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

Mathematics [65/1/1]
Date & Time: 21st March 2019, 10:30 am
Duration: 2h30m
• This question paper contains 29 questions divided into four sections A, B, C and D. Section A comprises of 4 questions of one mark each, Section B comprises of 8 questions of two marks each, Section C comprises of 11 questions of four marks each and Section D comprises of 6 questions of six marks each.
• All questions in Section A are to be answered in one word, one sentence, or as per the exact requirement of the question.
• 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
[1] 1

If A and B are square matrices of the same order 3, such that ∣A∣ = 2 and AB = 2I, write the value of ∣B∣.

Concept: Types of Matrices
Chapter: [2.02] Matrices
[1] 2

If f(x) = x + 1, find d/dx (fof) (x)

Concept: Concept of Differentiability
Chapter: [3.01] Continuity and Differentiability
[1] 3

Find the order and the degree of the differential equation x^2 (d^2y)/(dx^2) = { 1 + (dy/dx)^2}^4

Concept: Order and Degree of a Differential Equation
Chapter: [3.04] Differential Equations
[1] 4
[1] 4.1

If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.

Concept: Direction Cosines and Direction Ratios of a Line
Chapter: [4.01] Three - Dimensional Geometry
OR
[1] 4.2

Vector equation of a line which passes through a point (3, 4, 5) and parallels to the vector 2hati + 2hatj - 3hatk.

Concept: Vector and Cartesian Equation of a Plane
Chapter: [4.01] Three - Dimensional Geometry
SECTION - B
[2] 5

Examine whether the operation *defined on R by a * b = ab + 1 is (i) a binary or not. (ii) if a binary operation, is it associative or not?

Concept: Concept of Binary Operations
Chapter: [1.02] Relations and Functions
[2] 6

Find a matrix A such that 2A − 3B + 5C = 0, where B =[(-2, 2, 0), (3, 1, 4)] and  "C" = [(2, 0, -2),(7, 1, 6)].

Concept: Introduction of Operations on Matrices
Chapter: [2.02] Matrices
[2] 7

Find: int sec^2 x /sqrt(tan^2 x+4) dx.

Concept: Integration Using Trigonometric Identities
Chapter: [3.05] Integrals
[2] 8
[2] 8.1

Find: intsqrt(1 - sin 2x) dx, pi/4 < x < pi/2

Concept: Integration Using Trigonometric Identities
Chapter: [3.05] Integrals
OR
[2] 8.2

Find: int sin^-1 (2x) dx.

Concept: Integration Using Trigonometric Identities
Chapter: [3.05] Integrals
[2] 9

Form the differential equation representing the family of curves y = e2x (a + bx), where 'a' and 'b' are arbitrary constants.

Concept: Procedure to Form a Differential Equation that Will Represent a Given Family of Curves
Chapter: [3.04] Differential Equations
[2] 10
[2] 10.1

If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is sqrt(3).

Concept: Magnitude and Direction of a Vector
Chapter: [4.02] Vectors
OR
[2] 10.2

if  vec"a"= 2hat"i" + 3hat"j"+ hat"k", vec"b" = hat"i" -2hat"j" + hat"k" and vec"c" = -3hat"i" + hat"j" + 2hat"k", "find" [vec"a" vec"b" vec"c"]

Concept: Types of Matrices
Chapter: [2.02] Matrices
[2] 11

A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event "number obtained is even" and B be the event "number obtained is red". Find if A and B are independent events.

Concept: Independent Events
Chapter: [6.01] Probability
[2] 12
[2] 12.1

A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of
(i) 5 successes?
(ii) at least 5 successes?
(iii) at most 5 successes?

Concept: Probability Examples and Solutions
Chapter: [6.01] Probability
OR
[2] 12.2

The random variable X has probability distribution P(X) of the following form, where k is some number:

P(X = x) {(k, if x =0),(2k, if x =1),(3k, if x = 2),(0, "otherwise"):}

Determine the value of 'k'.

Concept: Random Variables and Its Probability Distributions
Chapter: [6.01] Probability
SECTION - C
[4] 13
[4] 13.1

Show that the relation R on R defined as R = {(a, b): a ≤ b}, is reflexive, and transitive but not symmetric.

Concept: Types of Relations
Chapter: [1.02] Relations and Functions
OR
[4] 13.2

Prove that the function f : N → N, defined by f(x) = x2 + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.

Concept: Increasing and Decreasing Functions
Chapter: [3.02] Applications of Derivatives
[4] 14

Solve: tan-1 4 x + tan-1 6x = π/(4).

Concept: Properties of Inverse Trigonometric Functions
Chapter: [1.01] Inverse Trigonometric Functions
[4] 15

Using properties of determinants, prove that

|(a^2 + 2a,2a + 1,1),(2a+1,a+2, 1),(3, 3, 1)| = (a - 1)^3

Concept: Properties of Determinants
Chapter: [2.01] Determinants
[4] 16
[4] 16.1

If log ("x"^2 + "y"^2) = 2 tan^-1 ("y"/"x"), "show that" (d"y")/(d"x") = ("x" + "y")/("x" - "y")

Concept: Derivatives of Inverse Trigonometric Functions
Chapter: [3.01] Continuity and Differentiability
OR
[4] 16.2

If xy - yx = ab, find (dy)/(dx).

Concept: Exponential and Logarithmic Functions
Chapter: [3.01] Continuity and Differentiability
[4] 17

If y = (sin^-1 x)^2, prove that (1-x^2) (d^2y)/dx^2 - x dy/dx -2 = 0.

Concept: Derivatives of Inverse Trigonometric Functions
Chapter: [3.01] Continuity and Differentiability
[4] 18

Find the equation of tangent to the curve y = sqrt(3x -2) which is parallel to the line 4x − 2y + 5 = 0. Also, write the equation of normal to the curve at the point of contact.

Concept: Tangents and Normals
Chapter: [3.02] Applications of Derivatives
[4] 19

Find: int (3x +5)/(x^2+3x-18)dx.

Concept: Indefinite Integral Problems
Chapter: [3.05] Integrals
[4] 20

Prove that int_0^"a" "f" ("x") "dx" = int_0^"a" "f" ("a" - "x") "d x", hence evaluate int_0^pi ("x" sin "x")/(1 + cos^2 "x") "dx"

Concept: Properties of Definite Integrals
Chapter: [3.05] Integrals
[4] 21
[4] 21.1

Solve the differential equation: x dy - y dx = sqrt(x^2 + y^2)dx, given that y = 0 when x = 1.

Concept: Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations
Chapter: [3.04] Differential Equations
OR
[4] 21.2

Solve the differential equation: (1 + x^2) dy/dx + 2xy - 4x^2 = 0, subject to the initial condition y(0) = 0.

Concept: Methods of Solving First Order, First Degree Differential Equations - Linear Differential Equations
Chapter: [3.04] Differential Equations
[4] 22

if hat"i" + hat"j" + hat"k", 2hat"i" + 5hat"j", 3hat"i" + 2 hat"j" - 3hat"k" and  hat"i" - 6hat"j" - hat"k" respectively are the position vectors A, B, C and D, then find the angle between the straight lines AB and CD. Find whether vec"AB" and vec"CD" are collinear or not.

Concept: Basic Concepts of Vector Algebra
Chapter: [4.02] Vectors
[4] 23

Find the value of λ, so that the lines (1-"x")/(3) = (7"y" -14)/(λ) = (z -3)/(2) and (7 -7"x")/(3λ) = ("y" - 5)/(1) = (6 -z)/(5) are at right angles. Also, find whether the lines are intersecting or not.

Concept: Equation of a Line in Space
Chapter: [4.01] Three - Dimensional Geometry
SECTION -D
[6] 24
[6] 24.1

If "A" = [(1,1,1),(1,0,2),(3,1,1)], find A-1. Hence, solve the system of equations x + y + z = 6, x + 2z = 7, 3x + y + z = 12.

Concept: Minors and Co-factors
Chapter: [2.01] Determinants
OR
[6] 24.2

Find the inverse of the following matrix using elementary operations.

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

Concept: Inverse of a Matrix - Inverse of a Nonsingular Matrix by Elementary Transformation
Chapter: [2.02] Matrices
[6] 25

A tank with rectangular base and rectangular sides, open at the top is to the constructed so that its depth is 2 m and volume is 8 m3. If building of tank cost 70 per square metre for the base and Rs 45 per square matre for sides, what is the cost of least expensive tank?

Concept: Graph of Maxima and Minima
Chapter: [3.02] Applications of Derivatives
[6] 26
[6] 26.1

Using integration, find the area of triangle ABC, whose vertices are A(2, 5), B(4, 7) and C(6, 2).

Concept: Area of a Triangle
Chapter: [2.01] Determinants
OR
[6] 26.2

Find the area of the region lying above x-axis and included between the circle x2 + y2 = 8x nd inside of the parabola y2 = 4x.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [3.03] Applications of the Integrals
[6] 27
[6] 27.1

Find the vector and Cartesian equations of the plane passing through the points (2, 2 –1), (3, 4, 2) and (7, 0, 6). Also find the vector equation of a plane passing through (4, 3, 1) and parallel to the plane obtained above.

Concept: Vector and Cartesian Equation of a Plane
Chapter: [4.01] Three - Dimensional Geometry
[6] 27.2

Find the vector equation of the plane that contains the lines vecr = (hat"i" + hat"j") + λ (hat"i" + 2hat"j" - hat"k") and the point (–1, 3, –4). Also, find the length of the perpendicular drawn from the point (2, 1, 4) to the plane thus obtained.

Concept: Vector and Cartesian Equation of a Plane
Chapter: [4.01] Three - Dimensional Geometry
[6] 28

A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that was produced by A?

Concept: Baye'S Theorem
Chapter: [6.01] Probability
[6] 29

A manufacturer has employed 5 skilled men and 10 semi-skilled men and makes two models A and B of an article. The making of one item of model A requires 2 hours of work by a skilled man and 2 hours work by a semi-skilled man. One item of model B requires 1 hour by a skilled man and 3 hours by a semi-skilled man. No man is expected to work more than 8 hours per day. The manufacturer's profit on an item of model A is ₹ 15 and on an item of model B is ₹ 10. How many items of each model should be made per day in order to maximize daily profit? Formulate the above LPP and solve it graphically and find the maximum profit.

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

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