HSC Arts (English Medium) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Find the derivative of 7^x w.r.t.x⁷

Suppose y = f(x) is differentiable function of x and y is one-one onto, `dy/dx ≠ 0`. Also, if x = f^–1(y) is differentiable, then prove that `dx/dy = 1/((dy/dx))`, where `dy/dx ≠ 0`

Hence, find `d/dx(tan^-1x)`.

Find the maximum and the minimum values of the function f(x) = x²e^x.

If the points (1, 1, λ) and (–3, 0, 1) are equidistant from the plane `barr*(3hati + 4hatj - 12hatk) + 13` = 0, find the value of λ.

The negation of (p → q) ∧ r is ______.

A right circular cylinder is to be made so that the sum of the radius and height is 6 metres. Find the maximum volume of the cylinder.

Find the equations of the planes parallel to the plane x – 2y + 2z – 4 = 0 which is a unit distance from the point (1, 2, 3).

If x + y = 8, then the maximum value of x²y is ______.

Divide the number 100 into two parts so that the sum of their squares is minimum.

Find the point on the curve y² = 4x, which is nearest to the point (2, 1).

If x = f(t) and y = g(t) are differentiable functions of t, so that y is function of x and `(dx)/dt ≠ 0` then prove that `dy/(dx) = (dy/dt)/((dx)/dt)`. Hence find `dy/(dx)`, if x = at², y = 2at.

A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?

Write converse, inverse and contrapositive of the following statement.

If x < y then x² < y² (x, y ∈ R)

Write converse, inverse and contrapositive of the following statement.

A family becomes literate if the woman in it is literate.

Write converse, inverse and contrapositive of the following statement.

If surface area decreases then pressure increases.

Write converse, inverse and contrapositive of the following statement. "If voltage increases then current decreases".

Find `bb((d^2y)/(dx^2))` of the following:

x = 2at², y = 4at

If y = `"e"^("m"tan^(-1)x`, show that `(1 + x^2) ("d"^2y)/("d"x^2) + (2x - "m")("d"y)/("d"x)` = 0

If y = cos(m cos^–1x), then show that `(1 - x^2) ("d"^2y)/("d"x^2) - x("d"y)/("d"x) + "m"^2y` = 0

If y = sin^–1x, then show that `(1 - x^2) (d^2y)/(dx^2) - x * dy/dx` = 0