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

Find `int (cos theta)/((4 + sin^2 theta)(5 - 4 cos^2 theta)) d theta`

Determine the product `[(-4,4,4),(-7,1,3),(5,-3,-1)][(1,-1,1),(1,-2,-2),(2,1,3)]` and use it to solve the system of equations x - y + z = 4, x- 2y- 2z = 9, 2x + y + 3z = 1.

Using the method of integration, find the area of the triangle ABC, coordinates of whose vertices are A (4 , 1), B (6, 6) and C (8, 4).

Find the area enclosed between the parabola 4y = 3x² and the straight line 3x - 2y + 12 = 0.

Find `(dy)/(dx)` at x = 1, y = `pi/4` if `sin^2 y + cos xy = K`

Show that the function f(x) = 4x³ - 18x² + 27x - 7 is always increasing on R.

Prove that `tan {pi/4 + 1/2 cos^(-1)  a/b} + tan {pi/4 - 1/2 cos^(-1)  a/b} = (2b)/a`

Let A = `((2,-1),(3,4))`, B = `((5,2),(7,4))`, C= `((2,5),(3,8))` find a matrix D such that CD − AB = O

Use product `[(1,-1,2),(0,2,-3),(3,-2,4)][(-2,0,1),(9,2,-3),(6,1,-2)]` to solve the system of equations x + 3z = 9, −x + 2y − 2z = 4, 2x − 3y + 4z = −3

Find the area bounded by the circle x² + y² = 16 and the line `sqrt3 y = x` in the first quadrant, using integration.

if A =  `((2,3,10),(4,-6,5),(6,9,-20))`, Find `A^(-1)`. Using `A^(-1)` Solve the system of equation `2/x + 3/y +10/z = 2`; `4/x - 6/y + 5/z = 5`; `6/x + 9/y - 20/z = -4`

if `y = sin^(-1) (6xsqrt(1-9x^2))`, `1/(3sqrt2) < x < 1/(3sqrt2)` then find `(dy)/(dx)`

Solve the following equation for x:  `cos (tan^(-1) x) = sin (cot^(-1)  3/4)`

Find the particular solution of the differential equation

`tan x * (dy)/(dx) = 2x tan x + x^2 - y`; `(tan x != 0)` given that y = 0 when `x = pi/2`

if the matrix A =`[(0,a,-3),(2,0,-1),(b,1,0)]` is skew symmetric, Find the value of 'a' and 'b'

Given `A = [(2,-3),(-4,7)]` compute `A^(-1)` and show that `2A^(-1) = 9I - A`

Prove that `3sin^(-1)x = sin^(-1) (3x - 4x^3)`, `x in [-1/2, 1/2]`

Evaluate `int (cos 2x + 2sin^2x)/(cos^2x) dx`

Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12`  is (a) strictly increasing, (b) strictly decreasing

A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 minutes on the hand-operated machines to manufacture a packet of screws 'A' while it takes 6 minutes on the automatic and 3 minutes on the hand-operated machine to manufacture a packet of screws 'B'. Each machine is available for at most 4 hours on any day. The manufacturer can sell a packet of screws 'A' at a profit of 70 paise and screws 'B' at a profit of Rs 1. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maximize his profit? Formulate the above LPP and solve it graphically and find the maximum profit.