HSC Arts 12th Board ExamMaharashtra State Board
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# Question Paper Solutions - Mathematics and Statistics 2014 - 2015 HSC Arts 12th Board Exam

SubjectMathematics and Statistics
Year2014 - 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

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

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

Chapter: [4] Pair of Straight Lines
Concept: Pair of Straight Lines - Condition for Parallel 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.”

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

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

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

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

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

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

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

Chapter: [7] Vectors
Concept: Vectors - Linear Combination of 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.

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

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

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

Chapter: [7] Vectors
Concept: Section formula
[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

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

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

Chapter: [2] Matrices
Concept: Elementary Operation (Transformation) of a Matrix
[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

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

Chapter: [4] Pair of Straight Lines
Concept: Pair of Straight Lines - Point of Intersection of Two Lines
[3]3.1.3

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

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

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

Chapter: [3] Trigonometric Functions
Concept: Trigonometric Functions - General Solution of Trigonometric Equation of the Type
[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).

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

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

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

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

(a) 0

(b) -2

(c) 2

(d) ±2

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

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

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

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

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

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

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

If y=eax ,show that  xdy/dx=ylogy

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

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

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

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

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

Chapter: [17] Differential Equation
Concept: Differential Equations - Linear 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.

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

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

Chapter: [15] Integration
Concept: Methods of Integration - Integration by Parts
[4]5.2.3

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

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

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

If  log_10((x^3-y^3)/(x^3+Y^3))=2

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

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

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

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

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

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