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

Read the following passage:

Based on the above information, answer the following questions:

1. Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
2. Prove that the function V(t) is an increasing function. (2)

If `d/dx f(x) = 2x + 3/x` and f(1) = 1, then f(x) is ______.

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

The interval in which the function f(x) = 2x³ + 9x² + 12x – 1 is decreasing is ______.

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

f(x) = tan^–1 x.

If f(x) = `{{:(x^2"," if x ≥ 1),(x"," if x < 1):}`, then show that f is not differentiable at x = 1.

The function f(x) = x | x |, x ∈ R is differentiable ______.

If A = `[(5, x),(y, 0)]` and A = A^T, where A^T is the transpose of the matrix A, then ______.

The function f(x) = x³ + 3x is increasing in interval ______.

Find the value of `tan^-1 [2 cos (2 sin^-1  1/2)] + tan^-1 1`.

If f(x) = | cos x |, then `f((3π)/4)` is ______.

The set of all points where the function f(x) = x + |x| is differentiable, is ______.

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 the interval/s in which the function f : R `rightarrow` R defined by f(x) = xe^x, is increasing.

Write the following function in the simplest form:

`tan^-1 ((cos x - sin x)/(cos x + sin x)), (-pi)/4 < x < (3 pi)/4`

`tan^-1 sqrt3 - cot^-1 (- sqrt3)` is equal to ______.

Prove that the greatest integer function defined by f(x) = [x], 0 < x < 3 is not differentiable at x = 1 and x = 2.

if `2[[3,4],[5,x]]+[[1,y],[0,1]]=[[7,0],[10,5]]` , find (x−y).