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Mathematics and Statistics 2014-2015 HSC Science (Computer Science) 12th Board Exam Question Paper Solution

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Mathematics and Statistics
2014-2015 March
Marks: 80

[12]1
[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  `A=[[2,0,0],[0,2,0],[0,0,2]]` then A6=  ......................

(a) 6A

(b) 12A

(c) 16A

(d) 32A

 

Concept: Operations on Matrices - Addition of Matrices
Chapter: [2] Matrices
[2]1.1.2

The principal solution of `cos^-1(-1/2)` is :

(a) π/3

(b) π/6

(c) 2π/3

(d) 3π/2

Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch
Chapter: [3] Trigonometric Functions
[2]1.1.3

If an equation hxy + gx + fy + c = 0 represents a pair of lines, then.........................

(a) fg = ch                       (b) gh = cf

(c) Jh = cg                     (d) hf= - eg

Concept: Pair of Straight Lines - Condition for Parallel Lines
Chapter: [4] Pair of Straight Lines
[6]1.2 | Attempt any THREE of the following
[2]1.2.1

Write the converse and contrapositive of the statement — “If two triangles are congruent, then their areas are equal.”

Concept: Mathematical Logic - Statement Patterns and Logical Equivalence
Chapter: [1] Mathematical Logic
[2]1.2.2

Find ‘k' if the sum of slopes of lines represented by equation x2+ kxy - 3y2 = 0 is twice their product.

Concept: Acute Angle Between the Lines
Chapter: [4] Pair of Straight Lines
[2]1.2.3

Find the angle between the planes `bar r.(2bar i+barj-bark)=3 and bar r.(hati+2hatj+hatk)=1`

Concept: Angle Between Two Planes
Chapter: [10] Plane
[2]1.2.4

The Cartesian equations of line are 3x -1 = 6y + 2 = 1 - z. Find the vector equation of line.

Concept: Equation of a Line in Space
Chapter: [9] Line
[2]1.2.5

If `bara=bari+2barj, barb=-2bari+barj,barc=4bari+3barj`,  find x and y such that `barc=xbara+ybarb`

 

Concept: Vectors - Linear Combination of Vectors
Chapter: [7] Vectors
[14]2
[6]2.1 | Attempt any TWO of the following
[3]2.1.1

If A, B, C, D are (1, 1, 1), (2, I, 3), (3, 2, 2), (3, 3, 4) respectively, then find the volume of parallelopiped with AB, AC and AD as the concurrent edges.

Concept: Scalar Triple Product of Vectors
Chapter: [7] Vectors
[3]2.1.2

Discuss the statement pattern, using truth table : ~(~p ∧ ~q) v q

 

Concept: Mathematical Logic - Truth Tables of Compound Statements
Chapter: [1] Mathematical Logic
[3]2.1.3

If point C `(barc)` divides the segment joining the points A(`bara`) and  B(`barb`) internally in the ratio m : n, then prove that `barc=(mbarb+nbara)/(m+n)`

 

 

Concept: Section formula
Chapter: [7] Vectors
[8]2.2 |  Attempt any TWO of the following
[4]2.2.1

Find the direction cosines of the line perpendicular to the lines whose direction ratios are -2, 1,-1 and -3, - 4, 1 

Concept: Direction Cosines and Direction Ratios of a Line
Chapter: [8] Three Dimensional Geometry
[4]2.2.2

In any ΔABC if  a2 , b2 , c2 are in arithmetic progression, then prove that Cot A, Cot B, Cot C are in arithmetic progression.

Concept: Trigonometric Functions - Solution of a Triangle
Chapter: [3] Trigonometric Functions
[4]2.2.3

The sum of three numbers is 6. When second number is subtracted from thrice the sum of first and third number, we get number 10. Four times the sum of third number is subtracted from five times the sum of first and second number, the result is 3. Using above information, find these three numbers by matrix method.

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

If θ is the acute angle between the lines represented by equation ax2 + 2hxy + by2 = 0  then prove that `tantheta=|(2sqrt(h^2-ab))/(a+b)|, a+b!=0`

 

 

 

Concept: Acute Angle Between the Lines
Chapter: [4] Pair of Straight Lines
[3]3.1.2

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]3.1.3

Construct the switching circuit for the following statement : [p v (~ p ∧ q)] v [(- q ∧ r) v ~ p]

Concept: Mathematical Logic - Application - Introduction to Switching Circuits
Chapter: [1] Mathematical Logic
[8]3.2 | Attempt any TWO of the following
[4]3.2.1

Find the general solution of : cos x - sin x = 1.

Concept: Trigonometric Functions - General Solution of Trigonometric Equation of the Type
Chapter: [3] Trigonometric Functions
[4]3.2.2

Find the equations of the planes parallel to the plane x-2y + 2z-4 = 0, which are at a unit distance from the point (1,2, 3).

Concept: Plane - Equation of a Plane Passing Through Three Non Collinear Points
Chapter: [10] Plane
[4]3.2.3

A diet of a sick person must contain at least 48 units of vitamin A and 64 units of vitamin B. Two foods F 1 and F2 are available. Food F1 costs Rs. 6 per unit and food F2 costs Rs. 10 per unit. One unit of food F1 contains 6 units of vitamin A and 7 units of vitamin B. One unit of food F2 contains 8 units of vitamin A and 12 units of vitamin B.Find the minimum cost for the diet that consists of mixture of these two foods and also meeting the minimal nutritional requirements.

Concept: Different Types of Linear Programming Problems
Chapter: [11] Linear Programming Problems
[12]4
[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

A random variable X has the following probability distribution:

then E(X)=....................

(a) 0.8

(b) 0.9

(c) 0.7

(d) 1.1

Concept: Random Variables and Its Probability Distributions
Chapter: [19] Probability Distribution
[2]4.1.2

If `int_0^alpha3x^2dx=8` then the value of α is :

(a) 0

(b) -2

(c) 2 

(d) ±2

Concept: Properties of Definite Integrals
Chapter: [15] Integration
[2]4.1.3

The differential equation of y=c/x+c2 is :

(a)`x^4(dy/dx)^2-xdy/dx=y`

(b)`(d^2y)/dx^2+xdy/dx+y=0`

(c)`x^3(dy/dx)^2+xdy/dx=y`

(d)`(d^2y)/dx^2+dy/dx-y=0`

Concept: General and Particular Solutions of a Differential Equation
Chapter: [17] Differential Equation
[6]4.2 | Attempt any THREE of the following
[2]4.2.1

Evaluate : `int e^x[(sqrt(1-x^2)sin^-1x+1)/(sqrt(1-x^2))]dx`

Concept: Properties of Definite Integrals
Chapter: [15] Integration
[2]4.2.2

If   `y=sqrt(sinx+sqrt(sinx+sqrt(sinx+..... oo))),` then show that `dy/dx=cosx/(2y-1)`

Concept: General and Particular Solutions of a Differential Equation
Chapter: [17] Differential Equation
[2]4.2.3

Evaluate :`int_0^(pi/2)1/(1+cosx)dx`

 

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [15] Integration
[2]4.2.4

If y=eax ,show that  `xdy/dx=ylogy`

Concept: Derivatives of Implicit Functions
Chapter: [13] Differentiation
[2]4.2.5

A fair coin is tossed five times. Find the probability that it shows exactly three times head.

Concept: Conditional Probability
Chapter: [19] Probability Distribution
[14]5
[6]5.1 | Attempt any TWO of the following
[3]5.1.1

Integrate : sec3 x w. r. t. x.

Concept: Methods of Integration - Integration by Parts
Chapter: [15] Integration
[3]5.1.2

If y = (tan-1 x)2, show that `(1+x^2)^2(d^2y)/dx^2+2x(1+x^2)dy/dx-2=0`

Concept: Differential Equations - Linear Differential Equation
Chapter: [17] Differential Equation
[3]5.1.3

If `f(x)=[tan(pi/4+x)]^(1/x), `

           = k                        ,for x=0

is continuous at x=0 , find k.

 

Concept: Continuity - Continuity of a Function at a Point
Chapter: [12] Continuity
[8]5.2 | Attempt any TWO of the following :
[4]5.2.1

Find the co-ordinates of the points on the curve y=x-(4/x) where the tangents are parallel to the line y=2x

Concept: Conics - Tangents from a Point Outside Conics
Chapter: [6] Conics
[4]5.2.2

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]5.2.3

Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`

Concept: Properties of Definite Integrals
Chapter: [15] Integration
[14]6
[6]6.1 | Attempt any two of the following
[3]6.1.1

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: Continuity - Continuity of a Function at a Point
Chapter: [12] Continuity
[3]6.1.2

If  `log_10((x^3-y^3)/(x^3+Y^3))=2 `

 

Concept: Derivatives of Functions in Parametric Forms
Chapter: [13] Differentiation
[3]6.1.3

Let the p. m. f. (probability mass function) of random variable x be

`p(x)=(4/x)(5/9)^x(4/9)^(4-x), x=0, 1, 2, 3, 4`

         =0 otherwise

find E(x) and var (x)

Concept: Probability Distribution - Probability Mass Function (P.M.F.)
Chapter: [19] Probability Distribution
[8]6.2 | Attempt any two of the following
[4]6.2.1

Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x). 

Concept: Maxima and Minima
Chapter: [14] Applications of Derivative
[4]6.2.2

Solve the differential equation (x2 + y2)dx- 2xydy = 0

Concept: Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations
Chapter: [17] Differential Equation
[4]6.2.3

Given the p. d. f. (probability density function) of a continuous random variable x as :

 `f(x)=x^2/3, -1`

         = 0 , otherwise

Determine the c. d. f. (cumulative distribution function) of x and hence find P(x < 1), P(x ≤ -2), P(x > 0), P(1 < x < 2)

Concept: Probability Distribution - Probability Density Function (P.D.F.)
Chapter: [19] Probability Distribution

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