# CBSE Question Paper 2019

2019 Mathematics Questions Set -1

**: 3 Hours**

**: 50**

- All questions are compulsory
- This 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. 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, two questions of 2 marks, four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternative in all such questions.
- Use of calculator is not permitted.

## SECTION A

- Find the coordinates of a point A, where AB is diameter of a circle whose centre is (2, -3) and B is the point (1,4).
- For what values of k, the roots of the equation x2 + 4x + k=0 are real?

OR

Find the value of k for which the roots of the equation 3x2 - 10x + k = 0 are reciprocal of each other. - Find A if tan 2A = cot (A -24°)

OR

Find the value of (sin233°+sin257°) - How many two digits numbers are divisible by 3?
- In Fig. 1, DE || BC, AD = 1 cm and BD = 2 cm. What is the ratio of the ar(? ABC) to the ar(?ADE)?
- Find a rational number between 2–√2 and 3–√3.

Find the common difference of the Arithmetic Progression (A.P.) 1/a, (3-a)/3a, (3-2a)/3a ,..(a≠0).

## SECTION B

- Find the HCF of 1260 and 7344 using Euclid's algorithm.

OR

Show that every positive odd integer is of the form (4q+1) or (4q+3), where q is some integer. - Which term of the AP 3, 15, 27, 39, .... will be 120 more than its 21st term?

OR

If Sn, the sum of first n terms of an AP is given by Sn = 3n2 - 4n, find the nth term. - Find the ratio in which the segment joining the points (1, -3) and (4, 5) is divided by x-axis? Also find the coordinates of this point on x-axis.
- A game consists of tossing a coin 3 times and noting the outcome each time. If getting the same result in all the tosses is a success, find the probability of losing the game.
- A die is thrown once. Find the probability of getting a number which (i) is a prime number (ii) lies between 2 and 6.
- Find c if the system of equations cx+ 3y +(3 - c) = 0; 12x + cy - c = 0 has infinitely many solutions?

## SECTION C

- Prove that 2–√2 is an irrational number.
- Find the value of k such that the polynomial x2 -(k + 6)x + 2(2k - 1) has sum of its zeros equal to half of their product.
- A father's age is three times the sum of the ages of his two children. After 5 years his age will be two times the sum of their ages. Find the present age of the father.

**OR**

A fraction becomes 1313 when 2 is subtracted from the numerator and it becomes 1212 when 1 is subtracted from the denominator. Find the fraction - Find the point on y-axis which is equidistant from the points (5,-2) and (-3, 2).

OR

The line segment joining the points A(2, 1) and B(5, -8) is trisected at the points P and Q such that P is nearer to A. If P also lies on the line given by 2x - y + k = 0, find the value of k. - Prove that (sinθ+cosecθ)2+ (cosθ+secθ) 2=7 + tan2θ + cot2θ

OR

Prove that (1 + cotA - cosec A) (1 + tan A + sec A) = 2 - In Fig. 2, PQ is a chord of length 8 cm of a circle of radius 5 cm and centre O. The tangents at P and Q intersect at point T. Find the length of TP.
- In Fig. 3, ACB = 90° and CD AB, prove that CD2 = BD ×× AD.

If P and Q are the points on side CA and CB respectively of A ABC, right angled at C, prove that (AQ2 +BP2)=(AB2 +PQ2). - Find the area of the shaded region in Fig. 4, if ABCD is a rectangle with sides 8 cm and 6 cm and O is the centre of circle. (Take ππ = 3.14)
- 21. Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/hour. How much area will it irrigate in 30 minutes; if 8 cm standing water is needed?
- Find the mode of the following frequency distribution.

Class | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |
---|---|---|---|---|---|---|---|

Frequency | 8 | 10 | 10 | 16 | 12 | 6 | 7 |

## SECTION D

Marks |
No. of Students |
---|---|

0-5 | 2 |

5-10 | 5 |

10-15 | 6 |

15-20 | 8 |

20-25 | 10 |

25-30 | 25 |

30-35 | 20 |

35-40 | 18 |

40-45 | 4 |

45-50 | 2 |

- Two water taps together can fill a tank in 178178 hours. The tap with longer diameter takes 2 hours less than the tap with smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately. Or A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water.
- If the sum of first four terms of an AP is 40 and that of first 14 terms is 280. Find the sum of its first n terms.
- Prove that
- .A man in a boat rowing away from a light house 100 m high takes 2 minutes to change the angle of elevation of the top of the light house from 60° to 30°. Find the speed of the boat in metres per minute. [ Use 3–√3 = 1.732].

OR

Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30° respectively. Find the height of the poles and the distances of the point from the poles. - In Fig. 1, DE || BC, AD = 1 cm and BD = 2 cm. What is the ratio of the ar(? ABC) to the ar(?ADE)?
- Construct a ??ABC in which CA = 6 cm, AB = 5 cm and ∠∠BAC = 45°. Then construct a triangle whose sides are of the corresponding sides of ??ABC.
- 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 of circular ends of the bucket are 20 cm and 12 cm respectively. Find the height of the bucket and also the area of the metal sheet used in making it.
- Prove that in a right angle triangle, the square of the hypotenuse is equal the sum of squares of other two sides.
- If the median of the following frequency distribution is 32.5. Find the values of f1 and f2.
Class 0-10 10-20 20-30 30-40 40-50 50-60 60-70 Frequency 8 10 10 16 12 6 7

OR

The marks obtained by 100 students of a class in an examination are given below. - Draw 'a less than' type cumulative frequency curves(ogive). Hence, find median.

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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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• 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.

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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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