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

The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?

Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle

`x^2+y^2=4 at (1, sqrt3)`

Find the particular solution of the differential equation `dy/dx=(xy)/(x^2+y^2)` given that y = 1, when x = 0.

If A is a square matrix, such that A²=A, then write the value of 7A−(I+A)³, where I is an identity matrix.

If y = P e^ax + Q e^bx, show that

`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`

Find the value(s) of x for which y = [x(x − 2)]² is an increasing function.

Find the particular solution of the differential equation dy/dx=1 + x + y + xy, given that y = 0 when x = 1.

Find the particular solution of the differential equation log(dy/dx)= 3x + 4y, given that y = 0 when x = 0.

Find the particular solution of the differential equation x (1 + y²) dx – y (1 + x²) dy = 0, given that y = 1 when x = 0.

If for any 2 x 2 square matrix A, `A("adj"  "A") = [(8,0), (0,8)]`, then write the value of |A|

Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`

Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R

Find `int dx/(5 - 8x - x^2)`

Show that the function f : R_* → R_* defined by f(x) = `1/x` is one-one and onto, where R_* is the set of all non-zero real numbers. Is the result true if the domain R_* is replaced by N, with the co-domain being the same as R?

Check the injectivity and surjectivity of the following function:

f : N → N given by f(x) = x²

Check the injectivity and surjectivity of the following function:

f : Z → Z given by f(x) = x²

Check the injectivity and surjectivity of the following function:

f : R → R given by f(x) = x²

Check the injectivity and surjectivity of the following function:

f : N → N given by f(x) = x³

Check the injectivity and surjectivity of the following function:

f : Z → Z given by f(x) = x³

Prove that the greatest integer function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.