If \(\quad \alpha=\frac{5}{2 ! 3}+\frac{5 \cdot 7}{3 ! 3^2}+\frac{5 \cdot 7 \cdot 9}{4 ! 3^3}+\ldots, \quad\) then \(\alpha^2+4 \alpha\) is equal to

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?

The rank of
\(A=\left[\begin{array}{ccc}
1 & x & x+1 \\
2 x & x^2-x & x^2+x \\
3 x(x-1) & x\left(x^2-3 x+2\right) & x\left(x^2-1\right)
\end{array}\right] \text { is }\)

Let \(f(x)=x^3+2 x^2-x\) be a real valued function. Then, the value of Lagrange's constant \(C\) in \((-1,2)\) is

If $\int \frac{2x^{12}+5x^9}{{\left(x^5+x^3+1\right)}^3}dx=\frac{1}{2}f\left(x\right)+C,$ then $f\left(1\right)-f\left(0\right)=$

The domain of the function \(f(x)=\frac{1}{\sqrt{[x]^2-[x]-2}}\) is Here \([x]\) denotes the greatest integer not exceeding the value of \([x]\)

If \(y=\left(\sin ^{-1} 2 x\right)^2+\left(\cos ^{-1} 2 x\right)^2\), then \(\left(1-4 x^2\right) y_2-4 x y_1=\)

Water flows through a horizontal pipe of variable crosssection at the rate of \(12 \pi\) litre per minute. The velocity of the water at the point where the diameter of the pipe becomes 2 cm is

At \(550 \mathrm{~K}\), the \(K_c\) for the following reaction is \(10^4 \mathrm{~mol}^{-1} \mathrm{~L} X(g)+Y(g) \rightleftharpoons Z(g)\) At equilibrium, it was observed that \([X]=\frac{1}{2}[Y]=\frac{1}{2}[Z]\) What is the value of [Z] (in \(\mathrm{mol} \mathrm{L}^{-1}\) ) at equilibrium?

Let $f\left(x\right)=(x-3{)}^{2018}(x-2{)}^{2019}, x\in \mathrm{ℝ}$. If $f\left(\alpha \right)$ is a relative maximum of $f$ at $x=\alpha$, then $2\alpha +3f\left(\alpha \right)=$

The period of \(\tan k y+\sin k y\), where \(k=1+4+9+\ldots 20\) terms, is

A reversible engine converts one-sixth of the heat supplied into work. When the temperature of the sink is reduced by \(62^{\circ} \mathrm{C}\), the efficiency of the engine is doubled. The temperatures of the source and sink are

Let \(a, b, c\) be three vectors such that \(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{c}}=5\) and \(|\vec{a}+\vec{b}-\vec{c}|^2+|\vec{b}+\vec{c}-\vec{a}|^2+|\vec{c}+\vec{a}-\vec{b}|^2=50\) then \(\vec{a} \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}=\)