Question

Find the equation of the tangent to the curve $\begin{array}{r c l}{y}&{=}&{{\sqrt{3x-2}}}\end{array}$ which is parallel to the line 4x – 2y + 5 = 0.

Find the equation of the normal at a point on the curve x² = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.

Prove that the curves x = y² and xy = k cut at right angles if 8k² = 1.

If A = $\left(\begin{array}{l l l }1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3\end{array}\right)$, prove that A³ – 6A² + 7A + 2I = 0

Consider the experiment of tossing a coin. If the coin shows head, toss is done again, but if it shows tail, then throw a die. Find the conditional probability of the events that ‘the die shows a number greater than 4’, given that ‘there is atleast one tail’.

If A and B are two independent events such that P(A' ∩ B)= 2/15 and P(A ∩ B')= 1/6, then find P(A) and P(B).

Solve the differential equation $x\;\frac{d y}{d x}\;+y=$ x cos x + sin x, given that y = 1 when $x={\frac{\pi}{2}}$