Show that f(x) f(y) = f(x + y), where f(x) =\(\begin{bmatrix} cos \ x & -sin \ x & 0 \\ sin x & cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}\).

Prove that |A| = \(\begin{vmatrix} (q+r)^2& p^2 &p^2 \\ q^2 & (r+p)^2 & q^2 \\ r^2 &r^2 & (p+q)^2 \end{vmatrix}\) = 2pqr(p + q + r)³.

Show that \(\begin{vmatrix} 1 &1 &1 \\ x & y & z \\ x^2 & y^2 & z^2 \end{vmatrix}\) = (x - y)( y - z)(z - x).

Prove that the smallar angle between any two diagonals of a cube is cos^-1 \(({1\over3})\).

Evaluate the following limits : \(lim_{x\rightarrow2}{2-\sqrt{x+2}\over 3\sqrt{2}-3\sqrt{4-x}}\)

According to Einstein’s theory of relativity, the mass m of a body moving with velocity v is m = \({m_O\over \sqrt{1-{v^2\over c^2}}},\)  where m₀ is the initial mass and c is the speed of light. What happens to m as \(v\rightarrow{c}^-\). Why is a left hand limit necessary?

Do the limits of following functions exist as x\(\rightarrow 0?\) State reasons for your answer.\(x \left\lfloor x \right\rfloor \over sin |x|\)

Show that the function \(\begin{cases}\frac{x^{3}-1}{x-1}, & \text { if } x \neq 1 \\ 3, & \text { if } x=1\end{cases}\) is continuous on \((-\infty,\infty)\)

For what value of \(\alpha\)is this function \(f(x)= \begin{cases}\frac{x^{4}-1}{x-1}, & \text { if } x \neq 1 \\ \alpha, & \text { if } x=1\end{cases}\)  continuous at x = 1?

Which of the following functions f has a removable discontinuity at x = x₀? If the discontinuity is removable, find a function g that agrees with f for x ≠ x0 and is continuous on R
\(f(x)={x^3+64\over x+4},x_o=-4\)

Show that the following functions are not differentiable at the indicated value of x.

If y = tan^-1\(({1+x\over 1-x}), \) find y'.

Find y' if x⁴ + y⁴ = 16.

If \(y={sin^{-1}x\over \sqrt{1-x^2}}\) , Show that (1 - x²) y₂ - 3x y₁ - y = 0.

A tree is growing so that, after t - years its height is increasing at a rate of \({18\over \sqrt{t}}\) cm per year Assume that when t = 0, the height is 5 cm.
(i) Find the height of the tree after 4 years.
(ii) After how many years will the height be 149 cm?

Evaluate : \(\int tan^{-1}({2x\over 1-x^2})dx\)

Integrate the following with respect to x: : \({x sin^{-1}\over \sqrt{1-x^2}}\)

Integrate the following with respect to x : \({log \ x \over (1+log x)^2}\)

Evaluate the following integrals : \(\int{3x+5\over x^2+4x+7}dx\)

Evaluate the following integrals : \(\int{2x+3\over \sqrt{x^2+x+1}}dx\)

There are two identical urns containing respectively 6 black and 4 red balls, 2 black and 2 red balls. An urn is chosen at random and a ball is drawn from it.
(i) find the probability that the ball is black
(ii) if the ball is black, what is the probability that it is from the first urn?

The chances of A, B, and C becoming manager of a certain company are 5 : 3: 2. The probabilities that the office canteen will be improved if A, B, and C become managers are 0.4, 0.5 and 0.3 respectively. If the office canteen has been improved, what is the probability that B was appointed as the manager?

X speaks truth in 70 percent of cases, and Y in 90 percent of cases. What is the probability that they likely to contradict each other in stating the same fact?

Three candidates X, Y, and Z are going to play in a chess competition to win FIDE (World chess Federation) cup this year. X is thrice as likely to win as Y and Y is twice as likely as to win Z. Find the respective probability of X,Y and Z to win the cup.

The probability that a girl, preparing for competitive examination will get a State Government service is 0.12, the probability that she will get a Central Government job is 0.25, and the probability that she will get both is 0.07. Find the probability that (i) she will get atleast one of the two jobs (ii) she will get only one of the two jobs.