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

Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.

The function f(x) = [x], where [x] denotes the greatest integer less than or equal to x; is continuous at ______.

`int secx/(secx - tanx)dx` equals ______.

Write the domain and range (principle value branch) of the following functions:

f(x) = tan^–1 x.

Find the distance between the lines:

`vecr = (hati + 2hatj - 4hatk) + λ(2hati + 3hatj + 6hatk)`;

`vecr = (3hati + 3hatj - 5hatk) + μ(4hati + 6hatj + 12hatk)`

In answering a question on a multiple choice test, a student either knows the answer or guesses. Let `3/5` be the probability that he knows the answer and `2/5` be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability `1/3`. What is the probability that the student knows the answer, given that he answered it correctly?

The lines `vecr = hati + hatj - hatk + λ(2hati + 3hatj - 6hatk)` and `vecr = 2hati - hatj - hatk + μ(6hati + 9hatj - 18hatk)`; (where λ and μ are scalars) are ______.

Let f(x) be a polynomial function of degree 6 such that `d/dx (f(x))` = (x – 1)³ (x – 3)², then

Assertion (A): f(x) has a minimum at x = 1.

Reason (R): When `d/dx (f(x)) < 0, ∀  x ∈ (a - h, a)` and `d/dx (f(x)) > 0, ∀  x ∈ (a, a + h)`; where 'h' is an infinitesimally small positive quantity, then f(x) has a minimum at x = a, provided f(x) is continuous at x = a.

ASSERTION (A): The relation f : {1, 2, 3, 4} `rightarrow` {x, y, z, p} defined by f = {(1, x), (2, y), (3, z)} is a bijective function.

REASON (R): The function f : {1, 2, 3} `rightarrow` {x, y, z, p} such that f = {(1, x), (2, y), (3, z)} is one-one.

Find the domain of sin^–1 (x² – 4).

Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.

An aeroplane is flying along the line `vecr = λ(hati - hatj + hatk)`; where 'λ' is a scalar and another aeroplane is flying along the line `vecr = hati - hatj + μ(-2hatj + hatk)`; where 'μ' is a scalar. At what points on the lines should they reach, so that the distance between them is the shortest? Find the shortest possible distance between them.

Evaluate :`int_(pi/6)^(pi/3) dx/(1+sqrtcotx)`

Evaluate : `intsin(x-a)/sin(x+a)dx`

Show that the differential equation 2y^x/y dx + (y − 2x e^x/y) dy = 0 is homogeneous. Find the particular solution of this differential equation, given that x = 0 when y = 1.

Solve the differential equation :

`y+x dy/dx=x−y dy/dx`

Show that the differential  equation `2xydy/dx=x^2+3y^2`  is homogeneous and solve it.

Find the particular solution of the differential equation:

2y e^x/y dx + (y - 2x e^x/y) dy = 0 given that x = 0 when y = 1.

If `A = [(1, 2, -3),(5, 0, 2),(1, -1, 1)], B = [(3, -1, 2),(4, 2, 5),(2, 0, 3)] and C = [(4, 1, 2),(0, 3, 2),(1, -2, 3)]` then compute (A + B) and (B – C). Also verify that A + (B – C) = (A + B) – C.

If ` A = [(2/3, 1, 5/3),(1/3, 2/3, 4/3),(7/3, 2, 2/3)]` and `B = [(2/5, 3/5, 1),(1/5, 2/5, 4/5),(7/5, 6/5, 2/5)]`, then compute 3A – 5B.