The common difference of the \(A.P.:a_{1},a_{2},....,a_{m}\) is 13 more than the common difference of the \(A.P.: b_{1}, b_{2},...,b_{n}.\) If \(b_{31}=-277, b_{43}=-385\) and \(a_{78}=327,\) then \(a_{1}\) is equal to

Let the equation \(\mathrm{x}(\mathrm{x}+2)(12-\mathrm{k})=2\) have equal roots. Then the distance of the point \(\left(\mathrm{k}, \frac{\mathrm{k}}{2}\right)\) from the line \(3 x+4 y+5=0\) is

If \({ }^{2 n+1} P_{n-1}:{ }^{2 n-1} P_n=11: 21\), then \(n^2+n+15\) is equal to:

Let a unit vector \(\hat{\mathrm{u}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}}\) make angles \(\frac{\pi}{2}, \frac{\pi}{3}\) and \(\frac{2 \pi}{3}\) with the vectors \(\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}, \frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}\) and \(\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{j}}\) respectively. If \(\overrightarrow{\mathrm{v}}=\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}\), then \(|\hat{\mathrm{u}}-\overrightarrow{\mathrm{v}}|^2\) is equal to

The equation of the plane passing through the line of intersection of the planes \(\vec{r} .(\hat{i}+\hat{j}+\hat{k})=1\) and \(\vec{r} \cdot(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}})+4=0\) and parallel to the \(\mathrm{x}\)-axis is:

Let the point \(P\) be the vertex of the parabola \(y = x^2 - 6x + 12\). If a line passing through the point \(P\) intersects the circle \(x^2 + y^2 - 2x - 4y + 3 = 0\) at the points \(R\) and \(S\), then the maximum value of \((PR + PS)^2\) is :

The coefficient of \(x ^{101}\) in the expression \((5+x)^{500}+x(5+x)^{499}+x^{2}(5+x)^{498}+\ldots . x^{500}\) \(x>0\), is

If the ratio of the fifth term from the begining to the fifth term from the end in the expansion of \(\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^n\) is \(\sqrt{6}: 1\), then the third term from the beginning is:

Let \(f: R \rightarrow R\) be a continuous function such that \(f(x)+f(x+1)=2,\) for all \(x \in R\). If \(I _{1}=\int_{0}^{8} f( x ) d x\) and \(I _{2}=\int_{-1}^{3} f( x ) d x ,\) then the value of \(I _{1}+2 I _{2}\) is equal to...........

Let \(\mathrm{A}=\)\(\left\{\theta \in[0,2 \pi]: 1+10 \operatorname{Re}\left(\frac{2 \cos \theta+i \sin \theta}{\cos \theta-3 i \sin \theta}\right)=0\right\} .\)
Then \(\sum_{\theta \in A} \theta^2\) is equal to

Let \(f(x)=\left\{\begin{array}{cc}-2, & -2 \leq x \leq 0 \\ x-2, & 0 < x \leq 2\end{array}\right.\) and \(h(x)=f(|x|)+|f(x)|\). Then \(\int_{-2}^2 \mathrm{~h}(\mathrm{x}) \mathrm{dx}\) is equal to :

Let the tangent to the circle \(C _{1}: x^{2}+y^{2}=2\) at the point \(M (-1,1)\) intersect the circle \(C _{2}\) : \(( x -3)^{2}+(y-2)^{2}=5\), at two distinct points \(A\) and \(B\). If the tangents to \(C _{2}\) at the points \(A\) and \(B\) intersect at \(N\), then the area of the triangle \(ANB\) is equal to

Let \(I\) be the identity matrix of order \(3 \times 3\) and for the matrix \(\mathrm{A}=\left[\begin{array}{ccc}\lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2\end{array}\right],|\mathrm{A}|=-1\). Let B be the inverse of the matrix \(\operatorname{adj}\left(\mathrm{A} \operatorname{adj}\left(\mathrm{A}^2\right)\right)\). Then \(|(\lambda B+1)|\) is equal to _____