[10 × 3] If the matrix is symmetric, find the value of . If touches the conic , find the value of …

If the matrix $\begin{pmatrix} 6 & -x^2 \\ 2x - 15 & 10 \end{pmatrix}$ is symmetric, find the value of $x$.

If $y - 2x - k = 0$ touches the conic $3x^2 - 5y^2 = 15$, find the value of $k$.

Prove that $\frac{1}{2}\cos^{-1}\left(\frac{1-x}{1+x}\right) = \tan^{-1}\sqrt{x}$.

Using L’Hospital’s Rule, evaluate: $\lim_{x \to \pi/2} \left(x \tan x - \frac{\pi}{2} \sec x\right)$

Evaluate: $\int \frac{1}{x^2}\sin^2\left(\frac{1}{x}\right) dx$

Evaluate: $\int_{0}^{\pi/4} \log(1 + \tan\theta) \, d\theta$

By using the data $\bar{x} = 25$, $\bar{y} = 30$, $b_{yx} = 1.6$ and $b_{xy} = 0.4$, find: (a) The regression equation $y$ on $x$. (b) What is the most likely value of $y$ when $x = 60$? (c) What is the coefficient of correlation between $x$ and $y$?

A problem is given to three students whose chances of solving it are $\frac{1}{4}, \frac{1}{5}$ and $\frac{1}{3}$ respectively. Find the probability that the problem is solved.

If $a + ib = \frac{x+iy}{x-iy}$, prove that $a^2 + b^2 = 1$ and $\frac{b}{a} = \frac{2xy}{x^2-y^2}$.

Solve: $\frac{dy}{dx} = 1 - xy + y - x$