A survey shows that \(73 \%\) of the persons working in an office like coffee, whereas \(65 \%\) like tea. If \(x\) denotes the percentage of them, who like both coffee and tea, then \(x\) cannot be

If the sum and product of the first three term in an \(A.P\). are \(33\) and \(1155\), respectively, then a value of its \(11^{th}\) tern is

Let \(X\) be a random variable having binomial distribution \(B (7, p )\). If \(P ( X =3)=5 P ( X =4)\), then the sum of the mean and the variance of \(X\) is

If one real root of the quadratic equation \(81x^2 + kx  + 256 = 0\) is cube of the other root, then a value of \(k\) is

The set of all possible values of \(\theta\) in the interval \((0, \pi)\) for which the points \((1,2)\) and \((\sin \theta,\) \(\cos \theta)\) lie on the same side of the line \(x+y= 1\) is

Let the equation of the plane, that passes through the point \((1,4,-3)\) and contains the line of intersection of the planes \(3 x-2 y+4 z-7=0\) and \(x+5 y-2 z+9=0\), be \(\alpha x+\beta y+\gamma z+3=0\), then \(\alpha+\beta+\gamma\) is equal to :

\(S = {\tan ^{ - 1}}\left( {\frac{1}{{{n^2} + n + 1}}} \right) + {\tan ^{ - 1}}\left( {\frac{1}{{{n^2} + 3n + 3}}} \right) + ..... + {\tan ^{ - 1}}\left( {\frac{1}{{1 + \left( {n + 19} \right)\left( {n + 20} \right)}}} \right)\) , then \(tan\,S\) is equal to 

Let \(\quad \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=-\hat{\mathrm{i}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{c}}=\beta \hat{\mathrm{j}}-\hat{\mathrm{k}}\), where \(\alpha\) and \(\beta\) are integers and \(\alpha \beta=-6\). Let the values of the ordered pair \((\alpha, \beta)\) for which the area of the parallelogram of diagonals \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) is \(\frac{\sqrt{21}}{2}\), be \(\left(\alpha_1, \beta_1\right)\) and \(\left(\alpha_2, \beta_2\right)\). Then \(\alpha_1^2+\beta_1^2-\alpha_2 \beta_2\) is equal to

\(\int \limits_{-\pi}^{\pi}|\pi-| x || d x\) is equal to :

If the domain of the function \(f(x)=\frac{\sqrt{x^2-25}}{\left(4-x^2\right)}\) \(+\log _{10}\left(x^2+2 x-15\right)\) is \((-\infty, \alpha) U[\beta, \infty)\), then \(\alpha^2+\beta^3\) is equal to :

The sum of the maximum and minimum values of the function \(f(x)=|5 x-7|+\left[x^{2}+2 x\right]\) is the interval \(\left[\frac{5}{4}, 2\right]\), where \([ t ]\) is the greatest integer \(\leq t\) is \(.......\)

In an examination of Mathematics paper, there are \(20\) questions of equal marks and the question paper is divided into three sections : \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\). A student is required to attempt total \(15\) questions taking at least \(4\) questions from each section. If section \(A\) has \(8\) questions, section \(\mathrm{B}\) has \(6\) questions and section \(\mathrm{C}\) has \(6\) questions, then the total number of ways a student can select \(15\) questions is

Let \(\alpha x=\exp \left(x^\beta y^\gamma\right)\) be the solution of the differential equation \(2 x^2 y d y-\left(1-x y^2\right) d x=0\), \(x >0, y (2)=\sqrt{\log _e 2}\). Then \(\alpha+\beta-\gamma\) equals :