A student has to answer a multiple-choice question having 5 alternatives in which two or more than two alternatives are correct. Then the number of ways in which the student can answer that question is

A plane passes through \((2,3,-1)\) and is perpendicular to the line having direction ratios \(3,-4,7\). The perpendicular distance from the origin to this plane is

The smaller side of the rectangle with the largest area, that can be inscribed inside a semi-circle of radius $2$ units is of length

If \(f: \mathbb{R} \rightarrow \mathbb{R}\) is defined by \(f(\mathrm{x})=2 \mathrm{x}+\sin \mathrm{x}, \mathrm{x} \in \mathrm{R}\), then \(f\) is

If the error in measuring the side \(l\) of an equilateral triangle is 0.01 , then the percentage error in the area of the triangle, in terms of its side \(l\) is

Let \(\sigma_1, \sigma_2\) be the standard deviations of two distributions \(D_1\) and \(D_2\) respectively and \(D_1\) be more consistent than \(D_2\). If the means of \(D_1\) and \(D_2\) are same, then the percentage increase in the standard deviation of \(D_2\) over the standard deviation of \(D_1\) is

In a triangle ABC , if \(a=5, b=3, c=7\), then \(\sqrt{\frac{\sin (A-B)}{\sin (A+B)}}=\)

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\) Assertion (A) : The identity \(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for \(\vec{a}\). Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\), \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\) Which of the following is correct?

If \(R\) is the set of all real numbers and \(f: R-\{2\} \rightarrow R\) is defined by \(f(x)=\frac{2+x}{2-x}\) for \(x \in R-\{2\}\)

Certain volume of oxygen gas diffuses through a porous pot in 20 seconds. Same volume of another gas (X) diffuses in \(\mathrm{Y}\) seconds as that of oxygen, then \((\mathrm{X})\) and \(\mathrm{Y}\) respectively are

The vapour pressure of pure \(\mathrm{CCl}_4\) (molar mass \(=154 \mathrm{~g} \mathrm{~mol}^{-1}\) ) and \(\mathrm{SnCl}_4\) (molar mass \(=170 \mathrm{~g} \mathrm{~mol}^{-1}\) ) at \(25^{\circ} \mathrm{C}\) are 115.0 and 238.0 torr respectively. Assuming ideal behaviour, calculate the total approximate vapour pressure in torr of a solution containing \(10 \mathrm{~g}\) of \(\mathrm{CCl}_4\) and \(15 \mathrm{~g}\) of \(\mathrm{SnCl}_4\).

Let \(f: R \rightarrow R\) be defined by
\(f(x)=\left\{\begin{array}{ccc}\alpha+\frac{\sin [x]}{x}, & \text { if } & x>0 \\ 2, & \text { if } & x=0 \\ \beta+\left[\frac{\sin x-x}{x^3}\right], & \text { if } & x < 0\end{array}\right.\)
equal to

For a reversible reaction \(\quad A \rightleftharpoons B\), which one of the following statements is wrong from the given energy profile diagram?