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# Question Paper Solutions - Mathematics and Statistics 2015 - 2016-H.S.C-12th Board Exam Maharashtra State Board (MSBSHSE)

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

The negation of p ∧ (q → r) is

1. p ∨ (~q ∨ r)
2. ~p ∧ (q → r)
3. ~p ∧ (~q → ~r)
4. ~p ∨ (q ∧ ~r)
Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Algebra of Statements
[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

Chapter: [1] Relations and Functions (Section A)
Concept: Basic Concepts of 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
Chapter: [4] Pair of Straight Lines
Concept: Pair of Straight Lines - Pair of Lines Not Passing Through Origin-combined Equation of Any Two 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)]

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

Chapter: [6] Three - Dimensional Geometry (Section B)
Concept: Vector and Cartesian Equation of a 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

Chapter: [4.01] Vectors
Concept: Section formula
[2]1.2.4

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

Chapter: [4] Pair of Straight Lines
Concept: Pair of Straight Lines - Pair of Lines Not Passing Through Origin-combined Equation of Any Two 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

Chapter: [4.02] Three - Dimensional Geometry
Concept: Shortest Distance Between Two Lines
[14]2
[6]2.1 | Attempt any TWO of the following
[5]2.1.1

Examine whether the following logical statement pattern is tautology, contradiction or contingency.

[(p → q) ∧ q] → p

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Statement Patterns and Logical Equivalence
[3]2.1.2

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

Chapter: [7] Vectors
Concept: Vectors - Medians of a Triangle Are Concurrent
[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

Chapter: [4.02] Three - Dimensional Geometry
Concept: Shortest Distance Between Two Lines
[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

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

Chapter: [11] Linear Programming Problems
Concept: Graphical Method of Solving 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.

Chapter: [2.01] Matrices and Determinants
Concept: Elementary Operation (Transformation) of a Matrix
[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

Chapter: [8] Three Dimensional Geometry
Concept: Three Dimensional Geometry - Problems
[3]3.1.2

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

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Algebra of Statements
[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.

Chapter: [4] Pair of Straight Lines
Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Homogenous Equation
[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.

Chapter: [6] Three - Dimensional Geometry (Section B)
Concept: Equation of a Line in Space
[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.

Chapter: [6] Three - Dimensional Geometry (Section B)
Concept: Vector and Cartesian Equation of a Plane
[4]3.2.3

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

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

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

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

Chapter: [6] Conics
Concept: Conics - Tangents and normals - equations of tangent and normal at a point
[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

Chapter: [4] Probability (Section A)
Concept: 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)

Chapter: [3.01] Continuity and Differentiability
Concept: Exponential and Logarithmic Functions
[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.

Chapter: [14] Applications of Derivative
Concept: Maxima and Minima in Closed Interval
[2]4.2.3

Find the area bounded by the curve y2 = 4axx-axis and the lines x = 0 and x = a.

Chapter: [3.04] Applications of the Integrals
Concept: Area of the Region Bounded by a Curve and a Line
[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)

Chapter: [19] Probability Distribution
Concept: Probability Distribution of a Discrete Random Variable
[2]4.2.5

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

Chapter: [3.03] Integrals
Concept: Methods of Integration - Integration by Substitution
[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)

Chapter: [13] Differentiation
Concept: Derivative - Every Differentiable Function is Continuous but Converse is Not True
[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.

Chapter: [19] Probability Distribution
Concept: Probability Distribution of a Discrete Random Variable
[3]5.1.3

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

Chapter: [3.03] Integrals
Concept: Methods of Integration - Integration by Substitution
[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

Chapter: [3.01] Continuity and Differentiability
Concept: Concept of 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

Chapter: [3.03] Integrals
Concept: Evaluation of Definite Integrals by Substitution
[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?

Chapter: [17] Differential Equation
Concept: Differential Equations - Applications of 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

Chapter: [7] Definite Integrals
Concept: Properties of Definite Integrals
[3]6.1.2

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

Chapter: [3.03] Integrals
Concept: Evaluation of Definite Integrals by Substitution
[3]6.1.3

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

Chapter: [4] Differentiation
Concept: Derivative - Derivative of Inverse Function
[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.

Chapter: [3.05] Differential Equations
Concept: General and Particular Solutions of a 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.

Chapter: [14] Applications of Derivative
Concept: Maxima and Minima - Introduction of Extrema and Extreme Values
[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.

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