# Mathematics 65/3/1 2018-2019 CBSE (Commerce) Class 12 Question Paper Solution

Mathematics [65/3/1]
Date & Time: 21st March 2019, 10:30 am
Duration: 2h30m

SECTION-A
 1

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

Concept: Types of Matrices
Chapter: [0.03] Matrices
 2

If y = sin-1 x + cos-1x find  (dy)/(dx).

Concept: Derivatives of Inverse Trigonometric Functions
Chapter: [0.05] Continuity and Differentiability
 3

Write the order and degree of the differential equation ((d^4"y")/(d"x"^4))^2 =  [ "x" + ((d"y")/(d"x"))^2]^3.

Concept: Order and Degree of a Differential Equation
Chapter: [0.09] Differential Equations
 4
 4.1

If a line has the direction ratios −18, 12, −4, then what are its direction cosines?

Concept: Direction Cosines and Direction Ratios of a Line
Chapter: [0.11] Three - Dimensional Geometry
OR
 4.2

Find the Cartesian equation of the line which passes through the point (−2, 4, −5) and is parallel to the line (x+3)/3=(4-y)/5=(z+8)/6

Concept: Equation of a Line in Space
Chapter: [0.11] Three - Dimensional Geometry
Section B
 5

If * is defined on the set R of all real number by *: a * b = sqrt(a^2 + b^2) find the identity element if exist in R with respect to *

Concept: Concept of Binary Operations
Chapter: [0.01] Relations and Functions
 6

If A = [[0 , 2],[3, -4]] and kA = [[0 , 3"a"],[2"b", 24]] then find the value of k,a and b.

Concept: Types of Matrices
Chapter: [0.03] Matrices
 7

Find int_  (sin "x" - cos "x" )/sqrt(1 + sin 2"x") d"x", 0 < "x" < π / 2

Concept: Integration Using Trigonometric Identities
Chapter: [0.07] Integrals
 8
 8.1

Find int_  sin ("x" - a)/(sin ("x" + a )) d"x"

Concept: Integration Using Trigonometric Identities
Chapter: [0.07] Integrals
OR
 8.2

Find int_  (log "x")^2 d"x"

Concept: Integration Using Trigonometric Identities
Chapter: [0.07] Integrals
 9

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: [0.09] Differential Equations
 10
 10.1

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: [0.1] Vectors
OR
 10.2

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: [0.1] Vectors
 11

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: [0.13] Probability
 12
 12.1

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: [0.13] Probability
OR
 12.2

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: [0.13] Probability
SECTION – C
 13
 13.1

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

Concept: Types of Relations
Chapter: [0.01] Relations and Functions
OR
 13.2

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: [0.01] Relations and Functions
 14

If tan-1 x - cot-1 x = tan-1 (1/sqrt(3)),x> 0 then find the value of x and hence find the value of sec-1 (2/x).

Concept: Properties of Inverse Trigonometric Functions
Chapter: [0.02] Inverse Trigonometric Functions
 15

Using properties of determinants, prove that

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

Concept: Properties of Determinants
Chapter: [0.04] Determinants
 16
 16.1

If sin y = xsin(a + y) prove that (dy)/(dx) = sin^2(a + y)/sin a

Concept: Concept of Differentiability
Chapter: [0.05] Continuity and Differentiability
OR
 16.2

If (sin "x")^"y" = "x" + "y", "find" (d"y")/(d"x")

Concept: Logarithmic Differentiation
Chapter: [0.05] Continuity and Differentiability
 17

"If y" = (sec^-1 "x")^2 , "x" > 0  "show that"  "x"^2 ("x"^2 - 1) (d^2"y")/(d"x"^2) + (2"x"^3 - "x") (d"y")/(d"x") - 2 = 0

Concept: Concept of Differentiability
Chapter: [0.05] Continuity and Differentiability
 18

Find the equation of a tangent and the normal to the curve "y" = (("x" - 7))/(("x"-2)("x"-3) at the point where it cuts the x-axis

Concept: Tangents and Normals
Chapter: [0.06] Applications of Derivatives
 19

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

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.07] Integrals
 20

Prove that int_a^b ƒ ("x") d"x" = int_a^bƒ(a + b - "x") d"x" and "hence evaluate" int_(π/6)^(π/3) (d"x")/(1+sqrt(tan "x")

Concept: Definite Integrals Problems
Chapter: [0.07] Integrals
 21
 21.1

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

Concept: Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations
Chapter: [0.09] Differential Equations
OR
 21.2

Solve the differential equation: (1 + x2) dy + 2xy dx = cot x dx

Concept: Solutions of Linear Differential Equation
Chapter: [0.09] Differential Equations
 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: [0.1] Vectors
 23

Find the value of λ for which the following lines are perpendicular to each other ("x"-5)/(5λ+2) = (2 -"y")/(5) = (1 -"z")/(-1); ("x")/(1) = ("y"+1/2)/(2λ) = ("z" -1)/(3)

hence, find whether the lines intersect or not

Concept: Plane - Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point
Chapter: [0.11] Three - Dimensional Geometry
SECTION – D
 24
 24.1

If A = [(1, 1, 1),(0, 1, 3),(1, -2, 1)],find A-1

hence, solve the following system of equations

x + y + z = 6
y + 3z =11
x- 2y + z = 0

Concept: Inverse of Matrix - Inverse of a Matrix by Elementary Transformation
Chapter: [0.03] Matrices
OR
 24.2

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: [0.04] Determinants
 25

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: Simple Problems on Applications of Derivatives
Chapter: [0.06] Applications of Derivatives
 26
 26.1

Find the area of the triangle whose vertices are (-1, 1), (0, 5) and (3, 2), using integration.

Concept: Integration Using Trigonometric Identities
Chapter: [0.07] Integrals
OR
 26.2

Find the area of the region bounded by the curves (x -1)2 + y2 = 1 and x2 + y2 = 1, using integration.

Concept: Procedure to Form a Differential Equation that Will Represent a Given Family of Curves
Chapter: [0.09] Differential Equations
 27
 27.1

Find the vector and cartesian equation of the plane passing through the point (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: [0.11] Three - Dimensional Geometry
OR
 27.2

Find the equation of the plane passing through the intersection of the planes vecr . (hati + hatj + hatk) and vecr.(2hati + 3hatj - hatk) + 4 = 0` and parallel to the x-axis. Hence, find the distance of the plane from the x-axis.

Concept: Plane - Plane Passing Through the Intersection of Two Given Planes
Chapter: [0.11] Three - Dimensional Geometry
 28

There are two boxes I and II. Box I contains 3 red and 6 Black balls. Box II contains 5 red and black balls. One of the two boxes, box I and box II is selected at random and a ball is drawn at random. The ball drawn is found to be red. If the probability that this red ball comes out from box II is ' a find the value of n

Concept: Probability Examples and Solutions
Chapter: [0.13] Probability
 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: [0.12] Linear Programming

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