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

Mathematics and Statistics
Date: March 2014

[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: [0.08] Three Dimensional Geometry
[2] 1.1.2

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

2

- 2

10

-10

Chapter: [0.02] Matrices
[2] 1.1.3

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

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

theta =npi+-pi/6, n in z

theta=npi+-pi/4, n in z

0=npi, n in z

Concept: Trigonometric Functions - General Solution of Trigonometric Equation of the Type
Chapter: [0.03] 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: [0.07] 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: [0.013999999999999999] Pair of Straight Lines [0.09] 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: [0.1] 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: [0.08] Three Dimensional Geometry
[2] 1.2.5

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

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

Write the dual of the following statements:

Madhuri has curly hair and brown eyes.

Concept: Statement Patterns and Logical Equivalence
Chapter: [0.01] Mathematical Logic [0.011000000000000001] 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: Distance of a Point from a Line
Chapter: [0.016] Line and Plane [0.09] Line
[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: Vector and Cartesian Equations of a Line - Conditions of Coplanarity of Three Vectors
Chapter: [0.07] Vectors
[3] 2.1.3

Using truth table prove that ∼p ˄ q ≡ (p ˅ q) ˄ ∼p

Concept: Logical Connective, Simple and Compound Statements
Chapter: [0.01] Mathematical Logic [0.01] Mathematical Logic [0.011000000000000001] 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: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions [0.03] 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: [0.04] 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 Transformations
Chapter: [0.02] 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: [0.04] 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: Application of Logic to Switching Circuits
Chapter: [0.01] Mathematical Logic [0.011000000000000001] 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: [0.015] Vectors [0.07] 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: [0.1] 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: [0.017] Linear Programming [0.11] 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: [0.03] 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 - cos θ , x = 1 - sin θ , then  dy/dx  at " "θ =pi/4  is ________

Concept: Derivatives of Functions in Parametric Forms
Chapter: [0.13] 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: [0.17] 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: [0.06] 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: [0.12] 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: [0.14] Applications of Derivative
[2] 4.2.3

Evaluate : intsec^nxtanxdx

Concept: Properties of Definite Integrals
Chapter: [0.15] 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: [0.19] 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: [0.18] 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: The Concept of Derivative - Every Differentiable Function is Continuous but Converse is Not True
Chapter: [0.13] 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: [0.17] 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: [0.023] Indefinite Integration [0.15] 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 turning up the sides. Find the maximum volume of the box.

Concept: Maxima and Minima
Chapter: [0.022000000000000002] Applications of Derivatives [0.14] 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: [0.15] 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: Definition of Continuity - Continuity in Interval - Definition
Chapter: [0.12] 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: [0.17] 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: [0.19] 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: [0.12] Continuity [0.13] 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: [0.16] 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: [0.023] Indefinite Integration [0.15] 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: [0.19] Probability Distribution

#### Request Question Paper

If you dont find a question paper, kindly write to us

View All Requests

#### Submit Question Paper

Help us maintain new question papers on Shaalaa.com, so we can continue to help students

only jpg, png and pdf files

## Maharashtra State Board previous year question papers 12th Board Exam Mathematics and Statistics with solutions 2013 - 2014

Maharashtra State Board 12th Board Exam Maths question paper solution is key to score more marks in final exams. Students who have used our past year paper solution have significantly improved in speed and boosted their confidence to solve any question in the examination. Our Maharashtra State Board 12th Board Exam Maths question paper 2014 serve as a catalyst to prepare for your Mathematics and Statistics board examination.
Previous year Question paper for Maharashtra State Board 12th Board Exam Maths-2014 is solved by experts. Solved question papers gives you the chance to check yourself after your mock test.
By referring the question paper Solutions for Mathematics and Statistics, you can scale your preparation level and work on your weak areas. It will also help the candidates in developing the time-management skills. Practice makes perfect, and there is no better way to practice than to attempt previous year question paper solutions of Maharashtra State Board 12th Board Exam.

How Maharashtra State Board 12th Board Exam Question Paper solutions Help Students ?
• Question paper solutions for Mathematics and Statistics will helps students to prepare for exam.
• Question paper with answer will boost students confidence in exam time and also give you an idea About the important questions and topics to be prepared for the board exam.
• For finding solution of question papers no need to refer so multiple sources like textbook or guides.