If \(0 <  x , y < \pi\) and \(\cos x +\cos y-\cos ( x + y )=\frac{3}{2},\) then \(\sin x+\cos y\) is equal to ...... .

Sum of squares of modulus of all the complex numbers \(z\) satisfying \(\bar{z}=i z^{2}+z^{2}-z\) is equal to

In a group of \(400\) people, \(160\) are smokers and nonvegetarian; \(100\) are smokers and vegetarian and the remaining \(140\) are non-smokers and vegetarian. Their chances of getting a particular chest disorder are \(35\, \%, 20 \,\%\) and \(10 \,\%\) respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is ...... .

A common tangent \(T\) to the curves \(C_{1}: \frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) and \(C_{2}: \frac{x^{2}}{42}-\frac{y^{2}}{143}=1\) does not pass through the fourth quadrant. If \(T\) touches \(C _{1}\) at ( \(\left.x _{1}, y _{1}\right)\) and \(C _{2}\) at \(\left( x _{2}, y _{2}\right)\), then \(\left|2 x _{1}+ x _{2}\right|\) is equal to \(......\)

Suppose \(f\) is a function satisfying \(f ( x + y )= f ( x )+ f ( y )\) for all \(x , y \in N\) and \(f (1)=\frac{1}{5}\). If \(\sum \limits_{n=1}^m \frac{f(n)}{n(n+1)(n+2)}=\frac{1}{12}\), then \(m\) is equal to \(...............\).

If the probability that the random variable X takes the value \(x\) is given by \(P(X=x)=k(x+1) 3^{-x}\), \(\mathrm{x}=0,1,2,3 \ldots \ldots\), where k is a constant, then \(\mathrm{P}(\mathrm{X} \geq 3)\) is equal to

If \(f(x)=x^3-x^2 f^{\prime}(1)+x f^{\prime \prime}(2)-f^{\prime \prime \prime}(3), x \in R\), then

Consider a line \(\mathrm{L}\) passing through the points \(\mathrm{P}(1,2,1)\) and \(\mathrm{Q}(2,1,-1)\). If the mirror image of the point \(\mathrm{A}(2,2,2)\) in the line \(\mathrm{L}\) is \((\alpha, \beta, \gamma)\), then \(\alpha+\beta+6 \gamma\) is equal to ...........

Let \(5\) digit numbers be constructed using the digits \(0,2,3,4,7,9\) with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number \(42923\) is \(...............\).

If \(y (x)\) is the solution of the differential equation \(\frac{{dy}}{{dx}} + \left( {\frac{{2x + 1}}{x}} \right)y = {e^{ - 2x}},x > 0\) where \(y\,\,(1)\, = \,\frac{1}{2}{e^{ - 2}},\) then

Let \(\vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{d}}=\vec{a} \times \overrightarrow{\mathrm{b}}\). If \(\overrightarrow{\mathrm{c}}\) is a vector such that \(\vec{a} \cdot \overrightarrow{\mathrm{c}}=|\overrightarrow{\mathrm{c}}|\), \(|\overrightarrow{\mathrm{c}}-2 \vec{a}|^2=8\) and the angle between \(\overrightarrow{\mathrm{d}}\) and \(\overrightarrow{\mathrm{c}}\) is \(\frac{\pi}{4}\), then \(|10-3 \overrightarrow{\mathrm{~b}} \cdot \overrightarrow{\mathrm{c}}|+|\overrightarrow{\mathrm{d}} \times \overrightarrow{\mathrm{c}}|^2\) is equal to

\(\lim _{n \rightarrow \infty}\left(1+\frac{1+\frac{1}{2}+\ldots \ldots .+\frac{1}{n}}{n^{2}}\right)^{n}=.........\)

The number of critical points of the function \(f(x) = \begin{cases} \left|\dfrac{\sin x}{x}\right|, & x \neq 0 \\ 1, & x = 0 \end{cases}\) in the interval \((-2\pi, 2\pi)\) is equal to :