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

Equation of a line passing through (1, 1, 1) and parallel to z-axis is ______.

The objective function Z = ax + by of an LPP has maximum vaiue 42 at (4, 6) and minimum value 19 at (3, 2). Which of the following is true?

The corner points of the feasible region of a linear programming problem are (0, 4), (8, 0) and `(20/3, 4/3)`. If Z = 30x + 24y is the objective function, then (maximum value of Z – minimum value of Z) is equal to ______.

Draw the graph of cos^–1 x, where x ∈ [–1, 0]. Also, write its range.

Find `int dx/sqrt(sin^3x cos(x - α))`.

Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.

Find the general solution of the differential equation:

`(x^2 + 1) dy/dx + 2xy = sqrt(x^2 + 4)`

Solve the following linear programming problem graphically:

Minimize: Z = 5x + 10y

Subject to constraints:

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

Find the equations of the diagonals of the parallelogram PQRS whose vertices are P(4, 2, – 6), Q(5, – 3, 1), R(12, 4, 5) and S(11, 9, – 2). Use these equations to find the point of intersection of diagonals.

Read the following passage:

Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore. One complete of a four-cylinder four-stroke engine. The volume displace is marked The cylinder bore in the form of circular cylinder open at the top is to be made from a metal sheet of area 75π cm2.

Based on the above information, answer the following questions:

1. If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (1)
2. Find `(dV)/(dr)`. (1)
3. (a) Find the radius of cylinder when its volume is maximum. (2)
   OR
   (b) For maximum volume, h > r. State true or false and justify. (2)

Solve the following linear programming problem graphically:

Maximize: Z = x + 2y

Subject to constraints:

x + 2y ≥ 100,

2x – y ≤ 0

2x + y ≤ 200,

x ≥ 0, y ≥ 0.

Solve the following Linear Programming problem graphically:

Maximize: Z = 3x + 3.5y

Subject to constraints:

x + 2y ≥ 240,

3x + 1.5y ≥ 270,

1.5x + 2y ≤ 310,

x ≥ 0, y ≥ 0.

Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.

Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.

If for a square matrix A, A² – A + I = 0, then A^–1 equals ______.

`int secx/(secx - tanx)dx` equals ______.

Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.

Solve the following Linear Programming Problem graphically:

Maximize: P = 70x + 40y

Subject to: 3x + 2y ≤ 9,

3x + y ≤ 9,

x ≥ 0,y ≥ 0.

Read the following passage:

Based on the above information, answer the following questions:

1. Convert the given above situation into a matrix equation of the form AX = B. (1)
2. Find | A |. (1)
3. Find A^–1. (2)
   OR
   Determine P = A² – 5A. (2)

Draw the graph of the principal branch of the function f(x) = cos^–1 x.