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# Question Paper Solutions - Mathematics 2013 - 2014-CBSE 12th-Class 12 CBSE (Central Board of Secondary Education)

SubjectMathematics
Year2013 - 2014 (March)

#### Alternate Sets

Marks: 100
[1]1

If R=[(x, y) : x+2y=8] is a relation on N, write the range of R.

Chapter: [1] Relations and Functions (Section A)
Concept: Types of Relations
[1]2

If tan-1x+tan-1y=π/4,xy<1, then write the value of x+y+xy.

Chapter: [1.02] Inverse Trigonometric Functions
Concept: Inverse Trigonometric Functions (Simplification and Examples)
[1]3

If A is a square matrix, such that A2=A, then write the value of 7A(I+A)3, where I is an identity matrix.

Chapter: [11] Matrices
Concept: Types of Matrices
[1]4

If [[x-y,z],[2x-y,w]]=[[-1,4],[0,5]] find the value of x+y.

Chapter: [2.01] Matrices
Concept: Equality of Matrices
[1]5

If [[3x,7],[-2,4]]=[[8,7],[6,4]], find the value of x

Chapter: [2.01] Matrices
Concept: Introduction of Operations on Matrices
[1]6

If f(x) =∫_0^xt sin t dt , then write the value of f ' (x).

Chapter: [3.03] Integrals
Concept: Integration as an Inverse Process of Differentiation
[1]7

find ∫_2^4 x/(x^2 + 1)dx

Chapter: [3.03] Integrals
Concept: Evaluation of Definite Integrals by Substitution
[1]8

Find the value of 'p' for which the vectors 3hati+2hatj+9hatk and hati-2phatj+3hatk are parallel

Chapter: [8] Vectors
Concept: Basic Concepts of Vector Algebra
[1]9

Find veca.(vecbxxvecc), " if " veca=2hati+hatj+3hatk, vecb=-hati+2hatj+hatk  " and " vecc=3hati+hatj+2hatk

Chapter: [4.01] Vectors
Concept: Vectors Examples and Solutions
[1]10

If the Cartesian equations of a line are  (3-x)/5=(y+4)/7=(2z-6)/4 , write the vector equation for the line.

Chapter: [6] Three - Dimensional Geometry (Section B)
Concept: Equation of a Line in Space
[4]11

If the function f : R → R be given by f[x] = x2 + 2 and g : R ​→ R be given by  g(x)=x/(x−1) , x1, find fog and gof and hence find fog (2) and gof (−3).

Chapter: [1.01] Relations and Functions
Concept: Inverse of a Function
[4]12 | Attempt any one
[4]12.1

Prove that

tan^(-1) [(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]=pi/4-1/2 cos^(-1)x,-1/sqrt2<=x<=1

Chapter: [1.02] Inverse Trigonometric Functions
Concept: Inverse Trigonometric Functions (Simplification and Examples)
[4]12.2

If tan^(-1)((x-2)/(x-4)) +tan^(-1)((x+2)/(x+4))=pi/4 ,find the value of x

Chapter: [1.02] Inverse Trigonometric Functions
Concept: Inverse Trigonometric Functions (Simplification and Examples)
[4]13

Using properties of determinants, prove that

|[x+y,x,x],[5x+4y,4x,2x],[10x+8y,8x,3x]|=x^3

Chapter: [2.02] Determinants
Concept: Properties of Determinants
[4]14

Find the value of dy/dx " at " theta =pi/4 if x=ae^theta (sintheta-costheta) and y=ae^theta(sintheta+cos theta)

Chapter: [4] Differentiation
Concept: Derivatives of Functions in Parametric Forms
[4]15

If y = P eax + Q ebx, show that

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

Chapter: [3.05] Differential Equations
Concept: General and Particular Solutions of a Differential Equation
[4]16
[4]16.1

Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.

Chapter: [5] Applications of Derivative
Concept: Increasing and Decreasing Functions
[4]16.2

Find the equations of the tangent and normal to the curve x^2/a^2−y^2/b^2=1 at the point (sqrt2a,b) .

Chapter: [14] Applications of Derivative
Concept: Tangents and Normals
[4]17
[4]17.1

Evaluate :

∫_0^π(4x sin x)/(1+cos^2 x) dx

Chapter: [3.03] Integrals
Concept: Evaluation of Definite Integrals by Substitution
[4]17.2

Evaluate :

∫(x+2)/sqrt(x^2+5x+6)dx

Chapter: [3.03] Integrals
Concept: Methods of Integration - Integration by Substitution
[4]18

Find the particular solution of the differential equation dy/dx=1 + x + y + xy, given that y = 0 when x = 1.

Chapter: [3.05] Differential Equations
Concept: General and Particular Solutions of a Differential Equation
[4]19

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

Chapter: [3.05] Differential Equations
Concept: Solutions of Linear Differential Equation
[4]20
[4]20.1

Show that four points A, B, C and D whose position vectors are

4hati+5hatj+hatk,-hatj-hatk-hatk, 3hati+9hatj+4hatk and 4(-hati+hatj+hatk) respectively are coplanar.

Chapter: [4.02] Three - Dimensional Geometry
Concept: Coplanarity of Two Lines
[4]20.2

The scalar product of the vector veca=hati+hatj+hatk with a unit vector along the sum of vectors vecb=2hati+4hatj−5hatk and vecc=λhati+2hatj+3hatk is equal to one. Find the value of λ and hence, find the unit vector along vecb +vecc

Chapter: [5] Vectors (Section B)
Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
[4]21

A line passes through (2, −1, 3) and is perpendicular to the lines vecr=(hati+hatj-hatk)+lambda(2hati-2hatj+hatk) and vecr=(2hati-hatj-3hatk)+mu(hati+2hatj+2hatk) . Obtain its equation in vector and Cartesian from.

Chapter: [6] Three - Dimensional Geometry (Section B)
Concept: Equation of a Line in Space
[4]22

An experiment succeeds thrice as often as it fails. Find the probability that in the next five trials, there will be at least 3 successes.

Chapter: [6.01] Probability
Concept: Probability Examples and Solutions
[6]23

Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. School A wants to award Rs x each, Rs y each and Rs z each for the three respective values to 3, 2 and 1 students, respectively with a total award money of Rs 1,600. School B wants to spend Rs 2,300 to award 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is Rs 900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for an award.

Chapter: [2.02] Determinants
Concept: Adjoint and Inverse of a Matrix
[6]24

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

Chapter: [14] Applications of Derivative
Concept: Maxima and Minima
[6]25

Evaluate:

∫1/(cos^4x+sin^4x)dx

Chapter: [3.03] Integrals
Concept: Methods of Integration - Integration by Substitution
[6]26

Using integration, find the area of the region bounded by the triangle whose vertices are (−1, 2), (1, 5) and (3, 4).

Chapter: [16] Applications of Definite Integral
Concept: Area Between Two Curves
[6]27 | Attempt any one
[6]27.1

Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x − y + z = 0. Also find the distance of the plane, obtained above, from the origin.

Chapter: [10] Plane
Concept: Distance of a Point from a Plane
[6]27.2

Find the distance of the point (2, 12, 5) from the point of intersection of the line

vecr=2hati-4hat+2hatk+lambda(3hati+4hatj+2hatk)

Chapter: [4.02] Three - Dimensional Geometry
Concept: Three - Dimensional Geometry Examples and Solutions
[6]28

A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30, respectively. The company makes a profit of Rs 80 on each piece of type A and Rs 120 on each piece of type B. How many pieces of type A and type B should be manufactured per week to get maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week?

Chapter: [11] Linear Programming Problems
Concept: Graphical Method of Solving Linear Programming Problems
[6]29 | Attempt any one
[6]29.1

There are three coins. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the times and the third is also a biased coin that comes up tails 40% of the time. One of the three coins is chosen at random and tossed and it shows heads. What is the probability that it was the two-headed coin?

Chapter: [22] Probability
Concept: Baye'S Theorem
[6]29.2

Two the numbers are selected at random (without replacement) from first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of X. Find the mean and variance of this distribution.

Chapter: [4] Probability (Section A)
Concept: Mean of a Random Variable
S