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Mathematics and Statistics 2015-2016 HSC Science (Computer Science) 12th Board Exam Question Paper Solution

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Mathematics and Statistics
Marks: 80 Academic Year: 2015-2016
Date: July 2016
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[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

Inverse of the statement pattern (p ∨ q) → (p ∧ q) is 

(A) (p ∧ q) → (p ∨ q)

(B) ∼ (p ∨ q) → (p ∧ q)

(C) (∼ p ∨ ∼ q) → (∼ p ∧ ∼ q)

(D) (∼ p ∧ ∼ q) → (∼ p ∨ ∼ q)

Concept: Mathematical Logic - Sentences and Statement in Logic
Chapter: [0.01] Mathematical Logic
[2] 1.1.2

If the vectors `2hati-qhatj+3hatk and 4hati-5hatj+6hatk` are collinear, then value of q is

(A) 5

(B) 10

(C) 5/2

(D) 5/4

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

If in ∆ABC with usual notations a = 18, b = 24, c = 30 then sin A/2 is equal to

(A) `1/sqrt5`

(B) `1/sqrt10`

(C) `1/sqrt15`

(D) `1/(2sqrt5)`

Concept: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions
[6] 1.2 | Attempt any THREE of the following:
[2] 1.2.1

Find the angle between the lines `barr=3hati+2hatj-4hatk+lambda(hati+2hatj+2hatk)` and `barr=5 hati-2hatk+mu(3hati+2hatj+6hatk)`

Concept: Acute Angle Between the Lines
Chapter: [0.04] Pair of Straight Lines
[2] 1.2.2

If p, q, r are the statements with truth values T, F, T, respectively then find the truth value of (r ∧ q) ↔ ∼ p

Concept: Truth Value of Statement
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[2] 1.2.3

If `A =[[2,-3],[3,5]]` then find A-1 by adjoint method. 

Concept: Determinants - Adjoint Method
Chapter: [0.02] Matrices
[2] 1.2.4

By vector method show that the quadrilateral with vertices A (1, 2, –1), B (8, –3, –4), C (5, –4, 1), D (–2, 1, 4) is a parallelogram.

Concept: Vector and Cartesian Equations of a Line - Diagonals of a Parallelogram Bisect Each Other and Converse
Chapter: [0.07] Vectors
[2] 1.2.5

Find the general solution of the equation sin x = tan x.

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

Find the joint equation of pair of lines passing through the origin and perpendicular to the lines represented by ax2+ 2hxy + by2= 0

Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Homogenous Equation
Chapter: [0.04] Pair of Straight Lines
[3] 2.1.2

Find the principal value of `sin^-1(1/sqrt2)`

Concept: Basic Concepts of Trigonometric Functions
Chapter: [0.03] Trigonometric Functions
[3] 2.1.3

Find the cartesian form of the equation of the plane `bar r=(hati+hatj)+s(hati-hatj+2hatk)+t(hati+2hatj+hatj)`

Concept: Vector and Cartesian Equation of a Plane
Chapter: [0.1] Plane
[8] 2.2 | Attempt any TWO of the following:
[4] 2.2.1

Simplify the following circuit so that new circuit has minimum number of switches. Also draw simplified circuit.

 

Concept: Application of Logic to Switching Circuits
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[4] 2.2.2

A line makes angles of measures 45° and 60° with positive direction of y and z axes respectively. Find the d.c.s. of the line and also find the vector of  magnitude 5 along the direction of line.

Concept: Concept of Line - Equation of Line Passing Through Given Point and Parallel to Given Vector
Chapter: [0.09] Line
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[4] 2.2.3

Solve the following LPP by graphical method:

Maximize: z = 3x + 5y
Subject to:  x + 4y ≤ 24
                  3x + y ≤ 21
                  x + y ≤ 9
                  x ≥ 0, y ≥ 0

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.017] Linear Programming [0.11] Linear Programming Problems
[14] 3
[6] 3.1 | Attempt any TWO of the following:
[3] 3.1.1

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: Shortest Distance Between Two Lines
Chapter: [0.09] Line
[3] 3.1.2

Show that the points (1, –1, 3) and (3, 4, 3) are equidistant from the plane 5x + 2y – 7z + 8 = 0

Concept: Distance of a Point from a Plane
Chapter: [0.016] Line and Plane [0.1] Plane
[3] 3.1.3

In any triangle ABC with usual notations prove c = a cos B + b cos A

Concept: Trigonometric Functions - General Solution of Trigonometric Equation of the Type
Chapter: [0.03] Trigonometric Functions
[8] 3.2 | Attempt any TWO of the following:
[4] 3.2.1

Find p and k if the equation px2 – 8xy + 3y+14x + 2y + k = 0 represents a pair of perpendicular lines.

Concept: Concept of Line - Equation of Line Passing Through Given Point and Parallel to Given Vector
Chapter: [0.09] Line
[4] 3.2.2

The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is Rs. 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is Rs. 90 whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is Rs. 70. Find the cost of each item per dozen by using matrices.

Concept: Elementary Transformations
Chapter: [0.02] Matrices
[4] 3.2.3

Find the volume of the parallelopiped whose coterminus edges are given by vectors `2hati+5hatj-4hatk, 5hati+7hatj+5hatk and 4hati+5hatj-2hatk`

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

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
[2] 4.1.2

`∫_4^9 1/sqrtxdx=`_____

(A) 1

(B) –2

(C) 2

(D) –1

Concept: Properties of Definite Integrals
Chapter: [0.15] Integration
[2] 4.1.3

If the p.d.f. of a continuous random variable X is given as

`f(x)=x^2/3` for -1< x<2

       =0   otherwise

then c.d.f. fo X is

(A) `x^3/9+1/9`

(B) `x^3/9-1/9`

(C) `x^2/4+1/4`

(D) `1/(9x^3)+1/9`

Concept: Probability Distribution - Probability Density Function (P.D.F.)
Chapter: [0.19] Probability Distribution
[6] 4.2 | Attempt any THREE of the following:
[2] 4.2.1

If `y = sec sqrtx` then find dy/dx.

Concept: The Concept of Derivative - Derivative of Functions in Product of Function Form
Chapter: [0.13] Differentiation
[2] 4.2.2

Evaluate : `∫(x+1)/((x+2)(x+3))dx`

Concept: Methods of Integration: Integration Using Partial Fractions
Chapter: [0.023] Indefinite Integration [0.15] Integration
[2] 4.2.3

Find the area of the region lying in the first quandrant bounded by the curve y2= 4x, X axis and the lines x = 1, x = 4

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

For the differential equations find the general solution:

sec2 x tan y dx + sec2 y tan x dy = 0

Concept: Methods of Solving First Order, First Degree Differential Equations - Differential Equations with Variables Separable Method
Chapter: [0.17] Differential Equation
[2] 4.2.5

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
[14] 5
[6] 5.1 | Attempt any TWO of the following:
[2] 5.1.1

If the function `f(x)=(4^sinx-1)^2/(xlog(1+2x))`  for x ≠ 0 is continuous at x = 0, find f (0).

Concept: Continuity of Some Standard Functions - Trigonometric Function
Chapter: [0.12] Continuity
[2] 5.1.2

Evaluate : `∫1/(3+2sinx+cosx)dx`

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.023] Indefinite Integration [0.15] Integration
[2] 5.1.3

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: The Concept of Derivative - Derivative of Inverse Function
Chapter: [0.13] Differentiation
[8] 5.2 | Attempt any TWO of the following:
[4] 5.2.1

A point source of light is hung 30 feet directly above a straight horizontal path on which a man of 6 feet in height is walking. How fast will the man’s shadow lengthen and how fast will the tip of shadow move when he is walking away from the light at the rate of 100 ft/min.

Concept: Rate of Change of Bodies Or Quantities
Chapter: [0.14] Applications of Derivative
[4] 5.2.2

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
[4] 5.2.3

Prove that `int_0^af(x)dx=int_0^af(a-x) dx`

hence evaluate `int_0^(pi/2)sinx/(sinx+cosx) dx`

Concept: Properties of Definite Integrals
Chapter: [0.15] Integration
[14] 6
[6] 6.1 | Attempt any TWO of the following:
[3] 6.1.1

If y = etan x+ (log x)tan x then find dy/dx

Concept: General and Particular Solutions of a Differential Equation
Chapter: [0.17] Differential Equation
[3] 6.1.2

If the probability that a fluorescent light has a useful life of at least 800 hours is 0.9, find the probabilities that among 20 such lights at least 2 will not have a useful life of at least 800 hours. [Given : (0⋅9)19 = 0⋅1348]

 

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027000000000000003] Probability Distributions [0.19] Probability Distribution
[3] 6.1.3

Find a and b, so that the function f(x) defined by

f(x)=-2sin x,       for -π≤ x ≤ -π/2

     =a sin x+b,  for -π/2≤ x ≤ π/2

     =cos x,        for π/2≤ x ≤ π

is continuous on [- π, π]

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 equation of a curve passing through the point (0, 2), given that the sum of the coordinates of any point on the curve exceeds the slope of the tangent to the curve at that point by 5

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.16] Applications of Definite Integral
[4] 6.2.2

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
[4] 6.2.3

Find the approximate value of log10 (1016) given that log10= 0⋅4343.

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