# Mathematics and Statistics Model set 2 shaalaa.com 2021-2022 HSC Science (Electronics) 12th Standard Board Exam Question Paper Solution

Mathematics and Statistics [Model set 2 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

For the random variable X, if V(X) = 4, E(X) = 3, then E(x2) is ______

9

13

12

7

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027000000000000003] Probability Distributions [0.19] Probability Distribution
 1.ii

The area of triangle ΔABC whose vertices are A(1, 1), B(2, 1) and C(3, 3) is ______ sq.units

1

2

3

4

Concept: Area Between Two Curves
Chapter: [0.025] Application of Definite Integration [0.16] Applications of Definite Integral
 1.iii

(hat"i" + hat"j" - hat"k")*(hat"i" - hat"j" + hat"k") = ______

hat"i" - hat"j" - hat"k"

1

−1

−hat"j" + hat"k"

Concept: Representation of Vector
Chapter: [0.015] Vectors
 1.iv

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
 1.v

Negation of p → (p ˅ ∼ q) is

∼ p → (∼ p ˅ q)

p ˄ (∼ p ˄ q)

∼ p ˅ (∼ p ˅ ∼ q)

∼ p → (∼ p → q)

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

Choose correct alternatives :

The direction ratios of the line 3x + 1 = 6y – 2 = 1 – z are ______

2, 1, 6

2, 1, – 6

2, – 1, 6

– 2, 1, 6

Concept: Distance Between Skew Lines and Parallel Lines
Chapter: [0.016] Line and Plane
 1.vii

If the tangent at (1, 1) on y2 = x(2 − x)2 meets the curve again at P, then P is

(4, 4)

(−1, 2)

(3, 6)

(9/4, 3/8)

Concept: Applications of Derivatives in Geometry
Chapter: [0.022000000000000002] Applications of Derivatives
 1.viii

int_0^(pi/2) log(tanx)  "d"x =

pi/8(log2)

0

- pi/8 (log2)

pi/2 (log2)

Concept: Methods of Evaluation and Properties of Definite Integral
Chapter: [0.024] Definite Integration
 2 | Answer the following questions:
 2.i

State if the following is not the probability mass function of a random variable. Give reasons for your answer.

 X 0 1 2 3 4 P(X) 0.1 0.5 0.2 − 0.1 0.2
Concept: Probability Distribution of Discrete Random Variables
Chapter: [0.027000000000000003] Probability Distributions
 2.ii

If bar("a") = 4hat"i" + 3hat"k" and bar("b") = -2hat"i" + hat"j" + 5hat"k", then find 2bar("a") + 5bar("b")

Concept: Vector Joining Two Points
Chapter: [0.015] Vectors
 2.iii

State the truth value of (p ˅ ∼p)

Concept: Truth Value of Statement
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
 2.iv

int e^x/x [x (log x)^2 + 2 log x] dx = ______________

ex log x + c

ex (log x)2 + c

e2x log x + c

e2x (log x)2 + c

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.023] Indefinite Integration [0.15] Integration
SECTION-B
 3 | Attempt any EIGHT of the following questions:

Find the area of the region bounded by the following curves, X-axis and the given lines: x = 2y, y = 0, y = 4

Concept: Area Bounded by the Curve, Axis and Line
Chapter: [0.025] Application of Definite Integration
 4

In ∆ABC, if a = 13, b = 14, c = 15, then find the value of cos B

Concept: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions
 5

Find the perpendicular distance of origin from the plane 6x − 2y + 3z - 7 = 0

Concept: Equation of a Plane
Chapter: [0.016] Line and Plane
 6

Evaluate: int_0^(pi/2) sqrt(1 - cos 4x)  "d"x

Concept: Methods of Evaluation and Properties of Definite Integral
Chapter: [0.024] Definite Integration
 7

Write the following compound statements symbolically.

Triangle is equilateral or isosceles

Concept: Logical Connective, Simple and Compound Statements
Chapter: [0.01] Mathematical Logic [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
 8

If f(x) = x2 − 2x − 3 then find f(A) when A = [(1, 2),(2, 1)]

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

int "e"^(3logx) (x^4 + 1)^(-1) "d"x

Concept: Methods of Integration: Integration Using Partial Fractions
Chapter: [0.023] Indefinite Integration [0.15] Integration
 10

If y = log [cos(x5)] then find ("d"y)/("d"x)

Concept: Logarithmic Differentiation
Chapter: [0.021] Differentiation
 11

Find the differential equation of family of all ellipse whose major axis is twice the minor axis

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

Evaluate : int_1^9(x + 1)/sqrt(x)*dx

Concept: Fundamental Theorem of Integral Calculus
Chapter: [0.024] Definite Integration
 13

If a vector has direction angles 45° and 60°, find the third direction angle.

Concept: Scalar Triple Product of Vectors
Chapter: [0.015] Vectors [0.07] Vectors
 14

Find the direction ratios of the line perpendicular to the lines

(x - 7)/2 = (y + 7)/(-3) = (z - 6)/1 and (x + 5)/1 = (y + 3)/2 = (z - 6)/(-2)

Concept: Vector and Cartesian Equations of a Line
Chapter: [0.016] Line and Plane
SECTION-C
 15 | Attempt any EIGHT of the following questions:

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
 16

Find the centroid of tetrahedron with vertices K(5, −7, 0), L(1, 5, 3), M(4, −6, 3), N(6, −4, 2)

Concept: Section Formula
Chapter: [0.015] Vectors [0.07] Vectors
 17

Find the condition that the line 4x + 5y = 0 coincides with one of the lines given by ax2 + 2hxy + by2 = 0

Concept: Homogeneous Equation of Degree Two
Chapter: [0.013999999999999999] Pair of Straight Lines
 18

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

n2 ≥ 1

Concept: Quantifier and Quantified Statements in Logic
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
 19

A ladder 10 meter long is leaning against a vertical wall. If the bottom of the ladder is pulled horizontally away from the wall at the rate of 1.2 meters per seconds, find how fast the top of the ladder is sliding down the wall when the bottom is 6 meters away from the wall

Concept: Derivatives as a Rate Measure
Chapter: [0.022000000000000002] Applications of Derivatives
 20

int sqrt((9 + x)/(9 - x))  "d"x

Concept: Methods of Integration: Integration Using Partial Fractions
Chapter: [0.023] Indefinite Integration [0.15] Integration
 21

Solve the differential equation ("d"y)/("d"x) + y = e−x

Concept: Differential Equations
Chapter: [0.026000000000000002] Differential Equations
 22

If the angles A, B, C of ΔABC are in A.P. and its sides a, b, c are in G.P., then show that a2, b2, c2 are in A.P.

Concept: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions
 23

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 represented by ax2 + 2hxy + by2 = 0
Chapter: [0.013999999999999999] Pair of Straight Lines
 24

Solve the differential equation (x2 – yx2)dy + (y2 + xy2)dx = 0

Concept: Differential Equations
Chapter: [0.026000000000000002] Differential Equations
 25

If y = log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))], find ("d"y)/("d"x)

Concept: Logarithmic Differentiation
Chapter: [0.021] Differentiation
 26

In ΔABC, if a cos A = b cos B, then prove that ΔABC is either a right angled or an isosceles triangle

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

Using vector method, find the incenter of the triangle whose vertices are A(0, 3, 0), B(0, 0, 4) and C(0, 3, 4)

Concept: Section Formula
Chapter: [0.015] Vectors [0.07] Vectors
 28

int (3x + 4)/sqrt(2x^2 + 2x + 1)  "d"x

Concept: Methods of Integration: Integration Using Partial Fractions
Chapter: [0.023] Indefinite Integration [0.15] Integration
 29

If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x such that the composite function y = f[g(x)] is a differentiable function of x, then ("d"y)/("d"x) = ("d"y)/("d"u)*("d"u)/("d"x). Hence find ("d"y)/("d"x) if y = sin2x

Concept: Derivatives of Composite Functions - Chain Rule
Chapter: [0.021] Differentiation [0.13] Differentiation
 30

Solve the following:

A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.

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

Find m, if the lines (1 - x)/3 =(7y - 14)/(2"m") = (z - 3)/2 and (7 - 7x)/(3"m") = (y - 5)/1 = (6 - z)/5 are at right angles

Concept: Vector and Cartesian Equations of a Line
Chapter: [0.016] Line and Plane
 32

Find the inverse of A = [(2, -3, 3),(2, 2, 3),(3, -2, 2)] by using elementary row transformations

Concept: Elementry Transformations
Chapter: [0.012] Matrics
 33

Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean, variance and standard deviation of the number of kings drawn

Concept: Variance of a Random Variable
Chapter: [0.027000000000000003] Probability Distributions
 34

Maximize z = 7x + 11y subject to 3x + 5y ≤ 26, 5x + 3y ≤ 30, x ≥ 0, y ≥ 0

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

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