# Mathematics and Statistics 2016-2017 HSC Science (General) 12th Board Exam Question Paper Solution

Mathematics and Statistics
Date: July 2017

 1
 1.1 | Select and write the correct answer from the given alternatives in each of the following sub-questions:
 1.1.1

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

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

(B) 1/5[[3,1],[-2,1]]

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

(D) 1/5[[3,-1],[2,-1]]

Concept: Matrices - Inverse of a Matrix Existance
Chapter: [0.02] Matrices
 1.1.2

If bara=3hati-hatj+4hatk, barb=2hati+3hatj-hatk, barc=-5hati+2hatj+3hatk then bara.(barbxxbarc)=

(A) 100

(B) 101

(C) 110

(D) 109

Concept: Scalar Triple Product of Vectors
Chapter: [0.015] Vectors [0.07] Vectors
 1.1.3

If a line makes angles 90°, 135°, 45° with the X, Y, and Z axes respectively, then its direction cosines are _______.

(A) 0,1/sqrt2,-1/sqrt2

(B) 0,-1/sqrt2,-1/sqrt2

(C) 1,1/sqrt2,1/sqrt2

(D) 0,-1/sqrt2,1/sqrt2

Concept: Direction Cosines and Direction Ratios of a Line
Chapter: [0.08] Three Dimensional Geometry
 1.2 | Attempt any THREE of the following:
 1.2.1

barr=(hati-2hatj+3hatk)+lambda(2hati+hatj+2hatk) is parallel to the plane barr.(3hati-2hatj+phatk)=10, find the value of p.

Concept: Plane - Equation of Plane Passing Through the Given Point and Parallel to Two Given Vectors
Chapter: [0.1] Plane
 1.2.2

If a line makes angles α, β, γ with co-ordinate axes, prove that cos 2α + cos2β + cos2γ+ 1 = 0.

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

Write the negations of the following statements:

a.forall n in N, n+7>6

b. The kitchen is neat and tidy.

Concept: Mathematical Logic - Sentences and Statement in Logic
Chapter: [0.01] Mathematical Logic
 1.2.4

Find the angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.

Concept: Direction Cosines and Direction Ratios of a Line
Chapter: [0.08] Three Dimensional Geometry
 1.2.5

If bara, barb, barc are position vectors of the points A, B, C respectively such that 3bara+ 5barb-8barc = 0, find the ratio in which A divides BC.

Concept: Basic Concepts of Vector Algebra
Chapter: [0.07] Vectors
 2
 2.1 | Attempt any TWO of the following:
 2.1.1

If tan^-1(2x)+tan^-1(3x)=pi/4, then find the value of ‘x’.

Concept: Properties of Inverse Trigonometric Functions
Chapter: [0.03] Trigonometric Functions
 2.1.2

Write the converse, inverse and contrapositive of the following statement.
“If it rains then the match will be cancelled.”

Concept: Mathematical Logic - Sentences and Statement in Logic
Chapter: [0.01] Mathematical Logic
 2.1.3

Find p and q, if the equation px^2-8xy+3y^2+14x+2y+q=0 represents a pair of prependicular lines.

Concept: Pair of Straight Lines - Condition for Perpendicular Lines
Chapter: [0.04] Pair of Straight Lines
 2.2 | Attempt any TWO of the following:
 2.2.1

Find the equation of the plane passing through the intersection of the planes 3x + 2y – z + 1 = 0 and x + y + z – 2 = 0 and the point (2, 2, 1).

Concept: Plane - Equation of Plane Passing Through the Intersection of Two Given Planes
Chapter: [0.1] Plane
 2.2.2

Let A(bara) and B(barb) be any two points in the space and R(barr) be a point on the line segment AB dividing it internally in the ratio m : n, then prove that bar r=(mbarb+nbara)/(m+n) . Hence find the position vector of R which divides the line segment joining the points A(1, –2, 1) and B(1, 4, –2) internally in the ratio 2 : 1.

Concept: Equation of a Line in Space
Chapter: [0.013999999999999999] Pair of Straight Lines [0.09] Line
 2.2.3

The angles of the ΔABC are in A.P. and b:c=sqrt3:sqrt2 then findangleA,angleB,angleC

Concept: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions
 3
 3.1 | Attempt any TWO of the following:
 3.1.1

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

Find the vector equation of a line passing through the points A(3, 4, –7) and B(6, –1, 1).

Concept: Vector and Cartesian Equation of a Plane
Chapter: [0.1] Plane
 3.1.2

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
 3.1.3

find the symbolic fom of the following switching circuit, construct its switching table and interpret it. Concept: Application of Logic to Switching Circuits
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
 3.2 | Attempt any TWO of the following:
 3.2.1

If A=[[1,-1,2],[3,0,-2],[1,0,3]] verify that A (adj A) = |A| I.

Chapter: [0.02] Matrices
 3.2.2

A company manufactures bicycles and tricycles each of which must be processed through machines A and B. Machine A has maximum of 120 hours available and machine B has maximum of 180 hours available. Manufacturing a bicycle requires 6 hours on machine A and 3 hours on machine B. Manufacturing a tricycle requires 4 hours on machine A and 10 hours on machine B.
If profits are Rs. 180 for a bicycle and Rs. 220 for a tricycle, formulate and solve the L.P.P. to determine the number of bicycles and tricycles that should be manufactured in order to maximize the profit.

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.017] Linear Programming [0.11] Linear Programming Problems
 3.2.3

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

find the acute angle between the lines
x2 – 4xy + y2 = 0.

Concept: Acute Angle Between the Lines
Chapter: [0.04] Pair of Straight Lines
 4
 4.1 | Select and write the correct answer from the given alternatives in each of the following sub-questions:
 4.1.1

Given f (x) = 2x, x < 0
= 0, x ≥ 0
then f (x) is _______ .

discontinuous and not differentiable at x = 0

continuous and differentiable at x = 0

discontinuous and differentiable at x = 0

continuous and not differentiable at x = 0

Concept: Definition of Continuity - Discontinuity of a Function
Chapter: [0.12] Continuity
 4.1.2

If int_0^alpha(3x^2+2x+1)dx=14 then alpha=

(A) 1

(B) 2

(C) –1

(D) –2

Concept: Properties of Definite Integrals
Chapter: [0.15] Integration
 4.1.3

The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.

(A) increasing

(B) decreasing

(C) increasing and decreasing

(D) neither increasing nor decreasing

Concept: Increasing and Decreasing Functions
Chapter: [0.022000000000000002] Applications of Derivatives [0.14] Applications of Derivative
 4.2 | Attempt any THREE of the following:
 4.2.1

Differentiate 3x w.r.t. log3x

Concept: Exponential and Logarithmic Functions
Chapter: [0.12] Continuity [0.13] Differentiation
 4.2.2

Check whether the conditions of Rolle’s theorem are satisfied by the function
f (x) = (x - 1) (x - 2) (x - 3), x ∈ [1, 3]

Concept: Mean Value Theorem
Chapter: [0.14] Applications of Derivative
 4.2.3

Evaluate: int sqrt(tanx)/(sinxcosx) dx

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.023] Indefinite Integration [0.15] Integration
 4.2.4

Find the area of the region bounded by the curve x2 = 16y, lines y = 2, y = 6 and Y-axis lying in the first quadrant.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.16] Applications of Definite Integral
 4.2.5

Given X ~ B (n, p)
If n = 10 and p = 0.4, find E(X) and var (X).

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

If the function f(x)=(5^sinx-1)^2/(xlog(1+2x))  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
 5.1.2

The probability mass function for X = number of major defects in a randomly selected
appliance of a certain type is

 X = x 0 1 2 3 4 P(X = x) 0.08 0.15 0.45 0.27 0.05

Find the expected value and variance of X.

Concept: Variance of Binomial Distribution (P.M.F.)
Chapter: [0.027999999999999997] Binomial Distribution [0.2] Bernoulli Trials and Binomial Distribution
 5.1.3

Suppose that 80% of all families own a television set. If 5 families are interviewed at  random, find the probability that
a. three families own a television set.
b. at least two families own a television set.

Concept: Conditional Probability
Chapter: [0.19] Probability Distribution
 5.2 | Attempt any TWO of the following:
 5.2.1

Find the approximate value of cos (60° 30').

(Given: 1° = 0.0175c, sin 60° = 0.8660)

Concept: Approximations
Chapter: [0.022000000000000002] Applications of Derivatives [0.14] Applications of Derivative
 5.2.2

The rate of growth of bacteria is proportional to the number present. If, initially, there were
1000 bacteria and the number doubles in one hour, find the number of bacteria after 2½
hours.

[Take sqrt2 = 1.414]

Concept: Rate of Change of Bodies Or Quantities
Chapter: [0.14] Applications of Derivative
 5.2.3

Prove that : int_-a^af(x)dx=2int_0^af(x)dx , if f (x) is an even function.

= 0,                   if f (x) is an odd function.

Concept: Methods of Integration: Integration by Parts
Chapter: [0.023] Indefinite Integration [0.15] Integration
 6
 6.1 | Attempt any TWO of the following
 6.1.1

If f (x) is continuous on [–4, 2] defined as

f (x) = 6b – 3ax, for -4 ≤ x < –2
= 4x + 1,    for –2 ≤ x ≤ 2

Show that a + b =-7/6

Concept: Algebra of Continuous Functions
Chapter: [0.12] Continuity
 6.1.2

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
 6.1.3

Probability distribution of X is given by

 X = x 1 2 3 4 P(X = x) 0.1 0.3 0.4 0.2

Find P(X ≥ 2) and obtain cumulative distribution function of X

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027000000000000003] Probability Distributions [0.19] Probability Distribution
 6.2 | Attempt any TWO of the following
 6.2.1

Solve the differential equation dy/dx -y =e^x

Concept: General and Particular Solutions of a Differential Equation
Chapter: [0.17] Differential Equation
 6.2.2

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/dxne0

Hence if y=sin^-1x, -1<=x<=1 , -pi/2<=y<=pi/2

then show that dy/dx=1/sqrt(1-x^2), where  |x|<1

Concept: The Concept of Derivative - Derivative of Inverse Function
Chapter: [0.13] Differentiation
 6.2.3

Evaluate: ∫8/((x+2)(x^2+4))dx

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

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