Commerce (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

An aeroplane can carry a maximum of 200 passengers. A profit of Rs 1000 is made on each executive class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. What is the maximum profit?

Find the angle between the following pair of lines:

`vecr = 2hati - 5hatj + hatk + lambda(3hati - 2hatj + 6hatk) and vecr = 7hati - 6hatk + mu(hati + 2hatj + 2hatk)`

Find the angle between the following pair of lines:

`vecr = 3hati + hatj - 2hatk + lambda(hati - hatj - 2hatk) and vecr = 2hati - hatj -56hatk + mu(3hati - 5hatj - 4hatk)`

Find the angle between the following pairs of lines: 

`(x-2)/2 = (y-1)/5 = (z+3)/(-3)` and `(x+2)/(-1) = (y-4)/8 = (z -5)/4`

Find the angle between the following pairs of lines:

`x/y = y/2 = z/1` and `(x-5)/4 = (y-2)/1 = (z - 3)/8`

Find the values of p so the line `(1-x)/3 = (7y-14)/2p = (z-3)/2` and `(7-7x)/(3p) = (y -5)/1 = (6-z)/5` are at right angles.

Find the angle between the lines whose direction ratios are a, b, c and b − c, c − a, a − b.

Find `bb(dy/dx)` in the following:

2x + 3y = sin x

Find `bb(dy/dx)` in the following:

2x + 3y = sin y

Find `bb(dy/dx)` in the following:

ax + by² = cos y

Find `bb(dy/dx)` in the following:

xy + y² = tan x + y

Find `bb(dy/dx)` in the following:

x² + xy + y² = 100

Find `bb(dy/dx)` in the following:

x³ + x²y + xy² + y³ = 81

Find `bb(dy/dx)` in the following:

sin² y + cos xy = k

Find `bb(dy/dx)` in the following:

sin² x + cos² y = 1

Find `bb(dy/dx)` in the following:

`y = sin^(-1)((2x)/(1+x^2))`

Show that the given differential equation is homogeneous and solve them.

(x² + xy) dy = (x² + y²) dx

Show that the given differential equation is homogeneous and solve them.

`y' = (x + y)/x`

Show that the given differential equation is homogeneous and solve them.

(x – y) dy – (x + y) dx = 0

Show that the given differential equation is homogeneous and solve them.

(x² – y²) dx + 2xy dy = 0