Let the plane \(a x+b y+c z+d=0\) bisect the line joining the points \((4,-3,1)\) and \((2,3,-5)\) at the right angles. If \(a , b , c , d\) are integers, then the minimum value of \(\left(a^{2}+b^{2}+c^{2}+d^{2}\right)\) is

If two different numbers are taken from the set \(\left\{ {0,1,2,3, \ldots ,10} \right\}\), then the probability that their sum as well as absolute difference are both multiple of \(4\), is 

Let \(\alpha\) and \(\beta\) be the roots of \(x^2+\sqrt{3 x}-16=0\), and \(\gamma\) and \(\delta\) be the roots of \(x^2+3 x-1=0\). If \(P_n=\alpha^n+\beta^n\) and \(Q_n=\gamma^n+\delta^n\), then \(\frac{\mathrm{P}_{25}+\sqrt{3 \mathrm{P}_{24}}}{2 \mathrm{P}_{23}}+\frac{\mathrm{Q}_{25}-\mathrm{Q}_{23}}{\mathrm{Q}_{24}}\) is equal to

If the vectors \(\vec{a}=\lambda \hat{i}+\mu \hat{j}+4 \hat{k}, \vec{b}=2 \hat{i}+4 \hat{j}-2 \hat{k}\) and \(\vec{c}=2 \hat{i}+3 \hat{j}+\hat{k}\) are coplanar and the projection of \(\vec{a}\) on the vector \(\vec{b}\) is \(\sqrt{54}\) units, then the sum of all possible values of \(\lambda+\mu\) is equal to

The term independent of \(x\) in the expansion of \(\left[\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right]^{10}, x \neq 1,\) is equal to ....... .

 Let \( f, \mathrm{~g}: \mathrm{R} \rightarrow \mathrm{R}\) be defined as : \(f(\mathrm{x})=|\mathrm{x}-1|\) and  \(g(x)=\left\{\begin{array}{cc}\mathrm{e}^{\mathrm{x}}, & \mathrm{x} \geq 0 \\ \mathrm{x}+1, & \mathrm{x} \leq 0\end{array}\right.\). Then the function \(f(\mathrm{~g}(\mathrm{x}))\) is

The sum of all rational terms in the expansion of \(\left(1+2^{1 / 3}+3^{1 / 2}\right)^6\) is equal to

\(\smallint \frac{{2{x^{12}} + 5{x^9}}}{{{{\left( {{x^5} + {x^3} + 1} \right)}^3}}}dx = \)

If the two roots of the equation \(\left( {a - 1} \right)\left( {{x^4} + {x^2} + 1} \right) + \left( {a + 1} \right){\left( {{x^2} + x + 1} \right)^2} = 0\) are real and distinct, then the set of all values of \('a'\) is

The foot of the perpendicular drawn from the point \((4,2,3)\) to the line joining the points \((1,-2,3)\) and \((1,1,0)\) lies on the plane

\(\mathop \smallint \limits_0^\pi \sqrt {1 + 4{{\sin }^2}\frac{x}{2} - 4\sin \frac{x}{2}} \;dx = \)

In a survey of \(220\) students of a higher secondary school, it was found that at least \(125\) and at most \(130\) students studied Mathematics; at least \(85\) and at most \(95\) studied Physics; at least \(75\) and at most \(90\) studied Chemistry; \(30\) studied both Physics and Chemistry; \(50\) studied both Chemistry and Mathematics; \(40\) studied both Mathematics and Physics and \(10\) studied none of these subjects. Let \(\mathrm{m}\) and \(\mathrm{n}\) respectively be the least and the most number of students who studied all the three subjects. Then \(\mathrm{m}+\mathrm{n}\) is equal to ...........

Let \(P(\alpha,\beta,\gamma)\) be the point on the line \(\frac{x-1}{2}=\frac{y+1}{-3}=z\) at a distance \(4\sqrt{14}\) from the point (1, -1, 0) and nearer to the origin. Then the shortest distance, between the lines \(\frac{x-\alpha}{1}=\frac{y-\beta}{2}=\frac{z-\gamma}{3}\) and \(\frac{x+5}{2}=\frac{y-10}{1}=\frac{z-3}{1}\), is equal to