# Mathematics and Statistics 2012-2013 HSC Arts 12th Board Exam Question Paper Solution

Mathematics and Statistics
Date: March 2013

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

The principal solution of the equation cot x=-sqrt 3  is

pi/6

pi/3

(5pi)/6

(-5pi)/6

Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch
Chapter: [0.03] Trigonometric Functions
[2]1.1.2

If the vectors -3hati+4hatj-2hatk, hati+2hatk, hati-phatj are coplanar, then the value of of p is

(A) -2

(B) 1

(C) -1

(D) 2

Concept: Vectors - Collinearity and Coplanarity of Vectors
Chapter: [0.07] Vectors
[2]1.1.3

If the line y =x+k  touches the hyperbola 9x2 -16y2 =144, then k = _________

7

-7

+-sqrt7

+-sqrt19

Concept: Conics - Tangents and normals - equations of tangent and normal at a point
Chapter: [0.06] Conics
[6]1.2 | Attempt any THREE of the following:
[2]1.2.1

Write down the following statements in symbolic form :

(A) A triangle is equilateral if and only if it is equiangular.
(B) Price increases and demand falls

Concept: Mathematical Logic - Logical Connectives
Chapter: [0.01] Mathematical Logic
[2]1.2.2

If A=[[2,-2],[4,3]] then find A^-1 by adjoint method.

Chapter: [0.02] Matrices
[2]1.2.3

Find the separate equations of the lines represented by the equation  3"x"^2-10"xy"-8"y"^2=0

Concept: Equation of a Line in Space
Chapter: [0.013999999999999999] Pair of Straight Lines [0.09] Line
[2]1.2.4

Find the equation of the director circle of a circle  x^2 + y^2 =100.

Concept: Circle - Director circle
Chapter: [0.05] Circle
[2]1.2.5

Find the general solution of the equation 4cos^2x=1

Concept: Trigonometric Functions - General Solution of Trigonometric Equation of the Type
Chapter: [0.03] Trigonometric Functions
[14]2
[6]2.1 | Attempt any TWO of the following:
[3]2.1.1

Without using the truth table show that P ↔ q ≡ (p ∧ q) ∨ (~ p ∧ ~ q)

Concept: Mathematical Logic - Algebra of Statements
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[3]2.1.2

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
[3]2.1.3

Show that the line x+ 2y + 8 = 0 is tangent to the parabola y2 = 8x. Hence find the point of contact

Concept: Conics - Tangents and normals - equations of tangent and normal at a point
Chapter: [0.06] Conics
[8]2.2 | Attempt any TWO of the following :
[4]2.2.1

The sum of three numbers is 9. If we multiply third number by 3 and add to the second number, we get 16. By adding the first and the third number and then subtracting twice the second number from this sum, we get 6. Use this information and find the system of linear equations. Hence, find the three numbers using matrices.

Concept: Elementary Operation (Transformation) of a Matrix
Chapter: [0.012] Matrics [0.02] Matrices
[4]2.2.2

Find the general solution of cos x +sin x =1.

Concept: Trigonometric Functions - General Solution of Trigonometric Equation of the Type
Chapter: [0.03] Trigonometric Functions
[4]2.2.3

If bar a and bar b are any two non-zero and non-collinear vectors then prove that any vector bar r  coplanar with  bar a and bar b can be uniquely expressed as bar r=t_1bara+t_2barb , where  t_1 and t_2  are scalars

Concept: Vectors - Collinearity and Coplanarity of Vectors
Chapter: [0.07] Vectors
[14]3
[6]3.1 | Attempt any TWO of the following :
[3]3.1.1

Using truth table examine whether the following statement pattern is tautology, contradiction or contingency (p^^~q) harr (p->q)

Concept: Mathematical Logic - Statement Patterns and Logical Equivalence
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[3]3.1.2

Find k if the length of the tangent segment from (8,-3) to the circle  x^2+y^2-2x+ky-23=0 is sqrt10  units.

Concept: Circle - Length of Tangent Segments to Circle
Chapter: [0.05] Circle
[3]3.1.3

Show that the lines given by (x+1)/-10=(y+3)/-1=(z-4)/1  and (x+10)/-1=(y+1)/-3=(z-1)/4 intersect. Also find the co-ordinates of the point of intersection.

Concept: Pair of Straight Lines - Point of Intersection of Two Lines
Chapter: [0.04] Pair of Straight Lines
[8]3.2 | Attempt any TWO of the following:
[4]3.2.1

Find the equation of the locus of the point of intersection of two tangents drawn to the hyperbola x^2/7-y^2/5=1 such that the sum of the cubes of their slopes is 8.

Concept: Conics - Locus of Points from Which Two Tangents Are Mutually Perpendicular
Chapter: [0.06] Conics
[4]3.2.2

Solve the following L.P.P graphically:

Maximize: Z = 10x + 25y
Subject to: x ≤ 3, y ≤ 3, x + y ≤ 5, x ≥ 0, y ≥ 0

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.017] Linear Programming [0.11] Linear Programming Problems
[4]3.2.3

Find the equation of the planes parallel to the plane x + 2y+ 2z + 8 =0 which are at the distance of 2  units from the point (1,1, 2)

Concept: Distance of a Point from a Plane
Chapter: [0.016] Line and Plane [0.1] Plane
[12]4
[6]4.1 | Select and write the correct answer from the given alternatives in each of the folloiwng:
[2]4.1.1

Function  f (x)= x^2 - 3x +4 has minimum value at

(A) 0

(B)-3/2

(C) 1

(D)3/2

Concept: Maxima and Minima - Introduction of Extrema and Extreme Values
Chapter: [0.14] Applications of Derivative
[2]4.1.2

int1/xlogxdx=...............

(A)log(log x)+ c

(B) 1/2 (logx )2+c

(C) 2log x + c

(D) log x + c

Concept: Methods of Integration - Integration by Parts
Chapter: [0.023] Indefinite Integration [0.15] Integration
[2]4.1.3

Order and degree of the differential equation [1+(dy/dx)^3]^(7/3)=7(d^2y)/(dx^2) are respectively

(A) 2, 3

(B) 3, 2

(C) 7, 2

(D) 3, 7

Concept: Order and Degree of a Differential Equation
Chapter: [0.026000000000000002] Differential Equations [0.17] Differential Equation
[6]4.2 | Attempt any THREE of the following:
[2]4.2.1

If x=at2, y= 2at , then find dy/dx.

Concept: Derivatives of Functions in Parametric Forms
Chapter: [0.13] Differentiation
[2]4.2.2

Find the approximate value of  sqrt8.95

Concept: Approximations
Chapter: [0.022000000000000002] Applications of Derivatives [0.14] Applications of Derivative
[2]4.2.3

Find the area of the region bounded by the parabola y2 = 16x and the line x = 3.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.16] Applications of Definite Integral
[2]4.2.4

For the bivariate data r = 0.3, cov(X, Y) = 18, σx = 3, find σy .

Concept: Statistics - Bivariate Frequency Distribution
Chapter: [0.18] Statistics
[2]4.2.5

triangle bounded by the lines y = 0, y = x and x = 4 is revolved about the X-axis. Find the volume of the solid of revolution.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.16] Applications of Definite Integral
[14]5
[6]5.1 | Attempt any Two of the following:
[3]5.1.1

A function f (x) is defined as
f (x) = x + a, x < 0
= x,       0 ≤x ≤ 1
= b- x,   x ≥1
is continuous in its domain.
Find a + b.

Concept: Algebra of Continuous Functions
Chapter: [0.12] Continuity
[3]5.1.2

If x=a(t-1/t),y=a(t+1/t), then show that dy/dx=x/y

Concept: Derivatives of Functions in Parametric Forms
Chapter: [0.13] Differentiation
[3]5.1.3

Evaluate : int1/(3+5cosx)dx

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.15] Integration
[8]5.2 | Attempt any TWO of the following:
[4]5.2.1

An insurance agent insures lives of 5 men, all of the same age and in good health. The probability that a man of this age will survive the next 30 years is known to be 2/3 . Find the probability that in the next 30 years at most 3 men will survive.

Concept: Conditional Probability
Chapter: [0.19] Probability Distribution
[4]5.2.2

The surface area of a spherical balloon is increasing at the rate of 2cm2 / sec. At what rate is the volume of the balloon is increasing when the radius of the balloon is 6 cm?

Concept: Rate of Change of Bodies Or Quantities
Chapter: [0.14] Applications of Derivative
[4]5.2.3

The slope of the tangent to the curve at any point is equal to y+ 2x. Find the equation of the curve passing through the origin.

Concept: Differential Equations - Linear Differential Equation
Chapter: [0.17] Differential Equation
[14]6
[6]6.1 | Attempt any TWO of the following :
[3]6.1.1

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]6.1.2

The time ( in minutes) for a lab assistant to prepare the equipment for a certain experiment is a random variable X taking values between 25 and 35 minutes with p.d.f

f(x)=1/10,25<=x<=35=0

 =0    "otherwise"

Wat is the probability that preparation time exceeds 33 minutes? Also find the c.d.f. of X.

Concept: Probability Distribution - Probability Density Function (P.D.F.)
Chapter: [0.19] Probability Distribution
[3]6.1.3

The probability that a certain kind of component will survive a check test is 0.6. Find the probability that exactly 2 of the next 4 tested components survive

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

If ax^2+2hxy+by^2=0 , show that (d^2y)/(dx^2)=0

Concept: Derivatives of Functions in Parametric Forms
Chapter: [0.13] Differentiation
[4]6.2.2

Find the area of the region common to the circle x2 + y2 =9 and the parabola y2 =8x

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.16] Applications of Definite Integral
[4]6.2.3

For 10 pairs of observations on two variables X and Y, the following data are available:

sum(x-2)=30, sum(y-5)=40, sum(x-2)^2=900, sum(y-5)^2=800, sum(x-2)(y-5)=480

Find the correlation coefficient between X and Y.

Concept: Statistics - Karl Pearson’s Coefficient of Correlation
Chapter: [0.18] Statistics

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