Science (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

Find the general solution of the differential equation:

`(dy)/(dx) = (3e^(2x) + 3e^(4x))/(e^x + e^-x)`

If `[(1, 2, 1),(2, 3, 1),(3, a, 1)]` is non-singular matrix and a ∈ A, then the set A is ______.

If f(x) = `{{:(ax + b; 0 < x ≤ 1),(2x^2 - x; 1 < x < 2):}` is a differentiable function in (0, 2), then find the values of a and b.

A function f : [– 4, 4] `rightarrow` [0, 4] is given by f(x) = `sqrt(16 - x^2)`. Show that f is an onto function but not a one-one function. Further, find all possible values of 'a' for which f(a) = `sqrt(7)`.

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.