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

SubjectMathematics and Statistics
Year2013 - 2014 (October)

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[bar a barb barc]!=0  then bar a . barp +bar b . bar q+bar c. bar r  is equal to

(a) 0

(b) 1

(c) 2

(d) 3

Chapter: [7] Vectors
Concept: Vectors - Linear Combination of Vectors
[2]1.1.2

The inverse of the matrix [[2,0,0],[0,1,0],[0,0,-1]]is --------

(a) [[1/2,0,0],[0,1,0],[0,0,-1]]

(b) [[-1/2,0,0],[0,-1,0],[0,0,1]]

(c) [[-1,0,0],[0,-1/2,0],[0,0,1/2]]

(d) 1/2[[-1/2,0,0],[0,-1,0],[0,0,-1]]

Chapter: [2] Matrices
Concept: Matrices - Inverse of a Matrix Existance
[2]1.1.3

Direction cosines of the line passing through the points A (- 4, 2, 3) and B (1, 3, -2) are.........

(a) +-1/sqrt51,+-5/sqrt51,+-1/sqrt51

(b) +-5/sqrt51,+-5/sqrt51,+-5/sqrt51

(c) +-sqrt5,+-1,+-5

(d) +-sqrt51,+-sqrt51+-sqrt51

Chapter: [8] Three Dimensional Geometry
Concept: Direction Cosines and Direction Ratios of a Line
[6]1.2 | Attempt any THREE of the following:
[2]1.2.1

Write truth values of the following statements :sqrt5 is an irrational number but 3 +sqrt 5 is a complex number.

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Truth Value of Statement in Logic

Write truth values of the following statements : ∃ n ∈ N such that n + 5 > 10.

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Truth Value of Statement in Logic
[2]1.2.2

If bar c = 3bara- 2bar b  Prove that [bar a bar b barc]=0

Chapter: [7] Vectors
Concept: Scalar Triple Product of Vectors
[2]1.2.3

Find the vector equation of the plane which is at a distance of 5 units from the origin and which is normal to the vector  2hati + hatj + 2hatk.

Chapter: [10] Plane
Concept: Vector and Cartesian Equation of a Plane
[2]1.2.4

The Cartesian equations of line are 3x+1=6y-2=1-z find its equation in vector form.

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

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are -2, 1, -1, and -3, -4, 1.

Chapter: [7] Vectors
Concept: Basic Concepts of Vector Algebra
[12]2
[6]2.1 | Attempt any TWO of the following:
[3]2.1.1

Using truth table, prove the following logical equivalence :

(p ∧ q)→r = p → (q→r)

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Truth Tables of Compound Statements
[3]2.1.2

Find the joint equation of the pair of lines through the origin each of which is making an angle of 30° with the line 3x + 2y - 11 = 0

Chapter: [4] Pair of Straight Lines
Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation
[3]2.1.3

Show that: 2sin^-1(3/5)=tan^-1(24/7)

Chapter: [3] Trigonometric Functions
Concept: Basic Concepts of Trigonometric Functions
[8]2.2 | Attempt any TWO of the following:
[4]2.2.1

Solve the following equations by the method of reduction :

2x-y + z=1,  x + 2y +3z = 8, 3x + y-4z=1.

Chapter: [2] Matrices
Concept: Elementary Operation (Transformation) of a Matrix
[4]2.2.2

Prove that the volume of a parallelopiped with coterminal edges as   bara ,bar b , barc

Hence find the volume of the parallelopiped with coterminal edges  bar i+barj, barj+bark

Chapter: [7] Vectors
Concept: Scalar Triple Product of Vectors
[4]2.2.3

Solve the following LPP by using graphical method.

Maximize : Z = 6x + 4y

Subject to x ≤ 2, x + y ≤  3, -2x + y ≤  1, x ≥  0, y ≥ 0.

Also find maximum value of Z.

Chapter: [11] Linear Programming Problems
Concept: Graphical Method of Solving Linear Programming Problems
[14]3
[6]3.1 | Attempt any TWO of the following
[3]3.1.1

In ΔABC with usual notations, prove that 2a {sin^2(C/2)+csin^2 (A/2)} = (a +   c - b)

Chapter: [3] Trigonometric Functions
Concept: Trigonometric Functions - Solution of a Triangle
[3]3.1.2

If p : It is a day time, q : It is warm, write the compound statements in verbal form

denoted by -

(a) p ∧ ~ q

(b)  ~ p  → q

(c)  q  ↔  p

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Compound Statement in Logic
[3]3.1.3

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
[8]3.2 | Attempt any TWO of the following:
[4]3.2.1

Parametric form of the equation of the plane is bar r=(2hati+hatk)+lambdahati+mu(hat i+2hatj+hatk) λ and μ are parameters. Find normal to the plane and hence equation of the plane in normal form. Write its Cartesian form.

Chapter: [10] Plane
Concept: Vector and Cartesian Equation of a Plane
[4]3.2.2

If the angle between the lines represented by ax2 + 2hxy + by2 = 0 is equal to the angle between the lines 2x2 - 5xy + 3y2 =0,

then show that 100(h2 - ab) = (a + b)2

Chapter: [8] Three Dimensional Geometry
Concept: Angle Between Two Lines
[4]3.2.3

Find the general solution of: sinx · tanx = tanx- sinx+ 1

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

The differential equation of the family of curves y=c1ex+c2e-x is......

(a)(d^2y)/dx^2+y=0

(b)(d^2y)/dx^2-y=0

(c)(d^2y)/dx^2+1=0

(d)(d^2y)/dx^2-1=0

Chapter: [17] Differential Equation
Concept: General and Particular Solutions of a Differential Equation
[2]4.1.2

If X is a random variable with probability mass function

P(x) = kx ,  x=1,2,3

= 0 ,     otherwise

then , k=..............

(a) 1/5

(b) 1/4

(c) 1/6

(d) 2/3

Chapter: [19] Probability Distribution
Concept: Probability Distribution - Probability Mass Function (P.M.F.)
[2]4.1.3

If sec((x+y)/(x-y))=a^2. " then " (d^2y)/dx^2=........

(a) y

(b) x

(c) y/x

(d) 0

Chapter: [13] Differentiation
Concept: Derivatives of Inverse Trigonometric Functions
[6]4.2 | Attempt any THREE of the following:
[2]4.2.1

If y=sin^-1(3x)+sec^-1(1/(3x)),   find dy/dx

Chapter: [13] Differentiation
Concept: Derivatives of Inverse Trigonometric Functions
[2]4.2.2

Evaluate :intxlogxdx

Chapter: [15] Integration
Concept: Methods of Integration - Integration by Substitution
[2]4.2.3

If int_0^h1/(2+8x^2)dx=pi/16 then find the value of h.

Chapter: [15] Integration
Concept: Fundamental Theorem of Calculus
[2]4.2.4

The probability that a certain kind of component will survive a check test is 0.5. Find the probability that exactly two of the next four components tested will survive.

Chapter: [19] Probability Distribution
Concept: Conditional Probability
[2]4.2.5

Find the area of the region bounded by the curve y = sinx, the lines x=-π/2 , x=π/2 and X-axis

Chapter: [16] Applications of Definite Integral
Concept: Area of the Region Bounded by a Curve and a Line
[14]5
[6]5.1 | Attempt any TWO of the following:
[3]5.1.1

Examine the continuity of the following function at given point:

f(x)=(logx-log8)/(x-8) ,

 =8,

Chapter: [12] Continuity
Concept: Continuity - Discontinuity of a Function
[3]5.1.2

If x = Φ(t) differentiable function of ‘ t ' then prove that int f(x) dx=intf[phi(t)]phi'(t)dt

Chapter: [17] Differential Equation
Concept: General and Particular Solutions of a Differential Equation
[3]5.1.3

Solve : 3ex tanydx + (1 +ex) sec2 ydy = 0

Also, find the particular solution when x = 0 and y = π.

Chapter: [17] Differential Equation
Concept: General and Particular Solutions of a Differential Equation
[8]5.2 | Attempt any TWO of the following:
[4]5.2.1

A point source of light is hung 30 feet directly above a straight horizontal path on which a man of 6 feet in height is walking. How fast will the man’s shadow lengthen and how fast will the tip of shadow move when he is walking away from the light at the rate of 100 ft/min.

Chapter: [14] Applications of Derivative
Concept: Rate of Change of Bodies Or Quantities
[4]5.2.2

Evaluate : intlogx/(1+logx)^2dx

Chapter: [15] Integration
Concept: Properties of Definite Integrals
[4]5.2.3

If x = f(t), y = g(t) are differentiable functions of parammeter ‘ t ’ then prove that y is a differentiable function of 'x' and  hence, find dy/dx if x=a cost, y=a sint

Chapter: [13] Differentiation
Concept: Derivatives of Functions in Parametric Forms
[14]6
[6]6.1 | Attempt any TWO of the following:
[3]6.1.1

Show that the function defined by f(x) =|cosx| is continuous function.

Chapter: [12] Continuity
Concept: Introduction of Continuity
[3]6.1.2

Solve the differential equation dy/dx=(y+sqrt(x^2+y^2))/x

Chapter: [17] Differential Equation
Concept: General and Particular Solutions of a Differential Equation
[3]6.1.3

Given X ~ B(n, p). If n = 20, E(X) = 10, find p, Var. (X) and   S.D. (X).

Chapter: [20] Bernoulli Trials and Binomial Distribution
Concept: Bernoulli Trials and Binomial Distribution - Normal Distribution (P.D.F)
[8]6.2 | Attempt any TWO of the following:
[4]6.2.1

A bakerman sells 5 types of cakes. Profits due to the sale of each type of cake is respectively Rs. 3, Rs. 2.5, Rs. 2, Rs. 1.5, Rs. 1. The demands for these cakes are 10%, 5%, 25%, 45% and 15% respectively. What is the expected profit per cake?

Chapter: [18] Statistics
Concept: Statistics - Bivariate Frequency Distribution
[4]6.2.2

Verify Lagrange’s mean value theorem for the function f(x)=x+1/x, x ∈ [1, 3]

Chapter: [14] Applications of Derivative
Concept: Mean Value Theorem
[4]6.2.3

Prove that int_a^bf(x)dx=f(a+b-x)dx. Hence evaluate : int_a^bf(x)/(f(x)+f(a-b-x))dx

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