# Mathematics and Statistics 2016-2017 HSC Arts 12th Board Exam Question Paper Solution

Mathematics and Statistics
Date & Time: 6th March 2017, 11:00 am
Duration: 3h

[12]1
[6]1.1 | Select and write the appropriate answer from the given alternatives in each of the following sub-questions:
[2]1.1.1

If the points A(2, 1, 1), B(0, -1, 4) and C(k, 3, -2) are collinear, then k

(A) 0

(B) 1

(C) 4

(D) -4

Concept: Relation Between Direction Ratio and Direction Cosines
Chapter: [0.08] Three Dimensional Geometry
[2]1.1.2

The inverse of the matrix [[-1,5],[-3,2]] is _______

1/13[[2,-5],[3,-1]]

1/13 [[-1,5],[-3,2]]

1/13 [[-1,-3],[5,2]]

1/13 [[1,5],[3,-2]]

Concept: Matrices - Inverse of a Matrix Existance
Chapter: [0.02] Matrices
[2]1.1.3

In Δ ABC, if a = 13, b = 14 and c = 15, then sin (A/2)= _______.

(A) 1/5

(B) sqrt(1/5)

(C) 4/5

(D) 2/5

Concept: Trigonometric Functions - Solution of a Triangle
Chapter: [0.03] Trigonometric Functions
[12]1.2 | Attempt any THREE of the following:
[2]1.2.1

Find the volume of the parallelopiped whose coterminus edges are given by vectors

2hati+3hatj-4hatk, 5hati+7hatj+5hatk and 4hati+5hatj-2hatk

Concept: Scalar Triple Product of Vectors
Chapter: [0.015] Vectors [0.07] Vectors
[2]1.2.2

In Δ ABC, prove that, a (b cos C - c cos B) = b2 - c2.

Concept: Trigonometric Functions - Trigonometric equations
Chapter: [0.03] Trigonometric Functions
[2]1.2.3

If from a point Q (a, b, c) perpendiculars QA and QB are drawn to the YZ and ZX planes respectively, then find the vector equation of the plane QAB.

Concept: Trigonometric Functions - Trigonometric equations
Chapter: [0.03] Trigonometric Functions
[2]1.2.4

Find the cartesian equation of the line passing throught the points A(3, 4, -7) and B(6,-1, 1).

Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation
Chapter: [0.04] Pair of Straight Lines
[2]1.2.5

Write the following statement in symbolic form and find its truth value:
∀ n ∈ N, n2 + n is an even number and n2 - n is an odd number.

Concept: Mathematical Logic - Truth Value of Statement in Logic
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[14]2
[6]2.1 | Attempt any TWO of the following:
[3]2.1.1

Using truth tables, examine whether the statement pattern (p ∧ q) ∨ (p ∧ r) is a tautology, contradiction or contingency.

Concept: Mathematical Logic - Truth Value of Statement in Logic
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[3]2.1.2

Find the shortest distance between the lines (x-1)/2=(y-2)/3=(z-3)/4 and (x-2)/3=(y-4)/4=(z-5)/5

Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation
Chapter: [0.04] Pair of Straight Lines
[3]2.1.3

Find the general solution of the equation sin 2x + sin 4x + sin 6x = 0

Concept: Trigonometric Functions - General Solution of Trigonometric Equation of the Type
Chapter: [0.03] Trigonometric Functions
[8]2.2 | Attempt any TWO of the following:
[4]2.2.1

Solve the following equations by method of reduction:

x-y + z = 4,

2x + y - 3z = 0,

x + y + z = 2

Concept: Matrices - Solving System of Linear Equations in Two Or Three Variables Using Reduction of a Matrix Or Reduction Method
Chapter: [0.02] Matrices
[4]2.2.2

If θ is the measure of acute angle between the pair of lines given by ax^2+2hxy+by^2=0, then prove that tantheta=|(2sqrt(h^2-ab))/(a+b)|,a+bne0

Concept: Acute Angle Between the Lines
Chapter: [0.04] Pair of Straight Lines
[4]2.2.3

Using vector method, find incentre of the triangle whoose vertices are P(0, 4, 0), Q(0, 0, 3)
and R(0, 4, 3).

Concept: Vectors - Application of Vectors to Geometry
Chapter: [0.07] Vectors
[14]3
[6]3.1 | Attempt any TWO of the following:
[3]3.1.1

Construct the switching circuit for the statement (p ∧ q) ∨ (~ p) ∨ (p ∧ ~ q).

Concept: Mathematical Logic - Application of Logic to Switching Circuits, Switching Table.
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[3]3.1.2

Find the joint equation of the pair of lines passing through the origin which are perpendicular respectively to the lines represented by 5x2 +2xy- 3y2 = 0.

Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Homogenous Equation
Chapter: [0.04] Pair of Straight Lines
[3]3.1.3

Show that:

cos^(-1)(4/5)+cos^(-1)(12/13)=cos^(-1)(33/65)

Concept: Basic Concepts of Trigonometric Functions
Chapter: [0.03] Trigonometric Functions
[8]3.2 | Attempt any TWO of the following
[4]3.2.1

If l, m, n are the direction cosines of a line, then prove that l2 + m2 + n2 = 1. Hence find the
direction angle of the line with the X axis which makes direction angles of 135° and 45° with Y and Z axes respectively.

Concept: Direction Cosines and Direction Ratios of a Line
Chapter: [0.08] Three Dimensional Geometry
[4]3.2.2

Find the vector and cartesian equations of the plane passing through the points A( 1, 1, -2), B(1, 2, 1) and C(2, -1, 1).

Concept: Plane - Equation of Plane Passing Through the Given Point and Perpendicular to Given Vector
Chapter: [0.1] Plane
[4]3.2.3

Solve the following LPP by using graphical method.

Maximize : Z = 6x + 4y

Subject to x ≤ 2, x + y ≤  3, -2x + y ≤  1, x ≥  0, y ≥ 0.

Also find maximum value of Z.

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.017] Linear Programming [0.11] Linear Programming Problems
[12]4
[6]4.1 | Select and write the appropriate answer from the given alternatives in each of the following sub-questions:
[2]4.1.1

Derivatives of  tan3θ with respect to sec3θ at θ=π/3 is

(A) 3/2

(B) sqrt3/2

(C) 1/2

(D) -sqrt3/2

Concept: Derivatives of Functions in Parametric Forms
Chapter: [0.13] Differentiation
[2]4.1.2

The equation of tangent to the curve y = 3x2 - x + 1 at the point (1, 3) is

(a) y=5x+2

(b)y=5x-2

(c)y=1/5x+2

(d)y=1/5x-2

Concept: Conics - Tangents and normals - equations of tangent and normal at a point
Chapter: [0.06] Conics
[2]4.1.3

The expected value of the number of heads obtained when three fair coins are tossed simultaneously is

(A) 1

(B) 1.5

(C) 0

(D) -1

Concept: Probability Distribution - Expected Value, Variance and Standard Deviation of a Discrete Random Variable
Chapter: [0.19] Probability Distribution
[6]4.2 | Attempt any THREE of the following:
[2]4.2.1

Find dy/dx if x sin y + y sin x = 0.

Concept: Derivatives of Implicit Functions
Chapter: [0.021] Differentiation [0.13] Differentiation
[2]4.2.2

Test whether the function is increasing or decreasing.

f(x) = "x" -1/"x", x ∈ R, x ≠ 0,

Concept: Increasing and Decreasing Functions
Chapter: [0.022000000000000002] Applications of Derivatives [0.14] Applications of Derivative
[2]4.2.3

Evaluate: intsinsqrtx/sqrtxdx

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.15] Integration
[2]4.2.4

Form the differential equation by eliminating arbitrary constants from the relation y=Ae^(5x)+Be^(-5x)

Concept: Formation of Differential Equation by Eliminating Arbitary Constant
Chapter: [0.17] Differential Equation
[2]4.2.5

The probability that a bomb will hit a target is 0.8. Find the probability that out of 10 bombs dropped, exactly 4 will hit the target.

Concept: Bernoulli Trials and Binomial Distribution
Chapter: [0.027999999999999997] Binomial Distribution [0.2] Bernoulli Trials and Binomial Distribution
[14]5
[6]5.1 | Attempt any TWO of the following:
[3]5.1.1

Solve: dy/dx = cos(x + y)

Concept: Methods of Integration - Integration by Substitution
Chapter: [0.023] Indefinite Integration [0.15] Integration
[3]5.1.2

If u and v are two functions of x then prove that

intuvdx=uintvdx-int[du/dxintvdx]dx

Hence evaluate, int xe^xdx

Concept: Methods of Integration - Integration by Parts
Chapter: [0.023] Indefinite Integration [0.15] Integration
[3]5.1.3

If f(x) =(e^(x^2)-cosx)/x^2, for x= 0, is continuous at x = 0, find f(0).

Concept: Definition of Continuity - Continuity of a Function at a Point
Chapter: [0.12] Continuity
[8]5.2 | Attempt any TWO of the following:
[4]5.2.1

If y = f(x) is a differentiable function of x such that inverse function x = f–1 (y) exists, then prove that x is a differentiable function of y and dx/dy=1/((dy/dx)) " where " dy/dx≠0

Concept: Derivative - Derivative of Inverse Function
Chapter: [0.13] Differentiation
[4]5.2.2

A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.

Concept: Maxima and Minima
Chapter: [0.022000000000000002] Applications of Derivatives [0.14] Applications of Derivative
[4]5.2.3

Evaluate: int_(-a)^asqrt((a-x)/(a+x)) dx

Concept: Properties of Definite Integrals
Chapter: [0.15] Integration
[14]6
[6]6.1 | Attempt any TWO of the following:
[3]6.1.1

Discuss the continuity of the following function, at x = 0.

f(x)=x/|x|,for x ne0

=1,for x=0

Concept: Definition of Continuity - Continuity of a Function at a Point
Chapter: [0.12] Continuity
[3]6.1.2

If the population of a country doubles in 60 years, in how many years will it be triple under
the assumption that the rate of increase in proportional to the number of inhabitants?
[Given : log 2 = 0.6912 and log 3 = 1.0986.]

Concept: Differential Equations - Applications of Differential Equation
Chapter: [0.17] Differential Equation
[3]6.1.3

A fair coin is tossed 8 times. Find the probability that it shows heads exactly 5 times.

Concept: Conditional Probability
Chapter: [0.19] Probability Distribution

A fair coin is tossed 8 times. Find the probability that it shows heads at least once

Concept: Conditional Probability
Chapter: [0.19] Probability Distribution
[8]6.2 | Attempt any TWO of the following:
[4]6.2.1

Find: I=intdx/(sinx+sin2x)

Concept: Methods of Integration - Integration Using Partial Fractions
Chapter: [0.023] Indefinite Integration [0.15] Integration
[4]6.2.2

Find the area of the region lying between the parabolas y2 = 4ax and x2 = 4ay.

Concept: Area Between Two Curves
Chapter: [0.025] Application of Definite Integration [0.16] Applications of Definite Integral
[4]6.2.3

Given the p. d. f. (probability density function) of a continuous random variable x as :

f(x)=x^2/3, -1

= 0 , otherwise

Determine the c. d. f. (cumulative distribution function) of x and hence find P(x < 1), P(x ≤ -2), P(x > 0), P(1 < x < 2)

Concept: Probability Distribution - Probability Density Function (P.D.F.)
Chapter: [0.19] Probability Distribution

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