HSC Science (General) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

A man of height 1.5 meters walks towards a lamp post of height 4.5 meters, at the rate of `(3/4)` meter/sec. Find the rate at which (i) his shadow is shortening (ii) the tip of shadow is moving.

A ladder 10 metres long is leaning against a vertical wall. If the bottom of the ladder is pulled horizontally away from the wall at the rate of 1.2 metres per second, find how fast the top of the ladder is sliding down the wall, when the bottom is 6 metres away from the wall.

Choose the correct option from the given alternatives :

A ladder 5 m in length is resting against vertical wall. The bottom of the ladder is pulled along the ground away from the wall at the rate of `(1.5 "m")/sec`. The length of the higher point of ladder when the foot of the ladder is 4.0 m away from the wall decreases at the rate of

Solve the following:

A water tank in the farm of an inverted cone is being emptied at the rate of 2 cubic feet per second. The height of the cone is 8 feet and the radius is 4 feet. Find the rate of change of the water level when the depth is 6 feet.

Solve the following : Find all points on the ellipse 9x² + 16y² = 400, at which the y-coordinate is decreasing and the coordinate is increasing at the same rate.

Solve the following : The position of a particle is given by the function s (t) = 2t² + 3t – 4. Find the time t = c in the interval 0 ≤ t ≤ 4 when the instantaneous velocity of the particle equal to its average velocity in this interval.

Integrate the following w.r.t. x : x³ + x2 – x + 1

Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`

Integrate the following w.r.t. x:

`3 sec^2x - 4/x + 1/(xsqrt(x)) - 7`

Integrate the following w.r.t. x:

`2x^3 - 5x + 3/x + 4/x^5`

Integrate the following w.r.t. x : `(3x^3 - 2x + 5)/(xsqrt(x)`

Evaluate the following integrals : tan²x dx

Evaluate the following integrals : `int (sin2x)/(cosx)dx`

Evaluate the following integrals : `int sin x/cos^2x dx`

Evaluate the following integrals:

`int (cos2x)/sin^2x dx` 

Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`

Evaluate the following integrals : `int sinx/(1 + sinx)dx`

Evaluate the following integrals : `int tanx/(sec x + tan x)dx`

Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`

Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`