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31. From a lot of 30 bulbs which include 6 defective bulbs, a sample of 2 bulbs is drawn at random one by one with replacement. Find the probability distribution of the number of defective bulbs and hence find the mean number of defective bulbs.

Show that the function $f(x)=\frac{16\sin x}{4+\cos x}-x$, is strictly decreasing in $(\frac{\pi}{2},\pi)$

Find: $\int x^{2}\log(x^{2}+1)dx$

Find: $\int\frac{\cos x}{\sin 3x}dx$

Evaluate $\int_{1}^{e}\frac{1}{\sqrt{4x^{2}-(x\log x)^{2}}}dx$

Evaluate: $\int_{1}^{3}\frac{\sqrt{4-x}}{\sqrt{x}+\sqrt{4-x}}dx$

28. (b) OR: Evaluate: $\int_{0}^{\pi/2}\sqrt{\sin x}\cos^{5}x~dx$

28. (a) Find: $\int\frac{e^{x}}{\sqrt{5-4e^{x}-e^{2x}}}dx$

27. (b) OR: Evaluate: $\int_{-2}^{2}\frac{x^{2}}{1+5^{x}}dx$

27. (a) Evaluate: $\int_{\pi/4}^{\pi/2}e^{2x}(\frac{1-\sin 2x}{1-\cos 2x})dx$

26. Find: $\int\frac{x^{2}+x+1}{(x+1)^{2}(x+2)}dx$

Evaluate : $\int_{0}^{\frac{\pi}{2}}e^{x }\sin x~dx$.
OR
Find: $\int\frac{1}{\cos(x-a)\cos(x-b)}dx$

Find:$\int \frac{1}{\sqrt{x}(\sqrt{x}+1)(\sqrt{x}+2)} \, dx$

Evaluate $\int_{0}^{\frac{\pi}{2}}[\log(\sin~x)-\log(2\cos~x)]dx.$

Evaluate: $\int_ 0 ^{π/2} [\sin 2x \tan⁻¹(\sin x)] dx$

(a) Evaluate: $\int_0^{2\pi} \frac{1}{1 + e^{\sin x}} dx $
OR
(b) Find: $\int \frac{x⁴} { ((x-1)(x²+1))}dx.$

If $A=\begin{bmatrix}1 & 0 & 2\\ 0 & 2 & 1\\ 2 & 0 & 3\end{bmatrix}$, then show that $A^{3}-6A^{2}+7A+2I=O$

If $A=\begin{bmatrix}1 & 2 & 3\\ 3 & -2 & 1\\ 4 & 2 & 1\end{bmatrix}$, then show that A³ - 23A - 40I = O.

If the angle between the lines $\frac{x-5}{\alpha}=\frac{y+2}{-5}=\frac{z+\frac{24}{5}}{\beta}$ and $\frac{x}{1}=\frac{y}{0}=\frac{z}{1}$ is $\frac{\pi}{4}$, find the relation between $\alpha$ and $\beta$.

25. (a) Find the vector equation of the line passing through the point $(2, 1, 3)$ and perpendicular to both the lines $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3} ; \frac{x}{-3}=\frac{y}{2}=\frac{z}{5}$