Let \(g_{i}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}, i=1,2\), and \(f:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}\) be functions such that \(g_{1}(x)=1, g_{2}(x)=|4 x-\pi|\) and \(f(x)=\sin ^{2} x\), for all \(x \in\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right]\)

Define

\(

S_{i}=\int_{\frac{\pi}{8}}^{\frac{3 \pi}{8}} f(x) \cdot g_{i}(x) d x, \quad i=1,2

\)

The value of $\frac{48S_2}{{\pi }^2}$ is ____.

Let $f:R\to R$ be defined by $f\left(x\right)=\frac{x^2-3x-6}{x^2+2x+4}$ Then which of the following statements is(are) TRUE?

Let the functions $f:R\to R$ and $g:R\to R$ be defined by
$f\left(x\right)=e^{x-1}-e^{-\left|x-1\right|}$ and $g\left(x\right)=\frac{1}{2}\left(e^{x-1}+e^{1-x}\right)$.
Then the area of the region in the first quadrant bounded by the curves $y=f\left(x\right), y=g\left(x\right)$ and $x=0$ is

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Let \(O\) be the origin, and \(\overrightarrow{O X}, \overrightarrow{O Y}, \overrightarrow{O Z}\) be three unit vectors in the directions of the sides \(\overrightarrow{Q R}, \overrightarrow{R P}\), \(\overrightarrow{P Q}\), respectively, of a triangle \(P Q R\).


Question:

If the triangle \(P Q R\) varies, then the minimum value of

\(\cos (P+Q)+\cos (Q+R)+\cos (R+P)\) is

Let $\mathrm{M}=\left[\begin{array}{ccc}0& 1& \mathrm{a}\\ 1& 2& 3\\ 3& b& 1\end{array}\right]$ and $\mathrm{a}\mathrm{d}\mathrm{j}\mathrm{M}=\left[\begin{array}{ccc}-1& 1& -1\\ 8& -6& 2\\ -5& 3& -1\end{array}\right]$ where $\mathrm{a}$ and $\mathrm{b}$ are real numbers. Which of the following options is/are correct?

If \(x^2-10 a x-11 b=0\) have roots \(c\) and \(d\) and \(x^2-10 c x-11 d=0\) have roots \(a\) and \(b\), then, find \(a+b+c+d\).

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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 \(g^{\prime}\left(\frac{1}{2}\right)\) is

Two loudspeakers M and N are located 20m apart and emit sound at frequencies 118 Hz and 121 Hz, respectively. A car is initially at a point P, 1800 m away from the midpoint Q of the line MN and moves towards Q constantly at 60 km/hr along the perpendicular bisector of MN. It crosses Q and eventually reaches a point R, 1800 m away from Q. Let $\upsilon \left(t\right)$ represent the beat frequency measured by a person sitting in the car at time t. Let ${\nu }_p, {\nu }_Q and {\nu }_R$ be the beat frequencies measured at locations P, Q and R, respectively. The speed of sound in air is $330 m{s}^{-1}.$ Which of the following statement(s) is(are) true regarding the sound heard by the person ?

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The concentration of potassium ions inside a biological cell is at least twenty times higher than the outside. The resulting potential difference across the cell is important in several processes such as transmission of nerve impulses and maintaining the ion balance. A simple model for such a concentration cell involving a metal \(\mathrm{M}\) is :
\(
M(s) \mid M^{+}(\alpha q ; 0.05 \text { molar }) \| M^{+}(a q ; 1 \text { molar })\) \(\mid M(s)
\)
For the above electrolytic cell the magnitude of the cell potential \(\left|E_{\text {cell }}\right|=70 \mathrm{mV}\).
Question:
If the \(0.05\) molar solution of \(M^{+}\)is replaced by a \(0.0025\) molar \(M^{+}\) solution, then the magnitude of the cell potential would be

For the function \(f(x)=x \cos \frac{1}{x}, x \geq 1\),

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?

Let \(P Q R\) be a triangle of area \(\Delta\) with \(a=2, b=\frac{7}{2}\) and \(c=\frac{5}{2} ;\) where \(a, b\), and \(c\) are the lengths of the sides of the triangle opposite to the angles at \(P, Q\) and \(R\) respectively. Then \(\frac{2 \sin P-\sin 2 P}{2 \sin P+\sin 2 P}\) equals.

Three boys and two girls stand in a queue. The probability that the number of boys ahead of every girl is atleast one more than the number of girls ahead of her is