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Mathematics and Statistics 2013-2014 HSC Science (Computer Science) 12th Board Exam Question Paper Solution

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Mathematics and Statistics
2013-2014 March
Marks: 80

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

Which of the following represents direction cosines of the line :

(a)`0,1/sqrt2,1/2`

(b)`0,-sqrt3/2,1/sqrt2`

(c)`0,sqrt3/2,1/2`

(d)`1/2,1/2,1/2`

Concept: Direction Cosines and Direction Ratios of a Line
Chapter: [8] Three Dimensional Geometry
[2]1.1.2

`A=[[1,2],[3,4]]` ans A(Adj A)=KI, then the value of 'K' is

(a) 2  

(b) -2

(c) 10

(d) -10

Concept: Determinants - Adjoint Method
Chapter: [2] Matrices
[2]1.1.3
 

The general solution of the trigonometric equation tan2 θ = 1 is..........................

(a)`theta =npi+-(pi/3),n in z`

(b)`theta =npi+-pi/6, n in z`

(c)`theta=npi+-pi/4, n in z`

(d) `0=npi, n in z`

 
Concept: Trigonometric Functions - General Solution of Trigonometric Equation of the Type
Chapter: [3] Trigonometric Functions
[6]1.2 | Attempt any THREE of the following :
[2]1.2.1

If `bara, barb, bar c` are the position vectors of the points A, B, C respectively and ` 2bara + 3barb - 5barc = 0` , then find the ratio in which the point C divides line segment  AB.

Concept: Basic Concepts of Vector Algebra
Chapter: [7] Vectors
[2]1.2.2

The Cartestation equation of  line is `(x-6)/2=(y+4)/7=(z-5)/3` find its vector equation.

Concept: Equation of a Line in Space
Chapter: [9] Line
[2]1.2.3

Equation of a plane is `vecr (3hati-4hatj+12hatk)=8`. Find the length of the perpendicular from the origin to the plane.

Concept: Plane - Equation of Plane Passing Through the Given Point and Perpendicular to Given Vector
Chapter: [10] Plane
[2]1.2.4

Find the acute angle between the lines whose direction ratios are 5, 12, -13 and 3, - 4, 5.

Concept: Angle Between Two Lines
Chapter: [8] Three Dimensional Geometry
[2]1.2.5

Write the dual of the following statements: (p ∨ q) ∧ T

Concept: Mathematical Logic - Statement Patterns and Logical Equivalence
Chapter: [1] Mathematical Logic

Write the dual of the following statements:

Madhuri has curly hair and brown eyes.

Concept: Mathematical Logic - Statement Patterns and Logical Equivalence
Chapter: [1] Mathematical Logic
[14]2
[6]2.1 | Attempt any TWO of the following
[3]2.1.1

If the lines `(x-1)/2=(y+1)/3=(z-1)/4 ` and `(x-3)/1=(y-k)/2=z/1` intersect each other then find value of k

Concept: Pair of Straight Lines - Point of Intersection of Two Lines
Chapter: [4] Pair of Straight Lines
[3]2.1.2

Prove that three vectors `bara, barb and barc ` are coplanar, if and only if, there exists a non-zero linear combination `xbara+ybarb +z barc=0`

Concept: Vectors - Conditions of Coplanarity of Three Vectors
Chapter: [7] Vectors
[3]2.1.3

Using truth table prove that :

`~p^^q-=(p vv q)^^~p`

Concept: Mathematical Logic - Truth Tables of Compound Statements
Chapter: [1] Mathematical Logic
[8]2.2 | Attempt any TWO of the following
[4]2.2.1

In any ΔABC, with usual notations, prove that `b^2=c^2+a^2-2ca cosB`.

Concept: Trigonometric Functions - Solution of a Triangle
Chapter: [3] Trigonometric Functions
[4]2.2.2

Show that the equation `x^2-6xy+5y^2+10x-14y+9=0 ` represents a pair of lines. Find the acute angle between them. Also find the point of intersection of the lines.

Concept: Acute Angle Between the Lines
Chapter: [4] Pair of Straight Lines
[4]2.2.3

Express the following equations in the matrix form and solve them by method of reduction :

2x- y + z = 1, x + 2y + 3z = 8, 3x + y - 4z =1

Concept: Elementary Operation (Transformation) of a Matrix
Chapter: [2] Matrices
[14]3
[6]3.1 | Attempt any TWO of the following :
[3]3.1.1

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 if h2ab0.

Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Homogenous Equation
Chapter: [4] Pair of Straight Lines
[3]3.1.2

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

Concept: Mathematical Logic - Application - Introduction to Switching Circuits
Chapter: [1] Mathematical Logic
[3]3.1.3

if A, B, C, D are (1, i, I), (2, l ,3), (3; 2, 2) and (3, 3, 4) respetivly., then find the volume of the parallepiped with AB, AC and AD as concurrent edges

Concept: Scalar Triple Product of Vectors
Chapter: [7] Vectors
[8]3.2 | Attempt any TWO of the follolving
[4]3.2.1

Find the equation of the plane passing through the line of intersection of planes 2x - y + z = 3 and 4x- 3y + 5z + 9 = 0 and parallel to the line

` (x+1)/2=(y+3)/4=(z-3)/5`

Concept: Plane - Equation of Plane Passing Through the Intersection of Two Given Planes
Chapter: [10] Plane
[4]3.2.2

Minimize :Z=6x+4y

Subject to : 3x+2y ≥12

x+y ≥5

0 ≤x ≤4

0 ≤ y ≤ 4 

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [11] Linear Programming Problems
[4]3.2.3

Show that:

`cos^(-1)(4/5)+cos^(-1)(12/13)=cos^(-1)(33/65)`

Concept: Basic Concepts of Trigonometric Functions
Chapter: [3] Trigonometric Functions
[12]4
[6]4.1 | Select an write the correct answer from the given alternatives in each of the following:
[2]4.1.1

If y =1-cosθ , x = 1-sinθ , then ` dy/dx at " "0 =pi/4`  is ............

Concept: Derivatives of Functions in Parametric Forms
Chapter: [13] Differentiation
[2]4.1.2

The integrating factor of linear differential equation `dy/dx+ysecx=tanx` is

(a)secx- tan x

(b) sec x · tan x

(c)sex+tanx

(d) secx.cotx

Concept: Differential Equations - Linear Differential Equation
Chapter: [17] Differential Equation
[2]4.1.3

The equation of tangent to the curve y = 3x2 - x + 1 at the point (1, 3) is 

(a) y=5x+2

(b)y=5x-2

(c)y=1/5x+2

(d)y=1/5x-2

 

Concept: Conics - Tangents and normals - equations of tangent and normal at a point
Chapter: [6] Conics
[6]4.2 | Attempt any THREE of the following:
[2]4.2.1

Examine the continuity of the function
f(x) =sin x- cos x, for x ≠ 0

      =- 1 ,forx=0

at the poinl x = 0

Concept: Introduction of Continuity
Chapter: [12] Continuity
[2]4.2.2

Verify Rolle's theorem for the function  

f(x)=x2-5x+9 on [1,4]

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

Evaluate : `intsec^nxtanxdx`

Concept: Properties of Definite Integrals
Chapter: [15] Integration
[2]4.2.4

The probability mass function (p.m.f.) of X is given below:

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

 find E(X2)

Concept: Probability Distribution - Probability Mass Function (P.M.F.)
Chapter: [19] Probability Distribution
[2]4.2.5

Given that X~ B(n = 10, p), if E(X) = 8. find the value of p.

Concept: Statistics - Bivariate Frequency Distribution
Chapter: [18] Statistics
[14]5
[6]5.1 | Attempt any TWO of' the following :
[3]5.1.1

Ify y=f(u) is a differentiable function of u and u = g(x) is a differentiable function of x then prove that y = f (g(x)) is a  differentiable function of x and

`(dy)/(dx)=(dy)/(du)*(du)/(dx)`

 

Concept: Derivative - Every Differentiable Function is Continuous but Converse is Not True
Chapter: [13] Differentiation
[3]5.1.2

Obtain the differential equation by eliminating arbitrary constants A, B from the equation -
y = A cos (log x) + B sin (log x)

Concept: Formation of Differential Equation by Eliminating Arbitary Constant
Chapter: [17] Differential Equation
[3]5.1.3

Evaluate : `int x^2/((x^2+2)(2x^2+1))dx` 

Concept: Methods of Integration - Integration Using Partial Fractions
Chapter: [15] Integration
[8]5.2 | Attempt any TWO of the following :
[4]5.2.1

An open box is to be made out of a piece of a square card board of sides 18 cms. by cutting off equal squares from  the comers and tumi11g up the sides. Find the maximum volume of the box.

Concept: Maxima and Minima
Chapter: [14] Applications of Derivative
[4]5.2.2

Prove that :

`int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2a-x)dx`

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [15] Integration
[4]5.2.3

If the function f (x) is continuous in the interval [-2, 2],find the values of a and b where

`f(x)=(sinax)/x-2, for-2<=x<=0`

`=2x+1, for 0<=x<=1`

`=2bsqrt(x^2+3)-1, for 1<x<=2`

Concept: Continuity - Continuity in Interval - Definition
Chapter: [12] Continuity
[14]6
[6]6.1 | Attempt any TWO of the following
[3]6.1.1

Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`

Concept: General and Particular Solutions of a Differential Equation
Chapter: [17] Differential Equation
[3]6.1.2

A fair coin is tossed 8 times. Find the probability that it shows heads at least once

Concept: Conditional Probability
Chapter: [19] Probability Distribution
[3]6.1.3

If xpyq=(x+y)p+q then Prove that `dy/dx=y/x`

 

Concept: Exponential and Logarithmic Functions
Chapter: [12] Continuity [13] Differentiation
[8]6.2 | Attempt any TWO of the following :
[4]6.2.1

Find the area of the sector of a circle bounded by the circle x2 + y2 = 16 and the line y = x in the ftrst quadrant.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [16] Applications of Definite Integral
[4]6.2.2

Prove that `int sqrt(x^2-a^2)dx=x/2sqrt(x^2-a^2)-a^2/2log|x+sqrt(x^2-a^2)|+c`

 

Concept: Methods of Integration - Integration by Parts
Chapter: [15] Integration
[4]6.2.3

A random variable X has the following probability distribution :

X=x 0 1 2 3 4 5 6
P[X=x] k 3k 5k 7k 9k 11k 13k

(a) Find k
(b) find P(O <X< 4)
(c) Obtain cumulative distribution function (c. d. f.) of X.

Concept: Probability Distribution - Cumulative Probability Distribution of a Discrete Random Variable
Chapter: [19] Probability Distribution

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