A seven digit number is formed using digits \(3 ,3,4,4,4,5,5 .\) The probability, that number so formed is divisible by \(2,\) is ..... .

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

Let \(f\) be a differential function such that \(f'\left( x \right) = 7 - \frac{3}{4}\frac{{f\left( x \right)}}{x},\left( {x > 0} \right)\) and \(f(1) \ne 4\). Then \(\mathop {\lim }\limits_{x \to {0^ + }} xf\left( {\frac{1}{x}} \right)\)

Let \(K\) be the set of all real values of \(x\) where the function \(f\left( x \right) = \sin \,\left| x \right| - \left| x \right| + 2\,\left( {x - \pi } \right)\,\cos \,\left| x \right|\) is not differentiable. Then the set \(K\) is equal to

If \(\overrightarrow{ a }=\hat{ i }+2 \hat{ k }, \overrightarrow{ b }=\hat{ i }+\hat{ j }+\hat{ k }, \overrightarrow{ c }=7 \hat{ i }-3 \hat{ k }+4 \hat{ k }\) \(\overrightarrow{ r } \times \overrightarrow{ b }+\overrightarrow{ b } \times \overrightarrow{ c }=\overrightarrow{0}\) and \(\overrightarrow{ r } \cdot \overrightarrow{ a }=0\) then \(\overrightarrow{ r } \cdot \overrightarrow{ c }\) is equal to :

Let the tangents at the points \(A (4,-11)\) and \(B (8,-5)\) on the circle \(x^2+y^2-3 x+10 y-15=0\), intersect at the point \(C\). Then the radius of the circle, whose centre is \(C\) and the line joining \(A\) and \(B\) is its tangent, 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

Let the number of elements in sets \(A\) and \(B\) be five and two respectively. Then the number of subsets of \(A \times B\) each having at least \(3\) and at most \(6\) element is :

The coefficient of \(x^{18}\) in the product \((1+ x)(1- x)^{10} (1+ x + x^2 )^9\) is 

If \(A\) is a square matrix of order \(3\) such that \( \operatorname{det}(\mathrm{A})=3 \text { and } \) \( \operatorname{det}\left(\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 \mathrm{~A})^{-1}\right)\right)\right)\right)\right)=2^{\mathrm{m}} 3^{\mathrm{n}},\) then \(m+ 2 n\) is equal to :

Let \(S_n\) be the sum to n-terms of an arithmetic progression \(3,7,11, \ldots\) If \(40<\left(\frac{6}{\mathrm{n}(\mathrm{n}+1)} \sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{S}_{\mathrm{k}}\right)<42\), then \(\mathrm{n}\) equals

Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common is :

For \(k \in N\), if the sum of the series \(1+\frac{4}{ k }+\frac{8}{ k ^2}+\frac{13}{ k ^3}+\frac{19}{ k ^4}+\ldots\) is 10 , then the value of \(k\) is