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

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

[0.033] Integrals
Chapter: [0.033] Integrals
Concept: Introduction of Integrals

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

[0.033] Integrals
Chapter: [0.033] Integrals
Concept: Introduction of Integrals

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. 

[0.033] Integrals
Chapter: [0.033] Integrals
Concept: Introduction of Integrals

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

[0.06] Three - Dimensional Geometry (Section B)
Chapter: [0.06] Three - Dimensional Geometry (Section B)
Concept: Direction Cosines and Direction Ratios of a Line

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

[0.033] Integrals
Chapter: [0.033] Integrals
Concept: Introduction of Integrals

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

[0.033] Integrals
Chapter: [0.033] Integrals
Concept: Introduction of Integrals

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

[0.033] Integrals
Chapter: [0.033] Integrals
Concept: Introduction of Integrals

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

[0.033] Integrals
Chapter: [0.033] Integrals
Concept: Introduction of Integrals

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. 

[0.1] Linear Programming (Section C)
Chapter: [0.1] Linear Programming (Section C)
Concept: Introduction of Linear Programming

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.

[0.06] Three - Dimensional Geometry (Section B)
Chapter: [0.06] Three - Dimensional Geometry (Section B)
Concept: Equation of a Line in Space

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)`

[0.07] Application of Integrals (Section B)
Chapter: [0.07] Application of Integrals (Section B)
Concept: Area Under Simple Curves
 

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.

 
[0.06] Three - Dimensional Geometry (Section B)
Chapter: [0.06] Three - Dimensional Geometry (Section B)
Concept: Equation of a Line in Space

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

[0.031] Continuity, Differentiability and Differentiation
Chapter: [0.031] Continuity, Differentiability and Differentiation
Concept: Concept of Continuity

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

[0.031] Continuity, Differentiability and Differentiation
Chapter: [0.031] Continuity, Differentiability and Differentiation
Concept: Concept of Continuity

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

[0.07] Application of Integrals (Section B)
Chapter: [0.07] Application of Integrals (Section B)
Concept: Area Under Simple Curves

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`

[0.034] Differential Equations
Chapter: [0.034] Differential Equations
Concept: Basic Concepts of Differential Equation

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

[0.034] Differential Equations
Chapter: [0.034] Differential Equations
Concept: Basic Concepts of Differential Equation

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.

[0.09] Linear Regression (Section C)
Chapter: [0.09] Linear Regression (Section C)
Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression

Solve the following initial value problem:-

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

[0.034] Differential Equations
Chapter: [0.034] Differential Equations
Concept: Basic Concepts of Differential Equation

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

[0.031] Continuity, Differentiability and Differentiation
Chapter: [0.031] Continuity, Differentiability and Differentiation
Concept: Concept of Continuity
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