For the following questions answer them individually
The integral $$\int_{1/4}^{3/4} \cos\left(2\cot^{-1}\sqrt{\frac{1-x}{1+x}}\right) dx$$ is equal to
Let the range of the function $$f(x) = \frac{1}{2 + \sin 3x + \cos 3x}, x \in \mathbb{R}$$ be $$[a, b]$$. If $$\alpha$$ and $$\beta$$ are respectively the A.M. and the G.M. of $$a$$ and $$b$$, then $$\frac{\alpha}{\beta}$$ is equal to
If $$\log_e y = 3\sin^{-1} x$$, then $$(1 - x^2)y'' - xy'$$ at $$x = \frac{1}{2}$$ is equal to
Let $$\int_0^x \sqrt{1 - (y'(t))^2} \, dt = \int_0^x y(t) \, dt, \, 0 \le x \le 3, \, y \ge 0, \, y(0) = 0$$. Then at $$x = 2$$, $$y'' + y + 1$$ is equal to
The value of the integral $$\int_{-1}^{2} \log_e\left(x + \sqrt{x^2 + 1}\right) dx$$ is
The area (in square units) of the region enclosed by the ellipse $$x^2 + 3y^2 = 18$$ in the first quadrant below the line $$y = x$$ is
Between the following two statements: Statement I : Let $$\vec{a} = \hat{i} + 2\hat{j} - 3\hat{k}$$ and $$\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$$. Then the vector $$\vec{r}$$ satisfying $$\vec{a} \times \vec{r} = \vec{a} \times \vec{b}$$ and $$\vec{a} \cdot \vec{r} = 0$$ is of magnitude $$\sqrt{10}$$. Statement II : In a triangle $$ABC$$, $$\cos 2A + \cos 2B + \cos 2C \ge -\frac{3}{2}$$.
Let $$\vec{a} = 2\hat{i} + \alpha\hat{j} + \hat{k}, \vec{b} = -\hat{i} + \hat{k}, \vec{c} = \beta\hat{j} - \hat{k}$$, where $$\alpha$$ and $$\beta$$ are integers and $$\alpha\beta = -6$$. Let the values of the ordered pair $$(\alpha, \beta)$$, for which the area of the parallelogram of diagonals $$\vec{a} + \vec{b}$$ and $$\vec{b} + \vec{c}$$ is $$\frac{\sqrt{21}}{2}$$, be $$(\alpha_1, \beta_1)$$ and $$(\alpha_2, \beta_2)$$. Then $$\alpha_1^2 + \beta_1^2 - \alpha_2\beta_2$$ is equal to
Consider the line $$L$$ passing through the points $$(1, 2, 3)$$ and $$(2, 3, 5)$$. The distance of the point $$\left(\frac{11}{3}, \frac{11}{3}, \frac{19}{3}\right)$$ from the line $$L$$ along the line $$\frac{3x-11}{2} = \frac{3y-11}{1} = \frac{3z-19}{2}$$ is equal to
If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $$i^{th}$$ roll than the number obtained in the $$(i-1)^{th}$$ roll, $$i = 2, 3$$, is equal to
