HSC Science (Computer Science) इयत्ता १२ वी - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Evaluate:

`int sin^3x cos^3x  dx`

The side of a square is increasing at the rate of 0.5 cm/sec. Find the rate of increase of the perimeter when the side of the square is 10 cm long.

Find the particular solution of the differential equation `x^2 dy/dx + y^2 = xy dy/dx`, if y = 1 when x = 1.

In ΔABC, a = 3, b = 1, cos(A – B) = `2/9`, find c.

Using truth table prove that:

~ (p `leftrightarrow` q) ≡ (p ∧ ~ q) ∨ (q ∧ ~ p)

Solve the differential equation

e^x tan y dx + (1 + e^x) sec² y dy = 0

Form the differential equation of all concentric circles having centre at the origin.

Show that the slopes of the lines represented by 3x² – 4xy + y² = 0 differ by 2.

If the angles A, B, C of a ΔABC are in A.P. and ∠A = 30°, c = 5, then find the values of ‘a’ and ‘b’.

Sketch the graph of the following inequation in XOY co-ordinate system.

x + y ≤ 0

Sketch the graph of the following inequation in XOY co-ordinate system.

2y - 5x ≥ 0

Find graphical solution for the following system of linear in equation:

x + 2y ≥ 4, 2x - y ≤ 6

A particle is moving along the X-axis. Its acceleration at time t is proportional to its velocity at that time. Find the differential equation of the motion of the particle.

Construct the truth table for the statement pattern:

[(p → q) ∧ q] → p

Solve the following system of equations by the method of reduction:

x + y + z = 6, y + 3z = 11, x + z = 2y.

Evaluate `int (1 + "x" + "x"^2/(2!))`dx

If `sin^-1(1-x) -2sin^-1x = pi/2` then x is

1. -1/2
2. 1
3. 0
4. 1/2

Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`

Integrate : sec³ x w. r. t. x.

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`