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

Solve the following differential equation:-

\[x \log x\frac{dy}{dx} + y = \frac{2}{x}\log x\]

Solve the following differential equation:-

(1 + x²) dy + 2xy dx = cot x dx

Solve the following differential equation:-

\[\left( x + y \right)\frac{dy}{dx} = 1\]

Solve the following differential equation:-

y dx + (x − y²) dy = 0

Solve the following differential equation:-

\[\left( x + 3 y^2 \right)\frac{dy}{dx} = y\]

Find a particular solution of the following differential equation:- \[\left( 1 + x^2 \right)\frac{dy}{dx} + 2xy = \frac{1}{1 + x^2}; y = 0,\text{ when }x = 1\]

Find a particular solution of the following differential equation:- (x + y) dy + (x − y) dx = 0; y = 1 when x = 1

Find a particular solution of the following differential equation:- x² dy + (xy + y²) dx = 0; y = 1 when x = 1

Find the equation of the curve passing through the point (1, 1) whose differential equation is x dy = (2x² + 1) dx, x ≠ 0.

Find the equation of a curve passing through the point (−2, 3), given that the slope of the tangent to the curve at any point (x, y) is `(2x)/y^2.`

Find the equation of a curve passing through the point (0, 0) and whose differential equation is \[\frac{dy}{dx} = e^x \sin x.\]

Find the equation of a curve passing through the point (0, 1). If the slope of the tangent to the curve at any point (x, y) is equal to the sum of the x-coordinate and the product of the x-coordinate and y-coordinate of that point.

If A is a square matrix of order 3 with |A| = 4 , then the write the value of |-2A| . 

Find the intervals in which function f given by f(x)  = 4x³ - 6x² - 72x + 30 is (a) strictly increasing, (b) strictly decresing .

Find the area of the region. 

{(x,y) : 0 ≤ y ≤ x² , 0 ≤ y ≤ x + 2 ,-1 ≤ x ≤ 3} .

If sin y = xsin(a + y) prove that `(dy)/(dx) = sin^2(a + y)/sin a`

If A = `[[0 , 2],[3, -4]]` and kA = `[[0 , 3"a"],[2"b", 24]]` then find the value of k,a and b.

If tan^-1 x - cot^-1 x = tan^-1 `(1/sqrt(3)),`x> 0 then find the value of x and hence find the value of sec^-1 `(2/x)`.

`"If y" = (sec^-1 "x")^2 , "x" > 0  "show that"  "x"^2 ("x"^2 - 1) (d^2"y")/(d"x"^2) + (2"x"^3 - "x") (d"y")/(d"x") - 2 = 0`

Find: ∫ sin x · log cos x dx