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

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Mathematics
All India Set 3
2013-2014 March
Marks: 100

[1]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
[1]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
[1]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
[1]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
[1]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
[1]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
[1]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
[1]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
[1]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
[1]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
[4]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
[4]12 | Attempt any one
[4]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
[4]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
[4]13 | Attempt any one
[4]13.1

Evaluate :

`∫_0^π(4x sin x)/(1+cos^2 x) dx`

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [3.05] Integrals
[4]13.2

Evaluate :

`∫(x+2)/sqrt(x^2+5x+6)dx`

Concept: Methods of Integration - Integration by Substitution
Chapter: [3.05] Integrals
[4]14 | Attempt any one
[4]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
[4]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
[4]15

If the function f : R → R be given by f[x] = x2 + 2 and g : R ‚Äč→ 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
[4]16 | Attempt any one
[4]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
[4]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
[4]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
[4]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
[4]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
[4]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
[4]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
[4]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
[6]23 | Attempt any one
[6]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
[6]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
[6]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
[6]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
[6]26 | Attempt any one
[6]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
[6]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
[6]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
[6]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
[6]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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