ISC (Science) Class 12 - CISCE Question Bank Solutions for Mathematics

 Evaluate :  `int sec^2x/(cosec^2x)dx`

Evaluate: `int_0^x (xtan x)/(sec x + tan x) dx`

Given three identical Boxes A, B and C, Box A contains 2 gold and 1 silver coin, Box B contains 1 gold and 2 silver coins and Box C contains 3 silver coins. A person choose a Box at random and takes out a coin. If the coin drawn is of silver, find the probability that it has been drawn from the Box which has the remaining two coins also of silver. 

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

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

Evaluate: `int_0^(pi/4) "log" (1 + "tan" theta) "d" theta`

Evaluate: ` int tan^3x "dx"`

Using De Moivre’s theorem, find the least positive integer n such that `((2i)/(1+i))^n`  is a positive integer.

A company manufactures two types of toys A and B. A toy of type A requires 5 minutes for cutting and 10 minutes for assembling. A toy of type B requires 8 minutes for cutting and 8 minutes for assembling. There are 3 hours available for cutting and 4 hours available for assembling the toys in a day. The profit is ₹ 50 each on a toy of type A and ₹ 60 each on a toy of type B. How many toys of each type should the company manufacture in a day to maximize the profit? Use linear programming to find the solution. 

Find the coordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XZ plane. Also find the angle which this line makes with the XZ plane.

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)`

Find the value of p, so that the lines `l_1:(1-x)/3=(7y-14)/p=(z-3)/2 and l_2=(7-7x)/3p=(y-5)/1=(6-z)/5 ` are perpendicular to each other. Also find the equations of a line passing through a point (3, 2, – 4) and parallel to line l₁.

Find the value of constant ‘k’ so that the function f (x) defined as

f(x) = `{((x^2 -2x-3)/(x+1), x != -1),(k, x != -1):}`

is continous at x = -1

Show that the function f(x) = `{(x^2, x<=1),(1/2, x>1):}` is continuous at x = 1 but not differentiable.

Find the equation of an ellipse whose latus rectum is 8 and eccentricity is `1/3`

If 1, `omega` and `omega^2` are the cube roots of unity, prove `(a + b omega + c omega^2)/(c + s omega +  b omega^2) =  omega^2`

Solve the equation for x: `sin^(-1)  5/x + sin^(-1)  12/x = pi/2, x != 0`

Given that the observations are: (9, -4), (10, -3), (11, -1), (12, 0), (13, 1), (14, 3), (15, 5), (16, 8). Find the two lines of regression and estimate the value of y when x = 13·5.

Solve the following initial value problem:-

\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]

 Show that the function `f(x) = |x-4|, x ∈ R` is continuous, but not diffrent at x = 4.