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Mathematics 65/3/3 2018-2019 CBSE (Commerce) Class 12 Question Paper Solution

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Mathematics [65/3/3]
Marks: 100Academic Year: 2018-2019
Date & Time: 21st March 2019, 10:30 am
Duration: 2h30m

1 . All questions are compulsory. 

2 . Use of calaculators is not permitted. You may ask for logarithmic tables, if required.

3 . There is no overall choice. However, internal choice has been provided in 1 question of Section A, 3 questions of Section B, 3 questions of Section C and 3 questions of Section D. You have to attempt only one of the alternatives in all such questions. 


SECTION - A (4 Marks)
[1]1
[1]1.a

Find the direction cosines of the line joining the points P(4,3,-5) and Q(-2,1,-8) . 

Concept: Direction Cosines and Direction Ratios of a Line
Chapter: [4.01] Three - Dimensional Geometry
OR
[1]1.b

Find the value of p for which the following lines are perpendicular : 

`(1-x)/3 = (2y-14)/(2p) = (z-3)/2 ; (1-x)/(3p) = (y-5)/1 = (6-z)/5`

Concept: Equation of a Line in Space
Chapter: [4.01] Three - Dimensional Geometry
[1]2

Find the integerating factor of the differential equation `xdy/dx - 2y = 2x^2` . 

Concept: Methods of Solving First Order, First Degree Differential Equations - Linear Differential Equations
Chapter: [3.04] Differential Equations
[1]3

If A is a square matrix of order 3 with |A| = 4 , then the write the value of |-2A| . 

Concept: Types of Matrices
Chapter: [2.02] Matrices
[1]4

If `y = sin^-1 x + cos^-1 x , "find"  dy/dx`

Concept: Logarithmic Differentiation
Chapter: [3.01] Continuity and Differentiability
SECTION -B (16 Marks)
[2]5

If A = `[[3,9,0] ,[1,8,-2], [7,5,4]]` and B =`[[4,0,2],[7,1,4],[2,2,6]]` , then find the matrix `B'A'` .

Concept: Operations on Matrices - Multiplication of Matrices
Chapter: [2.02] Matrices
[2]6

Find : `∫_a^b logx/x` dx

Concept: Definite Integrals Problems
Chapter: [3.05] Integrals
[2]7

Form the differential equation representing the family of curves `y2 = m(a2 - x2) by eliminating the arbitrary constants 'm' and 'a'. 

Concept: Procedure to Form a Differential Equation that Will Represent a Given Family of Curves
Chapter: [3.04] Differential Equations
[2]8
[2]8.a

Find : 

`∫ sin(x-a)/sin(x+a)dx`

Concept: Methods of Integration - Integration Using Partial Fractions
Chapter: [3.05] Integrals
OR
[2]8.b

Find : 

`∫(log x)^2 dx`

Concept: Methods of Integration - Integration by Parts
Chapter: [3.05] Integrals
[2]9
[2]9.a

Find a unit vector perpendicular to both the vectors `veca and vecb` , where `veca = hat i - 7 hatj +7hatk`  and  `vecb = 3hati - 2hatj + 2hatk` . 

Concept: Product of Two Vectors - Vector (Or Cross) Product of Two Vectors
Chapter: [4.02] Vectors
OR
[2]9.b

Show that the vectors `hat (i) - 2 hat(j) + 3 hat (k), - 2 hat(i) + 3 hat(j) - 4 hat(k) " and " hat(i) - 3 hat(j) + 5 hat(k) ` are coplanar.

Concept: Scalar Triple Product of Vectors
Chapter: [4.02] Vectors
[2]10

Mother, father and son line up at random for a family photo. If A and B are two events given by
A = Son on one end, B = Father in the middle, find P(B / A).

Concept: Probability Examples and Solutions
Chapter: [6.01] Probability
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[2]11
[2]11.a

Let X be a random variable which assumes values  x1 , x2, x3 , x4 such that  2P (X = x1) = 3P (X = x2) = P (X = x3) = 5P (X = x4).  the probability distribution of X.

Concept: Random Variables and Its Probability Distributions
Chapter: [6.01] Probability
[2]11.b

A coin is tossed 5 times. Find the probability of getting (i) at least 4 heads, and (ii) at most 4  heads. 

Concept: Probability Examples and Solutions
Chapter: [6.01] Probability
[2]12

If * is defined on the set R of all real numbers by *: a*b = `sqrt(a^2 + b^2 ) `, find the identity elements, if it exists in R with respect to * .

Concept: Concept of Binary Operations
Chapter: [1.02] Relations and Functions
SECTION - C ( 44 marks )
[4]13

Find the value of x, if tan `[sec^(-1) (1/x) ] = sin ( tan^(-1) 2) , x > 0 `.

Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [1.01] Inverse Trigonometric Functions
[4]14
[4]14.a

If ey ( x +1)  = 1, then show that  `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`

Concept: Logarithmic Differentiation
Chapter: [3.01] Continuity and Differentiability
OR
[4]14.b

Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`

Concept: Logarithmic Differentiation
Chapter: [3.01] Continuity and Differentiability
[4]15

Find the intervals in which function f given by f(x)  = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .

Concept: Increasing and Decreasing Functions
Chapter: [3.02] Applications of Derivatives
[4]16
[4]16.a

Show that the relation R on the set Z of integers, given by R = {(a,b):2divides (a - b)} is an equivalence relation. 

Concept: Types of Relations
Chapter: [1.02] Relations and Functions
[4]16.b

If f (x)  = `(4x + 3)/(6x - 4) , x ≠ 2/3`, show that fof (x) = x for all ` x ≠ 2/3` . Also, find the inverse of f.

Concept: Types of Relations
Chapter: [1.02] Relations and Functions
[4]17

Using properties of determinants, prove that

`|[b+c , a ,a  ] ,[ b , a+c, b ] ,[c , c, a+b ]|` = 4abc 

Concept: Properties of Determinants
Chapter: [2.01] Determinants
[4]18

If y = (sec-1 x )2 , x > 0, show that 

`x^2 (x^2 - 1) (d^2 y)/(dx^2) + (2x^3 - x ) dy/dx -2 = 0`

Concept: Derivatives of Inverse Trigonometric Functions
Chapter: [3.01] Continuity and Differentiability
[4]19

Prove that `int _a^b f(x) dx = int_a^b f (a + b -x ) dx`  and hence evaluate   `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tan x))` .   

Concept: Properties of Definite Integrals
Chapter: [3.05] Integrals
[4]20

Find : ` int  (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`

Concept: Methods of Integration - Integration by Substitution
Chapter: [3.05] Integrals
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[4]21

Find the value of  λ for which the following lines are perpendicular to each other: 

`(x - 5)/(5 lambda + 2 ) = ( 2 - y )/5 = (1 - z ) /-1 ; x /1 = ( y + 1/2)/(2 lambda ) = ( z -1 ) / 3`

Concept: Equation of a Line in Space
Chapter: [4.01] Three - Dimensional Geometry
[4]22

Let `veca` , `vecb` and `vecc` be three vectors such that `|veca| = 1,|vecb| = 2, |vecc| = 3.` If the projection of `vecb` along `veca` is equal to the projection of `vecc` along `veca`; and `vecb` , `vecc` are perpendicular to each other, then find `|3veca - 2vecb + 2vecc|`.

Concept: Product of Two Vectors - Projection of a Vector on a Line
Chapter: [4.02] Vectors
[4]23
[4]23.a

Solve the differential equation:  ` (dy)/(dx) = (x + y )/ (x - y )`

Concept: Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations
Chapter: [3.04] Differential Equations
(OR)
[4]23.b

Solve the differential equation: (1 +x) dy + 2xy dx = cot x dx 

Concept: Methods of Solving First Order, First Degree Differential Equations - Linear Differential Equations
Chapter: [3.04] Differential Equations
SECTION D ( 36 marks )
[6]24
[6]24.a

Find the area of the region. 

{(x,y) : 0 ≤ y ≤ x, 0 ≤ y ≤ x + 2 ,-1 ≤ x ≤ 3} .

Concept: Area Under Simple Curves
Chapter: [3.03] Applications of the Integrals
[6]24.b

Evaluate `int_1^4 ( 1+ x +e^(2x)) dx` as limit of sums.

Concept: Definite Integral as the Limit of a Sum
Chapter: [3.05] Integrals
[6]25

Find the mean and variance of the random variable X which denotes the number of doublets in four throws of a pair of dice.

Concept: Bernoulli Trials and Binomial Distribution
Chapter: [6.01] Probability
[6]26
[6]26.a

Find the vector and cartesian equations of the plane passing throuh the points (2,5,- 3), (-2, - 3,5) and (5,3,-3). Also, find the point of intersection of this plane with the line passing through points (3, 1, 5) and (–1, –3, –1).

Concept: Vector and Cartesian Equation of a Plane
Chapter: [4.01] Three - Dimensional Geometry
[6]26.b

Find the equation of the plane passing through the intersection of the planes `vec(r) .(hat(i) + hat(j) + hat(k)) = 1"and" vec(r) . (2 hat(i) + 3hat(j) - hat(k)) +4 = 0 `and parallel to x-axis. Hence, find the distance of the plane from x-axis.

Concept: Vector and Cartesian Equation of a Plane
Chapter: [4.01] Three - Dimensional Geometry
[6]27

Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base. 

Concept: Maxima and Minima
Chapter: [3.02] Applications of Derivatives
[6]28
[6]28.a

If A = `[[1,1,1],[0,1,3],[1,-2,1]]` , find A-1Hence, solve the system of equations: 

x +y + z = 6

y + 3z = 11

and x -2y +z = 0

Concept: Applications of Determinants and Matrices
Chapter: [2.01] Determinants
[6]28.b

Find the inverse of the following matrix, using elementary transformations: 

`A= [[2 , 3 , 1 ],[2 , 4 , 1],[3 , 7 ,2]]`

Concept: Applications of Determinants and Matrices
Chapter: [2.01] Determinants
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[6]29

A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A
require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours and 20 minutes available  for cutting and 4 hours available for assembling. The profit is Rs. 50 each for type A and Rs. 60 each  for type B souvenirs. How many souvenirs of each type should the company manufacture in order to  maximize profit? Formulate the above LPP and solve it graphically and also find the maximum profit. 

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [5.01] Linear Programming
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