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# Mathematics All India Set 3 2013-2014 CBSE (Commerce) Class 12 Question Paper Solution

SubjectMathematics
Year2013 - 2014 (March)
Mathematics
All India Set 3
2013-2014 March
Marks: 100

1

If A is a square matrix such that A2 = I, then find the simplified value of (A – I)3 + (A + I)3 – 7A.

Concept: Types of Matrices
Chapter: [2.02] Matrices
2

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

Concept: Equality of Matrices
Chapter: [2.02] Matrices
3

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

Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [1.01] Inverse Trigonometric Functions
4

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

Concept: Introduction of Operations on Matrices
Chapter: [2.02] Matrices
5

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

Concept: Integration as an Inverse Process of Differentiation
Chapter: [3.05] Integrals
6

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

Concept: Basic Concepts of Vector Algebra
Chapter: [4.02] Vectors
7

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

Concept: Types of Relations
Chapter: [1.02] Relations and Functions
8

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

Concept: Equation of a Line in Space
Chapter: [4.01] Three - Dimensional Geometry
9

If int_0^a1/(4+x^2)dx=pi/8 , find the value of a.

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [3.05] Integrals
10

If veca and vecb are perpendicular vectors, |veca+vecb| = 13 and |veca| = 5 ,find the value of |vecb|.

Concept: Introduction of Product of Two Vectors
Chapter: [4.02] Vectors
11

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

Concept: Solutions of Linear Differential Equation
Chapter: [3.04] Differential Equations
12 | Attempt any one
12.1

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.

Concept: Coplanarity of Two Lines
Chapter: [4.01] Three - Dimensional Geometry
12.2

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

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Chapter: [4.02] Vectors
13 | Attempt any one
13.1

Evaluate :

∫_0^π(4x sin x)/(1+cos^2 x) dx

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [3.05] Integrals
13.2

Evaluate :

∫(x+2)/sqrt(x^2+5x+6)dx

Concept: Methods of Integration - Integration by Substitution
Chapter: [3.05] Integrals
14 | Attempt any one
14.1

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

Concept: Increasing and Decreasing Functions
Chapter: [3.02] Applications of Derivatives
14.2

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) .

Concept: Tangents and Normals
Chapter: [3.02] Applications of Derivatives
15

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).

Concept: Inverse of a Function
Chapter: [1.02] Relations and Functions
16 | Attempt any one
16.1

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

Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [1.01] Inverse Trigonometric Functions
16.2

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

Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [1.01] Inverse Trigonometric Functions
17

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.

Concept: Probability Examples and Solutions
Chapter: [6.01] Probability
18

If y = P eax + Q ebx, show that

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

Concept: General and Particular Solutions of a Differential Equation
Chapter: [3.04] Differential Equations
19

Using properties of determinants, prove that :

|[1+a,1,1],[1,1+b,1],[1,1,1+c]|=abc + bc + ca + ab

Concept: Elementary Transformations
Chapter: [2.01] Determinants
20

If x = cos t (3 – 2 cos2 t) and y = sin t (3 – 2 sin2 t), find the value of dx/dy at t =4/π.

Concept: Derivatives of Functions in Parametric Forms
Chapter: [3.01] Continuity and Differentiability
21

Find the particular solution of the differential equation log(dy/dx)= 3x + 4y, given that y = 0 when x = 0.

Concept: General and Particular Solutions of a Differential Equation
Chapter: [3.04] Differential Equations
22

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.

Concept: Equation of a Line in Space
Chapter: [4.01] Three - Dimensional Geometry
23 | Attempt any one
23.1

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.

Concept: Distance of a Point from a Plane
Chapter: [4.01] Three - Dimensional Geometry
23.2

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)

Concept: Three - Dimensional Geometry Examples and Solutions
Chapter: [4.01] Three - Dimensional Geometry
24

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

Concept: Area Between Two Curves
Chapter: [3.03] Applications of the Integrals
25

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?

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [5.01] Linear Programming
26 | Attempt any one
26.1

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?

Concept: Baye'S Theorem
Chapter: [6.01] Probability
26.2

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.

Concept: Mean of a Random Variable
Chapter: [6.01] Probability
27

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.

Concept: Adjoint and Inverse of a Matrix
Chapter: [2.01] Determinants
28

If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum, when the angle between them is 60º.

Concept: Simple Problems on Applications of Derivatives
Chapter: [3.02] Applications of Derivatives
29

Evaluate :

int1/(sin^4x+sin^2xcos^2x+cos^4x)dx

Concept: Methods of Integration - Integration by Substitution
Chapter: [3.05] Integrals

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