Two Mark Questions

Q1. A person goes to office either by car, scooter, bus or train the probabilities for which are 1/7, 3/7, 2/7 and 1/7 respectively. The probabilities that he reaches office late, if he takes car, scooter, bus or train are 2/9, 1/9, 4/9 and 1/9 respectively. Given that he reached office in time, what is the probability that he travelled by a car?

Q2 Find the range of values of t for which
${\displaystyle 2\sin t={\frac {1-2x+5x^{2}}{3x^{2}-2x-1}}}$
Q3 Circles with radii 3,4 and 5 units touch each other externally. Let P be the point of intersection of tangents to these circles at their points of contact. Find the distance of P from the points of contact.

Q4 Find the equation of the plane containing the line 2x – y + z – 3 = 0, 3x + y + z = 5 and at a distance of ${\displaystyle {\frac {1}{\sqrt {6}}}}$ from the point (2,1,–1).

Q5 A function f(x) is such that | f (x1)-f (x2) | < (x1-x2)2 for all real x1 and x2. Find the equation of the tangent to the curve y = f (x) at the point (1,2)

Q6 The total number of runs scored in n matches is given by f (n) (n > 1) and the number of runs scored in the kth match is given by g (k) (1 ≤ k ≤ n). Find the value of n.
${\displaystyle f(n)=\left({\frac {n+1}{4}}\right)\left(2^{\left(n+1\right)}-n-2\right)andg(k)=k2^{\left(n+1-k\right)}}$