If the order and degree of the differential equation corresponding to the family of curves \(y^2=4 a(x+a)\) ( \(a\) is parameter) are \(m\) and \(n\) respectively, then \(m+n^2=\)

If a random variable \(\mathrm{X}\) follows a poisson distribution such that \(P(X=1)=3 P(X=2)\), then \(P(x=3)=\)

The functions \(f(x)=x e^{-x}, \forall(x \in R)\) attains a maximum value at \(x\) is equal to

\(\begin{aligned} & 1+x \log _e a+\frac{x^2}{2 !}\left(\log _e a\right)^2 \ & +\frac{x^3}{3 !}\left(\log _e a\right)^3+\ldots(a>0, x \in R) \text { is equal to }\end{aligned}\)

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\) Assertion (A) : The identity \(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for \(\vec{a}\). Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\), \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\) Which of the following is correct?

If the slope of the tangent drawn at any point \((x, y)\) on the curve \(y=f(x)\) is \(\left(6 x^2+10 x-9\right)\) and \(f(2)=0\), then \(f(-2)=\)

Let the direction cosines of two lines satisfy the equations \(31+2 m+n=0\) and \(2 m n-3 n l+5 l m=0\). If \(\theta\) is the angle between these two lines, then \(\cos \theta=\)

If \(4+3 x-7 x^2\) attains its maximum value \(M\) at \(x=\alpha\) and \(5 x^2-2 x+1\) attains its minimum value \(m\) at \(x=\beta\), then \(\frac{28(M-\alpha)}{5(m+\beta)}=\)

The specific rate constant of decomposition of a compound is given by \(\ln k=5.0-\frac{12000}{T}\). The activation energy of decomposition for this compound at \(300 \mathrm{~K}\) is

Which gas has a density of $1.24 \mathrm{g}/\mathrm{L}$ at $0{}^{\circ }\mathrm{C}$ and $1 \mathrm{atm}$ pressure?

The transformed equation of the curve $2x^2+y^2-3x+5y-8=0$ when the origin is translated to the point $(-1, 2)$ is

A block is between two surfaces as shown in the figure. Find the normal reaction at both surfaces. [Assume, \(g=10 \mathrm{~m} / \mathrm{s}^2\) ]

The point dividing the join of \((3,-2,1)\) and \((-2,3,11)\) in the ratio \(2: 3\) is