HSC Science (Computer Science) इयत्ता १२ वी - Maharashtra State Board Question Bank Solutions

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.

Show that:

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

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`

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.

Find the feasible solution for the following system of linear inequations:
0 ≤ x ≤ 3, 0 ≤ y ≤ 3, x + y ≤ 5, 2x + y ≥ 4

. Show that the lines represented by 3x² - 4xy - 3y² = 0 are perpendicular to each other.

Show that the lines represented by x² + 6xy + 9y² = 0 are coincident.

Find the value of k if the lines represented by kx² + 4xy – 4y² = 0 are perpendicular to each other. 

Find the measure of the acute angle between the line represented by `3"x"^2 - 4sqrt3"xy" + 3"y"^2 = 0` 

Find the measure of the acute angle between the line represented by:

2x² + 7xy + 3y² = 0

Find the measure of the acute angle between the line represented by:

4x² + 5xy + y² = 0

Find the measure of the acute angle between the line represented by:

(a² - 3b²)x² + 8abxy + (b² - 3a²)y² = 0

Find the combined equation of lines passing through the origin each of which making an angle of 30° with the line 3x + 2y - 11 = 0 

If the angle between the lines represented by ax² + 2hxy + by² = 0 is equal to the angle between the lines 2x² - 5xy + 3y² = 0, then show that 100 (h² - ab) = (a + b)².