HSC Science (Electronics) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

If y = f(u) is a differentiable function of u and u = g(x) is a differentiate function of x such that the composite function y = f[g(x)] is a differentiable function of x then prove that

`dy/dx = dy/(du) xx (du)/dx`

Hence find `dy/dx` if y = log(x² + 5)

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

Find the area of the region bounded by the curve y = x², and the lines x = 1, x = 2, and y = 0.

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

Examine whether the following logical statement pattern is a tautology, contradiction, or contingency.

[(p→q) ∧ q]→p

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

Write the converse and contrapositive of the statement -
“If two triangles are congruent, then their areas are equal.”

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`