HSC Science (Computer Science) इयत्ता १२ वी - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

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.

If the vectors `-3hati+4hatj-2hatk, hati+2hatk, hati-phatj` are coplanar, then the value of of p is

(A) -2

(B) 1

(C) -1

(D) 2

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

If the vectors `2hati-qhatj+3hatk and 4hati-5hatj+6hatk` are collinear, then value of q is

(A) 5

(B) 10

(C) 5/2

(D) 5/4

 Show that the points A (-7 , 4 , -2),B (-2 , 1 , 0)and C (3 ,-2 ,2) are collinear.

The lines `(x - 2)/(1) = (y - 3)/(1) = (z - 4)/(-k) and (x - 1)/k = (y - 4)/(2) = (z - 5)/(1)` are coplnar if ______.

The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.

Test whether the following functions are increasing or decreasing : f(x) = x³ – 6x² + 12x – 16, x ∈ R.

Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x² – x³, x ∈ R.

Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.