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

SubjectMathematics and Statistics
Year2016 - 2017 (July)

Marks: 70
[12]1
[6]1.1 | Select and write the correct answer from the given alternatives in each of the following sub-questions:
[2]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]]

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

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

Chapter: [8] Three Dimensional Geometry
Concept: Direction Cosines and Direction Ratios of a Line
[6]1.2 | Attempt any THREE of the following:
[2]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.

Chapter: [10] Plane
Concept: Plane - Equation of Plane Passing Through the Given Point and Parallel to Two Given Vectors
[2]1.2.2

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

Chapter: [10] Plane
Concept: Angle Between Line and a Plane
[2]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.

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Sentences and Statement in Logic
[2]1.2.4

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

Chapter: [8] Three Dimensional Geometry
Concept: Direction Cosines and Direction Ratios of a Line
[2]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.

Chapter: [7] Vectors
Concept: Basic Concepts of Vector Algebra
[14]2
[6]2.1 | Attempt any TWO of the following:
[3]2.1.1

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

Chapter: [3] Trigonometric Functions
Concept: Properties of Inverse Trigonometric Functions
[3]2.1.2

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

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Sentences and Statement in Logic
[3]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.

Chapter: [4] Pair of Straight Lines
Concept: Pair of Straight Lines - Condition for Perpendicular Lines
[8]2.2 | Attempt any TWO of the following:
[4]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).

Chapter: [10] Plane
Concept: Plane - Equation of Plane Passing Through the Intersection of Two Given Planes
[4]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.

Chapter: [9] Line
Concept: Equation of a Line in Space
[4]2.2.3

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

Chapter: [3] Trigonometric Functions
Concept: Trigonometric Functions - Solution of a Triangle
[14]3
[6]3.1 | Attempt any TWO of the following:
[3]3.1.1

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

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

Chapter: [10] Plane
Concept: Vector and Cartesian Equation of a Plane
[3]3.1.2

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
[3]3.1.3

find the symbolic fom of the following switching circuit, construct its switching table and interpret it.

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Application - Introduction to Switching Circuits
[8]3.2 | Attempt any TWO of the following:
[4]3.2.1

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

Chapter: [2] Matrices
[4]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.

Chapter: [11] Linear Programming Problems
Concept: Graphical Method of Solving Linear Programming Problems
[4]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

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

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

Chapter: [4] Pair of Straight Lines
Concept: Acute Angle Between the Lines
[12]4
[6]4.1 | Select and write the correct answer from the given alternatives in each of the following sub-questions:
[2]4.1.1

Given f (x) = 2x, x < 0

= 0, x ≥ 0

then f (x) is _______.

(A) discontinuous and not differentiable at x = 0
(B) continuous and differentiable at x = 0
(C) discontinuous and differentiable at x = 0
(D) continuous and not differentiable at x = 0

Chapter: [12] Continuity
Concept: Continuity - Discontinuity of a Function
[2]4.1.2

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

(A) 1

(B) 2

(C) –1

(D) –2

Chapter: [15] Integration
Concept: Properties of Definite Integrals
[2]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

Chapter: [14] Applications of Derivative
Concept: Increasing and Decreasing Functions
[6]4.2 | Attempt any THREE of the following:
[2]4.2.1

Differentiate 3x w.r.t. log3x

Chapter: [13] Differentiation
Concept: Exponential and Logarithmic Functions

Differentiate 3x w.r.t. log3x

Chapter: [12] Continuity
Concept: Exponential and Logarithmic Functions
[2]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]

Chapter: [14] Applications of Derivative
Concept: Mean Value Theorem
[2]4.2.3

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

Chapter: [15] Integration
Concept: Methods of Integration - Integration by Substitution
[2]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.

Chapter: [16] Applications of Definite Integral
Concept: Area of the Region Bounded by a Curve and a Line
[2]4.2.5

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

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

If the function f(x)=(5^sinx-1)^2/(xlog(1+2x))  for x ≠ 0 is continuous at x = 0, find f (0).

Chapter: [12] Continuity
Concept: Continuity - Continuity of a Function at a Point
[3]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.

Chapter: [20] Bernoulli Trials and Binomial Distribution
Concept: Variance of Binomial Distribution (P.M.F.)
[3]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.

Chapter: [19] Probability Distribution
Concept: Conditional Probability
[8]5.2 | Attempt any TWO of the following:
[4]5.2.1

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

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

Chapter: [14] Applications of Derivative
Concept: Approximations
[4]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]

Chapter: [14] Applications of Derivative
Concept: Rate of Change of Bodies Or Quantities
[4]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.

Chapter: [15] Integration
Concept: Methods of Integration - Integration by Parts
[14]6
[6]6.1 | Attempt any TWO of the following
[3]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

Chapter: [12] Continuity
Concept: Algebra of Continuous Functions
[3]6.1.2

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]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

Chapter: [19] Probability Distribution
Concept: Random Variables and Its Probability Distributions
[8]6.2 | Attempt any TWO of the following
[4]6.2.1

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

Chapter: [17] Differential Equation
Concept: General and Particular Solutions of a Differential Equation
[4]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

Chapter: [13] Differentiation
Concept: Derivative - Derivative of Inverse Function
[4]6.2.3

Evaluate: ∫8/((x+2)(x^2+4))dx

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

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