Commerce (English Medium) इयत्ता १२ - CBSE Question Bank Solutions for Mathematics

Find the value of λ, so that the lines `(1-"x")/(3) = (7"y" -14)/(λ) = (z -3)/(2) and (7 -7"x")/(3λ) = ("y" - 5)/(1) = (6 -z)/(5)` are at right angles. Also, find whether the lines are intersecting or not.

A manufacturer has employed 5 skilled men and 10 semi-skilled men and makes two models A and B of an article. The making of one item of model A requires 2 hours of work by a skilled man and 2 hours work by a semi-skilled man. One item of model B requires 1 hour by a skilled man and 3 hours by a semi-skilled man. No man is expected to work more than 8 hours per day. The manufacturer's profit on an item of model A is ₹ 15 and on an item of model B is ₹ 10. How many items of each model should be made per day in order to maximize daily profit? Formulate the above LPP and solve it graphically and find the maximum profit.

A company manufactures two types of cardigans: type A and type B. It costs ₹ 360 to make a type A cardigan and ₹ 120 to make a type B cardigan. The company can make at most 300 cardigans and spend at most ₹ 72000 a day. The number of cardigans of type B cannot exceed the number of cardigans of type A by more than 200. The company makes a profit of ₹ 100 for each cardigan of type A and ₹ 50 for every cardigan of type B. 

Formulate this problem as a linear programming problem to maximize the profit to the company. Solve it graphically and find the maximum profit.

Evaluate the following integrals : `int tanx/(sec x + tan x)dx`

Evaluate the following integrals: `int(x - 2)/sqrt(x + 5).dx`

Integrate the following functions w.r.t. x : `(e^(2x) + 1)/(e^(2x) - 1)`

If A = `[(x, 5, 2),(2, y, 3),(1, 1, z)]`, xyz = 80, 3x + 2y + 10z = 20, ten A adj. A = `[(81, 0, 0),(0, 81, 0),(0, 0, 81)]`

If A = `[(0, 1, 3),(1, 2, x),(2, 3, 1)]`, A^–1 = `[(1/2, -4, 5/2),(-1/2, 3, -3/2),(1/2, y, 1/2)]` then x = 1, y = ^–1.

If A and B are invertible matrices, then which of the following is not correct?

(A³)^–1 = (A^–1)³, where A is a square matrix and |A| ≠ 0.

`("aA")^-1 = 1/"a"  "A"^-1`, where a is any real number and A is a square matrix.

|A^–1| ≠ |A|^–1, where A is non-singular matrix.

|adj. A| = |A|², where A is a square matrix of order two.

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