The sum and product of the mean and variance of a binomial distribution are \(82.5\) and \(1350\) respectively. They the number of trials in the binomial distribution is.

Total numbers of \(3-\)digit numbers that are divisible by 6 and can be formed by using the digits \(1, 2, 3, 4,5\) with repetition, is \(.......\).

Let \(a, b, c \in R\) be all non-zero and satisfy \(a^{3}+b^{3}+c^{3}=2 .\) If the matrix \(A=\left(\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right)\) satisfies \(\mathrm{A}^{\mathrm{T}} \mathrm{A}=\mathrm{I},\) then a value of \(abc\) can be

Let \(f :R \to R\) be defined by \(f(x)\,\, = \,\,\frac{x}{{1 + {x^2}}},\,x\, \in \,R.\) Then the range of \(f\) is

Let \(f:(0, \infty) \rightarrow R\) and \(F(x)=\int_0^x t f(t) d t\). If \(F\left(x^2\right)=\) \(x^4+x^5\), then \(\sum_{r=1}^{12} f\left(r^2\right)\) is equal to :

If \({\cos ^{ - 1}}\,x\, - \,{\cos ^{ - 1}}\,\frac{y}{2}\, = \,\alpha ,\) where \( - {\kern 1pt} 1\, \le \,x\, \le \,1,\,\) \(- {\kern 1pt} 2\, \le \,y\, \le \,2,\) \(x\, \le \,\,\frac{y}{2},\) then for all \(x, y, 4x^2 -4xy\,\,cos\,\alpha  + y^2\) is equal to

If \(x = 3\,tan\,t\) and \(y = 3\,sec\,t,\) then the value of \(\frac{{{d^2}y}}{{d{x^2}}}\) at \(t = \frac {\pi }{4},\) is

Suppose for a differentiable function \(h, h(0)=0\), \(\mathrm{h}(1)=1\) and \(\mathrm{h}^{\prime}(0)=\mathrm{h}^{\prime}(1)=2\). If \(\mathrm{g}(\mathrm{x})=\mathrm{h}\left(\mathrm{e}^{\mathrm{x}}\right) \mathrm{e}^{\mathrm{h}(\mathrm{x})}\), then \(g^{\prime}(0)\) is equal to :

The value of the integral \(\int \limits_1^2\left(\frac{t^4+1}{t^6+1}\right) d t\) is \(..........\).

If the distance between the plane, \(23 \mathrm{x}-10 \mathrm{y}-2 \mathrm{z}+48=0\) and the plane containing the lines \(\frac{x+1}{2}=\frac{y-3}{4}=\frac{z+1}{3}\) and \(\frac{x+3}{2}=\frac{y+2}{6}=\frac{z-1}{\lambda}(\lambda \in R)\) is equal to \(\frac{\mathrm{k}}{\sqrt{633}},\) then \(\mathrm{k}\) 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

Let \(\lambda, \mu \in R\). If the system of equations \( 3 x+5 y+\lambda z=3 \) \( 7 x+11 y-9 z=2 \) \( 97 x+155 y-189 z=\mu\) has infinitely many solutions, then \(\mu+2 \lambda\) is equal to :

Let the coefficients of third, fourth and fifth terms in the expansion of \(\left(x+\frac{a}{x^{2}}\right)^{n}, x \neq 0,\) be in the ratio \(12: 8: 3 .\) Then the term independent of \(x\) in the expansion, is equal to ...... .