Arts (English Medium) इयत्ता १२ - CBSE Question Bank Solutions for Mathematics

Solve the following Linear Programming Problems graphically:

Maximise Z = 3x + 2y

subject to x + 2y ≤ 10, 3x + y ≤ 15, x, y ≥ 0.

Solve the following Linear Programming Problems graphically:

Minimise Z = x + 2y

subject to 2x + y ≥ 3, x + 2y ≥ 6, x, y ≥ 0.

Show that the minimum of Z occurs at more than two points.

Minimise and Maximise Z = 5x + 10 y

subject to x + 2y ≤ 120, x + y ≥ 60, x – 2y ≥ 0, x, y ≥ 0.

Show that the minimum of Z occurs at more than two points.

Minimise and Maximise Z = x + 2y 

subject to x + 2y ≥ 100, 2x – y ≤ 0, 2x + y ≤ 200; x, y ≥ 0.

Show that the minimum of Z occurs at more than two points.

Maximise Z = – x + 2y, Subject to the constraints:

x ≥ 3, x + y ≥ 5, x + 2y ≥ 6, y ≥ 0.

Show that the minimum of Z occurs at more than two points.

Maximise Z = x + y, subject to x – y ≤ –1, –x + y ≤ 0, x, y ≥ 0.

Refer to Example 9. How many packets of each food should be used to maximize the amount of vitamin A in the diet? What is the maximum amount of vitamin A in the diet?

A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs 250 per bag contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing Rs 200 per bag contains 1.5 units of nutritional elements A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin content of one kg food is given below:

One kg of food X costs Rs 16 and one kg of food Y costs Rs 20. Find the least cost of the mixture which will produce the required diet?

A manufacturer makes two types of toys A and B. Three machines are needed for this purpose and the time (in minutes) required for each toy on the machines is given below:

Each machine is available for a maximum of 6 hours per day. If the profit on each toy of type A is Rs 7.50 and that on each toy of type B is Rs 5, show that 15 toys of type A and 30 of type B should be manufactured in a day to get maximum profit.

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