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Mathematics 65/4/3 2018-2019 CBSE (Arts) Class 12 Question Paper Solution

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Mathematics [65/4/3]
Marks: 100 Academic Year: 2018-2019
Date & Time: 21st March 2019, 10:30 am
Duration: 2h30m
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(i) All questions are compulsory.

(ii) 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.

(iii) All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.

(iv) 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.

(v) Use of calculators is not permitted. You may ask logarithmic tables, if required.


[1] 1
[1] 1.1

Find the acute angle between the planes `vec"r". (hat"i" - 2hat"j" - 2hat"k") = 1` and `vec"r". (3hat"i" - 6hat"j" - 2hat "k") = 0`

Concept: Concept of Direction Cosines
Chapter: [4.02] Vectors
OR
[1] 1.2
Find the length of the intercept, cut off by the plane 2x + y − z = 5 on the x-axis
Concept: Plane - Intercept Form of the Equation of a Plane
Chapter: [4.01] Three - Dimensional Geometry
[1] 2

If y = log (cos ex) then find `"dy"/"dx".`

Concept: Derivatives of Composite Functions - Chain Rule
Chapter: [3.01] Continuity and Differentiability
[1] 3

A is a square matrix with ∣A∣ = 4. then find the value of ∣A. (adj A)∣.

Concept: Determinant of a Square Matrix
Chapter: [2.01] Determinants
[1] 4

Form the differential equation representing the family of curves y = A sin x, by eliminating the arbitrary constant A.

Concept: Procedure to Form a Differential Equation that Will Represent a Given Family of Curves
Chapter: [3.04] Differential Equations
Section B
[2] 5
[2] 5.1

Find:

`int"x".tan^-1 "x"  "dx"`

Concept: Comparison Between Differentiation and Integration
Chapter: [3.05] Integrals
OR
[2] 5.2

Find:
`int"dx"/sqrt(5-4"x" - 2"x"^2)`

Concept: Integration Using Trigonometric Identities
Chapter: [3.05] Integrals
[2] 6

Solve the following differential equation :

`"dy"/"dx" + "y" = cos"x" - sin"x"`

Concept: Solutions of Linear Differential Equation
Chapter: [3.04] Differential Equations
[2] 7

Find:
`int_(-pi/4)^0 (1+tan"x")/(1-tan"x") "dx"`

Concept: Integrals of Some Particular Functions
Chapter: [3.05] Integrals
[2] 8

Let * be an operation defined as *: R × R ⟶ R, a * b = 2a + b, a, b ∈ R. Check if * is a binary operation. If yes, find if it is associative too.

Concept: Concept of Binary Operations
Chapter: [1.02] Relations and Functions
[2] 9
[2] 9.1

X and Y are two points with position vectors `3vec("a") + vec("b")` and `vec("a")-3vec("b")`respectively. Write the position vector of a point Z which divides the line segment XY in the ratio 2 : 1 externally.

Concept: Position Vector of a Point Dividing a Line Segment in a Given Ratio
Chapter: [4.02] Vectors
OR
[2] 9.2

Let  `vec("a") = hat"i" + 2hat"j" - 3hat"k"` and `vec("b") = 3hat"i" -"j" +2hat("k")` be two vectors. Show that the vectors `(vec("a")+vec("b"))` and `(vec("a")-vec("b"))`are perpendicular to each other.

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Chapter: [4.02] Vectors
[2] 10
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[2] 10.1

Out of 8 outstanding students of a school, in which there are 3 boys and 5 girls, a team of 4 students is to be selected for a quiz competition. Find the probability that 2 boys and 2 girls are selected.

Concept: Probability Examples and Solutions
Chapter: [6.01] Probability
OR
[2] 10.2
In a multiple-choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?
Concept: Bernoulli Trials and Binomial Distribution
Chapter: [6.01] Probability
[2] 11

The probabilities of solving a specific problem independently by A and B are `1/3` and `1/5` respectively. If both try to solve the problem independently, find the probability that the problem is solved.

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

For the matrix A = `[(2,3),(5,7)]`, find (A + A') and verify that it is a symmetric matrix.

Concept: Types of Relations
Chapter: [1.02] Relations and Functions
Section C
[4] 13

A ladder 13 m long is leaning against a vertical wall. The bottom of the ladder is dragged away from the wall along the ground at the rate of 2 cm/sec. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall.

Concept: Rate of Change of Bodies Or Quantities
Chapter: [3.02] Applications of Derivatives
[4] 14

Prove that :

`cos^-1 (12/13)  + sin^-1(3/5) = sin^-1(56/65)`

Concept: Proof Derivative X^n Sin Cos Tan
Chapter: [3.01] Continuity and Differentiability
[4] 15

Prove that `int_0^"a" "f(x)" "dx" = int_0^"a" "f"("a"-"x")"dx"` ,and hence evaluate `int_0^1 "x"^2(1 - "x")^"n""dx"`.

Concept: Properties of Indefinite Integral
Chapter: [3.05] Integrals
[4] 16
[4] 16.1

If x = sin t, y = sin pt, prove that`(1-"x"^2)("d"^2"y")/"dx"^2 - "x" "dy"/"dx" + "p"^2"y" = 0`

Concept: Higher Order Derivative
Chapter: [3.01] Continuity and Differentiability
OR
[4] 16.2

Differentiate `tan^-1[(sqrt(1+"x"^2)-sqrt(1-"x"^2))/(sqrt(1+"x"^2) + sqrt(1-"x"^2))]`with respect to cos−1x2.

Concept: Comparison Between Differentiation and Integration
Chapter: [3.05] Integrals
[4] 17

Integrate the function `cos("x + a")/sin("x + b")` w.r.t. x.

Concept: Integration Using Trigonometric Identities
Chapter: [3.05] Integrals
[4] 18
[4] 18.1

Let A = R − (2) and B = R − (1). If f: A ⟶ B is a function defined by`"f(x)"=("x"-1)/("x"-2),` how that f is one-one and onto. Hence, find f−1

Concept: Types of Functions
Chapter: [1.02] Relations and Functions
OR
[4] 18.2

Show that the relation S in the set A = [x ∈ Z : 0 ≤ x ≤ 12] given by S = [(a, b) : a, b ∈ Z, ∣a − b∣ is divisible by 3] is an equivalence relation.

Concept: Types of Relations
Chapter: [1.02] Relations and Functions
[4] 19
[4] 19.1

Solve the differential equation `"dy"/"dx" = 1 + "x"^2 +  "y"^2  +"x"^2"y"^2`, given that y = 1 when x = 0.

Concept: Methods of Solving First Order, First Degree Differential Equations - Differential Equations with Variables Separable Method
Chapter: [3.04] Differential Equations
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OR
[4] 19.2

Find the particular solution of the differential equation `"dy"/"dx" = "xy"/("x"^2+"y"^2),`given that y = 1 when x = 0

Concept: Basic Concepts of Differential Equation
Chapter: [3.04] Differential Equations
[4] 20

Using properties of determinants, find the value of x for which
`|(4-"x",4+"x",4+"x"),(4+"x",4-"x",4+"x"),(4+"x",4+"x",4-"x")|= 0`

Concept: Properties of Determinants
Chapter: [2.01] Determinants
[4] 21

Find the vector equation of the plane which contains the line of intersection of the planes `vec("r").(hat"i"+2hat"j"+3hat"k"),-4=0, vec("r").(2hat"i"+hat"j"-hat"k")+5=0`and which is perpendicular to the plane`vec("r").(5hat"i"+3hat"j"-6hat"k"),+8=0`

Concept: Plane - Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point
Chapter: [4.01] Three - Dimensional Geometry
[4] 22

Find the value of x such that the four-point with position vectors,
`"A"(3hat"i"+2hat"j"+hat"k"),"B" (4hat"i"+"x"hat"j"+5hat"k"),"c" (4hat"i"+2hat"j"-2hat"k")`and`"D"(6hat"i"+5hat"j"-hat"k")`are coplaner.

Concept: Position Vector of a Point Dividing a Line Segment in a Given Ratio
Chapter: [4.02] Vectors
[4] 23

If y = (log x)x + xlog x, find `"dy"/"dx".`

Concept: Logarithmic Differentiation
Chapter: [3.01] Continuity and Differentiability
Section D
[6] 24
[6] 24.1

Find the vector equation of a line passing through the point (2, 3, 2) and parallel to the line `vec("r") = (-2hat"i"+3hat"j") +lambda(2hat"i"-3hat"j"+6hat"k").`Also, find the distance between these two lines.

Concept: Direction Cosines and Direction Ratios of a Line
Chapter: [4.01] Three - Dimensional Geometry
OR
[6] 24.2
Find the coordinates of the foot of the perpendicular Q  drawn from P(3, 2, 1) to the plane 2x − y + z + 1 = 0. Also, find the distance PQ and the image of the point P treating this plane as a mirror
Concept: Plane - Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point
Chapter: [4.01] Three - Dimensional Geometry
[6] 25
[6] 25.1

Using elementary row transformation, find the inverse of the matrix

`[(2,-3,5),(3,2,-4),(1,1,-2)]`

Concept: Inverse of a Matrix - Inverse of a Nonsingular Matrix by Elementary Transformation
Chapter: [2.02] Matrices
OR
[6] 25.2

Using matrices, solve the following system of linear equations :

x + 2y − 3z = −4
2x + 3y + 2z = 2
3x − 3y − 4z = 11

Concept: Determinant of a Matrix of Order 3 × 3
Chapter: [2.01] Determinants
[6] 26
[6] 26.1

Using integration, find the area of the region bounded by the parabola y= 4x and the circle 4x2 + 4y2 = 9.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [3.03] Applications of the Integrals
OR
[6] 26.2

Using the method of integration, find the area of the region bounded by the lines 3x − 2y + 1 = 0, 2x + 3y − 21 = 0 and x − 5y + 9 = 0

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

An insurance company insured 3000 cyclists, 6000 scooter drivers, and 9000 car drivers. The probability of an accident involving a cyclist, a scooter driver, and a car driver are 0⋅3, 0⋅05 and 0⋅02 respectively. One of the insured persons meets with an accident. What is the probability that he is a cyclist?

Concept: Probability Examples and Solutions
Chapter: [6.01] Probability
[6] 28

Using integration, find the area of the smaller region bounded by the ellipse `"x"^2/9+"y"^2/4=1`and the line `"x"/3+"y"/2=1.`

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [3.03] Applications of the Integrals
[6] 29

A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of ₹ 35 per package of nuts and ₹ 14 per package of bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates each machine for almost 12 hours a day? convert it into an LPP and solve graphically.

Concept: Different Types of Linear Programming Problems
Chapter: [5.01] Linear Programming
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