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# Question Paper Solutions - Mathematics and Statistics 2012 - 2013-H.S.C-12th Board Exam Maharashtra State Board (MSBSHSE)

Marks: 80
[12]1
[6]1.1 | Select and write the correct answer from the given alternatives in each of the following:
[2]1.1.1

The principal solution of the equation cot x=-sqrt 3  is

(A) pi/6

(B) pi/3

(C)(5pi)/6

(D)(-5pi)/6

Chapter: [1.02] Inverse Trigonometric Functions
Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch
[2]1.1.2

If the vectors -3hati+4hatj-2hatk, hati+2hatk, hati-phatj are coplanar, then the value of of p is

(A) -2

(B) 1

(C) -1

(D) 2

Chapter: [7] Vectors
Concept: Vectors - Collinearity and Coplanarity of Vectors
[2]1.1.3

If the line y =x+k  touches the hyperbola 9x2 -16y2 =144, then k = .............

(A) 7

(B) -7

(C)+-sqrt7

(D) +-sqrt19

Chapter: [6] Conics
Concept: Conics - Tangents and normals - equations of tangent and normal at a point
[6]1.2 | Attempt any THREE of the following:
[2]1.2.1

Write down the following statements in symbolic form :

(A) A triangle is equilateral if and only if it is equiangular.
(B) Price increases and demand falls

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Logical Connectives
[2]1.2.2

If A=[[2,-2],[4,3]] then find A^-1 by adjoint method.

Chapter: [2] Matrices
[2]1.2.3

Find the separate equations of the lines represented by the equation  3x^2-10xy-8y^2=0

Chapter: [6] Three - Dimensional Geometry (Section B)
Concept: Equation of a Line in Space
[2]1.2.4

Find the equation of the director circle of a circle  x^2 + y^2 =100.

Chapter: [5] Circle
Concept: Circle - Director circle
[2]1.2.5

Find the general solution of the equation 4cos^2x=1

Chapter: [3] Trigonometric Functions
Concept: Trigonometric Functions - General Solution of Trigonometric Equation of the Type
[14]2
[6]2.1 | Attempt any TWO of the following:
[3]2.1.1

Without using truth table show that P↔q ≡(P ∧ q) ∨ (~ p ∧ ~ q)

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Algebra of Statements
[3]2.1.2

If θ is the measure of acute angle between the pair of lines given by ax^2+2hxy+by^2=0, then prove that tantheta=|(2sqrt(h^2-ab))/(a+b)|,a+bne0

Chapter: [4] Pair of Straight Lines
Concept: Acute Angle Between the Lines
[3]2.1.3

Show that the line x+ 2y + 8 = 0 is tangent to the parabola y2 = 8x. Hence find the point of contact

Chapter: [6] Conics
Concept: Conics - Tangents and normals - equations of tangent and normal at a point
[8]2.2 | Attempt any TWO of the following :
[4]2.2.1

The sum of three numbers is 9. If we multiply third number by 3 and add to the second number, we get 16. By adding the first and the third number and then subtracting twice the second number from this sum, we get 6. Use this information and find the system of linear equations. Hence, find the three numbers using matrices.

Chapter: [2.01] Matrices and Determinants
Concept: Elementary Operation (Transformation) of a Matrix
[4]2.2.2

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]2.2.3

If bar a and bar b are any two non-zero and non-collinear vectors then prove that any vector bar r  coplanar with  bar a and bar b can be uniquely expressed as bar r=t_1bara+t_2barb , where  t_1 and t_2  are scalars

Chapter: [7] Vectors
Concept: Vectors - Collinearity and Coplanarity of Vectors
[14]3
[6]3.1 | Attempt any TWO of the following :
[3]3.1.1

Using truth table examine whether the following statement pattern is tautology, contradiction or contingency (p^^~q) harr (p->q)

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Statement Patterns and Logical Equivalence
[3]3.1.2

Find k if the length of the tangent segment from (8,-3) to the circle  x^2+y^2-2x+ky-23=0 is sqrt10  units.

Chapter: [5] Circle
Concept: Circle - Length of Tangent Segments to Circle
[3]3.1.3

Show that the lines given by (x+1)/-10=(y+3)/-1=(z-4)/1  and (x+10)/-1=(y+1)/-3=(z-1)/4 intersect. Also find the co-ordinates of the point of intersection.

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

Find the equation of the locus of the point of intersection of two tangents drawn to the hyperbola x^2/7-y^2/5=1 such that the sum of the cubes of their slopes is 8.

Chapter: [6] Conics
Concept: Conics - Locus of Points from Which Two Tangents Are Mutually Perpendicular
[4]3.2.2

Solve the following L.P.P graphically:

Maximize :Z = 10x + 25y
Subject to : x ≤ 3, y ≤ 3, x + y ≤ 5, x ≥ 0, y ≥ 0

Chapter: [11] Linear Programming Problems
Concept: Graphical Method of Solving Linear Programming Problems
[4]3.2.3

Find the equation of the planes parallel to the plane x + 2y+ 2z + 8 =0 which are at the distance of 2  units from the point (1,1, 2)

Chapter: [10] Plane
Concept: Distance of a Point from a Plane
[12]4
[6]4.1 | Select and write the correct answer from the given alternatives in each of the folloiwng:
[2]4.1.1

Function  f (x)= x^2 - 3x +4 has minimum value at

(A) 0

(B)-3/2

(C) 1

(D)3/2

Chapter: [14] Applications of Derivative
Concept: Maxima and Minima - Introduction of Extrema and Extreme Values
[2]4.1.2

int1/xlogxdx=...............

(A)log(log x)+ c

(B) 1/2 (logx )2+c

(C) 2log x + c

(D) log x + c

Chapter: [3.03] Integrals
Concept: Methods of Integration - Integration by Parts
[2]4.1.3

Order and degree of the differential equation [1+(dy/dx)^3]^(7/3)=7(d^2y)/(dx^2) are respectively

(A) 2, 3

(B) 3, 2

(C) 7, 2

(D) 3, 7

Chapter: [17] Differential Equation
Concept: Order and Degree of a Differential Equation
[6]4.2 | Attempt any THREE of the following:
[2]4.2.1

If x=at2, y= 2at , then find dy/dx.

Chapter: [4] Differentiation
Concept: Derivatives of Functions in Parametric Forms
[2]4.2.2

Find the approximate value of  sqrt8.95

Chapter: [3.02] Applications of Derivatives
Concept: Approximations
[2]4.2.3

Find the area of the region bounded by the parabola y2 = 16x and the line x = 3.

Chapter: [3.04] Applications of the Integrals
Concept: Area of the Region Bounded by a Curve and a Line
[2]4.2.4

For the bivariate data r = 0.3, cov(X, Y) = 18, σx = 3, find σy .

Chapter: [12] Bivariate Data and Correlation
Concept: Statistics - Bivariate Frequency Distribution
[2]4.2.5

triangle bounded by the lines y = 0, y = x and x = 4 is revolved about the X-axis. Find the volume of the solid of revolution.

Chapter: [3.04] Applications of the Integrals
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

A function f (x) is defined as
f (x) = x + a, x < 0
= x,       0 ≤x ≤ 1
= b- x,   x ≥1
is continuous in its domain.
Find a + b.

Chapter: [3.01] Continuity and Differentiability
Concept: Algebra of Continuous Functions
[3]5.1.2

If x=a(t-1/t),y=a(t+1/t), then show that dy/dx=x/y

Chapter: [4] Differentiation
Concept: Derivatives of Functions in Parametric Forms
[3]5.1.3

Evaluate : int1/(3+5cosx)dx

Chapter: [3.03] Integrals
Concept: Evaluation of Definite Integrals by Substitution
[8]5.2 | Attempt any TWO of the following:
[4]5.2.1

An insurance agent insures lives of 5 men, all of the same age and in good health. The probability that a man of this age will survive the next 30 years is known to be 2/3 . Find the probability that in the next 30 years at most 3 men will survive.

Chapter: [22] Probability
Concept: Conditional Probability
[4]5.2.2

The surface area of a spherical balloon is increasing at the rate of 2cm2 / sec. At what rate is the volume of the balloon is increasing when the radius of the balloon is 6 cm?

Chapter: [3.02] Applications of Derivatives
Concept: Rate of Change of Bodies Or Quantities
[4]5.2.3

The slope of the tangent to the curve at any point is equal to y+ 2x. Find the equation of the curve passing through the origin.

Chapter: [17] Differential Equation
Concept: Differential Equations - Linear Differential Equation
[14]6
[6]6.1 | Attempt any TWO of the following :
[3]6.1.1

If u and v are two functions of x then prove that

intuvdx=uintvdx-int[du/dxintvdx]dx

Chapter: [3.03] Integrals
Concept: Methods of Integration - Integration by Parts
[3]6.1.2

The time ( in minutes) for a lab assistant to prepare the equipment for a certain experiment is a random variable X taking values between 25 and 35 minutes with p.d.f

f(x)=1/10,25<=x<=35=0

 =0 "   " otherwise

What is the probability that preparation time exceeds 33 minutes? Also find the c.d.f. of X.

Chapter: [19] Probability Distribution
Concept: Probability Distribution - Probability Density Function (P.D.F.)
[3]6.1.3

The probability that a certain kind of component will survive a check test is 0.6. Find the probability that exactly 2 of the next 4 tested components survive

Chapter: [22] Probability
Concept: Conditional Probability
[8]6.2 | Attempt any TWO of the following:
[4]6.2.1

If ax^2+2hxy+by^2=0 , show that (d^2y)/(dx^2)=0

Chapter: [4] Differentiation
Concept: Derivatives of Functions in Parametric Forms
[4]6.2.2

Find the area of the region common to the circle x2 + y2 =9 and the parabola y2 =8x

Chapter: [3.04] Applications of the Integrals
Concept: Area of the Region Bounded by a Curve and a Line
[4]6.2.3

For 10 pairs of observations on two variables X and Y, the following data are available:

sum(x-2)=30, sum(y-5)=40, sum(x-2)^2=900, sum(y-5)^2=800, sum(x-2)(y-5)=480

Find the correlation coefficient between X and Y.

Chapter: [18] Statistics
Concept: Statistics - Karl Pearson’s Coefficient of Correlation
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