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

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

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 ______.

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 ______.

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

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

If `sin (sin^(−1)(1/5)+cos^(−1) x)=1`, then find the value of x.

Prove that `cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))=x/2;x in (0,pi/4) `

Prove that `2tan^(-1)(1/5)+sec^(-1)((5sqrt2)/7)+2tan^(-1)(1/8)=pi/4`

Evaluate : `int(x-3)sqrt(x^2+3x-18)  dx`

A dealer in rural area wishes to purchase a number of sewing machines. He has only Rs 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine cost him Rs 360 and a manually operated sewing machine Rs 240. He can sell an electronic sewing machine at a profit of Rs 22 and a manually operated sewing machine at a profit of Rs 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as a LPP and solve it graphically.

Find the Cartesian equation of the line which passes through the point (−2, 4, −5) and is parallel to the line `(x+3)/3=(4-y)/5=(z+8)/6`

Show that the function f in `A=R-{2/3} ` defined as `f(x)=(4x+3)/(6x-4)` is one-one and onto hence find f^-1

Prove that: `tan^(-1)(1/2)+tan^(-1)(1/5)+tan^(-1)(1/8)=pi/4`

Show that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is `(2R)/sqrt3.`  Also, find the maximum volume.