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

If y = 5^x. x⁵. x^x. 5⁵ , find `("d"y)/("d"x)`

If x⁷ . y⁵ = (x + y)¹², show that `("d"y)/("d"x) = y/x`

Find the acute angle between the lines x = y, z = 0 and x = 0, z = 0

Find acute angle between the lines `(x - 1)/1 = (y - 2)/(-1) = (z - 3)/2` and `(x - 1)/2 = (y - 1)/1 = (z - 3)/1`

In a Binomial distribution with n = 4, if 2P(X = 3) = 3P(X = 2), then value of p is ______.

If X ~ B(n, p) with n = 10, p = 0.4, then find E(X²).

Find the vector equation of the lines passing through the point having position vector `(-hati - hatj + 2hatk)` and parallel to the line `vecr = (hati + 2hatj + 3hatk) + λ(3hati + 2hatj + hatk)`.

Verify Lagrange’s mean value theorem for the function f(x) = `sqrt(x + 4)` on the interval [0, 5].

Find `dy/dx`, if y = (sin x)^{tan x} – x^{log x}.

If y = `log(x + sqrt(x^2 + 4))`, show that `dy/dx = 1/sqrt(x^2 + 4)`

Find the acute angle between the lines `(x - 1)/1 = (y - 2)/-1 = (z - 3)/2` and `barr = (hati + 2hatj + 3hatk) + λ(2hati + hatj + hatk)`.

If y = `9^(log_3x)`, find `dy/dx`.

Find the value of c for which the conclusion of the mean value theorem holds for the function f(x) = log x on the interval [1, 3]

Find `dy/dx`, if y = (log x)^x.

Evaluate:

`int log x dx`

Price P for demand D is given as P = 183 +120D - 3D² Find D for which the price is increasing

Show that the lines ` (x+1)/-3=(y-3)/2=(z+2)/1; ` are coplanar. Find the equation of the plane containing them.

Show that four points A, B, C and D whose position vectors are 

`4hati+5hatj+hatk,-hatj-hatk-hatk, 3hati+9hatj+4hatk and 4(-hati+hatj+hatk)` respectively are coplanar.

Test whether the function is increasing or decreasing. 

f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0, 

The function f (x) = x³ – 3x² + 3x – 100, x∈ R is _______.

(A) increasing

(B) decreasing

(C) increasing and decreasing

(D) neither increasing nor decreasing