Mathematics Outside Delhi Set 2 2019-2020 Commerce (English Medium) Class 12 Question Paper Solution

The value of `sin^-1 (cos  (3π)/5)` is ______.

`π/10`

`(3π)/5`

`- π/10`

`(-3π)/5`

If A = `[(2, -3, 4)]`, B = `[(3),(2),(2)]`, X = `[(1, 2, 3)]` and Y = `[(2),(3),(4)]`, then AB + XY equals

[28]

[24]

28

24

If `|(2, 3, 2),(x, x, x),(4, 9, 1)| + 3 = 0`, then the value of x is ______.

3

0

–1

1

`int_0^(π/8) tan^2 (2x)` is equal to ______.

`(4 - π)/8`

`(4 + π)/8`

`(4 - π)/4`

`(4 - π)/2`

If `veca * vecb = 1/2 |veca| |vecb|`, then the angle between `veca` and `vecb` is ______.

0°

30°

60°

90°

The two lines x = ay + b, z = cy + d; and x = a'y + b', z = c'y + d' are perpendicular to each other, if

`a/a^' + c/c^' = 1`

`a/a^' + c/c^' = -1`

aa' + cc' = 1

aa' + cc' = –1

The two planes x – 2y + 4z = 10 and 18x + 17y + kz = 50 are perpendicular, if k is equal to ______.

– 4

4

2

– 2

Let A = `[(200, 50),(10, 2)]` and B = `[(50, 40),(2, 3)]`, then |AB| is equal to ______.

460

2000

3000

–7000

Let `veca = hati - 2hatj + 3hatk`. If `vecb` is a vector such that `veca.vecb = |vecb|^2` and `|veca - vecb| = sqrt(7)` then `|vecb|` equals

7

14

`sqrt(7)`

21

Three dice are thrown simultaneously. The probability of obtaining a total score of 5 is ______.

`5/216`

`1/6`

`1/36`

`1/49`

If f : R → R be given by f(x) = (3 – x³)^1/3, then fof(x) = ______.

If `[(x + y, 7),(9, x - y)] = [(2, 7),(9, 4)]`, then x · y = ______.

The number of points of discontinuity of f defined by f(x) = |x| – |x + 1| is ______.

The slope of the tangent to the curve y = x³ – x at the point (2, 6) is ______.

The rate of change of the area of a circle with respect to its radius r, when r = 3 cm, is ______.

If f(x) = 2|x| + 3|sin x| + 6, then the right hand derivative of f(x) at x = 0 is ______.

Find adj A, if A = `[(2, -1),(4, 3)]`.

Find `int (2^(x + 1) - 5^(x - 1))/(10^x) dx`

Evaluate `int_0^(2π) |sin x| dx`

Find `int sin^5 (x/2).cos (x/2) dx`

If A = `[(1, 0),(1, 1)]`, then find A³.

Check if the relation R on the set A = {1, 2, 3, 4, 5, 6} defined as R = {(x, y) : y is divisible by x} is (i) symmetric (ii) transitive.

Prove that: `(9π)/8 - 9/4 sin^-1 (1/3) = 9/4 sin^-1 ((2sqrt(2))/3)`

Find the value of `dy/dx` at `θ = π/3`, if x = cos θ – cos 2θ, y = sin θ – sin 2θ.

Show that the function f defined by f(x) = (x – 1)e^x + 1 is an increasing function for all x > 0.

Find `|veca|` and `|vecb|`, if `|veca| = 2|vecb|` and `(veca + vecb).(veca - vecb) = 12`.

Find the unit vector perpendicular to each of the vectors `veca = 4hati + 3hatj + hatk` and `vecb = 2hati - hatj + 2hatk`.

Find the derivative of x^{log x} w.r.t. log x.

Find [P(B/A) + P(A/B)], if P(A) = `3/10`, P(B) = `2/5` and P(A ∪ B) = `3/5`.

Prove that the relation R on Z, defined by R = {(x, y) : (x – y) is divisible by 5} is an equivalence relation.

If `y = sin^-1 ((sqrt(1 + x) + sqrt(1 - x))/2)`, then show that `dy/dx = (-1)/(2sqrt(1 - x^2)`.

Verify the Rolle’s Theorem for the function f(x) = e^x cos x in `[- π/2, π/2]`

Evaluate: `int_0^π (x sin x)/(1 + cos^2x) dx`.

For the differential equation given below, find a particular solution satisfying the given condition `(x + 1) dy/dx = 2e^-y + 1; y = 0` when x = 0.

Show that the function f : R → R defined by `f(x) = x/(x^2 + 1), ∀  x ∈ R` is neither one-one nor onto.

Evaluate: `int_(-1)^2 |x^3 - x| dx`

Using properties of determinants prove that:

`|(a - b, b + c, a),(b - c, c + a, b),(c - a, a + b, c)| = a^3 + b^3 + c^3 - 3abc`.

If A = `[(1, 3, 2),(2, 0, -1),(1, 2, 3)]`, then show that A³ – 4A² – 3A + 11I = 0, Hence find A^–1 .

Find the intervals on which the function f(x) = (x – 1)³ (x – 2)² is (a) strictly increasing (b) strictly decreasing.

Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also, find the maximum volume.

Find the area of the region lying in the first quadrant and enclosed by the x-axis, the line y = x and the circle x² + y² = 32.

Using integration find the area of the region: 

{(x, y) : 0 ≤ y ≤ x², 0 ≤ y ≤ x, 0 ≤ x ≤ 2}