HSC Arts (English Medium) १२ वीं कक्षा - Maharashtra State Board Important Questions

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

Find the point on the curve y² = 4x, which is nearest to the point (2, 1).

Find the approximate value of tan⁻¹ (1.002).
[Given: π = 3.1416]

A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?

Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`

Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`

Integrate : sec³ x w. r. t. x.

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:  `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`

Evaluate :`intxlogxdx`

Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`

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

(A)log(log x)+ c

(B) 1/2 (logx )²+c

(C) 2log x + c

(D) log x + c

Evaluate : `int x^2/((x^2+2)(2x^2+1))dx` 

Find : `int((2x-5)e^(2x))/(2x-3)^3dx`

Evaluate: `int sqrt(tanx)/(sinxcosx) dx`

Evaluate: `∫8/((x+2)(x^2+4))dx` 

Evaluate : `∫(x+1)/((x+2)(x+3))dx`

Evaluate : `∫1/(3+2sinx+cosx)dx`

Evaluate: `int 1/(x(x-1)) dx`

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`