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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 correct answer from the given alternatives in each of the following sub-questions:
[2]1.1.1

Inverse of the statement pattern (p ∨ q) → (p ∧ q) is 

(A) (p ∧ q) → (p ∨ q)

(B) ∼ (p ∨ q) → (p ∧ q)

(C) (∼ p ∨ ∼ q) → (∼ p ∧ ∼ q)

(D) (∼ p ∧ ∼ q) → (∼ p ∨ ∼ q)

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Sentences and Statement in Logic
[2]1.1.2

If the vectors `2hati-qhatj+3hatk and 4hati-5hatj+6hatk` are collinear, then value of q is

(A) 5

(B) 10

(C) 5/2

(D) 5/4

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

If in ∆ABC with usual notations a = 18, b = 24, c = 30 then sin A/2 is equal to

(A) `1/sqrt5`

(B) `1/sqrt10`

(C) `1/sqrt15`

(D) `1/(2sqrt5)`

Chapter: [3] Trigonometric Functions
Concept: Trigonometric Functions - Solution of a Triangle
[6]1.2 | Attempt any THREE of the following:
[2]1.2.1

Find the angle between the lines `barr=3hati+2hatj-4hatk+lambda(hati+2hatj+2hatk)` and `barr=5 hati-2hatk+mu(3hati+2hatj+6hatk)`

Chapter: [4] Pair of Straight Lines
Concept: Acute Angle Between the Lines
[2]1.2.2

If p, q, r are the statements with truth values T, F, T, respectively then find the truth value of (r ∧ q) ↔ ∼ p

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Truth Value of Statement in Logic
[2]1.2.3

If `A =[[2,-3],[3,5]]` then find A-1 by adjoint method. 

Chapter: [2] Matrices
Concept: Determinants - Adjoint Method
[2]1.2.4

By vector method show that the quadrilateral with vertices A (1, 2, –1), B (8, –3, –4), C (5, –4, 1), D (–2, 1, 4) is a parallelogram.

Chapter: [7] Vectors
Concept: Vectors - Diagonals of a Parallelogram Bisect Each Other and Converse
[2]1.2.5

Find the general solution of the equation sin x = tan x.

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

Find the joint equation of pair of lines passing through the origin and perpendicular to the lines represented by ax2+ 2hxy + by2= 0

Chapter: [4] Pair of Straight Lines
Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Homogenous Equation
[3]2.1.2

Find the principal value of `sin^-1(1/sqrt2)`

Chapter: [1.02] Inverse Trigonometric Functions
Concept: Basic Concepts of Trigonometric Functions
[3]2.1.3

Find the cartesian form of the equation of the plane `bar r=(hati+hatj)+s(hati-hatj+2hatk)+t(hati+2hatj+hatj)`

Chapter: [4.02] Three - Dimensional Geometry
Concept: Vector and Cartesian Equation of a Plane
[8]2.2 | Attempt any TWO of the following:
[4]2.2.1

Simplify the following circuit so that new circuit has minimum number of switches. Also draw simplified circuit.

 

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Application - Introduction to Switching Circuits
[4]2.2.2

A line makes angles of measures 45° and 60° with positive direction of y and z axes respectively. Find the d.c.s. of the line and also find the vector of  magnitude 5 along the direction of line.

Chapter: [6] Three - Dimensional Geometry (Section B)
Concept: Line - Equation of Line Passing Through Given Point and Parallel to Given Vector
[4]2.2.3

Maximize:
z = 3x + 5y
Subject to:  x + 4y ≤ 24
                  3x + y ≤ 21
                  x + y ≤ 9
                  x ≥ 0, y ≥ 0

Chapter: [10] Linear Programming (Section C)
Concept: Graphical Method of Solving Linear Programming Problems
[14]3
[6]3.1 | Attempt any TWO of the following:
[3]3.1.1

Find the shortest distance between the lines `(x+1)/7=(y+1)/(-6)=(z+1)/1 and (x-3)/1=(y-5)/(-2)=(z-7)/1`

Chapter: [6] Three - Dimensional Geometry (Section B)
Concept: Shortest Distance Between Two Lines
[3]3.1.2

Show that the points (1, –1, 3) and (3, 4, 3) are equidistant from the plane 5x + 2y – 7z + 8 = 0

Chapter: [4.02] Three - Dimensional Geometry
Concept: Distance of a Point from a Plane
[3]3.1.3

In any triangle ABC with usual notations prove c = a cos B + b cos A

Chapter: [3] Trigonometric Functions
Concept: Trigonometric Functions - General Solution of Trigonometric Equation of the Type
[8]3.2 | Attempt any TWO of the following:
[4]3.2.1

Find p and k if the equation px2 – 8xy + 3y+14x + 2y + k = 0 represents a pair of perpendicular lines.

Chapter: [6] Three - Dimensional Geometry (Section B)
Concept: Line - Equation of Line Passing Through Given Point and Parallel to Given Vector
[4]3.2.2

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
[4]3.2.3

Find the volume of the parallelopiped whose coterminus edges are given by vectors `2hati+3hatj-4hatk, 5hati+7hatj+5hatk and 4hati+5hatj-2hatk`

Chapter: [4.01] Vectors
Concept: Scalar Triple Product of Vectors
[12]4
[6]4.1 | Select and write the correct answer from the given alternatives in each of the following sub-questions:
[2]4.1.1

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: [3.04] Differential Equations
Concept: Order and Degree of a Differential Equation
[2]4.1.2

`∫_4^9 1/sqrtxdx=`_____

(A) 1

(B) –2

(C) 2

(D) –1

Chapter: [3.03] Integrals
Concept: Properties of Definite Integrals
[2]4.1.3

If the p.d.f. of a continuous random variable X is given as

`f(x)=x^2/3` for -1< x<2

       =0   otherwise

then c.d.f. fo X is

(A) `x^3/9+1/9`

(B) `x^3/9-1/9`

(C) `x^2/4+1/4`

(D) `1/(9x^3)+1/9`

Chapter: [19] Probability Distribution
Concept: Probability Distribution - Probability Density Function (P.D.F.)
[6]4.2 | Attempt any THREE of the following:
[2]4.2.1

If `y = sec sqrtx` then find dy/dx.

Chapter: [13] Differentiation
Concept: Derivative - Derivative of Functions in Product of Function Form
[2]4.2.2

Evaluate : `∫(x+1)/((x+2)(x+3))dx`

Chapter: [3.03] Integrals
Concept: Methods of Integration - Integration Using Partial Fractions
[2]4.2.3

Find the area of the region lying in the first quandrant bounded by the curve y2= 4x, X axis and the lines x = 1, x = 4

Chapter: [16] Applications of Definite Integral
Concept: Area of the Region Bounded by a Curve and a Line
[2]4.2.4

For the differential equations find the general solution:

sec2 x tan y dx + sec2 y tan x dy = 0

Chapter: [3.04] Differential Equations
Concept: Methods of Solving First Order, First Degree Differential Equations - Differential Equations with Variables Separable
[2]4.2.5

Given is X ~ B (n, p). If E(X) = 6, and Var(X) = 4.2, find the value of n.

Chapter: [20] Bernoulli Trials and Binomial Distribution
Concept: Bernoulli Trials and Binomial Distribution - Calculation of Probabilities
[14]5
[6]5.1 | Attempt any TWO of the following:
[2]5.1.1

If the function `f(x)=(4^sinx-1)^2/(xlog(1+2x))`  for x ≠ 0 is continuous at x = 0, find f (0).

Chapter: [12] Continuity
Concept: Continuity of Some Standard Functions - Trigonometric Function
[2]5.1.2

Evaluate : `∫1/(3+2sinx+cosx)dx`

Chapter: [3.03] Integrals
Concept: Methods of Integration - Integration by Substitution
[2]5.1.3

If y = f(x) is a differentiable function of x such that inverse function x = f–1 (y) exists, then prove that x is a differentiable function of y and `dx/dy=1/((dy/dx)) " where " dy/dx≠0`

 

Chapter: [4] Differentiation
Concept: Derivative - Derivative of Inverse Function
[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: [3.02] Applications of Derivatives
Concept: Rate of Change of Bodies Or Quantities
[4]5.2.2

The probability mass function for X = number of major defects in a randomly selected
appliance of a certain type is 

X = x 0 1 2 3 4
P(X = x) 0.08 0.15 0.45 0.27 0.05

Find the expected value and variance of X.

Chapter: [4] Probability (Section A)
Concept: Variance of Binomial Distribution (P.M.F.)
[4]5.2.3

Prove that `int_0^af(x)dx=int_0^af(a-x) dx`

hence evaluate `int_0^(pi/2)sinx/(sinx+cosx) dx`

Chapter: [3.03] Integrals
Concept: Properties of Definite Integrals
[14]6
[6]6.1 | Attempt any TWO of the following:
[3]6.1.1

If y = etan x+ (log x)tan x then find dy/dx

Chapter: [3.04] Differential Equations
Concept: General and Particular Solutions of a Differential Equation
[3]6.1.2

If the probability that a fluorescent light has a useful life of at least 800 hours is 0.9, find the probabilities that among 20 such lights at least 2 will not have a useful life of at least 800 hours. [Given : (0⋅9)19 = 0⋅1348]

 

Chapter: [19] Probability Distribution
Concept: Random Variables and Its Probability Distributions
[3]6.1.3

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
[8]6.2 | Attempt any TWO of the following:
[4]6.2.1

Find the equation of a curve passing through the point (0, 2), given that the sum of the coordinates of any point on the curve exceeds the slope of the tangent to the curve at that point by 5

Chapter: [16] Applications of Definite Integral
Concept: Area of the Region Bounded by a Curve and a Line
[4]6.2.2

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

`intuvdx=uintvdx-int[du/dxintvdx]dx`

Chapter: [15] Integration
Concept: Methods of Integration - Integration by Parts
[4]6.2.3

Find the approximate value of log10 (1016) given that log10= 0⋅4343.

Chapter: [14] Applications of Derivative
Concept: Approximations
S