HSC Arts 12th Board ExamMaharashtra State Board
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# Question Paper Solutions - Mathematics and Statistics 2016 - 2017 HSC Arts 12th Board Exam

SubjectMathematics and Statistics
Year2016 - 2017 (March)

Marks: 70
[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

Chapter: [8] Three Dimensional Geometry
Concept: Relation Between Direction Ratio and Direction Cosines
[2]1.1.2

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

(A) 1/13[[2,-5],[3,-1]]

(B) 1/13 [[-1,5],[-3,2]]

(C) 1/13 [[-1,-3],[5,2]]

(D) 1/13 [[1,5],[3,-2]]

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

Chapter: [3] Trigonometric Functions
Concept: Trigonometric Functions - Solution of a Triangle
[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

Chapter: [7] Vectors
Concept: Scalar Triple Product of Vectors
[2]1.2.2

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

Chapter: [3] Trigonometric Functions
Concept: Trigonometric Functions - Trigonometric equations
[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.

Chapter: [3] Trigonometric Functions
Concept: Trigonometric Functions - Trigonometric equations
[2]1.2.4

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

Chapter: [4] Pair of Straight Lines
Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation
[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.

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Truth Value of Statement in 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.

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Truth Value of Statement in 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

Chapter: [4] Pair of Straight Lines
Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation
[3]2.1.3

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

Chapter: [3] Trigonometric Functions
Concept: Trigonometric Functions - General Solution of Trigonometric Equation of the Type
[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

Chapter: [2] Matrices
Concept: Matrices - Solving System of Linear Equations in Two Or Three Variables Using Reduction of a Matrix Or Reduction Method
[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

Chapter: [4] Pair of Straight Lines
Concept: Acute Angle Between the 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).

Chapter: [7] Vectors
Concept: Vectors - Application of Vectors to Geometry
[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).

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Application - Introduction to Switching Circuits
[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.

Chapter: [4] Pair of Straight Lines
Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Homogenous Equation
[3]3.1.3

Show that:

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

Chapter: [3] Trigonometric Functions
Concept: Basic Concepts of 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.

Chapter: [8] Three Dimensional Geometry
Concept: Direction Cosines and Direction Ratios of a Line
[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).

Chapter: [10] Plane
Concept: Plane - Equation of Plane Passing Through the Given Point and Perpendicular to Given Vector
[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.

Chapter: [11] Linear Programming Problems
Concept: Graphical Method of Solving 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

Chapter: [13] Differentiation
Concept: Derivatives of Functions in Parametric Forms
[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

Chapter: [6] Conics
Concept: Conics - Tangents and normals - equations of tangent and normal at a point
[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

Chapter: [19] Probability Distribution
Concept: Probability Distribution - Expected Value, Variance and Standard Deviation of a Discrete Random Variable
[6]4.2 | Attempt any THREE of the following:
[2]4.2.1

Find dy/dx if x sin y + y sin x = 0.

Chapter: [13] Differentiation
Concept: Derivatives of Implicit Functions
[2]4.2.2

Test whether the function, f(x) = x -1/x, x ∈ R, x ≠ 0, is increasing or decreasing

Chapter: [14] Applications of Derivative
Concept: Increasing and Decreasing Functions
[2]4.2.3

Evaluate: intsinsqrtx/sqrtxdx

Chapter: [15] Integration
Concept: Evaluation of Definite Integrals by Substitution
[2]4.2.4

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

Chapter: [17] Differential Equation
Concept: Formation of Differential Equation by Eliminating Arbitary Constant
[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.

Chapter: [20] Bernoulli Trials and Binomial Distribution
Concept: 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)

Chapter: [15] Integration
Concept: Methods of Integration - Integration by Substitution
[3]5.1.2

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

intuvdx=uintvdx-int[du/dxintvdx]dx

Chapter: [15] Integration
Concept: Methods of Integration - Integration by Parts
[3]5.1.3

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

Chapter: [12] Continuity
Concept: Continuity - Continuity of a Function at a Point
[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

Chapter: [13] Differentiation
Concept: Derivative - Derivative of Inverse Function
[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.

Chapter: [14] Applications of Derivative
Concept: Maxima and Minima
[4]5.2.3

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

Chapter: [15] Integration
Concept: Properties of Definite Integrals
[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

Chapter: [12] Continuity
Concept: Continuity - Continuity of a Function at a Point
[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.]

Chapter: [17] Differential Equation
Concept: Differential Equations - Applications of Differential Equation
[3]6.1.3

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

Chapter: [19] Probability Distribution
Concept: Conditional Probability

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

Chapter: [19] Probability Distribution
Concept: Conditional Probability
[8]6.2 | Attempt any TWO of the following:
[4]6.2.1

Find: I=intdx/(sinx+sin2x)

Chapter: [15] Integration
Concept: Methods of Integration - Integration Using Partial Fractions
[4]6.2.2

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

Chapter: [16] Applications of Definite Integral
Concept: Area Between Two Curves
[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)

Chapter: [19] Probability Distribution
Concept: Probability Distribution - Probability Density Function (P.D.F.)
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