Let \(A=\left\{n \in N \mid n^{2} \leq n+10,000\right\}, B=\{3 k+1 \mid k \in N\}\) and \(C=\{2 k \mid k \in N\}\), then the sum of all the elements of the set \(A \cap(B-C)\) is equal to \(.....\)

The square of the distance of the point of intersection of the line \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+1}{6}\) and the plane \(2 x-y+z=6\) from the point \((-1,-1,2)\) is .... .

Let \(A=\left[\begin{array}{cc}1 & \frac{1}{51} \\ 0 & 1\end{array}\right]\). If \(B=\left[\begin{array}{cc}1 & 2 \\ -1 & -1\end{array}\right] A \left[\begin{array}{cc}-1 & -2 \\ 1 & 1\end{array}\right]\) then the sum of all the elements of the matrix \(\sum \limits_{n=1}^{50} B^n\) is equal to

Let \(f\) be a function such that \(3 f(x)+2 f\left(\frac{m}{19 x}\right)=5 x\), \(x \neq 0\), where \(m=\sum_{i=1}^9(i)^2\). Then \(f(5)-f(2)\) is equal to

Let the coefficients of the middle terms in the expansion of \(\left(\frac{1}{\sqrt{6}}+\beta x\right)^{4},(1-3 \beta x)^{2}\) and \(\left(1-\frac{\beta}{2} x\right)^{6}, \beta>0\), respectively form the first three terms of an \(A.P.\) If \(d\) is the common difference of this \(A.P.\), then \(50-\frac{2 d}{\beta^{2}}\) is equal to.

Let \(A\) be a \(2 \times 2\) matrix with real entries such that \(A ^{\prime}=\alpha A + I\), where \(\alpha \in R -\{-1,1\}\). If det \(\left(A^2-A\right)=4\), then the sum of all possible values of \(\alpha\) is equal to

The sum of the abosolute maximum and minimum values of the function \(f(x)=\left|x^2-5 x+6\right|-3 x+2\) in the interval \([-1,3]\) is equal to :

Let a tangent be drawn to the ellipse \(\frac{x^{2}}{27}+y^{2}=1\) at \((3 \sqrt{3} \cos \theta, \sin \theta)\) where \(\theta \in\left(0, \frac{\pi}{2}\right)\). Then the value of \(\theta\) such that the sum of intercepts on axes made by this tangent is minimum is equal to ..... .

The solution of the differential equation \(ydx - \left( {x + 2{y^2}} \right)dy = 0\) is \(x\, = f(y)\). If \(f(-1)\, = 1\), then \(f(1)\) is equal to

Let \(A\) and \(B\) be \(3 \times 3\) real matrices such that \(A\) is symmetric matrix and \(B\) is skew-symmetric matrix. Then the system of linear equations \(\left( A ^{2} B ^{2}- B ^{2} A ^{2}\right) X = O ,\) where \(X\) is a \(3 \times 1\) column matrix of unknown variables and \(O\) is a \(3 \times 1\) null matrix, has ....... .

If \(f\left( x \right) + 2f\left( {\frac{1}{x}} \right) = 3x,x \ne 0\) and \(S = \left\{ {x \in R:f\left( x \right) = f\left( { - x} \right)} \right\}\);then \(S :\)

If a straight line drawn through the point of intersection of the lines \(4x+3y-1=0\) and \(3x+4y-1=0\), meets the co-ordinate axes at the points P and Q, then the locus of the mid point of PQ is:

A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate going by bus, scooter and car, respectively, are \(\dfrac{2}{5}\), \(\dfrac{1}{5}\) and \(\dfrac{2}{5}\). The probabilities that the candidate reaches late at the examination centre are \(\dfrac{1}{5}\), \(\dfrac{1}{3}\) and \(\dfrac{1}{4}\) if the candidate uses bus, scooter and car, respectively. Given that the candidate reached late at the examination centre, the probability that the candidate travelled by bus is: