# Mathematics and Statistics Shaalaa.com Model Set 3 2020-2021 HSC Arts 12th Board Exam Question Paper Solution

Mathematics and Statistics [Shaalaa.com Model Set 3]
Date: April 2021
Duration: 3h
1. The question paper is divided into four sections.
2. Section A: Q. No. 1 contains 8 multiple-choice type of questions carrying two marks each.
3. Section A: Q. No. 2 contains 4 very short answer type of questions carrying One mark each.
4. Section B: Q. No. 3 to Q. No. 14 contains Twelve short answer type of questions carrying Two marks each. (Attempt any Eight).
5. Section C: Q. No.15 to Q. No. 26 contains Twelve short answer type of questions carrying Three marks each. (Attempt any Eight).
6. Section D: Q.No. 27 to Q. No. 34 contains Five long answer type of questions carrying Four marks each. (Attempt any Five).
7. Use of log table is allowed. Use of calculator is not allowed.
8. Figures to the right indicate full marks.
9. For each MCQ, correct answer must be written along with its alphabet.
e.g., (a) ..... / (b ) .... / ( c ) .... / ( d) ..... Only first attempt will be considered for evaluation.
10. Use of graph paper is not necessary. Only rough sketch of graph is expected:
11. Start answers to each section on a new page.

Section A
[16]1 | Select and write the most appropriate answer from the given alternatives for each sub-question:
[2]1.i

Choose the correct alternative :

Which of the following is not a statement?

Smoking is injuries to health

2 + 2 = 4

2 is the only even prime number.

Come here

Concept: Mathematical Logic - Truth Value of Statement in Logic
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[2]1.ii

The value of x, y, z for the following system of equations x + y + z = 6, x − y+ 2z = 5, 2x + y − z = 1 are ______

x = 1, y = 2, z = 3

x = 2, y = 1, z = 3

x = −1, y = 2, z = 3

x = y = z = 3

Concept: Applications of Determinants and Matrices
Chapter: [0.012] Matrics
[2]1.iii

The principal solutions of sqrt(3) sec x − 2 = 0 are ______

pi/3, (11pi)/6

pi/6, (11pi)/6

pi/4, (11pi)/4

pi/6, (11pi)/3

Concept: Trigonometric Equations and Their Solutions
Chapter: [0.013000000000000001] Trigonometric Functions
[2]1.iv

Given that X ~ B(n = 10, p), E(X) = 8, then value of p = ______

0.4

0.8

0.6

0.7

Concept: Variance of Binomial Distribution (P.M.F.)
Chapter: [0.027999999999999997] Binomial Distribution [0.2] Bernoulli Trials and Binomial Distribution
[2]1.v

If a d.r.v. X has the following probability distribution:

 X 1 2 3 4 5 6 7 P(X = x) k 2k 2k 3k k2 2k2 7k2 + k

then k = ______

1/7

1/8

1/9

1/10

Concept: Probability Distribution of Discrete Random Variables
Chapter: [0.027000000000000003] Probability Distributions
[2]1.vi

Select and write the correct alternative from the given option for the question

The order and degree of (("d"y)/("d"x))^3 - ("d"^3y)/("d"x^3) + y"e"^x = 0 are respectively

3, 1

1, 3

3, 3

1, 1

Concept: Order and Degree of a Differential Equation
Chapter: [0.026000000000000002] Differential Equations [0.17] Differential Equation
[2]1.vii

int  ("e"^x(x - 1))/(x^2)  "d"x = ______

x"e"^(-x) + c

("e"^x)/(x^2) + c

(x - 1/x)"e"^x + c

("e"^x)/x + c

Concept: Methods of Integration - Integration by Substitution
Chapter: [0.023] Indefinite Integration [0.15] Integration
[2]1.viii

Select the correct option from the given alternatives:

If α, β, γ are direction angles of a line and α = 60°, β = 45°, γ = ______

30° or 90°

45° or 60°

90° or 30°

60° or 120°

Concept: Representation of Vector
Chapter: [0.015] Vectors
[1]2 | Answer the following questions:
[1]2.i

State the truth Value of x2 = 25

Concept: Mathematical Logic - Truth Value of Statement in Logic
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[1]2.ii

Find the area bounded by the curve y2 = 36x, the line x = 2 in first quadrant

Concept: Area Bounded by the Curve, Axis and Line
Chapter: [0.025] Application of Definite Integration
[1]2.iii

If y = "e"^(1 + logx) then find ("d"y)/("d"x)

Concept: Differentiation
Chapter: [0.021] Differentiation
[1]2.iv

The displacement of a particle at time t is given by s = 2t3 – 5t2 + 4t – 3. Find the velocity when ЁЭСб = 2 sec

Concept: Derivatives as a Rate Measure
Chapter: [0.022000000000000002] Applications of Derivatives
Section B
[2]3 | Attempt any Eight:

If statements p, q are true and r, s are false, determine the truth values of the following.

(p ∧ ~r) ∧ (~q ∨ s)

Concept: Mathematical Logic - Truth Value of Statement in Logic
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[2]4

If A = [(2, 4),(1, 3)] and B = [(1, 1),(0, 1)] then find (A−1 B−1)

Concept: Inverse of a Matrix
Chapter: [0.012] Matrics
[2]5

Find A–1 using adjoint method, where A = [(cos theta, sin theta),(-sin theta, cos theta)]

Concept: Inverse of a Matrix
Chapter: [0.012] Matrics
[2]6

If tan−1x + tan−1y + tan−1z = π, then show that 1/(xy) + 1/(yz) + 1/(zx) = 1

Concept: Inverse Trigonometric Functions
Chapter: [0.013000000000000001] Trigonometric Functions
[2]7

With usual notations, prove that (cos "A")/"a" + (cos "B")/"b" + (cos "C")/"c" = ("a"^2 + "b"^2 + "c"^2)/(2"abc")

Concept: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions
[2]8

Evaluate: int_0^(pi/2) (sin^2x)/(1 + cos x)^2 "d"x

Concept: Methods of Evaluation and Properties of Definite Integral
Chapter: [0.024] Definite Integration
[2]9

Verify y = log x + c is the solution of differential equation x ("d"^2y)/("d"x^2) + ("d"y)/("d"x) = 0

Concept: Formation of Differential Equations
Chapter: [0.026000000000000002] Differential Equations
[2]10

Find the derivative of the inverse of function y = 2x3 – 6x and calculate its value at x = −2

Concept: Derivatives of Inverse Functions
Chapter: [0.021] Differentiation
[2]11

Find the values of x, for which the function f(x) = x3 + 12x2 + 36ЁЭСе + 6 is monotonically decreasing

Concept: Increasing and Decreasing Functions
Chapter: [0.022000000000000002] Applications of Derivatives [0.14] Applications of Derivative
[2]12

A car is moving in such a way that the distance it covers, is given by the equation s = 4t2 + 3t, where s is in meters and t is in seconds. What would be the velocity and the acceleration of the car at time t = 20 seconds?

Concept: Derivatives as a Rate Measure
Chapter: [0.022000000000000002] Applications of Derivatives
[2]13

int (cos2x)/(sin^2x cos^2x)  "d"x

Concept: Methods of Integration - Integration by Parts
Chapter: [0.023] Indefinite Integration [0.15] Integration
[2]14

Find the position vector of point R which divides the line joining the points P and Q whose position vectors are 2hat"i" - hat"j" + 3hat"k" and -5hat"i" + 2hat"j" - 5hat"k" in the ratio 3:2
(i) internally
(ii) externally

Concept: Section formula
Chapter: [0.015] Vectors [0.07] Vectors
Section C
[3]15 | Attempt any Eight:

Use quantifiers to convert the following open sentences defined on N, into a true statement.

n2 ≥ 1

Concept: Mathematical Logic - Quantifier and Quantified Statements in Logic
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

Use quantifiers to convert the given open sentence defined on N into a true statement

3x – 4 < 9

Concept: Mathematical Logic - Quantifier and Quantified Statements in Logic
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

Use quantifiers to convert the given open sentence defined on N into a true statement

Y + 4 > 6

Concept: Mathematical Logic - Quantifier and Quantified Statements in Logic
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[3]16

Find the area of the region bounded by the curves x2 = 8y, y = 2, y = 4 and the Y-axis, lying in the first quadrant

Concept: Area Bounded by the Curve, Axis and Line
Chapter: [0.025] Application of Definite Integration
[3]17

The probability that 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

Concept: Mean of Binomial Distribution (P.M.F.)
Chapter: [0.027999999999999997] Binomial Distribution [0.2] Bernoulli Trials and Binomial Distribution
[3]18

Let the p.m.f. of r.v. X be P(x)  {{:(((3 - x)/10",", "for"  x = -1",", 0",", 1",", 2)),((0",", "otherwise")):} Calculate E(X) and Var(X)

Concept: Probability Distribution of Discrete Random Variables
Chapter: [0.027000000000000003] Probability Distributions
[3]19

Find the combined equation of the following pair of line passing through (−1, 2), one is parallel to x + 3y − 1 = 0 and other is perpendicular to 2x − 3y − 1 = 0

Concept: Combined Equation of a Pair Lines
Chapter: [0.013999999999999999] Pair of Straight Lines
[3]20

Find the measure of the acute angle between the line represented by 3"x"^2 - 4sqrt3"xy" + 3"y"^2 = 0

Concept: Angle Between Lines
Chapter: [0.013999999999999999] Pair of Straight Lines
[3]21

If y = sqrt(cos x + sqrt(cos x + sqrt(cos x + ...... ∞), show that ("d"y)/("d"x) = (sin x)/(1 - 2y)

Concept: Differentiation
Chapter: [0.021] Differentiation
[3]22

If x sin(a + y) + sin a cos(a + y) = 0 then show that ("d"y)/("d"x) = (sin^2("a" + y))/(sin"a")

Concept: Derivatives of Parametric Functions
Chapter: [0.021] Differentiation
[3]23

int sec^2x sqrt(tan^2x + tanx - 7)  "d"x

Concept: Methods of Integration - Integration Using Partial Fractions
Chapter: [0.023] Indefinite Integration [0.15] Integration
[3]24

int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x

Concept: Methods of Integration - Integration Using Partial Fractions
Chapter: [0.023] Indefinite Integration [0.15] Integration
[3]25

Find the Cartesian equation of the line passing through (−1, −1, 2) and parallel to the line 2x − 2 = 3y + 1 = 6z – 2

Concept: Vector and Cartesian Equations of a Line
Chapter: [0.016] Line and Plane
[3]26

Find acute angle between the lines (x - 1)/1 = (y - 2)/(-1) = (z - 3)/2 and (x - 1)/2 = (y - 1)/1 = (z - 3)/1

Concept: Angle Between Planes
Chapter: [0.016] Line and Plane
Section D
[4]27 | Attempt any Five:

In ΔABC, prove that ("b"^2 - "c"^2)/"a" cos"A" + ("c"^2 - "a"^2)/"b" cos"B" + ("a"^2 - "b"^2)/"c" cos "C" = 0

Concept: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions
[4]28

Maximize z = −x + 2y subjected to constraints x + y ≥ 5, x ≥ 3, x + 2y ≤ 6, y ≥ 0 is this LPP solvable? Justify your answer

Concept: Linear Programming Problem (L.P.P.)
Chapter: [0.017] Linear Programming
[4]29

Prove that: int_(-"a")^"a" "f"(x)  "d"x {:(= 2 int_0^"a" "f" (x)  "d"x ",",  "If"  "f"(x)  "is even function"),(= 0",", "if"  "f"(x)  "is odd function"):}

Concept: Methods of Evaluation and Properties of Definite Integral
Chapter: [0.024] Definite Integration
[4]30

If the population of a town increases at a rate proportional to the population at that time. If the population increases from 40 thousand to 60 thousand in 40 years, what will be the population in another 20 years? ("Given" sqrt(3/2) = 1.2247)

Concept: Application of Differential Equations
Chapter: [0.026000000000000002] Differential Equations
[4]31

A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?

Concept: Maxima and Minima
Chapter: [0.022000000000000002] Applications of Derivatives [0.14] Applications of Derivative
[4]32

Find the Cartesian and vector equation of the plane which makes intercepts 1, 1, 1 on the coordinate axes

Concept: Vector and Cartesian Equations of a Line
Chapter: [0.016] Line and Plane
[4]33

If four points "A"(bar"a"), "B"(bar"b"), "C"(bar"c") and "D"(bar"d") are coplanar, then show that [(bar"a", bar"b", bar"c")] + [(bar"b", bar"c", bar"d")] + [(bar"c", bar"a", bar"d")] = [(bar"a", bar"b", bar"c")].

Concept: Representation of Vector
Chapter: [0.015] Vectors
[4]34

If Q is the foot of the perpendicular from P(2, 4, 3) on the line joining the point A(1, 2, 4) and B(3, 4, 5), find coordinates of Q

Concept: Representation of Vector
Chapter: [0.015] Vectors

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