\(\left|\begin{array}{ccc}\sqrt{3} & 2 \sqrt{5} & \sqrt{5} \\ \sqrt{15} & 5 & \sqrt{10} \\ 3 & \sqrt{15} & 5\end{array}\right|=\)

\(\int(\log x)^2 d x=\)

If \(2 \sin \theta+3 \cos \theta=2\) and \(\theta \neq(2 n+1) \frac{\pi}{2}\) then \(\sin \theta+\cos \theta=\)

If \(f(x)=\frac{1}{1+\frac{1}{x}}\) and \(g(x)=\frac{1}{1+\frac{1}{f(x)}}\), then \(g^{\prime}(2)\) is equal to

Two sides of a triangle are given by the roots of the equation \(x^2-5 x+6=0\) and the angle between the sides is \(\frac{\pi}{3}\). Then, the perimeter of the triangle is

A student is asked to answer 10 out of 13 questions is an examination such that he must answer atleast four questions from the first five questions. The number of choices available to him is

If \(\int \frac{2 x^{12}+5 x^9}{\left(1+x^3+x^5\right)^3} d x=\frac{x^m}{l\left(1+x^3+x^5\right)^r}+C\) then \(\frac{m-l}{r}=\)

Let \(\mathbf{x}=\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\mathbf{y}=3 \hat{\mathbf{i}}-2 \hat{\mathbf{k}}\). Then, the vector \(\mathbf{r}\) of magnitude \(\sqrt{21}\) satisfying \(\mathbf{r} \times \mathbf{x}=\mathbf{y} \times \mathbf{x}\) and \(\mathbf{r} \times \mathbf{y}=\mathbf{x} \times \mathbf{y}\), is

There are four bulbs of power \(100 \mathrm{~W}, 200 \mathrm{~W}\), \(500 \mathrm{~W}\) and 1000W. Among these whose filament has high resistance? (Assuming, same voltage source)

If \(\omega\) is a complex cube root of unity, then \(\begin{aligned} & \left(\frac{1-\sqrt{3} i}{2}\right)^{2020}+\left(\frac{1+\sqrt{3} i}{2}\right)^{2026} \ & +\sin \left(\sum_{j=1}^6(j+\omega)\left(j+\omega^2\right) \frac{3 \pi}{152}\right)=\end{aligned}\)

If \(\sinh ^{-1}(-\sqrt{3})+\cosh ^{-1}(2)=K\), then \(\cosh K=\)

If \(\sqrt{\frac{y}{x}}+\sqrt{\frac{x}{y}}=2\), then \(\frac{d y}{d x}\) is equal to

\[ \begin{aligned} & \text { Given } \mathrm{H}_2(\mathrm{~g})+\frac{1}{2} \mathrm{O}_2(\mathrm{~g}) \rightarrow \mathrm{H}_2 \mathrm{O}(l) ; \Delta \mathrm{H}=-285 \mathrm{~kJ} \ & \mathrm{~N}_2 \mathrm{O}_5(\mathrm{~g})+\mathrm{H}_2 \mathrm{O}(l) \rightarrow 2 \mathrm{HNO}_3(l) ; \Delta \mathrm{H}=-76.6 \mathrm{~kJ} \ & \mathrm{~N}_2(\mathrm{~g})+3 \mathrm{O}_2(\mathrm{~g})+\mathrm{H}_2(\mathrm{~g}) \rightarrow 2 \mathrm{HNO}_3(l) ; \Delta \mathrm{H}=-348.2 \mathrm{~kJ} \ & \text { Calculate the } \Delta \mathrm{H} \text { of } 2 \mathrm{~N}_2(\mathrm{~g})+5 \mathrm{O}_2(\mathrm{~g}) \rightarrow 2 \mathrm{~N}_2 \mathrm{O}_5(\mathrm{~g}) \end{aligned} \] Calculate the \(\Delta \mathrm{H}\) of \(2 \mathrm{~N}_2(\mathrm{~g})+5 \mathrm{O}_2(\mathrm{~g}) \rightarrow 2 \mathrm{~N}_2 \mathrm{O}_5(\mathrm{~g})\).