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

The value of cot^–1(–x) for all x ∈ R in terms of cot^–1x is ______.

If A is square matrix such that A² = A, show that (I + A)³ = 7A + I..

If A is a square matrix such that A² = I, then (A – I)³ + (A + I)³ –7A is equal to ______.

Matrix addition is associative as well as commutative.

Matrix multiplication is commutative.

If A and B are two square matrices of the same order, then A + B = B + A.

(AB)^–1 = A^–1. B^–1, where A and B are invertible matrices satisfying commutative property with respect to multiplication.

Find A^–1 if A = `[(0, 1, 1),(1, 0, 1),(1, 1, 0)]` and show that A^–1 = `("A"^2 - 3"I")/2`.

If A = `[(1, 2, 0),(-2, -1, -2),(0, -1, 1)]`, find A^–1. Using A^–1, solve the system of linear equations x – 2y = 10, 2x – y – z = 8, –2y + z = 7.

Using matrix method, solve the system of equations
3x + 2y – 2z = 3, x + 2y + 3z = 6, 2x – y + z = 2.

Given A = `[(2, 2, -4),(-4, 2, -4),(2, -1, 5)]`, B = `[(1, -1, 0),(2, 3, 4),(0, 1, 2)]`, find BA and use this to solve the system of equations y + 2z = 7, x – y = 3, 2x + 3y + 4z = 17.

If A is a matrix of order 3 × 3, then number of minors in determinant of A are ______.

The sum of the products of elements of any row with the co-factors of corresponding elements is equal to ______.

Show that the function f(x) = 4x³ – 18x² + 27x – 7 has neither maxima nor minima.

Find all the points of local maxima and local minima of the function f(x) = `- 3/4 x^4 - 8x^3 - 45/2 x^2 + 105`

Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.

If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is `pi/3`

Find the points of local maxima, local minima and the points of inflection of the function f(x) = x⁵ – 5x⁴ + 5x³ – 1. Also find the corresponding local maximum and local minimum values.

A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Re 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?

An open box with square base is to be made of a given quantity of cardboard of area c². Show that the maximum volume of the box is `"c"^3/(6sqrt(3))` cubic units