# Mathematics and Statistics Model set 1 shaalaa.com 2021-2022 HSC Science (Computer Science) 12th Board Exam Question Paper Solution

Mathematics and Statistics [Model set 1 shaalaa.com]
Date: March 2022

General instructions:

• The question paper is divided into four sections:
1. Section A: Question No. 1 contains 8 multiple choice type questions carrying two marks each. Question No. 2 contains 4 very short answer type questions carrying one mark each.
2. Section B: Question No. 3 to Question No. 14 are 12 short answer-I type questions carrying two marks each. Attempt any eight questions.
3. Section C: Question No. 15 to Question No. 26 are 12 short answer-II type questions carrying three marks each. Attempt any eight questions.
4. Section D: Question No. 27 to Question No. 34 are 8 long answer type questions carrying four marks each. Attempt any five questions.
• Start each section on a new page.
• Figures to the right indicate full marks.
• 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.
• Use of graph paper is not necessary. Only rough sketch of graph is expected
• Use of log table is necessary. Use of calculator is not allowed.

SECTION-A
 1 | Select and write the correct answers to the following questions:
 1.i

If p → q is an implication, then the implication ∼ q → ∼ p is called its

Converse

Contrapositive

Inverse

Alternative

Concept: Logical Connective, Simple and Compound Statements
Chapter: [0.01] Mathematical Logic [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
 1.ii

If |bar("a")| = 2, |bar("b")| = 5, and bar("a")*bar("b") = 8 then |bar("a") - bar("b")| = ______

13

12

sqrt(13)

sqrt(21)

Concept: Algebra of Vectors
Chapter: [0.015] Vectors
 1.iii

The displacement of a particle at time t is given by s = 2t3 – 5t2 + 4t – 3. The time when the acceleration is 14 ft/sec2, is

1 sec

2 sec

3 sec

4 sec

Concept: Derivatives as a Rate Measure
Chapter: [0.022000000000000002] Applications of Derivatives
 1.iv

If f(x) = logx (log x) then f'(e) is ______

1

e

1/"e"

0

Concept: Logarithmic Differentiation
Chapter: [0.021] Differentiation
 1.v

The general solution of ("d"y)/("d"x) = e−x is

y = ex + c

y = e–x + c

y = – e–x + c

y = e2x + c

Concept: Formation of Differential Equations
Chapter: [0.026000000000000002] Differential Equations
 1.vi

If A = [(4, -1),(-1, "k")] such that A2 − 6A + 7I = 0, then K = ______

1

3

2

4

Concept: Inverse of Matrix
Chapter: [0.012] Matrics
 1.vii

The area bounded by the parabola y2 = x along the X-axis and the lines x = 0, x = 2 is ______ sq.units

4/3

(4sqrt(2))/3

2/3

(2sqrt(2))/3

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

The combined equation of the lines through origin and perpendicular to the pair of lines 3x2 + 4xy − 5y2 = 0 is ______

5x2 + 4xy − 3y2 = 0

3x2 + 4xy − 5y2 = 0

3x2 - 4xy + 5y2 = 0

5x2 + 4xy + 3y2 = 0

Concept: Combined Equation of a Pair Lines
Chapter: [0.013999999999999999] Pair of Straight Lines
 2 | Answer the following questions:
 2.i

Evaluate: int_(pi/6)^(pi/3) cosx  "d"x

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

Find the distance from (4, −2, 6) to the XZ- plane

Concept: Three Dimensional (3-D) Coordinate System
Chapter: [0.015] Vectors
 2.iii

Find the separate equation of the line represented by the following equation:

3y2 + 7xy = 0

Concept: Combined Equation of a Pair Lines
Chapter: [0.013999999999999999] Pair of Straight Lines
 2.iv

Find the principal solutions of cosec x = 2

Concept: Trigonometric Equations and Their Solutions
Chapter: [0.013000000000000001] Trigonometric Functions
SECTION-B
 3 | Attempt any EIGHT of the following questions:

Write the following compound statement symbolically.

Nagpur is in Maharashtra and Chennai is in Tamil Nadu.

Concept: Logical Connective, Simple and Compound Statements
Chapter: [0.01] Mathematical Logic [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
 4

Find the Cartesian equation of the line passing through A(1, 2, 3) and B(2, 3, 4)

Concept: Vector and Cartesian Equations of a Line
Chapter: [0.016] Line and Plane
 5

Given X ~ B(n, p). If E(X) = 6, V(X) = 4.2, find n and p

Concept: Binomial Distribution
Chapter: [0.027999999999999997] Binomial Distribution
 6

If bar("a") and bar("b") are two vectors perpendicular each other, prove that (bar("a") + bar("b"))^2 = (bar("a") - bar("b"))^2

Concept: Scalar Product of Vectors (Dot)
Chapter: [0.015] Vectors
 7

Find the principal solutions of tan x = -sqrt(3)

Concept: Trigonometric Equations and Their Solutions
Chapter: [0.013000000000000001] Trigonometric Functions
 8

Find the adjoint of matrix A = [(6, 5),(3, 4)]

Concept: Inverse of Matrix
Chapter: [0.012] Matrics
 9

The probability distribution of X is as follows:

 X 0 1 2 3 4 P(X = x) 0.1 k 2k 2k k

Find k and P[X < 2]

Concept: Probability Distribution of Discrete Random Variables
Chapter: [0.027000000000000003] Probability Distributions
 10

Find the acute angle between the lines x = y, z = 0 and x = 0, z = 0

Concept: Angle Between Planes
Chapter: [0.016] Line and Plane
 11

Form the differential equation of y = (c1 + c2)ex

Concept: Formation of Differential Equations
Chapter: [0.026000000000000002] Differential Equations
 12

If A = [(-1),(2),(3)], B = [(3, 1, -2)], find B'A'

Concept: Inverse of Matrix
Chapter: [0.012] Matrics
 13

Evaluate: int_0^(pi/2) cos^3x  "d"x

Concept: Methods of Evaluation and Properties of Definite Integral
Chapter: [0.024] Definite Integration
 14

Find the area of the region bounded by the following curves, X-axis and the given lines : x = 0, x = 5, y = 0, y = 4

Concept: Area Bounded by the Curve, Axis and Line
Chapter: [0.025] Application of Definite Integration
SECTION-C
 15 | Attempt any EIGHT of the following questions:

int sin(logx)  "d"x

Concept: Methods of Integration: Integration Using Partial Fractions
Chapter: [0.023] Indefinite Integration [0.15] Integration
 16

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
 17

Prove that: int_"a"^"b" "f"(x)  "d"x = int_"a"^"c""f"(x)  "d"x + int_"c"^"b"  "f"(x)  "d"x, where a < c < b

Concept: Fundamental Theorem of Integral Calculus
Chapter: [0.024] Definite Integration
 18

A spherical soap bubble is expanding so that its radius is increasing at the rate of 0.02 cm/sec. At what rate is the surface area is increasing, when its radius is 5 cm?

Concept: Derivatives as a Rate Measure
Chapter: [0.022000000000000002] Applications of Derivatives
 19

Differentiate cot^-1((cos x)/(1 + sinx)) w.r. to x

Concept: Differentiation
Chapter: [0.021] Differentiation
 20

A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027000000000000003] Probability Distributions [0.19] Probability Distribution
 21

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

Concept: Equation of a Line in Space
Chapter: [0.013999999999999999] Pair of Straight Lines [0.09] Line
 22

Find the coordinates of the foot of perpendicular from the origin to the plane 2x + 6y − 3z = 63

Concept: Equation of a Plane
Chapter: [0.016] Line and Plane
 23

Prove that medians of a triangle are concurrent

Concept: Section Formula
Chapter: [0.015] Vectors [0.07] Vectors
 24

int (x^2 + x -1)/(x^2 + x - 6)  "d"x

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

A fair coin is tossed 8 times. Find the probability that it shows heads exactly 5 times

Concept: Mean of Binomial Distribution (P.M.F.)
Chapter: [0.027999999999999997] Binomial Distribution [0.2] Bernoulli Trials and Binomial Distribution
 26

In ∆ABC, prove that sin  (("A" - "B")/2) = (("a" - "b")/"c") cos ("C"/2)

Concept: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions
SECTION-D
 27 | Attempt any FIVE of the following questions:

int (x + sinx)/(1 - cosx)  "d"x

Concept: Methods of Integration: Integration Using Partial Fractions
Chapter: [0.023] Indefinite Integration [0.15] Integration
 28

Maximize z = 5x + 2y subject to 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0

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

Find the local maximum and local minimum value of  f(x) = x3 − 3x2 − 24x + 5

Concept: Maxima and Minima
Chapter: [0.022000000000000002] Applications of Derivatives [0.14] Applications of Derivative
 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
 31

Examine whether the statement pattern

[p → (∼q ˅ r)] ↔ ∼[p → (q → r)] is a tautology, contradiction or contingency.

Chapter: [0.011000000000000001] Mathematical Logic
 32

Prove that 2 tan^-1 (1/8) + tan^-1 (1/7) + 2tan^-1 (1/5) = pi/4

Concept: Inverse Trigonometric Functions
Chapter: [0.013000000000000001] Trigonometric Functions
 33

If y = cos(m cos–1x), then show that (1 - x^2) ("d"^2y)/("d"x^2) - x("d"y)/("d"x) + "m"^2y = 0

Concept: Derivatives of Parametric Functions
Chapter: [0.021] Differentiation
 34

Express - hat"i" - 3hat"j" + 4hat"k" as the linear combination of the vectors 2hat"i" + hat"j" - 4hat"k", 2hat"i" - hat"j" + 3hat"k" and 3hat"i" + hat"j" - 2hat"k"

Concept: Vectors and Their Types
Chapter: [0.015] Vectors

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## Maharashtra State Board previous year question papers 12th Board Exam Mathematics and Statistics with solutions 2021 - 2022

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