Let \(\mathrm{X} \sim \mathrm{B}\left(6, \frac{1}{2}\right)\), then \(\mathrm{P}[|x-4| \leqslant 2]\) is

\(\int \mathrm{e}^{x}\left(\frac{1-x}{1+x^{2}}\right)^{2} \mathrm{~d} x=\)

The value of \(\int \cos \left(\log _e(x)\right) \mathrm{d} x\) is equal to (where \(C\) is a constant of integration.)

Let \(a, b \in(a \neq 0)\). If the function \(f\) is defined as
\(f(x)=\left\{\begin{array}{cc}\frac{2 x^2}{\mathrm{a}} & , 0 \leq x \lt 1 \ \mathrm{a} & , 1 \leq x \lt \sqrt{2} \ \frac{2 \mathrm{~b}^2-4 \mathrm{~b}}{x} ,\end{array}\right.\) \( \sqrt{2} \leq x \lt \infty\)
is continuous in the interval \([0, \infty)\), then an ordered pair \((\mathrm{a}, \mathrm{b})\) is

\(\int \frac{x}{1+x^4} \mathrm{~d} x=\)

If the plane \(\frac{x}{3}+\frac{y}{2}-\frac{z}{4}=1\) cuts the co-ordinate axes at points \(\mathrm{A}, \mathrm{B}\) and C , then the area of the triangle \(A B C\) is

If \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}\) are three non coplanar vectors and \(\overrightarrow{\mathbf{p}}, \overrightarrow{\mathbf{q}}, \overrightarrow{\mathbf{r}}\) are defined by the relations
\(\overrightarrow{\mathbf{p}}=\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}, \quad \overrightarrow{\mathbf{q}}=\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}\) and \(\overrightarrow{\mathbf{r}}=\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}\)
then \(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{p}}+\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{q}}+\overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{r}}\) is equal to

If \(A\) and \(B\) are independent events and \(P(A)=\frac{2}{3}, P(B)=\frac{3}{5}\), then \(P\left(A^{\prime} \cap B\right)=\)

The p.d.f. of a random variable \(\mathrm{X}\) is given by \(f(x)=\frac{k}{\sqrt{x}}\) if \(0 \leq x \leq 4\) \(=0\) otherwise, then \(\mathrm{P}(1 < X < 4)=\)

The function \(f(x)=\frac{x+1}{9 x+x^{3}}\) is

If \(A=\left[\begin{array}{ll}3 & 2 \\ 0 & 1\end{array}\right]\), then \(\left(A^{-1}\right)^3=\)

The bond enthalpies of \(\mathrm{C}-\mathrm{C}, \mathrm{C}=\mathrm{C}, \mathrm{H}-\mathrm{H}\) and \(\mathrm{C}-\mathrm{H}\) bonds are \(360,600,400\) and \(410 \mathrm{~kJ} \mathrm{~mol}^{-1}\) respectively. What is the heat of hydrogenation of ethylene?

The majority charge carriers in p-type and n-type semiconductor are respectively