Mathematics Outside Delhi Set 1 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

In an LPP, if the objective function z = ax + by has the same maximum value on two corner points of the feasible region, then the number of points at which z_max oocurs is ______.

0

2

finite

infinite

From the set {1, 2, 3, 4, 5}, two numbers a and b (a ≠ b) are chosen at random. The probability that `a/b` is an integer is:

`1/3`

`1/4`

`1/2`

`3/5`

A bag contains 3 white, 4 black and 2 red balls. If 2 balls are drawn at random (without replacement), then the probability that both the balls are white is ______.

`1/18`

`1/36`

`1/12`

`1/24`

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 `veca` is a non-zero vector, then `(veca.hati).hati + (veca.hatj).hatj + (veca.hatk).hatk` ______.

The projection of the vector `hati - hatj` on the vector `hati + hatj` 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`

If `int_0^a dx/(1 + 4x^2) = π/8`, then find the value of a.

Find `int dx/(sqrt(x) + x)`

Show that the function y = ax + 2a² is a solution of the differential equation `2(dy/dx)^2 + x(dy/dx) - y = 0`.

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 equation of the plane with intercept 3 on the y-axis and parallel to xz – plane.

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.

A manufacturer has three machines I, II and III installed in his factory. Machine I and II are capable of being operated for atmost 12 hours whereas machine III must be operated for atleast 5 hours a day. He produces only two items M and N each requiring the use of all the three machines.

The number of hours required for producing 1 unit of M and N on three machines are given in the following table:

He makes a profit of ₹ 600 and ₹ 400 on one unit of items M and N respectively. How many units of each item should he produce so as to maximize his profit assuming that he can sell all the items that he produced. What will be the maximum profit?

A coin is biased so that the head is three times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails. Hence find the mean of the number of tails.

Suppose that 5 men out of 100 and 25 women out of 1000 are good orators. Assuming that there are equal number of men and women, find the probability of choosing a good orator.

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.

Show that the lines `vecr = veca + λvecb` and `vecr = vecb + μveca` are coplanar and the plane containing them is given by `vecr.(veca xx vecb) = 0`.