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

The angle between the lines 2x = 3y = – z and 6x = – y = – 4z is ______.

Five fair coins are tossed simultaneously. The probability of the events that at least one head comes up is ______.

If f(x) = `{{:(x^2"," if x ≥ 1),(x"," if x < 1):}`, then show that f is not differentiable at x = 1.

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.

The function f(x) = x | x |, x ∈ R is differentiable ______.

Find the angle between the following two lines:

`vecr = 2hati - 5hatj + hatk + λ(3hati + 2hatj + 6hatk)`

`vecr = 7hati - 6hatk + μ(hati + 2hatj + 2hatk)`

If A = `[(5, x),(y, 0)]` and A = A^T, where A^T is the transpose of the matrix A, then ______.

Solve the following Linear Programming Problem graphically:

Minimize: Z = 60x + 80y

Subject to constraints:

3x + 4y ≥ 8

5x + 2y ≥ 11

x, y ≥ 0

If f(x) = | cos x |, then `f((3π)/4)` is ______.

The feasible region corresponding to the linear constraints of a Linear Programming Problem is given below.

Which of the following is not a constraint to the given Linear Programming Problem?

A problem in Mathematics is given to three students whose chances of solving it are `1/2, 1/3, 1/4` respectively. If the events of their solving the problem are independent then the probability that the problem will be solved, is ______.

The set of all points where the function f(x) = x + |x| is differentiable, is ______.

If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).

Find the maximum profit that a company can make, if the profit function is given by P(x) = 72 + 42x – x², where x is the number of units and P is the profit in rupees.

Check whether the function f : R `rightarrow` R defined by f(x) = x³ + x, has any critical point/s or not ? If yes, then find the point/s.

Solve the following Linear Programming Problem graphically:

Minimize: z = x + 2y,

Subject to the constraints: x + 2y ≥ 100, 2x – y ≤ 0, 2x + y ≤ 200, x, y ≥ 0.

Solve the following Linear Programming Problem graphically:

Maximize: z = – x + 2y,

Subject to the constraints: x ≥ 3, x + y ≥ 5, x + 2y ≥ 6, y ≥ 0.

Prove that the greatest integer function defined by f(x) = [x], 0 < x < 3 is not differentiable at x = 1 and x = 2.

Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:

f(x) `= x sqrt(1 - x), 0 < x < 1`