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

Using elementary transformations, find the inverse of the matrix A =  `((8,4,3),(2,1,1),(1,2,2))`and use it to solve the following system of linear equations :

8x + 4y + 3z = 19

2x + y + z = 5

x + 2y + 2z = 7

Find the integrating factor of the differential equation.

`((e^(-2^sqrtx))/sqrtx-y/sqrtx)dy/dx=1`

If `y=tan^(−1) ((sqrt(1+x^2)+sqrt(1−x^2))/(sqrt(1+x^2)−sqrt(1−x^2)))` , x²≤1, then find dy/dx.

If the function f : R → R be given by f[x] = x² + 2 and g : R ​→ R be given by  `g(x)=x/(x−1)` , x≠1, find fog and gof and hence find fog (2) and gof (−3).

Solve the differential equation ` (1 + x2) dy/dx+y=e^(tan^(−1))x.`

If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum, when the angle between them is 60º.

Using properties of determinants, prove that :

`|[1+a,1,1],[1,1+b,1],[1,1,1+c]|=abc + bc + ca + ab`

Find the distance between the planes 2x - y +  2z = 5 and 5x - 2.5y + 5z = 20

Find the shortest distance between the lines: 

`vecr = (hati+2hatj+hatk) + lambda(hati-hatj+hatk)` and `vecr = 2hati - hatj - hatk + mu(2hati + hatj + 2hatk)`

Find the shortest distance between the lines.

`(x + 1)/7 = (y + 1)/(- 6) = (z + 1)/1` and `(x - 3)/1 = (y - 5)/(- 2) = (z - 7)/1`.

Find the shortest distance between the lines whose vector equations are `vecr = (hati + 2hatj + 3hatk) + lambda(hati - 3hatj + 2hatk)` and `vecr = 4hati + 5hatj + 6hatk + mu(2hati + 3hatj + hatk)`.

Find the shortest distance between the lines whose vector equations are `vecr = (1-t)hati + (t - 2)hatj + (3 -2t)hatk` and `vecr = (s+1)hati + (2s + 1)hatk`.

Find the shortest distance between lines `vecr = 6hati + 2hatj + 2hatk + lambda(hati - 2hatj + 2hatk)` and `vecr =-4hati - hatk + mu(3hati - 2hatj - 2hatk)`.

Find `bb(dy/dx)` in the following:

`y = tan^(-1) ((3x -x^3)/(1 - 3x^2)), - 1/sqrt3 < x < 1/sqrt3`

Find `bb(dy/dx)` in the following:

y = `cos^(-1) ((1-x^2)/(1+x^2))`, 0 < x < 1

Find `bb(dy/dx)` in the following:

y = `sin^(-1) ((1-x^2)/(1+x^2))`, 0 < x < 1

Find `bb(dy/dx)` in the following:

y = `cos^(-1) ((2x)/(1+x^2))`, −1 < x < 1

Find `bb(dy/dx)` in the following:

y = `sin^(-1)(2xsqrt(1-x^2)), -1/sqrt2 < x < 1/sqrt2`

Find `bb(dy/dx)` in the following:

y = `sec^(-1) (1/(2x^2 - 1)), 0 < x < 1/sqrt2`

Differentiate the function with respect to x:

`cot^(-1) [(sqrt(1+sinx) + sqrt(1-sinx))/(sqrt(1+sinx) - sqrt(1-sinx))], 0 < x < pi/2`