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

`int (cos4x)/(sin2x + cos2x)dx` = ______.

Write the negation of p ↔ q.

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’.

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

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

If the lines

`(x-1)/-3=(y-2)/(2k)=(z-3)/2 and (x-1)/(3k)=(y-5)/1=(z-6)/-5`

are at right angle then find the value of k

Find the shortest distance between the lines

`bar r = (4 hat i - hat j) + lambda(hat i + 2 hat j - 3 hat k)`

and

`bar r = (hat i - hat j + 2 hat k) + mu(hat i + 4 hat j -5 hat k)`

where λ and μ are parameters

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`

Solve the differential equation (x² + y²)dx- 2xydy = 0