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

Mathematics and Statistics
Date & Time: 26th February 2016, 11:00 am
Duration: 3h

[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

The negation of p ∧ (q → r) is ______________.

p ∨ ( ~q ∨ r )

~p ∧ ( q → r )

~p ∧ ( ~q → ~r )

~p ∨ ( q ∧ ~r )

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

If sin^-1(1-x) -2sin^-1x = pi/2 then x is

1. -1/2
2. 1
3. 0
4. 1/2

Concept: Basic Concepts of Trigonometric Functions
Chapter: [0.03] Trigonometric Functions
[2] 1.1.3

The joint equation of the pair of lines passing through (2,3) and parallel to the coordinate axes is

1.  xy -3x - 2y + 6 = 0
2. xy +3x + 2y + 6 = 0
3. xy = 0
4. xy - 3x - 2y - 6 = 0
Concept: Pair of Straight Lines - Pair of Lines Not Passing Through Origin-combined Equation of Any Two Lines
Chapter: [0.04] Pair of Straight Lines
[6] 1.2 | Attempt any 3 of the following
[2] 1.2.1

Find (AB)-1 if

A=[(1,2,3),(1,-2,-3)], B=[(1,-1),(1,2),(1,-2)]

Concept: Matrices - Inverse by Elementary Transformation
Chapter: [0.02] Matrices
[2] 1.2.2

Find the vector equation of the plane passing through a point having position vector 3 hat i- 2 hat j + hat k and perpendicular to the vector 4 hat i + 3 hat j + 2 hat k

Concept: Vector and Cartesian Equation of a Plane
Chapter: [0.1] Plane
[2] 1.2.3

If  bar p = hat i - 2 hat j + hat k and bar q = hat i + 4 hat j - 2 hat k are position vector (P.V.) of points P and Q, find the position vector of the point R which divides segment PQ internally in the ratio 2:1

Concept: Section Formula
Chapter: [0.015] Vectors [0.07] Vectors
[2] 1.2.4

Find k, if one of the lines given by 6x2 + kxy + y2 = 0 is 2x + y = 0

Concept: Pair of Straight Lines - Pair of Lines Not Passing Through Origin-combined Equation of Any Two Lines
Chapter: [0.04] Pair of Straight Lines
[2] 1.2.5

If the lines

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

are at right angle then find the value of k

Concept: Shortest Distance Between Two Lines
Chapter: [0.09] Line
[14] 2
[6] 2.1 | Attempt any TWO of the following
[5] 2.1.1

Examine whether the following logical statement pattern is a tautology, contradiction, or contingency.
[(p → q) ∧ q] → p

Concept: Statement Patterns and Logical Equivalence
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[3] 2.1.2

By vector method prove that the medians of a triangle are concurrent.

Concept: Vector and Cartesian Equations of a Line - Medians of a Triangle Are Concurrent
Chapter: [0.07] Vectors
[3] 2.1.3

Find the shortest distance between the lines

bar r = (4 hat i - hat j) + lambda(hat i + 2 hat j - 3 hat k)

and

bar r = (hat i - hat j + 2 hat k) + mu(hat i + 4 hat j -5 hat k)

where λ and μ are parameters

Concept: Shortest Distance Between Two Lines
Chapter: [0.09] Line
[8] 2.2 | Attempt any TWO of the following :
[4] 2.2.1

In Δ ABC with the usual notations prove that (a-b)^2 cos^2(C/2)+(a+b)^2sin^2(C/2)=c^2

Concept: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions
[4] 2.2.2

Minimize z=4x+5y  subject to 2x+y>=7, 2x+3y<=15, x<=3,x>=0, y>=0 solve using graphical method.

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.017] Linear Programming [0.11] Linear Programming Problems
[4] 2.2.3

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
[14] 3
[6] 3.1 | Attempt any TWO of the following:
[3] 3.1.1

Find the volume of tetrahedron whose coterminus edges are 7hat i+hatk; 2hati+5hatj-3hatk and 4 hat i+3hatj+hat k

Concept: Three Dimensional Geometry - Problems
Chapter: [0.08] Three Dimensional Geometry
[3] 3.1.2

Without using truth tabic show that ~(p v q)v(~p ∧ q) = ~p

Concept: Algebra of Statements
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[3] 3.1.3

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: [0.04] Pair of Straight Lines
[8] 3.2 | Attempt any TWO of the following
[4] 3.2.1

If a line drawn from the point A( 1, 2, 1) is perpendicular to the line joining P(1, 4, 6) and Q(5, 4, 4) then find the co-ordinates of the foot of the perpendicular.

Concept: Equation of a Line in Space
Chapter: [0.013999999999999999] Pair of Straight Lines [0.09] Line
[4] 3.2.2

Find the vector equation of the plane passing through the points hati +hatj-2hatk, hati+2hatj+hatk,2hati-hatj+hatk. Hence find the cartesian equation of the plane.

Concept: Vector and Cartesian Equation of a Plane
Chapter: [0.1] Plane
[4] 3.2.3

Find the general solution of sin x+sin3x+sin5x=0

Concept: Trigonometric Functions - General Solution of Trigonometric Equation of the Type
Chapter: [0.03] Trigonometric Functions
[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

if the function

f(x)=k+x, for x<1

=4x+3, for x>=1

id continuous at x=1 then k=

(a) 7

(b) 8

(c) 6

(d) -6

Concept: Definition of Continuity - Continuity of a Function at a Point
Chapter: [0.12] Continuity
[2] 4.1.2

The equation of tangent to the curve y=y=x^2+4x+1 at

(-1,-2) is...............

(a)  2x -y = 0                        (b)  2x+y-5 = 0

(c)  2x-y-1=0                        (d)  x+y-1=0

Concept: Conics - Tangents and normals - equations of tangent and normal at a point
Chapter: [0.06] Conics
[2] 4.1.3

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

p is ...........

(a) 0.6

(b) 0.7

(c) 0.8

(d) 0.4

Concept: Bernoulli Trials and Binomial Distribution
Chapter: [0.027999999999999997] Binomial Distribution [0.2] Bernoulli Trials and Binomial Distribution
[6] 4.2 | Attempt any THREE of the following:
[2] 4.2.1

if y=x^x find (dy)/(dx)

Concept: Derivative - Exponential and Log
Chapter:
[2] 4.2.2

The displacement 's' of a moving particle at time 't' is given by s = 5 + 20t — 2t2. Find its acceleration when the velocity is zero.

Concept: Maxima and Minima in Closed Interval
Chapter: [0.14] Applications of Derivative
[2] 4.2.3

Find the area bounded by the curve y2 = 4axx-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] 4.2.4

The probability distribution of a discrete random variable X is:

 X=x 1 2 3 4 5 P(X=x) k 2k 3k 4k 5k

find P(X≤4)

Concept: Probability Distribution of a Discrete Random Variable
Chapter: [0.19] Probability Distribution
[2] 4.2.5

Evaluate : int (sinx)/sqrt(36-cos^2x)dx

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.023] Indefinite Integration [0.15] Integration
[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: The Concept of Derivative - Every Differentiable Function is Continuous but Converse is Not True
Chapter: [0.13] Differentiation
[3] 5.1.2

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
[3] 5.1.3

Evaluate : int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.023] Indefinite Integration [0.15] Integration
[8] 5.2 | Attempt any TWO of the following
[4] 5.2.1

Discuss the continuity of the following functions. If the function have a removable discontinuity, redefine the function so as to remove the discontinuity

f(x)=(4^x-e^x)/(6^x-1)  for x ≠ 0

=log(2/3)  for x=0

Concept: Concept of Continuity
Chapter: [0.12] Continuity
[4] 5.2.2

Prove that : int sqrt(a^2-x^2)dx=x/2sqrt(a^2-x^2)=a^2/2sin^-1(x/a)+c

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.15] Integration
[4] 5.2.3

A body is heated at 110°C and placed in air at 10°C. After 1 hour its temperature is 60°C. How much additional time is required for it to cool to 35°C?

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

Prove that: int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2a-x)dx

Concept: Properties of Definite Integrals
Chapter: [0.15] Integration
[3] 6.1.2

Evaluate:  int (1+logx)/(x(2+logx)(3+logx))dx

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.15] Integration
[3] 6.1.3

If y=cos^-1(2xsqrt(1-x^2)), find dy/dx

Concept: The Concept of Derivative - Derivative of Inverse Function
Chapter: [0.13] Differentiation
[8] 6.2 | Attempt any TWO of the following :
[4] 6.2.1

Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.

Concept: General and Particular Solutions of a Differential Equation
Chapter: [0.17] Differential Equation
[4] 6.2.2

A wire of length l is cut into two parts. One part is bent into a circle and other into a square. Show that the sum of areas of the circle and square is the least, if the radius of circle is half the side of the square.

Concept: Maxima and Minima - Introduction of Extrema and Extreme Values
Chapter: [0.14] Applications of Derivative
[4] 6.2.3

The following is the p.d.f. (ProbabiIity Density Function) of a continuous random variable X :

f(x)=x/32,0<x<8

= 0 otherwise

(a) Find the expression for c.d.f. (Cumulative Distribution Function) of X.

(b) Also find its value at x = 0.5 and 9.

Concept: Probability Distribution - Probability Density Function (P.D.F.)
Chapter: [0.19] Probability Distribution

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