HSC Arts (English Medium) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

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

Express the following statement in symbolic form and write its truth value.

"If 4 is an odd number, then 6 is divisible by 3 "

Prove that the following statement pattern is equivalent :

(p ∨ q) →  r and (p → r) ∧ (q → r)

If `tan^-1((x-1)/(x-2))+cot^-1((x+2)/(x+1))=pi/4; `

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.