitted to station B. The probability of each station receiving the signal correctly is \(\frac{3}{4}\). If the signal received at station \(B\) is green, then the probability that the original signal green is

For nonnegative integers $s$ and $r$, let $\left(\begin{array}{l}s\\ r\end{array}\right)=\left\{\begin{array}{ll}\frac{s!}{r!\left(s-r\right)!}& \text{ if }r\le s\\ 0& \text{ if }r>s\end{array}\right.$. For positive integers $m$ and $n,$ let $g\left(m,n\right)=\sum _{p=0}^{m+n}\frac{f\left(m,n,p\right)}{\left(\begin{array}{c}n+p\\ p\end{array}\right)}$, where for any nonnegative integer $p$,
$f\left(m,n,p\right)=\sum _{i=0}^p\left(\begin{array}{c}m\\ i\end{array}\right)\left(\begin{array}{c}n+i\\ p\end{array}\right)\left(\begin{array}{c}p+n\\ p-i\end{array}\right)$. Then which of the following statements is/are TRUE?

A single slit diffraction experiment is performed to determine the slit width using the equation, \(\frac{b d}{D}=m \lambda\), where \(b\) is the slit width, \(D\) the shortest distance between the slit and the screen, \(d\) the distance between the \(m^{\text {th }}\) diffraction maximum and the central maximum, and \(\lambda\) is the wavelength. \(D\) and \(d\) are measured with scales of least count of 1 cm and 1 mm, respectively. The values of \(\lambda\) and \(m\) are known precisely to be 600 nm and 3, respectively. The maximum absolute error (in \(\mu \mathrm{m}\)) in the value of \(b\) estimated using the diffraction maximum that occurs for \(m=3\) with \(d=5 \mathrm{~mm}\) and \(D=1 \mathrm{~m}\) is ______.

During sodium nitroprusside test of sulphide ion in an aqueous solution, one of the ligands coordinated to the metal ion is converted to

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Let \(a\), b and \(c\) be three real numbers satisfying \(\left[\begin{array}{lll}a & b & c\end{array}\right]\left[\begin{array}{lll}1 & 9 & 7 \\ 8 & 2 & 7 \\ 7 & 3 & 7\end{array}\right]=\left[\begin{array}{lll}0 & 0 & 0\end{array}\right]\)Question:
Let \(b=6\), with \(a\) and \(c\) satisfying Eq. (E). If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(a x^2+b x+c=0\), then \(\sum_{n=0}^{\infty}\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)^n\) is

The total number of real solutions of the equation \(\theta=\tan ^{-1}(2 \tan \theta)-\frac{1}{2} \sin ^{-1}\left(\frac{6 \tan \theta}{9+\tan ^2 \theta}\right)\) is (Here, the inverse trigonometric functions \(\sin ^{-1} x\) and \(\tan ^{-1} x\) assume values in \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) and \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), respectively.)

Positive and negative point charges of equal magnitude are kept at \(\left(0,0, \frac{a}{2}\right)\) and \(\left(0,0, \frac{-a}{2}\right)\), respectively. The work done by the electric field when another positive point charge is moved from \((-a, 0,0)\) to \((0, a, 0)\) is

The correct statement(s) pertaining to the adsorption of a gas on a solid surface is(are)