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

Mathematics and Statistics
Date: October 2012

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

If A = {2, 3, 4, 5, 6}, then which of the following is not true?

(A) ∃ x ∈ A such that x + 3 = 8

(B) ∃ x ∈ A such that x + 2 < 5

(C) ∃ x ∈ A such that x + 2 < 9

(D) ∀ x ∈ A such that x + 6 ≥ 9

Concept: Mathematical Logic - Algebra of Statements
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[2]1.1.2

If 2x + y = 0 is one of the lines represented by 3x2 + kxy + 2y2 = 0, then the value of k is

1/2

11/2

5/2

(-11)/2

Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation
Chapter: [0.04] Pair of Straight Lines
[2]1.1.3

If a line is inclined at 60° and 30° with the X and Y-axes respectively, then the angle which it makes with Z-axis is

(A) 0

(B) pi/4

(C) pi/2

(D) pi/6

Concept: Acute Angle Between the Lines
Chapter: [0.04] Pair of Straight Lines
[6]1.2 | Attempt any THREE of the following
[2]1.2.1

If A = [(1,2),(3,4)] and AX = I then find X by using elementary transformations

Concept: Matrices - Elementary Transformation of a Matrix Revision of Cofactor and Minor
Chapter: [0.02] Matrices
[2]1.2.2

With usual notations, in ΔABC, prove that a(b cos C − c cos B) = b2 − c2

Concept: Trigonometric Functions - Solution of a Triangle
Chapter: [0.03] Trigonometric Functions
[2]1.2.3

Show that the equation of a tangent to the circle x2 + y2 = a2 at the point P(x1,y1) on it is xx1 + yy1 = a2

Concept: Circle - Tangents to a Circle from a Point Outside the Circle
Chapter: [0.05] Circle
[2]1.2.4

Find k, if the line 2x − 3y + k = 0 touches the ellipse 5x2 + 9y2 = 45.

Concept: Circle - Tangent of a Circle - Equation of a Tangent at a Point to General Circle
Chapter: [0.05] Circle
[2]1.2.5

Find the co-ordinates of the point, which divides the line segment joining the points A(2, − 6, 8) and B(− 1, 3, − 4) externally in the ratio 1 : 3.

Concept: Concept of Line - Distance of a Point from a Line
Chapter: [0.016] Line and Plane [0.09] Line
[14]2
[6]2.1 | Attempt any TWO of the following:
[3]2.1.1

Using truth table, prove that ~ p ∧ q ≡ (p ∨ q) ∧ ~ p

Concept: Mathematical Logic - Truth Tables of Compound Statements
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[3]2.1.2

Find the values of p and q, if the following equation represents a pair of perpendicular lines:
px2 − 8xy + 3y2 + 14x + 2y + q = 0.

Concept: Pair of Straight Lines - Condition for Perpendicular Lines
Chapter: [0.04] Pair of Straight Lines
[3]2.1.3

Find the equations of tangents to the parabola y2 = 12x from the point (2, 5).

Concept: Circle - Condition of tangency
Chapter: [0.05] Circle
[8]2.2 | Attempt any TWO of the following:
[4]2.2.1

The cost of 2 books, 6 notebooks and 3 pens is  Rs 40. The cost of 3 books, 4 notebooks and 2 pens is Rs 35, while the cost of 5 books, 7 notebooks and 4 pens is Rs 61. Using this information and matrix method, find the cost of 1 book, 1 notebook and 1 pen separately.

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

Prove that sin^(−1) (-1/2) + cos^(-1) (-sqrt3/2) = cos^(-1) (-1/2)

Concept: Trigonometric Functions - Trigonometric equations
Chapter: [0.03] Trigonometric Functions
[4]2.2.3

Show that the product of lengths of perpendicular segments drawn from the foci to any tangent to the hyperbola x^2/25 + y^2/16 = 1 is equal to 16.

Concept: Conics - Tangents and normals - equations of tangent and normal at a point
Chapter: [0.06] Conics
[14]3
[6]3.1 | Attempt any TWO of the following:
[3]3.1.1

Construct the new switching circuit for the following circuit with only one switch by simplifying the given circuit:

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 locus of a point, the tangents from which to the circle x2 + y2 = a2 are mutually perpendicular

Concept: Conics - Locus of Points from Which Two Tangents Are Mutually Perpendicular
Chapter: [0.06] Conics
[3]3.1.3

Find the shortest distance between the lines

(x+1)/7 = (y + 1)/(-6) = (z + 1)/1 and (x - 3)/1 = (y - 5)/(-2) = (z - 7)/1

Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation
Chapter: [0.04] Pair of Straight Lines
[8]3.2 | Attempt any TWO of the following:
[4]3.2.1

Find the angle between the line (x - 1)/3 = (y + 1)/2 = (z + 2)/4 and the plane 2x + y − 3z + 4 = 0.

Concept: Angle Between Line and a Plane
Chapter: [0.1] Plane
[4]3.2.2

Solve the following L. P. P. graphically:Linear Programming

Minimize Z = 6x + 2y

Subject to

5x + 9y ≤ 90

x + y ≥ 4

y ≤ 8

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 volume of a tetrahedron whose vertices are A(−1, 2, 3), B(3, −2, 1), C(2, 1, 3) and D(−1, −2, 4).

Concept: Vectors - Collinearity and Coplanarity of Vectors
Chapter: [0.07] Vectors
[12]4
[6]4.1 | Select and write the correct answer from the given alternatives in each of the following
[2]4.1.1

If xy = ex−y , then dy/dx = ______

A) (1+x)/(1 + log x)

B) log x/(1 + log x)^2

C) (1 - log x)/(1 + log x)

D) (1-x)/(1 + log x)

Concept: Continuity of Some Standard Functions - Trigonometric Function
Chapter: [0.12] Continuity
[2]4.1.2

int 1/(1 + cos x) dx = _____

A) tan(x/2) + c

B) 2 tan (x/2) + c

C) -cot (x/2) + c

D) -2 cot (x/2) + c

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.15] Integration
[2]4.1.3

If X ~ B (n, p) and E(X) = 12, Var(X) = 4, then the value of n is _______

(A) 3

(B) 48

(C) 18

(D) 36

Concept: Bernoulli Trial - Calculation of Probabilities
Chapter: [0.2] Bernoulli Trials and Binomial Distribution
[6]4.2 | Attempt any THREE of the following
[2]4.2.1

Find the equation of tangent to the curve y = 3x2 − x + 1 at P(1, 3).

Concept: Conics - Tangents and normals - equations of tangent and normal at a point
Chapter: [0.06] Conics
[2]4.2.2

Evaluate: int 1/(x(x-1)) dx

Concept: Methods of Integration - Integration by Substitution
Chapter: [0.023] Indefinite Integration [0.15] Integration
[2]4.2.3

Solve the differential equation y − x = dy/dx = 0

Concept: Differential Equations - Applications of Differential Equation
Chapter: [0.17] Differential Equation
[2]4.2.4

In a bivariate data, n = 10, bar x = 25, bary = 30 and sum xy = 7900. Find cov(X,Y)

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

A random variable X ~ N (0, 1). Find P(X > 0) and P(X < 0).

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027000000000000003] Probability Distributions [0.19] Probability Distribution
[14]5
[6]5.1 | Attempt any TWO of the following:
[3]5.1.1

Examine the function for maximum and minimum f(x) = x3 − 9x2 + 24x.

Concept: Maxima and Minima - Introduction of Extrema and Extreme Values
Chapter: [0.14] Applications of Derivative
[3]5.1.2

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/dx≠0

Concept: Derivative - Derivative of Inverse Function
Chapter: [0.13] Differentiation
[3]5.1.3

The probability distribution of X, the number of defects per 10 metres of a fabric is given by

 x 0 1 2 3 4 P(X = x) 0.45 0.35 0.15 0.03 0.02

Find the variance of X

Concept: Probability Distribution - Expected Value, Variance and Standard Deviation of a Discrete Random Variable
Chapter: [0.19] Probability Distribution
[8]5.2 | Attempt any TWO of the following:
[4]5.2.1

If sqrt(1-x^2)  + sqrt(1- y^2) =  a(x − y), show that dy/dx = sqrt((1-y^2)/(1-x^2))

Concept: Derivatives of Inverse Trigonometric Functions
Chapter: [0.13] Differentiation
[4]5.2.2

Solve the differential equation cos^2 x dy/dx + y = tan x

Concept: General and Particular Solutions of a Differential Equation
Chapter: [0.17] Differential Equation
[4]5.2.3

Find the area of the region bounded by the curves y2 = 4x and 4x2 + 4y2 = 9 with x >= 0.

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

Find the approximate value of tan−1 (1.001).

Concept: Derivatives of Inverse Trigonometric Functions
Chapter: [0.13] Differentiation
[3]6.1.2

Examine continuity of the function f(x) at x = 0, where

f(x) = (10^x + 7^x - 14^x - 5^x)/(1-cos 4x) , " for " x != 0

= 10/7 , " for"  x = 0

Concept: Definition of Continuity - Continuity of a Function at a Point
Chapter: [0.12] Continuity
[3]6.1.3

The probability that a person who undergoes kidney operation will recover is 0.5. Find the probability that of the six patients who undergo similar operations,

(a) None will recover

(b) Half of them will recover.

Concept: Probability Distribution of a Discrete Random Variable
Chapter: [0.19] Probability Distribution
[8]6.2 | Attempt any TWO of the following:
[4]6.2.1

Prove that:

int sqrt(a^2 +x^2)dx = x/2 sqrt(a^2 + x^2) + a^2/2 log |x + sqrt(x^2 + a^2)| + c

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

Find the volume of the solid generated, when the area between ellipse 4x2 + 9y2 = 36 and the chord AB, with A (3, 0), B (0, 2), is revolved about X-axis.

Concept: Circle - Tangent of a Circle - Equation of a Tangent at a Point to Standard Circle
Chapter: [0.05] Circle
[4]6.2.3

Find Karl Pearson’s coefficient of correlation between the variables X and Y for the following data

 X 11 7 9 5 8 6 10 Y 10 8 6 5 9 7 11
Concept: Statistics - Karl Pearson’s Coefficient of Correlation
Chapter: [0.18] Statistics

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