HSC Arts (English Medium) इयत्ता १२ वी - Maharashtra State Board Important Questions for Mathematics and Statistics

Find the condition that the line 4x + 5y = 0 coincides with one of the lines given by ax² + 2hxy + by² = 0 

Find the value of k if the lines represented by kx² + 4xy – 4y² = 0 are perpendicular to each other. 

Find the measure of the acute angle between the line represented by `3"x"^2 - 4sqrt3"xy" + 3"y"^2 = 0` 

If the foot of the perpendicular drawn from the origin to the plane is (4, −2, -5), then the equation of the plane is ______ 

Find the vector equation of the line passing through the point having position vector `4hat i - hat j + 2hat"k"` and parallel to the vector `-2hat i - hat j + hat k`.

Reduce the equation `bar"r"*(3hat"i" + 4hat"j" + 12hat"k")` = 8 to normal form

Find the Cartesian equation of the line passing through A(1, 2, 3) and B(2, 3, 4)

Find acute angle between the lines `(x - 1)/1 = (y - 2)/(-1) = (z - 3)/2` and `(x - 1)/2 = (y - 1)/1 = (z - 3)/1`

Find the cartesian equation of the plane passing through the point A(–1, 2, 3), the direction ratios of whose normal are 0, 2, 5.

Solve the following LPP by using graphical method.

Maximize : Z = 6x + 4y

Subject to x ≤ 2, x + y ≤  3, -2x + y ≤  1, x ≥  0, y ≥ 0.

Also find maximum value of Z.

Solve the following LPP by graphical method:

Maximize: z = 3x + 5y
Subject to: x + 4y ≤ 24
                  3x + y ≤ 21
                  x + y ≤ 9
                  x ≥ 0, y ≥ 0 

Also find the maximum value of z.

Solve the following L.P.P. by graphical method:

Minimize: z = 8x + 10y

Subject to: 2x + y ≥ 7, 2x + 3y ≥ 15, y ≥ 2, x ≥ 0, y ≥ 0.

If the corner points of the feasible solution are (0, 10), (2, 2) and (4, 0), then the point of minimum z = 3x + 2y is ______.

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

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

If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`

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). 

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.

The surface area of a spherical balloon is increasing at the rate of 2cm²/sec. At what rate the volume of the balloon is increasing when radius of the balloon is 6 cm?

A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.