HSC Arts 12th Board ExamMaharashtra State Board
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Question Paper Solutions - Mathematics and Statistics 2012 - 2013 HSC Arts 12th Board Exam


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

If A = {2, 3, 4, 5, 6}, then which of the following is not true?

(A) ∃ x ∈ A such that x + 3 = 8

(B) ∃ x ∈ A such that x + 2 < 5

(C) ∃ x ∈ A such that x + 2 < 9

(D) ∀ x ∈ A such that x + 6 ≥ 9

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Algebra of Statements
[2]1.1.2

If 2x + y = 0 is one of the lines represented by 3x2 + kxy + 2y2 = 0, then the value of k is

A) `1/2`

B) `11/2`

C) `5/2`

D) `(-11)/2`

Chapter: [4] Pair of Straight Lines
Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation
[2]1.1.3

If a line is inclined at 60° and 30° with the X and Y-axes respectively, then the angle which it makes with Z-axis is

(A) 0

(B) `pi/4`

(C) `pi/2`

(D) `pi/6`

Chapter: [4] Pair of Straight Lines
Concept: Acute Angle Between the Lines
[6]1.2 | Attempt any THREE of the following
[2]1.2.1

If A = `[(1,2),(3,4)]` and AX = I then find X by using elementary transformations

Chapter: [2] Matrices
Concept: Matrices - Elementary Transformation of a Matrix Revision of Cofactor and Minor
[2]1.2.2

With usual notations, in ΔABC, prove that a(b cos C − c cos B) = b2 − c2

Chapter: [3] Trigonometric Functions
Concept: Trigonometric Functions - Solution of a Triangle
[2]1.2.3

Show that the equation of a tangent to the circle x2 + y2 = a2 at the point P(x1,y1) on it is xx1 + yy1 = a2

Chapter: [5] Circle
Concept: Circle - Tangents to a Circle from a Point Outside the Circle
[2]1.2.4

Find k, if the line 2x − 3y + k = 0 touches the ellipse 5x2 + 9y2 = 45.

Chapter: [5] Circle
Concept: Circle - Tangent of a Circle - Equation of a Tangent at a Point to General Circle
[2]1.2.5

Find the co-ordinates of the point, which divides the line segment joining the points A(2, − 6, 8) and B(− 1, 3, − 4) externally in the ratio 1 : 3.

Chapter: [9] Line
Concept: Line - Distance of a Point from a Line
[14]2
[6]2.1 | Attempt any TWO of the following:
[3]2.1.1

Using truth table, prove that ~p ∧ q ≡ (p ∨ q) ∧ ~ p

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

Find the values of p and q, if the following equation represents a pair of perpendicular lines:
px2 − 8xy + 3y2 + 14x + 2y + q = 0.

Chapter: [4] Pair of Straight Lines
Concept: Pair of Straight Lines - Condition for Perpendicular Lines
[3]2.1.3

Find the equations of tangents to the parabola y2 = 12x from the point (2, 5).

Chapter: [5] Circle
Concept: Circle - Condition of tangency
[8]2.2 | Attempt any TWO of the following:
[4]2.2.1

The cost of 2 books, 6 notebooks and 3 pens is  Rs 40. The cost of 3 books, 4 notebooks and 2 pens is Rs 35, while the cost of 5 books, 7 notebooks and 4 pens is Rs 61. Using this information and matrix method, find the cost of 1 book, 1 notebook and 1 pen separately.

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

Prove that `sin^(−1) (-1/2) + cos^(-1) (-sqrt3/2) = cos^(-1) (-1/2)`

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

Show that the product of lengths of perpendicular segments drawn from the foci to any tangent to the hyperbola `x^2/25 + y^2/16 = 1` is equal to 16.

Chapter: [6] Conics
Concept: Conics - Tangents and normals - equations of tangent and normal at a point
[14]3
[6]3.1 | Attempt any TWO of the following:
[3]3.1.1

Construct the new switching circuit for the following circuit with only one switch by simplifying the given circuit:

Chapter: [1] Mathematical Logic
Concept: Mathematical Logic - Application - Introduction to Switching Circuits
[3]3.1.2

Find the locus of a point, the tangents from which to the circle x2 + y2 = a2 are mutually perpendicular

Chapter: [6] Conics
Concept: Conics - Locus of Points from Which Two Tangents Are Mutually Perpendicular
[3]3.1.3

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: [4] Pair of Straight Lines
Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation
[8]3.2 | Attempt any TWO of the following:
[4]3.2.1

Find the angle between the line `(x - 1)/3 = (y + 1)/2 = (z + 2)/4` and the plane 2x + y − 3z + 4 = 0.

Chapter: [10] Plane
Concept: Angle Between Line and a Plane
[4]3.2.2

Solve the following L. P. P. graphically:Linear Programming

Minimize Z = 6x + 2y

Subject to

5x + 9y ≤ 90

x + y ≥ 4

y ≤ 8

x ≥ 0, y ≥ 0

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

Find the volume of a tetrahedron whose vertices are A(−1, 2, 3), B(3, −2, 1), C(2, 1, 3) and D(−1, −2, 4).

Chapter: [7] Vectors
Concept: Vectors - Collinearity and Coplanarity of Vectors
[12]4
[6]4.1 | Select and write the correct answer from the given alternatives in each of the following
[2]4.1.1

If xy = ex−y , then `dy/dx` = ______

A) `(1+x)/(1 + log x)`

B) `log x/(1 + log x)^2`

C) `(1 - log x)/(1 + log x)`

D) `(1-x)/(1 + log x)`

Chapter: [12] Continuity
Concept: Continuity of Some Standard Functions - Trigonometric Function
[2]4.1.2

`int 1/(1 + cos x)` dx = _____

A) `tan(x/2) + c`

B) `2 tan (x/2) + c`

C) -`cot (x/2) + c`

D) -2 `cot (x/2)` + c

Chapter: [15] Integration
Concept: Evaluation of Definite Integrals by Substitution
[2]4.1.3

If X ~ B (n, p) and E(X) = 12, Var(X) = 4, then the value of n is _______

(A) 3

(B) 48

(C) 18

(D) 36

Chapter: [20] Bernoulli Trials and Binomial Distribution
Concept: Bernoulli Trials and Binomial Distribution - Calculation of Probabilities
[6]4.2 | Attempt any THREE of the following
[2]4.2.1

Find the equation of tangent to the curve y = 3x2 − x + 1 at P(1, 3).

Chapter: [6] Conics
Concept: Conics - Tangents and normals - equations of tangent and normal at a point
[2]4.2.2

Evaluate: `int 1/(x(x-1)) dx`

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

Solve the differential equation y − x = `dy/dx = 0`

Chapter: [17] Differential Equation
Concept: Differential Equations - Applications of Differential Equation
[2]4.2.4

In a bivariate data, n = 10, `bar x` = 25, `bary` = 30 and `sum xy` = 7900. Find cov(X,Y)

Chapter: [18] Statistics
Concept: Statistics - Bivariate Frequency Distribution
[2]4.2.5

A random variable X ~ N (0, 1). Find P(X > 0) and P(X < 0).

Chapter: [19] Probability Distribution
Concept: Random Variables and Its Probability Distributions
[14]5
[6]5.1 | Attempt any TWO of the following:
[3]5.1.1

Examine the function for maximum and minimum f(x) = x3 − 9x2 + 24x.

Chapter: [14] Applications of Derivative
Concept: Maxima and Minima - Introduction of Extrema and Extreme Values
[3]5.1.2

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: [13] Differentiation
Concept: Derivative - Derivative of Inverse Function
[3]5.1.3

The probability distribution of X, the number of defects per 10 metres of a fabric is given by

x 0 1 2 3 4
P(X = x) 0.45 0.35 0.15 0.03 0.02

Find the variance of X

 

Chapter: [19] Probability Distribution
Concept: Probability Distribution - Expected Value, Variance and Standard Deviation of a Discrete Random Variable
[8]5.2 | Attempt any TWO of the following:
[4]5.2.1

If `sqrt(1-x^2)  + sqrt(1- y^2)` =  a(x − y), show that dy/dx = `sqrt((1-y^2)/(1-x^2))`

Chapter: [13] Differentiation
Concept: Derivatives of Inverse Trigonometric Functions
[4]5.2.2

Solve the differential equation `cos^2 x dy/dx` + y = tan x

Chapter: [17] Differential Equation
Concept: General and Particular Solutions of a Differential Equation
[4]5.2.3

Find the area of the region bounded by the curves y2 = 4x and 4x2 + 4y2 = 9 with x >= 0.

Chapter: [4] Pair of Straight Lines
Concept: Pair of Straight Lines - Point of Intersection of Two Lines
[14]6
[6]6.1 | Attempt any TWO of the following
[3]6.1.1

Find the approximate value of tan−1 (1.001).

Chapter: [3] Trigonometric Functions
Concept: Trigonometric Functions - Trigonometric equations
[3]6.1.2

Examine continuity of the function f(x) at x = 0, where

`f(x) = (10^x + 7^x - 14^x - 5^x)/(1-cos 4x) , " for " x != 0`

`= 10/7 , " for"  x = 0`

Chapter: [12] Continuity
Concept: Continuity - Continuity of a Function at a Point
[3]6.1.3

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

Prove that:

`int sqrt(a^2 +x^2)dx = x/2 sqrt(a^2 + x^2) + a^2/2 log |x + sqrt(x^2 + a^2)| + c`

Chapter: [15] Integration
Concept: Methods of Integration - Integration Using Partial Fractions
[4]6.2.2

Find the volume of the solid generated, when the area between ellipse 4x2 + 9y2 = 36 and the chord AB, with A (3, 0), B (0, 2), is revolved about X-axis.

Chapter: [5] Circle
Concept: Circle - Tangent of a Circle - Equation of a Tangent at a Point to Standard Circle
[4]6.2.3

Find Karl Pearson’s coefficient of correlation between the variables X and Y for the following data

X 11 7 9 5 8 6 10
Y 10 8 6 5 9 7 11
Chapter: [18] Statistics
Concept: Statistics - Karl Pearson’s Coefficient of Correlation
S