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Find the area of the following region using integration: {(x,y): y² ≤ 2x and y ≥ x-4}

Sketch the region bounded by the lines 2x+y=8, y=2, y=4 and the y-axis. Hence, obtain its area using integration.

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 $\frac{dy}{dx}$

If $y=(x+\sqrt{x^{2}-1})^{2}$;, then show that $(x^{2}-1)(\frac{dy}{dx})^{2}=4y^{2}.$

(a) Differentiate $\text{sec}^{-1}\left(\frac{1}{\sqrt{1-x^2}}\right)$ w.r.t. $\sin^{-1}\left(2x\sqrt{1-x^2}\right)$.
OR
(b) If $y = \tan x + \sec x$, then prove that $\frac{d^2y}{dx^2} = \frac{\cos x}{(1-\sin x)^2}$.

(a) If $f(x) = \begin{cases} x^2, & \text{if } x \geq 1 \\ x, & \text{if } x < 1 \end{cases}$, then show that $f$ is not differentiable at $x=1$.
OR
(b) Find the value(s) of '$\lambda$', if the function $f(x) = \begin{cases} \frac{\sin^2 \lambda x}{x^2} & \text{if } x \neq 0 \\ 1 & \text{if } x=0 \end{cases}$ 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 : $3\sin^{-1}(\frac{1}{\sqrt{2}})+2\cos^{-1}(\frac{\sqrt{3}}{2})+\cos^{-1}(0)$

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

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

In $\Delta ABC$, $\vec{AB}=\hat{i}+\hat{j}+2\hat{k}$ and $\vec{AC}=3\hat{i}-\hat{j}+4\hat{k}$. If D is mid-point of BC, then vector $\vec{AD}$ is equal to :

$\vec{a}$ 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 $\vec{b}$:

If a vector makes an angle of $\frac{\pi}{4}$ with the positive directions of both x-axis and y-axis, then the angle which it makes with positive z-axis is :

13. If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b}$ then $\vec{a} \cdot \vec{b} \ge 0$ only when:

A unit vector along the vector $4\hat{i}-3\hat{k}$ is:

The number of corner points of the feasible region determined by the constraints x-y\ge0, 2y\le x+2, x\ge0, y\ge0 is:

The corner points of the feasible region in the graphical representation of a linear programming problem are (2, 72), (15, 20) and (40, 15). If z=18x+9y be the objective function, then :

Let A be the area of a triangle having vertices $(x_1, y_1), (x_2, y_2)$ and $(x_3, y_3)$. Which of the following is correct?