Without expanding show that \(\Delta =\left| \begin{matrix} { cosec }^{ 2 }\theta & { cot }^{ 2 }\theta & 1 \\ { cot }^{ 2 }\theta & { cosec }^{ 2 }\theta & -1 \\ 42 & 40 & 2 \end{matrix} \right| =0\)

In how many ways can the following prizes be given away to a class of 30 students, first and second in mathematics, first and second in physics, first in chemistry and first in English?

Prove that the tangents to the circle x² + y²  = 169 at (5,12) and (12,-5) are perpendicular to each other.

Prove that (sin 3x + sin x) sin x + (cos 3x - cos x) cos x = 0.

If the function \(f\left( x \right) =\begin{cases} 6ax+3b\quad if\quad x>1 \\ ax-2b\quad if\quad x<1\quad is\quad continuous\quad at\quad x=1 \\ 15\quad if\quad x=1 \end{cases}\) Find the value of a and b.

Find the maximum and minimum values of x³-6x²+7

A man, deposits Rs.75 at the end of 6 months in a bank which pays interest at 8% compounded semiannually. How much is to his credit at the end of 10 years?

A factory has 3 machines A₁, A₂, A₃ producing 1000, 2000, 3000 bolts per day respectively. A₁ produces 1% defectives, A₂ produces 1.5% and A₃ produces 2% defectives. A bolt is chosen at random and found defective. What is the probability that it comes from machine A₁?

Obtain the two regression lines from the following

X	6	2	10	4	8
Y	9	11	5	8	7

Reshma wishes to mix two types of food P and Q in such a way that the Vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs.60/kg and Food Q costs Rs.80/kg. Food P contains 3 units 1 kg of vitamin A and 5 units 1 kg of vitamin B while food Q contains 4 units 1 kg of vitamin A and 2 units 1 kg of vitamin B. Determine the minimum cost of the mixture.