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Mathematics and Statistics Shaalaa.com Model Set 2 2020-2021 HSC Science (General) 12th Board Exam Question Paper Solution

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Mathematics and Statistics [Shaalaa.com Model Set 2]
Marks: 80 Academic Year: 2020-2021
Date: April 2021
Duration: 3h
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  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

A biconditional statement is the conjunction of two ______ statements

Negative

Compound

Connective

Conditional

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

If polar co-ordinates of a point are `(3/4, (3pi)/4)`, then its Cartesian co-ordinate are ______

`(3/(4sqrt(2)), -3/(4sqrt(2)))`

`(3/(4sqrt(2)), 3/(4sqrt(2)))`

`(-3/(4sqrt(2)), 3/(4sqrt(2)))`

`(-3/(4sqrt(2)), -3/(4sqrt(2)))`

Concept: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions
[2] 1.iii

The feasible region is the set of point which satisfy.

The object functions

All the given constraints

Some of the given constraints

Only one constraint

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

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

Differential equation of the function c + 4yx = 0 is

`xy + ("d"y)/("d"x)` = 0

`x  ("d"y)/("d"x) + y` = 0

`("d"y)/("d"x) - 4xy` = 0

`x ("d"y)/("d"x) + 1` = 0

Concept: Differential Equations
Chapter: [0.026000000000000002] Differential Equations
[2] 1.v

If x = cos−1(t), y = `sqrt(1 - "t"^2)` then `("d"y)/("d"x)` = ______

t

– t

`(-1)/"t"`

`1/"t"`

Concept: Derivatives of Composite Functions - Chain Rule
Chapter: [0.021] Differentiation [0.13] Differentiation
[2] 1.vi

A ladder 5 m in length is resting against vertical wall. The bottom of the ladder is pulled along the ground, away from the wall at the rate of 1.5 m /sec. The length of the higher point of the when foot of the ladder is 4 m away from the wall decreases at the rate of ______

1

2

2.5

3

Concept: Rolle's Theorem
Chapter: [0.022000000000000002] Applications of Derivatives
[2] 1.vii

Select the correct option from the given alternatives:

If l, m, n are direction cosines of a line then `"l"hat
"i" + "m"hat"j" + "n"hat"k"` is ______ 

null vector

the unit vector along the line

any vector along the line

a vector perpendicular to the line

Concept: Vectors and Their Types
Chapter: [0.015] Vectors
[2] 1.viii

Select the correct option from the given alternatives:

The 2 vectors `hat"j" + hat"k"` and `3hat"i" - hat"j" + 4hat"k"` represents the two sides AB and AC respectively of a ΔABC. The length of the median through A is

`sqrt(34)/2`

`sqrt(48)/2`

`sqrt(18)`

of the median through A is

Concept: Position Vector of a Point P(X, Y, Z) in Space
Chapter: [0.015] Vectors
[1] 2 | Answer the following questions:
[1] 2.i

State the truth Value of x2 = 25

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

Evaluate: `int_0^1 ""^x/sqrt("e"^x - 1)  "d"x`

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

An urn contains 5 red and 2 black balls. Two balls are drawn at random. X denotes number of black balls drawn. What are possible values of X?

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

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

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

Write the converse and contrapositive of the following statements.

“If a function is differentiable then it is continuous”

Concept: Statement Patterns and Logical Equivalence
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
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[2] 4

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 [0.03] Trigonometric Functions
[2] 5

Find the graphical solution for the system of linear inequation 2x + y ≤ 2, x − y ≤ 1

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.017] Linear Programming [0.11] Linear Programming Problems
[2] 6

Find the area enclosed between the X-axis and the curve y = sin x for values of x between 0 to 2π

Concept: Area Between Two Curves
Chapter: [0.025] Application of Definite Integration [0.16] Applications of Definite Integral
[2] 7

Find the area of the ellipse `x^2/36 + y^2/64` = 1, using integration

Concept: Area Between Two Curves
Chapter: [0.025] Application of Definite Integration [0.16] Applications of Definite Integral
[2] 8

Let the p.m.f. of r.v. X be P(x) = `""^4"C"_x (5/9)^x (4/9)^(4 - x)`, x = 0, 1, 2, 3, 4. Find E(X) and Var(X)

Concept: Variance of Binomial Distribution (P.M.F.)
Chapter: [0.027999999999999997] Binomial Distribution [0.2] Bernoulli Trials and Binomial Distribution
[2] 9

Find the value of h, if the measure of the angle between the lines 3x2 + 2hxy + 2y2 = 0 is 45°. 

Concept: Angle between lines represented by ax2 + 2hxy + by2 = 0
Chapter: [0.013999999999999999] Pair of Straight Lines
[2] 10

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] 11

Water is being poured at the rate of 36 m3/sec in to a cylindrical vessel of base radius 3 meters. Find the rate at which water level is rising

Concept: Derivatives as a Rate Measure
Chapter: [0.022000000000000002] Applications of Derivatives
[2] 12

`int ["cosec"(logx)][1 - cot(logx)]  "d"x`

Concept: Methods of Integration: Integration by Parts
Chapter: [0.023] Indefinite Integration [0.15] Integration
[2] 13

If `bar("a"), bar("b")` and `bar("c")` are position vectors of the points A, B, C respectively and `5bar("a") - 3bar("b") - 2bar("c") = bar(0)`, then find the ratio in which the point C divides the line segement BA

Concept: Section Formula
Chapter: [0.015] Vectors [0.07] Vectors
[2] 14

If a line has the direction ratios 4, −12, 18, then find its direction cosines

Concept: Scalar Triple Product of Vectors
Chapter: [0.015] Vectors [0.07] Vectors
Section C
[3] 15 | Attempt any Eight:

Write the following statements in symbolic form

Milk is white if and only if the sky is not blue

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

Write the following statements in symbolic form

If Kutab – Minar is in Delhi then Taj - Mahal is in Agra

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

Write the following statements in symbolic form

Even though it is not cloudy, it is still raining

Concept: Logical Connective, Simple and Compound Statements
Chapter: [0.01] Mathematical Logic [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
[3] 16

Three chairs and two tables cost ₹ 1850. Five chairs and three tables cost ₹2850. Find the cost of four chairs and one table by using matrices

Concept: Applications of Determinants and Matrices
Chapter: [0.012] Matrics
[3] 17

Transform `[(1, 2, 4),(3, -1, 5),(2, 4, 6)]` into an upper triangular matrix by using suitable row transformations

Concept: Applications of Determinants and Matrices
Chapter: [0.012] Matrics
[3] 18

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
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[3] 19

In ∆ABC, prove that `(cos 2"A")/"a"^2 - (cos 2"c")/"c"^2 = 1/"a"^2 - 1/"c"^2`

Concept: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions
[3] 20

The probability that a person who undergoes a kidney operation will be recovered is 0.5. Find the probability that out of 6 patients who undergo similar operation none will recover

Concept: Mean of Binomial Distribution (P.M.F.)
Chapter: [0.027999999999999997] Binomial Distribution [0.2] Bernoulli Trials and Binomial Distribution
[3] 21

Prove that: `int_0^"a" "f"(x)  "d"x = int_0^"a" "f"("a" - x)  "d"x`. Hence find `int_0^(pi/2) sin^2x  "d"x` 

Concept: Fundamental Theorem of Integral Calculus
Chapter: [0.024] Definite Integration
[3] 22

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
[3] 23

Find the differential equation by eliminating arbitrary constants from the relation x2 + y2 = 2ax

Concept: Formation of Differential Equations
Chapter: [0.026000000000000002] Differential Equations
[3] 24

If logs `((x^4 + y^4)/(x^4 - y^4))` = 2, show that `("d"y)/("d"x) = (12x^2)/(13y^3)`

Concept: Logarithmic Differentiation
Chapter: [0.021] Differentiation
[3] 25

Find the vector equation of the line passing through the point having position vector `-hat"i"- hat"j" + 2hat"k"` and parallel to the line `bar"r" = (hat"i" + 2hat"j" + 3hat"k") + mu(3hat"i" + 2hat"j" + hat"k")`, µ is a parameter

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

Find the equation of the plane passing through the point (7, 8, 6) and parallel to the plane `bar"r"*(6hat"i" + 8hat"j" + 7hat"k")` = 0

Concept: Equation of a Plane
Chapter: [0.016] Line and Plane
Section D
[4] 27 | Attempt any Five:

The following is the c.d.f. of r.v. X:

X −3 −2 −1 0 1 2 3 4
F(X) 0.1 0.3 0.5 0.65 0.75 0.85 0.9 1

Find p.m.f. of X.
i. P(–1 ≤ X ≤ 2)
ii. P(X ≤ 3 / X > 0).

Concept: Probability Distribution of Discrete Random Variables
Chapter: [0.027000000000000003] Probability Distributions
[4] 28

Show that the combined equation of pair of lines passing through the origin is a homogeneous equation of degree 2 in x and y. Hence find the combined equation of the lines 2x + 3y = 0 and x − 2y = 0

Concept: Combined Equation of a Pair Lines
Chapter: [0.013999999999999999] Pair of Straight Lines
[4] 29

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
[4] 30

A man of height 180 cm is moving away from a lamp post at the rate of 1.2 meters per second. If the height of the lamp post is 4.5 meters, find the rate at which
(i) his shadow is lengthening
(ii) the tip of the shadow is moving

Concept: Derivatives as a Rate Measure
Chapter: [0.022000000000000002] Applications of Derivatives
[4] 31

`int  ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1])  "d"x`

Concept: Methods of Integration: Integration Using Partial Fractions
Chapter: [0.023] Indefinite Integration [0.15] Integration
[4] 32

`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
[4] 33

A(– 2, 3, 4), B(1, 1, 2) and C(4, –1, 0) are three points. Find the Cartesian equations of the line AB and show that points A, B, C are collinear

Concept: Vector and Cartesian Equations of a Line
Chapter: [0.016] Line and Plane
[4] 34

Let `"A" (bar"a")` and `"B" (bar"b")` are any two points in the space and `"R"(bar"r")` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `bar "r" = ("m"bar"b" + "n"bar"a")/("m" + "n") `

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