Let the plane \(2 x+3 y+z+20=0\) be rotated through a right angle about its line of intersection with the plane \(x-3 y+5 z=8\). If the mirror image of the point \(\left(2,-\frac{1}{2}, 2\right)\) in the rotated plane is \(B ( a , b , c )\), then

The probability of selecting integers \(a \in[-5,30]\) such that \(x^{2}+2(a+4) x-5 a+64>0\), for all \(x \in R\), is:

The solution of the differential equaiton \(\frac{{dy}}{{dx}} + \frac{y}{2}\sec \,x = \frac{{\tan \,x}}{{2y}}\) , where \(0 \le x < \frac{\pi }{2}\) , and \(y(0) = 1\) , is given by

Statement \(-1\) : The system of linear equations \(x + \left( {\sin \,\alpha } \right)y + \left( {\cos \,\alpha } \right)z = 0\) \(x + \left( {\cos \,\alpha } \right)y + \left( {\sin \alpha } \right)z = 0\) \(x - \left( {\sin \,\alpha } \right)y - \left( {\cos \alpha } \right)z = 0\) has a non-trivial solution for only one value of \(\alpha \) lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)  Statement \(-2\) : The equation in \(\alpha \) \(\left| {\begin{array}{*{20}{c}}
  {\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha } \\ 
  {\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha } \\ 
  {\cos {\mkern 1mu} \alpha }&{ - \sin {\mkern 1mu} \alpha }&{ - \cos {\mkern 1mu} \alpha } 
\end{array}} \right| = 0\) has only one solution lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)

Suppose \(f(x)=\frac{\left(2^x+2^{-x}\right) \tan x \sqrt{\tan ^{-1}\left(x^2-x+1\right)}}{\left(7 x^2+3 x+1\right)^3}\). Then the value of \(f^{\prime}(0)\) is equal to

\(\cos ^{-1}(\cos (-5))+\sin ^{-1}(\sin (6))-\tan ^{-1}(\tan (12))\) is equal to : (The inverse trigonometric functions take the principal values)

Let \(\vec{a}=4 \hat{i}-\hat{j}+\hat{k}, \vec{b}=11 \hat{i}-\hat{j}+\hat{k}\) and \(\vec{c}\) be a vector such that \((\vec{a}+\vec{b}) \times \vec{c}=\vec{c} \times(-2 \vec{a}+3 \vec{b}) \text {. }\) If \((2 \vec{a}+3 \vec{b}) \cdot \vec{c}=1670\), then \(|\vec{c}|^2\) is equal to :

Let a die be rolled \(n\) times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is \(\frac{ k }{2^{15}}\), then \(k\) is equal to:

The number of natural numbers, between 212 and 999 , such that the sum of their digits is 15 , is

The area of the region \(\{(x, y):|x-y| \leq y \leq 4 \sqrt{x}\}\) is

If the value of the integral \(\int \limits_{0}^{\frac{1}{2}} \frac{x^{2}}{\left(1-x^{2}\right)^{3 / 2}} d x\) is \(\frac{ k }{6},\) then \(k\) is equal to

Let \(A =\{2,3,4,5, \ldots ., 30\}\) and \(^{\prime} \simeq ^{\prime}\) be an equivalence relation on \(A \times A ,\) defined by \((a, b) \simeq (c, d),\) if and only if \(a d=b c .\) Then the number of ordered pairs which satisfy this equivalence relation with ordered pair \((4,3)\) is equal to :

If the area of an equilateral triangle inscribed in the circle, \(x^2 + y^2 + 10x + 12y + c = 0\) is \(27\sqrt 3 \,sq.\,units\) then \(c\) is equal to