Mathematics Delhi Set 3 2019-2020 Commerce (English Medium) Class 12 Question Paper Solution

If A is skew symmetric matrix of order 3, then the value of |A| is ______.

3

0

9

27

If A is a 3 × 3 matrix such that |A| = 8, then |3A| equals

8

24

72

216

If y = Ae^5x + Be^–5x, then `(d^2y)/(dx^2)` is equal to ______.

25y

5y

–25y

15y

`int x^2 e^(x^3) dx` equals

`1/3 e^(x^3) + C`

`1/3 e^(x^4) + C`

`1/2 e^(x^3) + C`

`1/2 e^(x^2) + C`

If `hati, hatj, hatk` are unit vectors along three mutually perpendicular directions, then

`hati * hatj = 1`

`hati xx hatj = 1`

`hati * hatk = 0`

`hati xx hatk = 0`

If `y = log_e (x^2/e^2)`, then `(d^2y)/(dx^2)` equals

`- 1/x`

`- 1/x^2`

`2/x^2`

`- 2/x^2`

The lines `(x - 2)/1 = (y - 3)/1 = (4 - z)/k` and `(x - 1)/k = (y - 4)/2 = (z - 5)/(-2)` are mutually perpendicular if the value of k is ______.

`- 2/3`

`2/3`

– 2

2

The graph of the inequality 2x + 3y > 6 is ______.

half plane that contains the origin.

half plane that neither contains the origin nor the points of the line 2x + 3y = 6.

whole XOY-plane excluding the points on the line 2x + 3y = 6.

entire XOY plane.

The distance of the origin (0, 0, 0) from the plane –2x + 6y – 3z = –7 is ______.

1 unit

`sqrt(2)` units

`2sqrt(2)` units

3 units

A die is thrown once. Let A be the event that the number obtained is greater than 3. Let B be the event that the number obtained is less than 5. Then P(A ∪ B) is ______.

`2/5`

`3/5`

0

1

If A and B are square matrices each of order 3 and |A| = 5, |B| = 3, then the value of |3AB| is ______.

If A + B = `[(1, 0),(1, 1)]` and A – 2B = `[(-1, 1),(0, -1)]`, then A = ______.

The least value of the function `f(x) = ax + b/x (a > 0, b > 0, x > 0)` is ______.

The integrating factor of the differential equation `x dy/dx + 2y = x^2` is ______.

The degree of the differential equation `1 + (dy/dx)^2 = x` is ______.

The vector equation of a line which passes through the point (3, 4, –7) and (1, –1, 6) is ______.

The line of shortest distance between two skew lines is ______ to both the lines.

Find the cofactors of all the elements of `[(1, -2),(4, 3)]`.

Let f(x) = x |x|, for all x ∈ R check its differentiability at x = 0.

If the function f defined as

`f(x) = {{:((x^2  -  9)/(x  -  3)",", x ≠ 3),(k",", x = 3):}`

is continuous at x = 3, find the value of k.

If f(x) = x⁴ – 10, then find the approximate value of f(2.1).

Find the slope of the tangent to the curve y = 2sin²(3x) at `x = π/6`.

Find the value of `int_1^4 |x - 5|dx`.

Find `int (x + 1)/((x + 2)(x + 3)) dx`.

Find `int x/(x^2 + 3x + 2) dx`

If x = a cos θ; y = b sin θ, then find `(d^2y)/(dx^2)`.

Find the differential of sin²x w.r.t. e^cosx.

Evaluate: `int_1^2 [1/x - 1/(2x^2)]e^(2x) dx`

Find the value of `int_0^1 x(1 - x)^n dx`.

Find the value of `int_0^1 tan^-1 ((1 - 2x)/(1 + x - x^2)) dx`.

Solve the equation for x: `sin^(-1)  5/x + sin^(-1)  12/x = π/2, x ≠ 0`

Find the general solution of the differential equation 

ye^x/y dx = (xe^x/y + y²) dy, y ≠ 0

Solve the differential equation:

`x sin (y/x) dy/dx + x - y sin (y/x) = 0`

Given that x = 1 when y = `π/2`.

If `veca = hati + 2hatj + 3hatk` and `vecb = 2hati + 4hatj - 5hatk` represent two adjacent sides of a parallelogram, find unit vectors parallel to the diagonals of the parallelogram.

Using vectors, find the area of the triangle ABC with vertices A(1, 2, 3), B(2, – 1, 4) and C(4, 5, – 1).

A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A requires 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. Given that total time for cutting is 3 hours 20 minutes and for assembling 4 hours. The profit for type A souvenir is ₹ 100 each and for type B souvenir, profit is ₹ 120 each. How many souvenirs of each type should the company manufacture in order to maximize the profit? Formulate the problem as an LPP and solve it graphically.

Three rotten apples are mixed with seven fresh apples. Find the probability distribution of the number of rotten apples, if three apples are drawn one by one with replacement. Find the mean of the number of rotten apples.

In a shop X, 30 tins of ghee of type A and 40 tins of ghee of type B which look alike, are kept for sale. While in shop Y, similar 50 tins of ghee of type A and 60 tins of ghee of type B are there. One tin of ghee is purchased from one of the randomly selected shop and is found to be of type B. Find the probability that it is purchased from shop Y.

Find the distance of the point P(3, 4, 4) from the point, where the line joining the points A(3, –4, –5) and B(2, –3, 1) intersects the plane 2x + y + z = 7.

Using integration find the area of the region bounded between the two circles x² + y² = 9 and (x – 3)² + y² = 9.

Evaluate the following integral as the limit of sums `int_1^4 (x^2 - x)dx`.

Find the minimum value of (ax + by), where xy = c².

If a, b, c are p^th, q^th and r^th terms respectively of a G.P., then prove that

`|(log a, p, 1),(log b, q, 1),(log c, r, 1)| = 0`

If A = `[(2, -3, 5),(3, 2, -4),(1, 1, -2)]`, then find A^–1. Using A^–1, solve the following system of equations:

2x – 3y + 5z = 11

3x + 2y – 4z = –5

x + y – 2z = –3