# CBSE Question Paper 2019

2019 Mathematics Questions Set -3

**: 3 Hours**

**: 100**

- All questions are compulsory.
- The question paper consists of 30 questions divided into four sections: A, B, C and D.
- Section A contains 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 8 questions of 4 marks each.
- There is no overall choice. However, an internal choice has been provided in two questions of 1 mark each, two questions of 2 marks each, four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions.
- Use of calculators is not permitted.

### SECTION - A

1. Write the discriminant of the quadratic equation (x + 5)2= 2 (5x – 3).

2. Find after how many places of decimal the decimal form of the number will terminate.

OR

Express 429 as a product of its prime factors.

3. Find the sum of first 10 multiples of 6.

4. Find the value(s) of x, if the distance between the points A(0, 0) and B(x, – 4) is 5 units.

5. Two concentric circles of radii a and b (a > b) are given. Find the length of the chord of the larger circle which touches the smaller circle.

6. In Figure 1, PS = 3 cm, QS = 4 cm, ∠PRQ=θ, ∠PSQ=90?∠PRQ=θ, ∠PSQ=90?, PQ ⊥ RQ and RQ = 9 cm. Evaluate tanθ.

OR

If tanα=, find the value of sec α.

### SECTION - B

7. Points A(3, 1), B(5, 1), C(a, b) and D(4, 3) are vertices of a parallelogram ABCD. Find the values of a and b.

OR

Points P and Q trisect the line segment joining the points A(– 2, 0) and B(0, 8) such that P is near to A. Find the coordinates of points P and Q.

8. Solve the following pair of linear equations :

3x – 5y = 4

2y + 7 = 9x

9. If HCF of 65 and 117 is expressible in the form 65n – 117, then find the value of n.

OR

On a morning walk, three persons step out together and their steps measure 30 cm, 36 cm and 40 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?

10 .A die is thrown once. Find the probability of getting (i) a composite number, (ii) a prime number.

11. Using completing the square method, show that the equation x2 – 8x + 18 = 0 has no solution.

12. Cards numbered 7 to 40 were put in a box. Poonam selects a card at random. What is the probability that Poonam selects a card which is a multiple of 7?

### SECTION - C

13. The perpendicular from A on side BC of a ?ABC meets BC at D such that DB = 3CD. Prove that 2AB2 = 2AC2 + BC2.

OR

AD and PM are medians of triangles ABC and PQR respectively where ΔABC∼ΔPQR. Prove that =

14. Check whether g(x) is a factor of p(x) by dividing polynomial p(x) by polynomial g(x), where p(x) = x5– 4x3 + x2 + 3x + 1, g(x) = x3 – 3x + 1.

15. Find the area of the triangle formed by joining the mid-points of the sides of the triangle ABC, whose vertices are A(0, – 1), B(2, 1) and C(0, 3).

16. Draw the graph of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Using this graph, find the values of x and y which satisfy both the equations.

17 .Prove that √3 is an irrational number.

OR

Find the largest number which on dividing 1251, 9377 and 15628 leaves remainders 1, 2 and 3 respectively.

18. A, B and C are interior angles of a triangle ABC. Show that

- sin()=cos
- If ∠A=90, then find the value of tan().

OR

If tan (A + B) = 1 and tan (A – B) = , 0 < A + B < 90°, A > B, then find the values of A and B.

19. In Figure 2, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP.

OR

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

20. Water in a canal, 6 m wide and 1·5 m deep, is flowing with a speed of 10 km/h. How much area will it irrigate in 30 minutes if 8 cm of standing water is needed?

21. A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent

Number of Days

0 - 6

6 - 12

12 - 18

18 - 24

24 - 30

30 - 36

36 - 42

Number of students

10

11

7

4

4

3

1

22. A car has two wipers which do not overlap. Each wiper has a blade of length 21 cm sweeping through an angle 120°. Find the total area cleaned at each sweep of the blades.(Take π=)

### SECTION - D

23. A pole has to be erected at a point on the boundary of a circular park of diameter 13 m in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 m. Is it possible to do so? If yes, at what distances from the two gates should the pole be erected?

24. If m times the mth term of an Arithmetic Progression is equal to n times its nth term and m≠ n, show that the (m + n)th term of the A.P. is zero.

OR

The sum of the first three numbers in an Arithmetic Progression is 18. If the product of the first and the third term is 5 times the common difference, find the three numbers.

25. Construct a triangle ABC with side BC = 6 cm, AB = 5 cm and ∠∠ABC = 60°. Then construct another triangle whose sides are of the corresponding sides of the triangle ABC.

26. In Figure 3, a decorative block is shown which is made of two solids, a cube and a hemisphere. The base of the block is a cube with edge 6 cm and the hemisphere fixed on the top has a diameter of 4·2 cm. Find

- the total surface area of the block.
- the volume of the block formed. (Take π=)

27. A bucket open at the top is in the form of a frustum of a cone with a capacity of 12308.8 cm3. The radii of the top and bottom circular ends are 20 cm and 12 cm respectively. Find the height of the bucket and the area of metal sheet used in making the bucket. If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio.

OR

Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

28. If 1 + sin2θ=3sinθcosθ, then prove that tanθ=1 or tanθ= .

29. Change the following distribution to a ‘more than type’ distribution. Hence draw the ‘more than type’ ogive for this distribution.

Class Interval

20 - 30

30 - 40

40 - 50

50 - 60

60 - 70

70 - 80

80 - 90

Frequency

10

8

12

24

6

25

15

30. The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun’s altitude is 30° than when it was 60°. Find the height of the tower. (Given √3=1.732)

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- Chapter 4 - Quadratic equations
- Chapter 5 - Arithmetic progressions
- Chapter 6 - Triangles
- Chapter 7 - Coordinate Geometry
- Chapter 8 - Introduction to trigonometry
- Chapter 9 - Some applications of trigonometry
- Chapter 10 - Circles
- Chapter 11 - Constructions
- Chapter 12 - Area Related to circles
- Chapter 13 - Surface Area and volumes
- Chapter 14 - Statistics
- Chapter 15 - Probability

**DETAILED SYLLABUS :**

**1 Real Number**

1.1 Real Number

1.1.1 Introduction

1.1.2 Euclid’s Division Lemma

1.1.3 The Fundamental Theorem of Arithmetic

1.1.4 Revisiting Irrational Numbers

1.1.5 Revisiting Rational Numbers and Their Decimal Expansions

**2 Polynomial**

2.1 Polynomial

2.1.1 Introduction

2.1.2 Geometrical Meaning of the Zeroes of a Polynomial

2.1.3 Relationship between Zeroes and Coefficients of a Polynomial

2.1.4 Division Algorithm for Polynomials

**3 Linear Equation in Two Variable**

3.1 Pair of Linear Equations in Two Variables

3.1.1 Introduction

3.1.2 Pair of Linear Equations in Two Variables

3.1.3 Graphical Method of Solution of a Pair of Linear Equations

3.2 Solving a Pair of Linear Equations

3.2.1 Substitution Method

3.2.2 Elimination Method

3.2.3 Cross - Multiplication Method

3.3 Equations Reducible to a Pair of Linear Equations in Two Variables

**4 Quadratic Equation**

4.1 Quadratic Equations

4.1.1 Introduction

4.1.2 Quadratic Equations

4.1.3 Solution of a Quadratic Equation by Factorisation

4.1.4 Solution of a Quadratic Equation by Completing the Square

4.1.5 Nature of Roots

4.1.6 Applications of quadratic equations

**5 Arithmetic Progression**

5.1 Arithmetic Progression

5.1.1 Introduction

5.1.2 Introduction

5.1.3 Arithmetic Progression

5.1.4 nth Term of an AP

5.1.5 Sum of First n Terms of an AP

**6 Triangles**

6.1 Triangles

6.1.1 Introduction

6.1.2 Similar Figures

6.1.3 Similarity of Triangles

6.1.4 Criteria for Similarity of Triangles

6.1.5 Areas of Similar Triangles

6.1.6 Pythagoras Theorem

**7 Coordinate Geometry**

7.1 Coordinate Geometry

7.1.1 Introduction

7.1.2 Distance Formula

7.1.3 Section Formula

7.1.4 Area of a Triangle

**8 Trigonometry**

8.1 Trigonometry

8.1.1 Introduction

8.1.2 Trigonometric Ratios

8.1.3 Trigonometric Ratios of Some Specific Angles

8.1.4 Trigonometric Ratios of Complementary Angles

8.1.5 Trigonometric Identities

**9 Application of Trigonometry**

9.1 Some applications of Trigonometry

9.1.1 Introduction

9.1.2 Heights and Distances

**10 Circles**

10.1 Circles

10.1.1 Introduction

10.1.2 Tangent to a Circle

10.1.3 Number of Tangents from a Point on a Circle

**11 Construction**

11.1 Construction

11.1.1 Introduction

11.1.2 Division of a Line Segment

11.1.3 Construction of Tangents to a circle

**12 Areas Related to Circle**

12.1 Areas related to circle

12.1.1 Introduction

12.1.2 Perimeter and Area of a Circle — A Review

12.1.3 Areas of Sector and Segment of a Circle

12.1.4 Areas of Combinations of Plane Figures

**13 Surface Area and Volume**

13.1 Surface Area & volume

13.1.1 Introduction

13.1.2 Surface Area of a Combination of Solids

13.2 Volumes

13.2.1 Volume of a Combination of Solids

13.3 Surface Area & volume

13.3.1 Conversion of Solid from One Shape to Another

13.3.2 Frustum of a Cone

**14 Statistics**

14.1 Statistics

14.1.1 Introduction

14.1.2 Mean of Grouped Data

14.1.3 Mode of Grouped Data

14.1.4 Median of Grouped Data

14.1.5 Graphical Representation of Cumulative Frequency Distribution

**15 Probability**

15.1 Probability — A Theoretical Approach

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**FEATURES OF THE COURSE :**

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• Easy tracking of progress – We provide easy tracking facility to help you understand how you are progressing with every step in the course.

• Self-paced learning – Every individual has a different learning pace. Keeping that in mind, we have designed the course to be self-paced.

• Identify focus areas – You may be better in certain concepts than others. This course will facilitate you for a better understanding of the areas you need to focus more.

• Build strong foundations – With this course, we aim to help you understand every concept of the subject so that your foundation in Science is very strong. Solid foundation right from the formative years will help build your interest in the subject further.

• Simple parent’s supervision – Parents will be able to monitor their children’s performance easily with the features provided in the course.

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- Chapter 3 - Metals and Non- Metals
- Chapter 4 - Carbon and Its Compounds
- Chapter 5 - Periodic Classification Of elements
- Chapter 6 - Life Process
- Chapter 7 - Control and Coordination
- Chapter 8 - How do Organisms Reproduce?
- Chapter 9 - Heredity and Evolution
- Chapter 10 - Light, Reflection and refraction
- Chapter 11 - Human Eye and colourful world
- Chapter 12 - Electricity
- Chapter 13 - Magnetic Effect of an electric current
- Chapter 14 - Sources of energy
- Chapter 15 - Sources of Energy
- Chapter 16 - Our Environment
- Chapter 17 - Sustainable Management of Natural Resources

**DETAILED SYLLABUS :**

**1. CHEMICAL REACTIONS AND EQUATIONS**

1.1 Chemical equations

1.1.1 Writing a chemical equation

1.1.2 Balanced chemical equations

1.2 Types of chemical reactions

1.2.1 Combination reaction.

1.2.2 Decomposition reaction

1.2.3 Displacement reactions

1.2.4 Double displacement reaction

1.2.5 Oxidation and reduction

1.3 Have you observed the effects of oxidation reactions in everyday life?

1.3.1 Corrosion

1.3.2 Rancidity

**2. ACIDS, BASES AND SALTS**

2.1 Understanding the chemical properties of acids and bases

2.1.1 Acids and bases in the laboratory

2.1.2 How do acids and bases react with metals?

2.1.3 How do metal carbonates and metal hydrogencarbonates react with acids?

2.1.4 How do acids and bases react with each other?

2.1.5 Reaction of metallic oxides with acids

2.1.6 Reaction of a non-metallic oxide with base

2.2 What do all acids and all bases have in common

2.2.1 What happens to an acid or a base in a water solution?

2.3 How strong are acid or base solutions?

2.3.1 Importance of ph in everyday life

2.3.2 Are plants and animals ph sensitive?

2.3.3 What is the ph of the soil in your backyard?

2.3.4 ph in our digestive system

2.3.5 ph change as the cause of tooth decay

2.3.6 Self defence by animals and plants through chemical warfare

2.4 More about salts

2.4.1 Family of salts

2.4.2 ph of salts

2.4.3 Chemicals from common salt

2.4.4 Are the crystals of salts really dry?

**3. METALS AND NON-METALS**

3.1 Physical properties

3.1.1 Metals

3.1.2 Non-metals

3.2 Chemical properties of metals

3.2.1 What happens when metals are burnt in air?

3.2.2 What happens when metals react with water?

3.2.3 What happens when metals react with acids?

3.2.4 How do metals react with solutions of other metal salts?

3.2.5 The reactivity series

3.3 How do metals and non-metals react?

3.3.1 Properties of ionic compounds

3.4 Occurrence of metals

3.4.1 Extraction of metals

3.4.2 Enrichment of ores

3.4.3 Extracting metals low in the activity series

3.4.4 Extracting metals in the middle of the activity series

3.4.5 Extracting metals towards the top of the activity series

3.4. 5 Refining of metals

3.5 Corrosion

3.5.1 Prevention of corrosion

**4. CARBON AND ITS COMPOUNDS**

4.1 Bonding in carbon – the covalent bond

4.2 Versatile nature of carbon

4.2.1 Saturated and unsaturated carbon compounds

4.2.2 Chains, branches and rings

4.2.3 Will you be my friend?

4.2.4 Homologous series

4.2.5 Nomenclature of carbon compound

4.3 Chemical properties of carbon compounds

4.3.1 Combustion

4.3.2 Oxidation

4.3.3 Addition reaction

4.3.4 Substitution reaction

4.4 Some important carbon compounds – ethanol and ethanoic acid

4.4.1 Properties of ethanol

4.4.2 Properties of ethanoic acid

4.5 Soaps and detergents

**5. PERIODIC CLASSIFICATION OF ELEMENTS**

5.1 Making order out of chaos – early attempts at the classification of elements

5.1.1 Döbereiner’s triads

5.1.2 Newlands law of octaves

5.2 Making order out of chaos – Mendeléev’s periodic table

5.2.1 Achievements of Mendeléev’s periodic table

5.2.2 Limitations of Mendeléev’s classification

5.3 Making order out of chaos – the modern periodic table

5.3.1 Position of elements in the modern periodic table

5.3.2 Trends in the modern periodic table

**6. LIFE PROCESSES**

6.1 What are life processes?

6.2 Nutrition

6.2.1 How do living things get their food?

6.2.1 Autotrophic nutrition

6.2.2 Heterotrophic nutrition

6.2.3 How do organisms obtain their nutrition?

6.2.4 Nutrition in human beings

6.3 Respiration

6.4 Transportation

6.4.1 Transportation in human being

6.4.2 Transportation in plants

6.5 Excretion

6.5.1 Excretion in human beings

6.5.2 Excretion in plants

**7. CONTROL AND COORDINATION**

7.1 Animals – nervous system

7.1.1 What happens in reflex actions?

7.1.2 Human brain

7.1.3 How are these tissues protected?

7.1.4 How does the nervous tissue cause action?

7.2 Coordination in plants

7.2.1 Immediate response to stimulus

7.2.2 Movement due to growth

7.3 Hormones in animals

**8. HOW DO ORGANISMS REPRODUCE?**

8.1 Do organisms create exact copies of themselves?

8.1.1 The importance of variation

8.2 Modes of reproduction used by single organisms

8.2.1 Fission

8.2.2 Fragmentation

8.2.3 Regeneration

8.2.4 Budding

8.2.5 Vegetative propagation

8.2.6 Spore formation

8.3 Sexual reproduction

8.3.1 Why the sexual mode of reproduction

8.3.2 Sexual reproduction in flowering plants

8.3.3 Reproduction in human beings

8.3.3 (a) Male reproductive system

8.3.3 (b) Female reproductive system

8.3.3 (c) What happens when the egg is not fertilised?

8.3.3 (d) Reproductive health

**9. HEREDITY AND EVOLUTION**

9.1 Accumulation of variation during reproduction

9.2 Heredity

9.2.1 Inherited traits

9.2.2 Rules for the inheritance of traits – mendel’s contributions

9.2.3 How do these traits get expressed?

9.2.4 Sex determination

9.3 Evolution

9.3.1 An illustration

9.3.2 Acquired and inherited traits

9.4 Speciation

9.5 Evolution and classification

9.5.1 Tracing evolutionary relationships

9.5.2 Fossils

9.5.3 Evolution by stages

9.6 Evolution should not be equated with ‘progress’

9.6.1 Human evolution

**10. LIGHT – REFLECTION AND REFRACTION**

10.1 Reflection of light

10.2 Spherical mirrors

10.2.1 Image formation by spherical mirrors

10.2.2 Representation of images formed by spherical mirrors using ray diagrams

10.2.3 Sign convention for reflection by spherical mirrors

10.2.4 Mirror formula and magnification

10.3 Refraction of light

10.3.1 Refraction through a rectangular glass slab

10.3.2 The refractive index

10.3.3 Refraction by spherical lenses

10.3.4 Image formation by lenses

10.3.5 Image formation in lenses using ray diagrams

10.3.6 Sign convention for spherical lenses

10.3.7 Lens formula and magnification

10.3.8 Power of a lens

**11. THE HUMAN EYE AND THE COLOURFUL WORLD**

11.1 The human eye

11.1.1 Power of accommodation

11.2 Defects of vision and their correction

11.3 Refraction of light through a prism

11.4 Dispersion of white light by a glass prism

11.5 Atmospheric refraction

11.6 Scattering of light

11.6.1 Tyndall effect

11.6.2 Why is the colour of the clear sky blue?

11.6.3 Colour of the sun at sunrise and sunset

**12.ELECTRICITY**

12.1 Electric current and circuit

12.2 Electric potential and potential difference

12.3 Circuit diagram

12.4 Ohm’s law

12.5 Factors on which the resistance of a conductor depends

12.6 Resistance of a system of resistors

12.6.1 Resistors in series

12.6.2 Resistors in parallel

12.7 Heating effect of electric current

12.7.1 Practical applications of heating effect of electric current

12.8 Electric power

**13. MAGNETIC EFFECTS OF ELECTRIC CURRENT**

13.1 Magnetic field and field lines

13.2 Magnetic field due to a current-carrying conductor

13.2.1 Magnetic field due to a current through a straight conductor

13.2.2 Right-hand thumb rule

13.2.3 Magnetic field due to a current through a circular loop

13.2.4 Magnetic field due to a current in a solenoid

13.3 Force on a current-carrying conductor in a magnetic field

13.4 Electric motor

13.5 Electromagnetic induction

13.6 Electric generator

13.7 Domestic electric circuits

**14. SOURCES OF ENERGY**

14.1 What is a good source of energy?

14.2 Conventional sources of energy

14.2.1 Fossil fuels

14.2.2 Thermal power plant

14.2.3 Hydro power plants

14.2.4 Improvements in the technology for using conventional sources of energy

14.3 Alternative or non-conventional sources of energy

14.3.1 Solar energy

14.3.2 Energy from the sea

14.3.3 Geothermal energy

14.3.4 Nuclear energy

14.4 Environmental consequences

14.5 How long will an energy source last us?

**15. OUR ENVIRONMENT**

15.1 What happen when we add our waste to the environment

15.2 Eco-system — what are its components?

15.2.1 Food chains and webs

15.3 How do our activities affect the environment?

15.3.1 Ozone layer and how it is getting depleted

15.3.2 Managing the garbage we produce

**16. MANAGEMENT OF NATURAL RESOURCES**

16.1 Why do we need to manage our resources?

16.2 Forests and wild life

16.2.1 Stakeholders

16.2.2 Sustainable management

16.3 Water for all

16.3.1 Dams

16.3.2 Water harvesting

16.4 Coal and petroleum

16.5 An overview of natural resource management

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