Paper Generator

(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=[(1, 2, 3), (3, -2, 1), (4, 2, 1)], 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 $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3} ; \frac{x}{-3}=\frac{y}{2}=\frac{z}{5}$

Find the distance between the lines: →r = (î + 2î - 4ê) + λ(2î + 3î + 6ê); →r = (3î + 3î - 5ê) + μ(4î + 6î + 12ê)

(a) Find the coordinates of the foot of the perpendicular drawn from the point P(0,2,3) to the line (x+3)/5 = (y-1)/2 = (z+4)/3. OR (b) Three vectors →a, →b and →c satisfy the condition →a + →b + →c = →0. Evaluate the quantity μ = →a·→b + →b·→c + →c·→a, if |→a|=3, |→b|=4 and |→c|=2.

Find the vector and the cartesian equations of a line that passes through the point A(1,2,-1) and parallel to the line 5x-25=14-7y=35z.

Find all the vectors of magnitude $3\sqrt{3}$ which are collinear to vector $\hat{i}+\hat{j}+\hat{k}.$

(a) If the vectors →a and →b are such that |→a|=3, |→b|=2/3 and →a x →b is a unit vector, then find the angle between →a and →b. OR (b) Find the area of a parallelogram whose adjacent sides are determined by the vectors →a = î - î + 3ê and →b = 2î - 7î + ê.

If $x=a\sin 2t, y=a(\cos 2t+\log\tan t)$ then find $\frac{dy}{dx}$

If $y=x^{\frac{1}{x}}$ then find $\frac{dy}{dx}$ at $x=1$.

22. If $(x^{2}+y^{2})^{2}=xy$, then find $\frac{dy}{dx}$

(a) If f(x) = { x², if x≥1; x, if x<1 }, then show that f is not differentiable at x=1. OR (b) Find the value(s) of 'λ', if the function f(x) = { (sin²λx)/x², if x≠0; 1, if x=0 } is continuous at x=0.

Draw the graph of $f(x)=\sin^{-1}x, x\in[-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}]$. Also, write range of $f(x)$.

Evaluate : $3\sin^{-1}(\frac{1}{\sqrt{2}})+2\cos^{-1}(\frac{\sqrt{3}}{2})+\cos^{-1}(0)$

21. (b) OR: Evaluate : $\cos^{-1}[\cos(-\frac{7\pi}{3})]$

21. (a) Find the domain of $y=\sin^{-1}(x^{2}-4)$.

Write the domain and range (principle value branch) of the following functions: f(x)=tan⁻¹x

$\vec{a}$ and $\vec{b}$ are two non-zero vectors such that the projection of $\vec{a}$ on $\vec{b}$ is 0. The angle between $\vec{a}$ and $\vec{b}$: