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

If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:

(A) 0

(B) π

(C) π/2

(D) π/4

Show that `2sin^-1(3/5) = tan^-1(24/7)`

Using truth table examine whether the following statement pattern is tautology, contradiction or contingency `(p^^~q) harr (p->q)`

`int1/xlogxdx=...............`

(A)log(log x)+ c

(B) 1/2 (logx )²+c

(C) 2log x + c

(D) log x + c

Find the approximate value of ` sqrt8.95 `

If u and v are two functions of x then prove that

`intuvdx=uintvdx-int[du/dxintvdx]dx`

Hence evaluate, `int xe^xdx`

The time (in minutes) for a lab assistant to prepare the equipment for a certain experiment is a random variable taking values between 25 and 35 minutes with p.d.f 

`f(x) = {{:(1/10",", 25 ≤ x ≤ 35),(0",", "otherwise"):}`

What is the probability that preparation time exceeds 33 minutes? Also, find the c.d.f. of X.

Write the dual of the following statements: (p ∨ q) ∧ T

Show that:

`cos^(-1)(4/5)+cos^(-1)(12/13)=cos^(-1)(33/65)`

Find the area of the region lying between the parabolas y² = 4ax and x² = 4ay.

Find the approximate value of cos (60° 30').

(Given: 1° = 0.0175c, sin 60° = 0.8660)

Find the principal value of the following:

`tan^(-1) (-sqrt3)`

Prove that:

`tan^-1 ((sqrt(1 + x) - sqrt(1 - x))/(sqrt(1 + x) + sqrt(1 - x))) = pi/4 - 1/2 cos^-1 x`, for `- 1/sqrt2 ≤ x ≤ 1`

[Hint: Put x =  cos 2θ]

Find the principal value of `sin^-1(1/sqrt2)`

Find the approximate value of log₁₀ (1016), given that log₁₀e = 0⋅4343.

Prove that:

`int sqrt(x^2 + a^2)dx = x/2 sqrt(x^2 + a^2) + a^2/2 log |x + sqrt(x^2 + a^2)| + c`

Write the dual of the following statements:

Madhuri has curly hair and brown eyes.

The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1. By adding second and third number to five times the first number, we get 6. Find the three numbers by using matrices.

Write converse and inverse of the following statement: 
“If a man is a bachelor then he is unhappy.” 

If   p : It is raining
     q : It is humid

Write the following statements in symbolic form:

(a) It is raining or humid.
(b) If it is raining then it is humid.
(c) It is raining but not humid.