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

Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle

`x^2+y^2=4 at (1, sqrt3)`

A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at Rs 7 profit and  B at a profit of Rs 4. Find the production level per day for maximum profit graphically.

If f : R → R, f(x) = x³  and g: R → R , g(x) =  2x² + 1, and R is the set of real numbers, then find fog(x) and gof (x)

Solve the differential equation  `dy/dx = (x + y+2)/(2(x+y)-1)`

The index number by the method of aggregates for the year 2010, taking 2000 as the base year, was found to be 116. If sum of the prices in the year 2000 is ₹ 300, find the values of x and y in the data given below

Using the matrix method, solve the following system of linear equations:

`2/x + 3/y + 10/z` = 4, `4/x - 6/y + 5/z` = 1, `6/x + 9/y - 20/z` = 2.

 If x=a sin 2t(1+cos 2t) and y=b cos 2t(1−cos 2t), find `dy/dx `

If y = e^ax. cos bx, then prove that

`(d^2y)/(dx^2) - 2ady/dx + (a^2 + b^2)y` = 0

Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `cos^(-1)(1/sqrt3)`

Show that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is `(2R)/sqrt3.`  Also, find the maximum volume.

Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .

If logy = tan^–1 x, then show that `(1+x^2) (d^2y)/(dx^2) + (2x - 1) dy/dx = 0 .`

Evaluate `∫_0^(3/2)|x cosπx|dx`

Evaluate:

\[\int \cos^{-1} \left(\sin x \right) \text{dx}\]

Find the particular solution of the differential equation:

2y e^x/y dx + (y - 2x e^x/y) dy = 0 given that x = 0 when y = 1.

Prove by vector method, that the angle subtended on semicircle is a right angle.

If the direction cosines of a line are `(1/c, 1/c, 1/c)` then ______.

 Find the equation of the lines passing through the point (2, 1, 3) and perpendicular to the lines

A monopolist's demand function is `x = 60 - p/5`. At what level of output will marginal revenue be zero?

For 50 students of a class, the regression equation of marks in statistics (X) on the marks in accountancy (Y) is 3y – 5x + 180 = 0. The mean marks in accountancy is 44 and the variance of marks in statistics is `(9/16)^(th)` of the variance of marks in accountancy. Find the mean marks in statistics and the correlation coefficient between marks in the two subjects.