For the natural numbers \(m, n\), if \((1-y)^{m}(1+y)^{n}=1+a_{1} y+a_{2} y^{2}+\ldots .+a_{m+n} y^{m+n}\) and \(a_{1}=a_{2}\) \(=10\), then the value of \((m+n)\) is equal to:

Let \(P\) be the plane passing through the points \((5,3\), \(0),(13,3,-2)\) and \((1,6,2)\). For \(\alpha \in N\), if the distances of the points \(A (3,4, \alpha)\) and \(B (2, \alpha\), a) from the plane \(P\) are \(2\) and \(3\) respectively, then the positive value of a is

Let \(u=\frac{2 z+i}{z-k i}, z=x+i y\) and \(k>0 .\) If the curve represented by \(\operatorname{Re}( u )+\operatorname{Im}( u )=1\) intersects the \(y\)-axis at the points \(P\) and \(Q\) where \(P Q=5,\) then the value of \(k\) is 

If the lines \(x -2y = 12\) is tangent to the ellipse \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\) at the point \(\left( {3,\frac{-9}{2}} \right)\), then the length of the latus rectum of the ellipse is

The value of the integral \(\displaystyle\int_{\pi/6}^{\pi/3} \left(\dfrac{4 - \csc^2 x}{\cos^4 x}\right) dx\) 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,

Numbers are to be formed between \(1000\) and \(3000\), which are divisible by \(4\), using the digits \(1,2,3,4,5\) and \(6\) without repetition of digits. Then the total number of such numbers is.

Let \(A=\{a, b, c\}\) and \(B=\{1,2,3,4\}\) Then the number of elements in the set \(C =\{ f : A \rightarrow B \mid 2 \in f ( A )\) and \(f\) is not one-one \(\}\) is

Let \(\mathrm{a}=1+\frac{{ }^2 \mathrm{C}_2}{3 !}+\frac{{ }^3 \mathrm{C}_2}{4 !}+\frac{{ }^4 \mathrm{C}_2}{5 !}+\ldots\), \(\mathrm{b}=1+\frac{{ }^1 \mathrm{C}_0+{ }^1 \mathrm{C}_1}{1 !}+\frac{{ }^2 \mathrm{C}_0+{ }^2 \mathrm{C}_1+{ }^2 \mathrm{C}_2}{2 !}+\frac{{ }^3 \mathrm{C}_0+{ }^3 \mathrm{C}_1+{ }^3 \mathrm{C}_2+{ }^3 \mathrm{C}_3}{3 !}+\ldots\) Then \(\frac{2 b}{a^2}\) is equal to ...........

Let \(f : R \rightarrow R\) be continuous function satisfying \(f ( x )+ f ( x + k )= n\), for all \(x \in R\) where \(k >0\) and \(n\)is a positive integer. If \(I _{1}=\int\limits_{0}^{4 n k} f ( x ) dx\) and \(I _{2}=\int\limits_{- k }^{3 k } f ( x ) dx\), then

Given an \(A.P.\) whose terms are all positive integers. The sum of its first nine terms is greater than \(200\) and less than \(220\). If the second term in it is \(12\), then its \(4^{th}\) term is

Let the mean and variance of four numbers \(3,7, x\) and \(y(x>y)\) be \(5\) and \(10\) respectively. Then the mean of four numbers \(3+2 \mathrm{x}, 7+2 \mathrm{y}, \mathrm{x}+\mathrm{y}\) and \(x-y\) is ..... .

If \(y=y(x)\) is the solution curve of the differential equation \(\frac{d y}{d x}+y \tan x=x \sec x, \quad 0 \leq x \leq \frac{\pi}{3}\), \(y (0)=1\), then \(y \left(\frac{\pi}{6}\right)\) is equal to