The foot of the perpendicular drawn from the origin, on the line, \(3x + y = \lambda \,\left( {\lambda  \ne 0} \right)\) is \(P\). If the line meets \(x-\) axis at \(A\) and \(y-\) axis at \(B\), then the ratio \(BP : PA\) is

Let \(f(x) =  - 1 + \left| {x - 2} \right|,\) and \(g(x) = 1 - \left| x \right|;\) then the set of all points where \(fog\) is discontinuous is

Let \(A\) be a \(2 \times 2\) symmetric matrix such that \(A\left[\begin{array}{l}1 \\ 1\end{array}\right]=\left[\begin{array}{l}3 \\ 7\end{array}\right]\) and the determinant of \(A\) be 1. If \(A^{-1}=\alpha A+\beta I\), where \(I\) is an identity matrix of order \(2 \times 2\), then \(\alpha+\beta\) equals ...........

\(\lim _{x \rightarrow \frac{\pi}{4}} \frac{8 \sqrt{2}-(\cos x+\sin x)^{7}}{\sqrt{2}-\sqrt{2} \sin 2 x}\) is equal to

If \(\sum \limits_{i=1}^{n}\left(x_{i}-a\right)=n\) and \(\sum \limits_{i=1}^{n}\left(x_{i}-a\right)^{2}=n a,(n, a>1)\) then the standard deviation of \(n\) observations \(x _{1}, x _{2}, \ldots, x _{ n }\) is

\(\smallint \frac{{dx}}{{{x^2}{{\left( {{x^4} + 1} \right)}^{\frac{3}{4}}}}} = \)

The area bounded by the lines \(y=\| x-1|-2 |\) is

The equation \(\sqrt {3 {x^2} + x + 5} = x - 3\) , where \(x\) is real, has

Let \(\vec{a}\) and \(\vec{b}\) be two vectors. Let \(|\vec{a}|=1,|\vec{b}|=4\) and \(\vec{a} \cdot \vec{b}=2\). If \(\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}\), then the value of \(\overrightarrow{ b } \cdot \overrightarrow{ c }\) is

Consider the following two statements : Statement \(I\) : For any two non-zero complex numbers \(\mathrm{z}_1, \mathrm{z}_2\) \(\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)\) and Statement \(II\) : If \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) are three distinct complex numbers and a, b, c are three positive real numbers such that \(\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}\), then \(\frac{\mathrm{a}^2}{\mathrm{y}-\mathrm{z}}+\frac{\mathrm{b}^2}{\mathrm{z}-\mathrm{x}}+\frac{\mathrm{c}^2}{\mathrm{x}-\mathrm{y}}=1\) Between the above two statements,

Let \(S\) be the set of all \((\lambda, \mu)\) for which the vectors \(\lambda \hat{ i }-\hat{ j }+\hat{ k }, \hat{ i }+2 \hat{ j }+\mu \hat{ k }\) and \(3 \hat{ i }-4 \hat{ j }+5 \hat{ k }\), where \(\lambda-\mu=5\), are coplanar, then \(\sum_{(\lambda, \mu) \in S} 80\left(\lambda^2+\mu^2\right)\) is equal to :

If \(\sum_{\mathrm{r}=0}^{10}\left(\frac{10^{\mathrm{r}+1}-1}{10^{\mathrm{r}}}\right) \cdot{ }^{11} \mathrm{C}_{\mathrm{r}+1}=\frac{\alpha^{11}-11^{11}}{10^{10}}\), then \(\alpha\) is equal to :

If the variance of the first \(n\) natural numbers is \(10\) and the variance of the first m even natural numbers is \(16\), then \(m + n\) is equal to