Mathematics Official 2019-2020 ISC (Commerce) Class 12 Question Paper Solution

Determine whether the binary operation ∗ on R defined by a ∗ b = |a – b| is commutative. Also, find the value of (–3) ∗ 2.

Prove that: tan²(sec^–1 2) + cot² (cosec^–1 3) = 11.

Without expanding at any stage, find the value of the determinant:

`Δ = |(20, a, b + c),(20, b, a + c),(20, c, a + b)|`

If `((2, 3),(5, 7))((1, -3),(-2, 4)) = ((-4, 6),(-9, x))`, find x.

Find `dy/dx` if x³ + y³ = 3axy

The edge of a variable cube is increasing at the rate of 10 cm/sec. How fast is the volume of the cube increasing when the edge is 5 cm long?

Evaluate:

`int_4^5 |x - 5| dx`

Form a differential equation of the family of the curves y² = 4ax.

A bag contains 5 white, 7 red and 4 black balls. If four balls are drawn one by one with replacement, what is the probability that none is white?

Let A and B be two events such that `P(A) = 1/2, P(B) = p` and `P(A ∪ B) = 3/5` find ‘p’ if A and B are independent events.

If the function f : R→ R be defined as `f(x) = (3x + 4)/(5x - 7), (x ≠ 7/5)` and g : R→ R be defined as `g(x) = (7x + 4)/(5x - 3), (x ≠ 3/5)` show that (gof)(x) = (fog)(x).

If `cos^-1  x/2 + cos^-1  y/3 = θ`, then prove that 9x² – 12xy cos θ + 4y² = 36 sin² θ

Evaluate: `cos(2cos^-1x + sin^-1x)` at `x = 1/5`.

Using properties of determinants, show that `|(x, p, q),(p, x, q),(q, q, x)| = (x - p)(x^2 + px - 2q)^2`

Verify Rolle’s theorem for the function, f(x) = –1 + cos x in the interval [0, 2π]

If `y = e^(msin^-1x)`, prove that `(1 - x^2) (d^2y)/(dx^2) - x dy/dx = m^2y`

The equation of tangent at (2, 3) on the curve y² = px³ + q is y = 4x – 7. Find the values of  ‘p’ and ‘q’.

Using L’Hospital’s rule, evaluate: 

`lim_(x → 0) (xe^x - log(1 + x))/x^2`

Evaluate:

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

Evaluate:

`int sin^3x cos^4x  dx`

Solve the differential equation:

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

Three persons A, B and C shoot to hit a target. Their probabilities of hitting the target are `5/6, 4/5` and `3/4` respectively. Find the probability that:

1. Exactly two persons hit the target.
2. At least one person hits the target.

Solve the following system of linear equations using matrices:

x – 2y = 10, 2x – у – z = 8, –2y + z = 7

Show that the radius of a closed right circular cylinder of given surface area and maximum volume is equal to half of its height.

Prove that the area of right-angled triangle of given hypotenuse is maximum when the triangle is isosceles.

Evaluate:

`int tan^-1 sqrt((1 - x)/(1 + x)) dx`

Evaluate:

`int (2x + 7)/(x^2 - x - 2) dx`

The probability that a bulb produced in a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs:

1. None will fuse after 150 days of use.
2. Not more than one will fuse after 150 days of use.
3. More than one will fuse after 150 days of use.
4. At least one will fuse after 150 days of use.

Write a vector of magnitude of 18 units in the direction of the vector `hati - 2hatj - 2hatk`.

Find the angle between the two lines:

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

Find the equation of the plane passing through the point (2, –3, 1) and perpendicular to the line joining the points (4, 5, 0) and (1, –2, 4).

Prove that `veca . [(vecb + vecc) xx (veca + 3vecb + 4vecc)] = [(veca, vecb, vecc)]`

Find the area of the triangle whose vertices are A(3, –1, 2), B(1, –1, –3) and C(4, –3, 1)

Find the image of the point (3, –2, 1) in the plane 3x – y + 4z = 2.

Determine the equation of the line passing through the point (–1, 3, –2) perpendicular to the lines:

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

Draw a rough sketch of the curves y² = x and y² = 4 – 3x and find the area enclosed between them.

The selling price of a commodity is fixed at ₹ 60 and its cost function is C(x) = 35x + 250

1. Determine the profit function.
2. Find the break even points.

The revenue function is given by R(x) = 100x – x² – x³. Find

1. The demand function.
2. Marginal revenue function.

For the lines of regression 4x – 2y = 4 and 2x – 3y + 6 = 0, find the mean of ‘x’ and the mean of ‘y’.

The correlation coefficient between x and y is 0.6. If the variance of x is 225, the variance of y is 400, mean of x is 10 and mean of y is 20, find

1. the equations of two regression lines.
2. the expected value of y when x = 2.

Find the regression coefficients b_yx, b_xy and correlation coefficient ‘r’ for the following data:

(2, 8), (6, 8), (4, 5), (7, 6), (5, 2)

The marginal cost of the production of the commodity is 30 + 2x, it is known that fixed costs are ₹ 200, find

1. The total cost.
2. The cost of increasing output from 100 to 200 units.

The total cost function of a firm is given by `C(x) = 1/3 x^3 - 5x^2 + 30x - 15` where the selling price per unit is given as ₹ 6. Find for what value of x will the profit be maximum.