HSC Arts 12th Board ExamMaharashtra State Board
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Question Paper Solutions - Mathematics and Statistics 2017 - 2018 HSC Arts 12th Board Exam


Marks: 80
[12]1
[6]1.1 | Select and write the appropriate answer from the given alternative in each of the following sub-question
[2]1.1.1

If A = `[(2,-3),(4,1)]`, then adjoint of matrix A is

(A) `[(1,3),(-4,2)]`

(B) `[(1,-3),(-4,2)]`

(C)  `[(1,3),(4,-2)]`

(D) `[(-1,-3),(-4,2)]`

Chapter: [2] Matrices
Concept: Determinants - Adjoint Method
[2]1.1.2

The principal solutions of sec x = `2/sqrt3` are _____

(A) `pi/3,(11pi)/6`

(B) `pi/6, (11pi)/6`

(C)`pi/4,(11pi)/4`

(D) `pi/6,(11pi)/4`

Chapter: [3] Trigonometric Functions
Concept: Trigonometric Functions - General Solution of Trigonometric Equation of the Type
[2]1.1.3

The measure of the acute angle between the lines whose direction ratios are 3, 2, 6 and –2, 1, 2 is ______.

Chapter: [8] Three Dimensional Geometry
Concept: Angle Between Two Lines
[6]1.2 | Attempt Any Three of the Following
[2]1.2.1

Write the negations of the following statements :

1) All students of this college live in the hostel

2) 6 is an even number or 36 is a perfect square.

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Sentences and Statement in Logic
[2]1.2.2

If a line makes angles α, β, γ with co-ordinate axes, prove that cos 2α + cos2β + cos2γ+ 1 = 0.

Chapter: [10] Plane
Concept: Angle Between Line and a Plane
[2]1.2.3

Find the distance of the point (1, 2, –1) from the plane x - 2y + 4z - 10 = 0 .

Chapter: [10] Plane
Concept: Distance of a Point from a Plane
[2]1.2.4

Find the vector equation of the lines which passes through the point with position vector `4hati - hatj +2hatk` and is in the direction of `-2hati + hatj + hatk`

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

if `bara = 3hati - 2hatj+7hatk`, `barb  = 5hati + hatj -2hatk`and `barc = hati + hatj - hatk` then find `bara.(barbxxbarc)`

Chapter: [7] Vectors
Concept: Scalar Triple Product of Vectors
[14]2
[6]2.1 | Attempt Any Two of the Following
[3]2.1.1

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

Chapter: [7] Vectors
Concept: Vectors - Medians of a Triangle Are Concurrent
[3]2.1.2

Using the truth table, prove the following logical equivalence :

p ↔ q ≡ (p ∧ q) ∨ (~p ∧ ~q )

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Truth Tables of Compound Statements
[3]2.1.3

If the origin is the centroid of the triangle whose vertices are A(2, p, –3), B(q, –2, 5) and R(–5, 1, r), then find the values of p, q, r.

Chapter: [7] Vectors
Concept: Section formula
[8]2.2 | Attempt Any Two of Following
[4]2.2.1

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
[4]2.2.2

In `triangle ABC` prove that `tan((C-A)/2) = ((c-a)/(c+a))cot  B/2`

Chapter: [3] Trigonometric Functions
Concept: Trigonometric Functions - Trigonometric equations
[4]2.2.3

Find the inverse of the matrix `A = [(1,2,-2),(-1,3,0),(0,-2,1)]`using elementary row transformations.

Chapter: [2] Matrices
Concept: Matrices - Inverse of a Matrix Existance
[14]3
[6]3.1 | Attempt Any Two of the Following
[3]3.1.1

Find the joint equation of the pair of lines passing through the origin which are perpendicular respectively to the lines represented by 5x2 +2xy- 3y2 = 0.

Chapter: [4] Pair of Straight Lines
Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Homogenous Equation
[3]3.1.2

Find the angle between the lines `(x -1)/4 = (y - 3)/1 = z/8`  and `(x-2)/2 = (y + 1)/2 = (z-4)/1`

Chapter: [9] Line
Concept: Line - Equation of Line Passing Through Given Point and Parallel to Given Vector
[3]3.1.3

Write converse, inverse and contrapositive of the following conditional statement :

If an angle is a right angle then its measure is 90°.

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Difference Between Converse, Contrapositive, Contradiction
[8]3.2 | Attempt Any Two of the Following
[4]3.2.1

Prove that `sin^(-1) (3/5) + cos^(-1) (12/13) = sin^(-1) (56/65)`

Chapter: [3] Trigonometric Functions
Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch
[4]3.2.2

Find the vector equation of the plane passing through the points A(1, 0, 1), B(1, –1, 1) and C(4, –3, 2).

Chapter: [10] Plane
Concept: Plane - Equation of Plane Passing Through the Given Point and Perpendicular to Given Vector
[4]3.2.3

Minimize Z = 7x + y subject to `5x + y >= 5, x + y >= 3, x>= 0, y >= 0`

 

Chapter: [11] Linear Programming Problems
Concept: Graphical Method of Solving Linear Programming Problems
[12]4
[6]4.1 | Select and write the appropriate answer from the given alternatives in each of the following sub-questions :
[2]4.1.1

Let the p. m. f. of a random variable X be __

P(x) = `(3-x)/10` for x = -1,0,1,2

= 0                        otherwise

Then E(X ) is ________.

(A) 1

(B) 2

(C) 0

(D) – 1

Chapter: [19] Probability Distribution
Concept: Probability Distribution - Expected Value, Variance and Standard Deviation of a Discrete Random Variable
[2]4.1.2

if `int_0^k 1/(2+ 8x^2) dx = pi/16` then the value of k is ________.

(A) `1/2`

(B) `1/3`

(C) `1/4`

(D) `1/5`

Chapter: [15] Integration
Concept: Definite Integral as the Limit of a Sum
[2]4.1.3

Integrating factor of linear differential equation `x (dy)/(dx) + 2y =x^2 log x` is ____________

(A) `1/x^2`

(B) `1/x`

(C) x

(D) `x^2`

Chapter: [17] Differential Equation
Concept: Differential Equations - Linear Differential Equation
[6]4.2 | Attempt Any Three of The Following
[2]4.2.1

Evaluate `int e^x [(cosx - sin x)/sin^2 x]dx`

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

if `y = tan^2(log x^3)`, find `(dy)/(dx)`

Chapter: [13] Differentiation
Concept: Derivatives of Composite Functions - Chain Rule
[2]4.2.3

Find the area of ellipse `x^2/1 + y^2/4 = 1`

 

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

Obtain the differential equation by eliminating the arbitrary constants from the following equation :

`y = c_1e^(2x) + c_2e^(-2x)`

Chapter: [17] Differential Equation
Concept: Formation of Differential Equation by Eliminating Arbitary Constant
[2]4.2.5

Given X ~ B (n, p)
If n = 10 and p = 0.4, find E(X) and var (X).

Chapter: [20] Bernoulli Trials and Binomial Distribution
Concept: Bernoulli Trials and Binomial Distribution
[14]5
[6]5.1 | Attempt any TWO of the following
[3]5.1.1

Evaluate `int 1/(3+ 2 sinx + cosx) dx`

Chapter: [15] Integration
Concept: Methods of Integration - Integration by Substitution
[3]5.1.2

If `x = acos^3t`, `y = asin^3 t`,

Show that `(dy)/(dx) =- (y/x)^(1/3)`

Chapter: [13] Differentiation
Concept: Derivatives of Functions in Parametric Forms
[3]5.1.3

Examine the continuity of the function:

f(x) = `(log100 + log(0.01+x))/"3x"`

= 100/3        for x = 0; at x = 0

Chapter: [12] Continuity
Concept: Continuity - Defination of Continuity of a Function at a Point
[8]5.2 | Attempt any TWO of the following
[4]5.2.1

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]5.2.2

Prove that `int 1/(a^2 - x^2) dx = 1/"2a" log|(a +x)/(a-x)| + c`

Chapter: [15] Integration
Concept: Fundamental Theorem of Calculus
[4]5.2.3

Prove that : `int_-a^af(x)dx=2int_0^af(x)dx` , if f (x) is an even function.

                      = 0,                   if f (x) is an odd function.

Chapter: [15] Integration
Concept: Methods of Integration - Integration by Parts
[14]6
[6]6.1 | Attempt any TWO of the following
[3]6.1.1

if  `f(x) = (x^2-9)/(x-3) + alpha`               for x> 3

           =5,                                     for x = 3

          `=2x^2+3x+beta`,             for x < 3

is continuous at x  = 3, find α and β.

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

Find `dy/dx` if `y = tan^(-1) ((5x+ 1)/(3-x-6x^2))`

Chapter: [13] Differentiation
Concept: Derivative - Derivative of Inverse Function
[3]6.1.3

A fair coin is tossed 9 times. Find the probability that it shows head exactly 5 times.

Chapter: [20] Bernoulli Trials and Binomial Distribution
Concept: Bernoulli Trials and Binomial Distribution
[8]6.2 | Attempt any TWO of the following
[4]6.2.1

Verify Rolle’s theorem for the following function:

f (x) = x2 - 4x + 10 on [0, 4]

Chapter: [14] Applications of Derivative
Concept: Mean Value Theorem
[4]6.2.2

Find the particular solution of the differential equation:

`y(1+logx) dx/dy - xlogx = 0`

when y = e2 and x = e

Chapter: [17] Differential Equation
Concept: Methods of Solving First Order, First Degree Differential Equations - Differential Equations with Variables Separable
[4]6.2.3

Find the variance and standard deviation of the random variable X whose probability distribution is given below :

x 0 1 2 3
P(X = x) `1/8` `3/8` `3/8` `1/8`
Chapter: [19] Probability Distribution
Concept: Probability Distribution - Expected Value, Variance and Standard Deviation of a Discrete Random Variable
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