Let $O$ be the origin and let $PQR$ be an arbitrary triangle. The point$S$ is such that $\overrightarrow{OP}.\overrightarrow{OQ}+\overrightarrow{OR}.\overrightarrow{OS}=\overrightarrow{OR}.\overrightarrow{OP}+\overrightarrow{OQ}.\overrightarrow{OS}=\overrightarrow{OQ}.\overrightarrow{OR}+\overrightarrow{OP}.\overrightarrow{OS}$ then triangle $PQR$ has $S$ as its

For \(x>0, \lim _{x \rightarrow 0}\left((\sin x)^{1 / x}+\left(\frac{1}{x}\right)^{\sin x}\right)\) is

Let $f :\left(0, \infty \right)\to R$ be given by $f\left(x\right)= \underset{\frac{1}{x}}{\overset{x}{\int }}{e}^{-\left(t+\frac{1}{t}\right)}\frac{dt}{t},$ then

A factory has a total of three manufacturing units, \(M_1, M_2\), and \(M_3\), which produce bulbs independent of each other. The units \(M_1, M_2\), and \(M_3\) produce bulbs in the proportions of 2:2:1, respectively. It is known that \(20 \%\) of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by \(M_1, 15 \%\) are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by \(M_2\) is \(\frac{2}{5}\).
If a bulb is chosen randomly from the bulbs produced by \(M_3\), then the probability that it is defective is ____.

Let \(k \in \mathbb{R}\). If \(\lim _{x \rightarrow 0^+}(\sin (\sin k x)+\cos x+x)^{\frac{2}{x}}=e^6\), then the value of \(k\) is

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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:
The sum \(V_1+V_2+\ldots+V_n\) is

The vector(s) which is/are coplanar with vectors \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and \(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\), are perpendicular to the vector \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\) is/are

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A spray gun is shown in the figure where a piston pushes air out of a nozzle. A thin tube of uniform cross section is connected to the nozzle. The other end of the tube is in a small liquid container. As the piston pushes air through the nozzle, the liquid from the container rises into the nozzle and is sprayed out. For the spray gun shown, the radii of the piston and the nozzle are \(20 \mathrm{~mm}\) and \(1 \mathrm{~mm}\), respectively. The upper end of the container is open to the atmosphere.

Question :
If the piston is pushed at a speed of \(5 \mathrm{~mms}^{-1}\), the air comes out of the nozzle with a speed of

The reaction sequence given below is carried out with 16 moles of \(\mathbf{X}\). The yield of the major product in each step is given below the product in parentheses. The amount (in grams) of \(\mathbf{S}\) produced is ______ .

Use: Atomic mass (in amu) : \(\mathrm{H}=1, \mathrm{C}=\) \(12, \mathrm{O}=16, \mathrm{Br}=80\)

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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
For the above cell

Let \((x, y)\) be such that \(\sin ^{-1}(a x)+\cos ^{-1}(y)+\cos ^{-1}(b x y)=\frac{\pi}{2}\). Match the statements in Column I with the values in Column II.

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There are \(n\) urns each containing \((n+1)\) balls such that the ith urn contains \(i\) white balls and \((n+1-i)\) red balls. Let \(u_i\) be the event of selecting ith urn, \(i=1,2,3, \ldots, n\) and \(W\) denotes the event of getting a white balls.Question:
If \(n\) is even and \(E\) denotes the event of choosing even numbered urn \(\left(P\left(u_i\right)=\frac{1}{n}\right)\), then the value of \(P(W / E)\) is

Let \(\mathbb{R}\) denote the set of all real numbers. Define the function \(f: \mathbb{R \rightarrow \mathbb { R }}\) by
\(f(\mathrm{x})=\left\{\begin{array}{cc}2-2 x^2-x^2 \sin \frac{1}{x} & \text { if } x \neq 0 \\ 2 & \text { if } x=0\end{array}\right.\)
Then which one of the following statements is TRUE ?