HSC Science (Electronics) इयत्ता १२ वी - Maharashtra State Board Important Questions for Mathematics and Statistics

 If x=a sin 2t(1+cos 2t) and y=b cos 2t(1−cos 2t), find `dy/dx `

If y = e^ax. cos bx, then prove that

`(d^2y)/(dx^2) - 2ady/dx + (a^2 + b^2)y` = 0

If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x such that the composite function y = f[g(x)] is a differentiable function of x, then `("d"y)/("d"x) = ("d"y)/("d"u)*("d"u)/("d"x)`. Hence find `("d"y)/("d"x)` if y = sin²x

Examine the maxima and minima of the function f(x) = 2x³ - 21x² + 36x - 20 . Also, find the maximum and minimum values of f(x). 

A point source of light is hung 30 feet directly above a straight horizontal path on which a man of 6 feet in height is walking. How fast will the man’s shadow lengthen and how fast will the tip of shadow move when he is walking away from the light at the rate of 100 ft/min.

Show that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is `(2R)/sqrt3.`  Also, find the maximum volume.

Find the values of x, for which the function f(x) = x³ + 12x² + 36𝑥 + 6 is monotonically decreasing

Prove that:

`int sqrt(x^2 - a^2)dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c`

Evaluate `∫_0^(3/2)|x cosπx|dx`

`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`

`int sec^2x sqrt(tan^2x + tanx - 7)  "d"x`

`int (3x + 4)/sqrt(2x^2 + 2x + 1)  "d"x`

Find the area bounded by the curve y² = 4ax, x-axis and the lines x = 0 and x = a.

Find the area of the region lying between the parabolas y² = 4ax and x² = 4ay.

Find the area of the region lying between the parabolas 4y² = 9x and 3x² = 16y

Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`

Order and degree of the differential equation `[1+(dy/dx)^3]^(7/3)=7(d^2y)/(dx^2)` are respectively 

(A) 2, 3

(B) 3, 2

(C) 7, 2

(D) 3, 7

Prove that:

`int_0^(2a)f(x)dx = int_0^af(x)dx + int_0^af(2a - x)dx`

Solve the differential equation `y - x dy/dx = 0`

Solve the differential equation `cos^2 x dy/dx` + y = tan x