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

If x + y = 8, then the maximum value of x²y is ______.

Divide the number 100 into two parts so that the sum of their squares is minimum.

Find the point on the curve y² = 4x, which is nearest to the point (2, 1).

If x = f(t) and y = g(t) are differentiable functions of t, so that y is function of x and `(dx)/dt ≠ 0` then prove that `dy/(dx) = (dy/dt)/((dx)/dt)`. Hence find `dy/(dx)`, if x = at², y = 2at.

A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?

Write converse, inverse and contrapositive of the following statement.

If x < y then x² < y² (x, y ∈ R)

Write converse, inverse and contrapositive of the following statement.

A family becomes literate if the woman in it is literate.

Write converse, inverse and contrapositive of the following statement.

If surface area decreases then pressure increases.

Write converse, inverse and contrapositive of the following statement. "If voltage increases then current decreases".

Find `bb((d^2y)/(dx^2))` of the following:

x = 2at², y = 4at

If y = `"e"^("m"tan^(-1)x`, show that `(1 + x^2) ("d"^2y)/("d"x^2) + (2x - "m")("d"y)/("d"x)` = 0

If y = cos(m cos^–1x), then show that `(1 - x^2) ("d"^2y)/("d"x^2) - x("d"y)/("d"x) + "m"^2y` = 0

If y = sin^–1x, then show that `(1 - x^2) (d^2y)/(dx^2) - x * dy/dx` = 0

If x = cos θ and y = sin³θ, show that `y(d^2y)/(dx^2) + (dy/dx)^2` = 3sin²θ(5cos²θ – 1)

The negation of p ∧ (q → r) is ______________.

Without using truth tabic show that ~(p v q)v(~p ∧ q) = ~p

Without using the truth table show that P ↔ q ≡ (p ∧ q) ∨ (~ p ∧ ~ q)

If A = {2, 3, 4, 5, 6}, then which of the following is not true?

(A) ∃ x ∈ A such that x + 3 = 8

(B) ∃ x ∈ A such that x + 2 < 5

(C) ∃ x ∈ A such that x + 2 < 9

(D) ∀ x ∈ A such that x + 6 ≥ 9

Rewrite the following statement without using if ...... then.

If a man is a judge then he is honest.

Rewrite the following statement without using if ...... then.

It 2 is a rational number then `sqrt2` is irrational number.