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

Mathematics and Statistics
Date & Time: 28th February 2015, 11:00 am
Duration: 3h

Section I
 1
 1.1 | Select and write the most appropriate answer from the given alternatives in each of the following sub-questions :
 1.1.1

if  A=[[2,0,0],[0,2,0],[0,0,2]] then A6=  ......................

6A

12A

16A

32A

Concept: Algebraic Operations on Matrices - Addition of Matrices
Chapter: [0.02] Matrices
 1.1.2

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

pi/3

pi/6

(2pi)/3

(3pi)/2

Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch
Chapter: [0.03] Trigonometric Functions
 1.1.3

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

fg = ch

gh = cf

Jh = cg

hf= - eg

Concept: Pair of Straight Lines - Condition for Parallel Lines
Chapter: [0.04] Pair of Straight Lines
 1.2 | Attempt any THREE of the following
 1.2.1

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

Concept: Statement Patterns and Logical Equivalence
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
 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: [0.04] Pair of Straight Lines
 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: [0.1] Plane
 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: [0.013999999999999999] Pair of Straight Lines [0.09] Line
 1.2.5

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

Concept: Vector and Cartesian Equations of a Line - Linear Combination of Vectors
Chapter: [0.07] Vectors
 2
 2.1 | Attempt any TWO of the following
 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: [0.015] Vectors [0.07] Vectors
 2.1.2

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

Concept: Logical Connective, Simple and Compound Statements
Chapter: [0.01] Mathematical Logic [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
 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: [0.015] Vectors [0.07] Vectors
 2.2 |  Attempt any TWO of the following
 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: [0.08] Three Dimensional Geometry
 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: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions
 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 Transformations
Chapter: [0.02] Matrices
 3
 3.1 | Attempt any TWO of the following
 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: [0.04] Pair of Straight Lines
 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: Distance of a Point from a Line
Chapter: [0.016] Line and Plane [0.09] Line
 3.1.3

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

Concept: Application of Logic to Switching Circuits
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
 3.2 | Attempt any TWO of the following
 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: [0.03] Trigonometric Functions
 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: [0.1] Plane
 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: [0.11] Linear Programming Problems
Section II
 4
 4.1 | Select and write the most appropriate answer from the given alternatives in each of the following sub-questions
 4.1.1

A random variable X has the following probability distribution:

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

0.8

0.9

0.7

1.1

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027000000000000003] Probability Distributions [0.19] Probability Distribution
 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: [0.15] Integration
 4.1.3

The differential equation of y=c/x+c^2 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: [0.17] Differential Equation
 4.2 | Attempt any THREE of the following
 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: [0.15] Integration
 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: [0.17] Differential Equation
 4.2.3

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

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.15] Integration
 4.2.4

If y=eax ,show that  xdy/dx=ylogy

Concept: Derivatives of Implicit Functions
Chapter: [0.021] Differentiation [0.13] Differentiation
 4.2.5

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

Concept: Conditional Probability
Chapter: [0.19] Probability Distribution
 5
 5.1 | Attempt any TWO of the following
 5.1.1

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

Concept: Methods of Integration: Integration by Parts
Chapter: [0.023] Indefinite Integration [0.15] Integration
 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: [0.17] Differential Equation
 5.1.3

If f(x)=[tan(pi/4+x)]^(1/x),

= k                        ,for x=0

is continuous at x=0 , find k.

Concept: Definition of Continuity - Continuity of a Function at a Point
Chapter: [0.12] Continuity
 5.2 | Attempt any TWO of the following :
 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: [0.06] Conics
 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: [0.023] Indefinite Integration [0.15] Integration
 5.2.3

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

Concept: Properties of Definite Integrals
Chapter: [0.15] Integration
 6
 6.1 | Attempt any two of the following
 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: Definition of Continuity - Continuity of a Function at a Point
Chapter: [0.12] Continuity
 6.1.2

If  log_10((x^3-y^3)/(x^3+y^3))=2 "then show that"  dy/dx = [-99x^2]/[101y^2]

Concept: Derivatives of Functions in Parametric Forms
Chapter: [0.13] Differentiation
 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: [0.19] Probability Distribution
 6.2 | Attempt any two of the following
 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: [0.022000000000000002] Applications of Derivatives [0.14] Applications of Derivative
 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: [0.026000000000000002] Differential Equations [0.17] Differential Equation
 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: [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 Standard Board Exam Mathematics and Statistics with solutions 2014 - 2015

Maharashtra State Board 12th Standard Board Exam Maths and Stats 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 Standard Board Exam Maths and Stats question paper 2015 serve as a catalyst to prepare for your Mathematics and Statistics board examination.
Previous year Question paper for Maharashtra State Board 12th Standard Board Exam Maths and Stats-2015 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 Standard Board Exam.

How Maharashtra State Board 12th Standard 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.