Let \(\mathrm{A}=\left[\mathrm{a}_{i j}\right]\) be a \(2 \times 2\) matrix such that \(\mathrm{a}_{i j} \in\{0,1\}\) for all \(i\) and \(j\). Let the random variable X denote the possible values of the determinant of the matrix \(A\). Then, the variance of \(X\) is :

Let \(\vec{a}=\hat{i}-3 \hat{j}+7 \hat{k}, \vec{b}=2 \hat{i}-\hat{j}+\hat{k}\) and \(\vec{c}\) be a vector such that \((\vec{a}+2 \vec{b}) \times \vec{c}=3(\vec{c} \times \vec{a})\). If \(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=130\), then \(\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}\) is equal to ....................

The \(4^{\text {tht }}\) term of \(GP\) is \(500\) and its common ratio is \(\frac{1}{m}, m \in N\). Let \(S_n\) denote the sum of the first \(n\) terms of this GP. If \(S_6 > S_5+1\) and \(S_7 < S_6+\frac{1}{2}\), then the number of possible values of \(m\) is \(..........\)

Let \(X\) be a binomially distributed random variable with mean \(4\) and variance \(\frac{4}{3}\). Then \(54 P ( X \leq 2)\) is equal to.

Numbers are to be formed between \(1000\) and \(3000\), which are divisible by \(4\), using the digits \(1,2,3,4,5\) and \(6\) without repetition of digits. Then the total number of such numbers is.

If \(f(x)=\left\{\begin{array}{ll}{\frac{\sin (a+2) x+\sin x}{x}} & {; x<0} \\ {b} & {; x=0} \\ {\frac{\left(x+3 x^{2}\right)^{\frac{1}{3}}-x^{\frac{1}{3}}}{x^{\frac{4}{3}}}} & {; x>0}\end{array}\right.\) is continuous at \(x=0,\) then \(a+2 b\) is equal to

The angle of elevation of the top \(P\) of a vertical tower \(PQ\) of height \(10\) from a point \(A\) on the horizontal ground is \(45^{\circ}\). Let \(R\) be a point on \(AQ\) and from a point \(B\), vertically above \(R\), the angle of elevation of \(P\) is \(60^{\circ}\). If \(\angle BAQ =30^{\circ}, AB = d\) and the area of the trapezium \(PQRB\) is \(\alpha\), then the ordered pair \(( d , \alpha)\) is.

Let \(y=y(x)\) be the solution of the differential equation \(\left(2 x \log _e x\right) \frac{d y}{d x}+2 y=\frac{3}{x} \log _e x, x>0\) and \(y\left(e^{-1}\right)=0\). Then, \(y(e)\) is equal to

If the chord joining the points \(P_{1}(x_{1},y_{1})\) and \(P_{2}(x_{2},y_{2})\) on the parabola \(y^{2}=12x\) subtends a right angle at the vertex of the parabola, then \(x_{1}x_{2}-y_{1}y_{2}\) 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

\(\lim \limits_{x \rightarrow 0}\left(\tan \left(\frac{\pi}{4}+x\right)\right)^{\frac{1}{x}}\) is equal to 

For \(\mathrm{n} \geq 2\), let \(S_n\) denote the set of all subsets of \(\{1,2 \ldots . . ., n\}\) with no two consecutive numbers. For example \(\{1,3,5\} \in \mathrm{S}_6\), but \(\{1,2,4\} \notin \mathrm{S}_6\). Then \(n\left(\mathrm{~S}_5\right)\) is equal to ________

Five numbers are in \(A.P.\), whose sum is \(25\) and product is \(2520 .\) If one of these five numbers is \(-\frac{1}{2},\) then the greatest number amongst them is