ISC (Science) ISC Class 12 - CISCE Important Questions for Mathematics

If y = x^x, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3`. Also find maximum volume in terms of volume of the sphere

Find : `int(x+3)sqrt(3-4x-x^2dx)`

Solve the differential equation:

(1 + y²) dx = (tan⁻¹ y − x) dy

The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves Rs 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?

If ` f(x)=|[a,-1,0],[ax,a,-1],[ax^2,ax,a]| ` , using properties of determinants find the value of f(2x) − f(x).

Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs 25, Rs 100 and Rs 50 each. The number of articles sold are given below:

Find the funds collected by each school separately by selling the above articles. Also find the total funds collected for the purpose.

Write one value generated by the above situation.

Using properties of determinants, prove that

`|[b+c , a ,a  ] ,[ b , a+c, b ] ,[c , c, a+b ]|` = 4abc 

If x = a sin 2t (1 + cos2t) and y = b cos 2t (1 – cos 2t), find the values of  `dy/dx `at t = `pi/4`

Determine the value of 'k' for which the following function is continuous at x = 3

`f(x) = {(((x + 3)^2 - 36)/(x - 3),  x != 3), (k,  x = 3):}`

Find `intsqrtx/sqrt(a^3-x^3)dx`

Integrate the following w.r.t. x `(x^3-3x+1)/sqrt(1-x^2)`

Evaluate :

`∫_(-pi)^pi (cos ax−sin bx)^2 dx`

Evaluate: `int_0^(2π) (1)/(1 + e^(sin x)`dx

Find the integrating factor of the differential equation.

`((e^(-2^sqrtx))/sqrtx-y/sqrtx)dy/dx=1`

If y = P e^ax + Q e^bx, show that

`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`

Solve the differential equation ` (1 + x2) dy/dx+y=e^(tan^(−1))x.`

A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards are drawn at random (without replacement) and are found to be all spades. Find the probability of the lost card being a spade.

The two vectors `hatj+hatk " and " 3hati-hatj+4hatk` represent the two sides AB and AC, respectively of a ∆ABC. Find the length of the median through A

Show that the vectors `veca, vecb` are coplanar if `veca+vecb, vecb+vecc ` are coplanar.