2015 Hsc Mathematics General 2 Answers

Mathematics and Statistics

Marks: 80 Academic Year: 2014-2015
Date & Time: 8th October 2015, 4:00 pm
Duration: 3h

[6] 1.1 | Select and write the most appropriate answer from the given alternatives in each of the following sub-questions:

[2] 1.1.1

If p ˄ q = F, p → q = F, then the truth value of p and q is :

Concept: Truth Value of Statement
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

[2] 1.1.2

If `A^-1=1/3[[1,4,-2],[-2,-5,4],[1,-2,1]]` and | A | = 3, then (adj. A) = _______

`1/9[[1,4,-2],[-2,-5,4],[1,-2,1]]`

`[[1,-2,1],[4,-5,-2],[-2,4,1]]`

`[[1,4,-2],[-2,-5,4],[1,-2,1]]`

`[[-1,-4,2],[2,5,-4],[1,-2,1]]`

Concept: Determinants - Adjoint Method
Chapter: [0.02] Matrices

[2] 1.1.3

The slopes of the lines given by 12x² + bxy + y² = 0 differ by 7. Then the value of b is :

(A) 2

(B) ± 2

(D) 1

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

In a Δ ABC, with usual notations prove that:` (a -bcos C) /(b -a cos C )= cos B/ cos A`

Concept: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions

[2] 1.2.2

Find 'k', if the equation kxy + 10x + 6y + 4 = 0 represents a pair of straight lines.

Concept: Acute Angle Between the Lines
Chapter: [0.04] Pair of Straight Lines

[2] 1.2.3

If A, B, C, D are four non-collinear points in the plane such that `bar(AD)+bar( BD)+bar( CD)=bar O` then prove that point D is the centroid of the ΔABC.

Concept: Vector and Cartesian Equations of a Line - Centroid Formula for Vector
Chapter: [0.07] Vectors

[2] 1.2.4

Find the direction cosines of the line

`(x+2)/2=(2y-5)/3; z=-1`

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

[2] 1.2.5

Show that the points (1, 1, 1) and (-3, 0, 1) are equidistant from the plane `bar r (3bari+4barj-12bark)+13=0`

Concept: Distance of a Point from a Plane
Chapter: [0.016] Line and Plane [0.1] Plane

[6] 2.1 | Attempt any TWO of the following:

[3] 2.1.1

Using truth table prove that p ↔ q = (p ∧ q) ∨ (~p ∧ ~q).

Concept: Logical Connective, Simple and Compound Statements
Chapter: [0.01] Mathematical Logic [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

[3] 2.1.2

Show that every homogeneous equation of degree two in x and y, i.e., ax² + 2hxy + by² = 0 represents a pair of lines passing through origin ifh ² − a b ≥ 0.

Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Homogenous Equation
Chapter: [0.04] Pair of Straight Lines

[3] 2.1.3

Prove that the volume of a parallelopiped with coterminal edges as ` bara ,bar b , barc `

Hence find the volume of the parallelopiped with coterminal edges `bar i+barj, barj+bark `

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

[8] 2.2 | Attempt any TWO of the following:

[4] 2.2.1

Find the inverse of the matrix, `A=[[1,3,3],[1,4,3],[1,3,4]]`by using column transformations.

Concept: Elementary Transformations
Chapter: [0.02] Matrices

[4] 2.2.2

In ΔABC, prove that : `tan((a-b)/2)=(a-b)/(a+b)cotC/2`

Concept: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions

[4] 2.2.3

Show that the lines ` (x+1)/-3=(y-3)/2=(z+2)/1; ` are coplanar. Find the equation of the plane containing them.

Concept: Coplanarity of Two Lines
Chapter: [0.016] Line and Plane [0.1] Plane

[6] 3.1 | Attempt any TWO of the following:

[3] 3.1.1

Construct the simplified circuit for the following circuit:

Concept: Application of Logic to Switching Circuits
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

[3] 3.1.2

Express `-bari-3barj+4bark ` as a linear combination of vectors `2bari+barj-4bark,2bari-barj+3bark`

Concept: Vector and Cartesian Equations of a Line - Linear Combination of Vectors
Chapter: [0.07] Vectors

[3] 3.1.3

Find the length of the perpendicular from the point (3, 2, 1) to the line `(x-7)/2=(y-7)/2=(z-6)/3`

Concept: Three - Dimensional Geometry - Condition for Perpendicular Lines
Chapter: [0.08] Three Dimensional Geometry

[8] 3.2 | Attempt any TWO of the following

[4] 3.2.1

Show that the angle between any two diagonals of a cube is `cos^-1(1/3)`

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

[4] 3.2.2

Minimize : Z = 6x + 4y

Subject to the conditions:

3x + 2y ≥ 12,

x + y ≥ 5,

0 ≤ x ≤ 4,

0 ≤ y ≤ 4

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

[4] 3.2.3

If `tan^-1((x-1)/(x-2))+cot^-1((x+2)/(x+1))=pi/4; `

Concept: Basic Concepts of Trigonometric Functions
Chapter: [0.03] Trigonometric Functions

[6] 4.1 | Select and write the most appropriate answer from the given alternatives in each of the following sub-questions

[2] 4.1.1

If `y=sec^-1((sqrtx-1)/(x+sqrtx))+sin_1((x+sqrtx)/(sqrtx-1)), `

(A) x

(B) 1/x

(D) 0

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

[2] 4.1.2

If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:

(A) 0

(B) π

(D) π/4

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

[2] 4.1.3

The solution of the differential equation dy/dx = sec x – y tan x is:

(A) y sec x = tan x + c

(B) y sec x + tan x = c

(D) sec x + y tan x = c

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

[6] 4.2 | Attempt any THREE of the following:

[2] 4.2.1

Evaluate: `int1/(xlogxlog(logx))dx`

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

[2] 4.2.2

Find the area bounded by the curve y² = 4ax,x-axis and the lines x = 0 and x = a.

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

[2] 4.2.3

Find k, such that the function P(x)=k(4/x) ;x=0,1,2,3,4 k>0

=0 ,otherwise

Concept: Standard Deviation of Binomial Distribution (P.M.F.)
Chapter: [0.2] Bernoulli Trials and Binomial Distribution

[2] 4.2.4

Given is X ~ B (n, p). If E(X) = 6, and Var(X) = 4.2, find the value of n.

Concept: Bernoulli Trial - Calculation of Probabilities
Chapter: [0.2] Bernoulli Trials and Binomial Distribution

[2] 4.2.5

Solve the differential equation `y-xdy/dx=0`

Concept: Methods of Solving First Order, First Degree Differential Equations - Differential Equations with Variables Separable Method
Chapter: [0.17] Differential Equation

[6] 5.1 | Attempt any TWO of the following:

[3] 5.1.1

Discuss the continuity of the function

`f(x)=(1-sinx)/(pi/2-x)^2, `

= 3, for x=π/2

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

[3] 5.1.2

If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).

Concept: Maxima and Minima
Chapter: [0.022000000000000002] Applications of Derivatives [0.14] Applications of Derivative

[3] 5.1.3

Differentiate `cos^-1((3cosx-2sinx)/sqrt13)` w. r. t. x.

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

[8] 5.2 | Attempt any TWO of the following:

[4] 5.2.1

Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`

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

[4] 5.2.2

A rectangle has area 50 cm² . Find its dimensions when its perimeter is the least

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

[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.

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

[6] 6.1 | Attempt any TWO of the following:

[3] 6.1.1

If y = f (u) is a differential function of u and u = g(x) is a differential function of x, then prove that y = f [g(x)] is a differential function of x and `dy/dx=dy/(du) xx (du)/dx`

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

[3] 6.1.2

Each of the total five questions in a multiple choice examination has four choices, only one of which is correct. A student is attempting to guess the answer. The random variable x is the number of questions answered correctly. What is the probability that the student will give atleast one correct answer?

Concept: Probability Distribution of a Discrete Random Variable
Chapter: [0.19] Probability Distribution

[3] 6.1.3

If f (x) = x 2 + a, for x ≥ 0 ` =2sqrt(x^2+1)+b, ` is continuous at x = 0, find a and b.

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

[8] 6.2 | Attempt any TWO of the following

[4] 6.2.1

Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]

Concept: Maxima and Minima
Chapter: [0.022000000000000002] Applications of Derivatives [0.14] Applications of Derivative

[4] 6.2.2

Solve the differential equation: `x+ydy/dx=sec(x^2+y^2)` Also find the particular solution if x = y = 0.

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

[4] 6.2.3

Find the expected value, variance and standard deviation of random variable X whose probability mass function (p.m.f.) is given below:

X=x	1	2	3
P(X=x)	1/5	2/5	2/5

Concept: Probability Distribution - Expected Value, Variance and Standard Deviation of a Discrete Random Variable
Chapter: [0.19] Probability Distribution