integrated syllabus (free sample) · 2018-05-08 · design & typing: m. anjani & yousuf...

CLASS - VII

IIT F

oundatio

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Olym

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rer - M

ath

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atic

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ss - VII

FOUNDATION OLYMPIAD&

IntegratedSyllabus

UNIQUE ATTRACTIONS●

● Cross word Puzzles

● Graded Exercise

Basic Practice■

Further Practice■

Brain Works■

● Multiple Answer Questions

● Paragraph Questions

Rs. 85Detailed solutionsfor all problems

of IIT Foundation &Olympiad Explorer

are available in this book

CLASS - X

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� Simple, clear and systematic presentation

� Concept maps provided for every chapter

� Set of objective and subjective questions at the

end of each chapter

� Previous contest questions at the end of each

chapter

� Designed to fulfill the preparation needs for

international/national talent exams, olympiads

and all competitive exams

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YOUR

COACH

India’s FIRST scientifically designed portalfor Olympiad preparation• Olympiad & Talent Exams preparation packages

Analysis Reports Previous question papers• •Free Demo Packages Free Android Mobile App• •

Get 15% discount on all packages by using the discount coupon code: KR157N

A unique opportunity to take about 50 tests per subject.

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CLASS - VII

FOUNDATION & OLYMPIAD

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Published by:

Brain Mapping Academy#16–11–16/1/B, First Floor,Farhat Hospital Road,Saleem Nagar, Malakpet,Hyderabad–500 036Andhra Pradesh, India.✆ 040–65165169, 66135169E–mail: [email protected]: www.bmatalent.com

© Brain Mapping Academy

ALL RIGHTS RESERVEDNo part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher.

Publication Team

Authors: M. Gurunadham & Y.S. Srinivasu

Design & Typing: M. Anjani & Yousuf Nawaz Ali Khan

ISBN: 978-81-907285-2-2

Disclaimer

Every care has been taken by the compilers andpublishers to give correct, complete and updated information. In case there is any omission, printing mistake or anyother error which might have crept in inadvertently,neither the compiler / publisher nor any of thedistributors take any legal responsibility.

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Preface

Speed and accuracy play an important role in climbing the competitive ladder. Students

have to integrate the habit of being able to calculate and function quickly as well as efficiently

in order to excel in the learning culture. They need to think on their feet, understand basic

requirements, identify appropriate information sources and use that to their best advantage.

The preparation required for the tough competitive examinations is fundamentally different

from that of qualifying ones like the board examinations. A student can emerge successful in

a qualifying examination by merely scoring the minimum percentage of marks, whereas in a

competitive examination, he has to score high and perform better than the others taking the

examination.

This book provides all types of questions that a student would be required to tackle at the

foundation level. The questions in the exercises are sequenced as Basic Practice, Further Practice,

Brainworks, Multiple Answer Questions and Paragraph Questions. Simple questions involving

a direct application of the concepts are given in Basic Practice. More challenging questions

on direct application are given in Further Practice. Questions involving higher order thinking

or an open-ended approach to problems are given in Brainworks. These questions encourage

students to think analytically, to be creative and to come up with solutions of their own.

Constant practice and familiarity with these questions will not only make him/her

conceptually sound, but will also give the student the confidence to face any entrance

examination with ease.

Valuable suggestions as well as criticism from the teacher and student community are most

welcome and will be incorporated in the ensuing edition.

Publisher

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1. Sets .................................................... 1

2. Rational Numbers .............................. 24

3. Factors and Multiples ........................ 49This page is intentionally left blank. 62

4. Cubes and Cube Roots ...................... 63

5. Ratio and Proportion ......................... 73

6. Percentage ........................................ 92This page is intentionally left blank. 109

7. Profit and Loss ................................... 110

8. Simple Interest................................... 129

9. Compound Interest ........................... 142This page is intentionally left blank. 155

10. Identities ............................................ 156

11. Factorisation ...................................... 189

12. Simple Equations ............................... 201This page is intentionally left blank. 223

13. Triangles & Polygons ......................... 224

14. Circles................................................. 266This page is intentionally left blank. 300

15. Mensuration ...................................... 301

16. Solid Geometry.................................. 348

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IIT Foundation & Olympiad Explorer Class - VII / Mathematics

© Brain Mapping Academy1. Sets 1

SYNOPSIS

Representing sets using Venn diagrams

Sets can also be represented by geometrical diagrams such as circles, squares, rectanglesor ovals. The elements of a set are then written inside the enclosed shape. This type ofrepresentation is called a Venn diagram.For example, A = {a, b, c, d, e}. Set A can be represented as follows:

A

be

a

cd

A dot in the Venn diagram

represents an element.

Listing elements and stating the number of elements in a set

The number of elements in set A is written as n(A).For example, if A = {1, 3, 5, 7} Then n(A) = 4.

Determining subset and using the symbol ⊂ or ⊂

Set A is a subset of set B if all the elements of set A are found in set B. It is written as,A ⊂ B. The symbol � ⊂ � represents subset. Let :P = {1, 2, 3} Q = {3, 4, 5, 6} R = {3, 5, 6} Therefore, we say R is a subset of Q. It is written as R ⊂ Q. On the other hand, not allthe elements of set P are found in set R. Therefore, P is not a subset of R and it iswritten as P ⊂ R.

Representing subsets using Venn diagrams

The relationship of sets, K, L and M in example can be represented in the Venn diagramsbelow.

K

s

p

rq

u

w

t

L

M

sp q

t

L

(a) (b)

1 Mathematical Induction

Chapter

1SetsSets

Chapter

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Listing subsets

Mr. Sinha wants to select some students to take part in a competition. Gowtham, theteacher of class 7 submitted three names to him: �A, B, C�.What are the possible combinations that can be formed from three students?Let P = {A, B, C}.The possible combinations are:

{A}, {B}, {C}, {A, B}, {B, C}, {C, A}, {A, B, C}, φ

Note : The empty set, ∅ is also a subset of P because none of them may be chosen.

Number of subsets for set P = 8.Relationship between sets and universal set

A universal set consists of all the elements under discussion.

It is denoted by the symbol ξ . This mens that all the sets under discussion are subsetsof the universal set.

ξ

A

s p

r q

v w

x

d

t

B

In the diagram above, ξ = {d, p, q, r, s, t, v, w, x},

A = {q, r, v}, and B = {d, t}.

All the elements in A and B are found in the universal set ξ . Therefore, A ⊂ ξ and

B ⊂ ξ .

The complement of a set

The complement of set A is the set that consists of all the elements of the universal setthat are not elements of A. It is written as �A�.

For example, if ξ = {all the letters in the alphabets} and

A = {vowels}, then A' = {consonants}.A' is represented by the shaded area in the Venn diagram below.

ξ

A

A′

Operation on Sets: Intersection and Union

We can perform operations such as plus, minus, multiply and divide on numbers. Similarlywe can perform operations such as intersection and union on sets.

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The intersection of two sets and three sets

During Reading Campaign Week, Sweeti and Saritha were asked to indicate the types ofbooks they often read. The picture below shows their choices.Let set A and set B be the choices of Saritha and Sweeti respectively.A = {science fiction, detective stories, romance}B = {science fiction, biographies, romance, horror stores}Intersection of set A and set B is a set which consists of common elements that belong toboth set A and set B. It is denoted by A ∩ B.Note that both Sweeti and Saritha like science fiction and romance. Science fiction andromance are the common elements of both sets. The symbol � ∩ � is used to denoteintersection of the two sets. Hence, A ∩ B = {science fiction, romance}.

Relationship between (a) A ∩ B and A

(b) A ∩ B and Bξ

A B

A B∩All the elements of A ∩ B are in A.Therefore, (A ∩ B) ⊂ A.Likewise, all the elements of A ∩ B are in B. Therefore, (A ∩ B) ⊂ B.

The complement of the intersection of sets

The complement of the intersection of sets A and B is the set consisting of all the elementsin the universal set that are not elements of the intersection.

The union of two sets and three sets

The union of set A and set B is the which consists of all the elements in set A or set B orboth. It is denoted by A ∪ B.

For example, set A = {a, c, d, g} and set B = {d, f, h}. All the elements of A and B are a, c,d, f, g and h. The symbol � ∪ � is used to denote union of 2 sets. Hence, A ∪ B = {a, c, d,f, g, h}.

Representing union of sets using Venn diagrams

If A = {a, c, d, g} and B = {d, f, h}, the union of set A and set B can be represented by theVenn diagram below. The shaded region represents A ∪ B.

f

h

a

c d

ξA B

g

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Relationship between (a) A ∪ B and A

(b) A ∪ B and B

All the elements of A are in A ∪ B.Therefore, A ⊂ (A ∪ B).All the elements of B are also in A ∪ B.Likewise, B ⊂ (A ∪ B).

ξA B

A B∪

The complement of the union of sets

The complement of the union of sets M and N is written as (M ∪ N)'. In this examplebelow, 14 and 16 are two elements that do not belong to set M ∪ N.Therefore, (M ∪ N)' = {14, 16}

18

12

15

16

14

1311

1719

NMξ

(M N)∪ ′

SOLVED EXAMPLES

Example 1:

Represent the following sets using Venn diagrams.

(a) G = {red, yellow, blue, white}

(b) H = {x : x is an even number, 1 ≤ x ≤ 10}.

Solution:

(a) G = {red, yellow, blue, white}

red yellow

bluewhite

G

(b) H = {2, 4, 6, 8, 10}

4

8

2

6

10

H

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Solution:

(a) All the elements in L are found in K. Therefore, L ⊂ K.

(b) The element w is not found in K. Therefore, M ⊂ K.

(c) All the elements in L are found in M. Therefore, L ⊂ M.

Example 6:

List all the subsets for the following.

(a) A = {x, y}

(b) B = {3, 4, 5, 6}

Solution:

(a) {x}, {y}, {x, y}, ∅ .

(b) {3}, {4}, {5}, {6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 6},

{3, 4, 5}, {3, 4, 6}, {3, 5, 6}, {4, 5, 6}, {3, 4, 5, 6}, ∅ .

Example 7:

Use a Venn diagram to illustrate the relationship between the sets and the universalset.

ξ = {x : x is an integer, 1 ≤ x ≤ 10}

A = {x : x is a square number}

B = {x : x is a prime number}

Solution:

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {1, 4, 9}

B = {2, 3, 5, 7}

Example 8:

Given ξ = {integers from 1 to 10}. Determine the complement of the following sets.

(a) A = {even numbers}

(b) B = {square numbers}

Solution:

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(a) A = {2, 4, 6, 8, 10}

A' = {1, 3, 5, 7, 9}

(b) B = {1, 4, 9}

B' = {2, 3, 5, 6, 7, 8, 10}

ξA B

41

9

8

2

37

5

10

6

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Example 9:

From the Venn diagram, determine

(a) n(C') (b) n(D')

Solution:

(a) C' = {m, s, r, t, v}, n(C') = 5

(b) D' = {p, q, u, r, t, v}, n(D') = 6

Example 10:

In the given Venn diagram,

(a) determine (i) n( ξ ) (ii) n(H) (iii) n(G').

(b) state the relationship between sets H, K and ξ .

Solution:

(a) (i) n( ξ ) = 6 + 3 + 2 + 5 = 16

(ii) n(H) = 3 + 2 = 5

(iii) n(G') = 5 + 3 + 2 = 10

(b) K ⊂ H ⊂ ξ

Example 11:

Given ξ = {integers from 1 to 20},

A = {two-digit even numbers},

B = {multiples of 6},

and C = {multiples of 4}.

Determine (a) A ∩ B

(b) A ∩ C

(c) A ∩ B ∩ C.Solution:

ξ = {1, 2, 3, 4, ...., 20}

A = {10, 12, 14, 16, 18, 20}

B = {6, 12, 18}

C = {4, 8, 12, 16, 20}

ξ

v

u

r

q p

t

m

S

C D

where G = {members of Gopi group}

k = {members of krishna group}

H = {members of Harsha group}

ξ

3

GK

2

H

5

6

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(a) A = {10, 12 , 14, 16, 18 , 20} (b) A = {10, 12 , 14, 16 , 18, 20 }

B = {6, 12 , 18 ,} C = {4, 8, 12 , 16 , 20 }

A ∩ B = {12, 18} A ∩ C = {12, 16, 20}

(c) A = {10, 12 , 14, 16, 18, 20}

B = {6, 12 , 18}

C = {4, 8, 12 , 16, 20}

A ∩ B ∩ C = {12}

Example 12:

Given ξ = {e, f, g, h, i, j, k, l, m},

P = {e, i, j, k},

Q = {e, g, k, m}

and R = {h, j, k, l, m}.

Use Venn diagrams to represent the following intersections.

(a) P ∩ Q (b) P ∩ Q ∩ R

Solution:

(a) P = { e , i, j, k } (b) P = {e, i, j, k }

Q = { e , g, k , m} Q = {e, g, k , m}

P ∩ Q = {e, k} R = {h, j, k l, m}

P ∩ Q ∩ R = {k}

h

j

eg

km

iP

ξ

fl

Q

h

j

e g

km

i

Pξ

fl

Q

R

Example 13:

Given ξ = {x : 10 ≤ x ≤ 20, x ∈ integers},

A = {multiples of 3} and B = {multiples of 4},

(a) Represent set ξ , A and B in a Venn diagram.

(b) Determine (i) A ∩ B (ii) n(A ∩ B)'.

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Solution:

ξ = {10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}

(a) A = {12, 15, 18}

B = {12, 16, 20}

1615

13

19

12

20

17

11

A

ξ

10

18

B14

(b) (i) A = { 12 , 15, 18}

B = { 12 , 16, 20}

A ∩ B = {12}

(ii) n(A ∩ B) = 1, n( ξ ) = 11

Therefore, n(A ∩ B)'

= 11 � 1

= 10

Example 14:

In each Venn diagram below, shade the region that represents

(a) (C ∩ D)' (b) (E ∩ F ∩ G)'.

ξ

C

D

ξ

G

E F

Solution:

(a) (C ∩ D)' (b) (E ∩ F ∩ G)'

ξ

C

D

ξ

G

E F

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OPERATIONS SETS: We can performoperations such as intersection andunion of sets.

SET : A well definedcollection of objects is

called a set.

SUBSET: Set A is a subsetof B if all the elements ofset A are found in set B. Itis written as A ⊂ B. Thesymbol � ⊂ � representssubset.

COMPLEMENT OF A SET: The complement of set A is the setthat consists of all the elements of the universal set that are notelements of A. It is written as A'.

UNION: The union of set A andB is the set which consists of allthe elements in set A or set B orboth. It is denoted by A ∪ B.

The complement of theintersection of set A and B isthe set consisting of all theelements in the universal setthat are not elements of theintersection and is written as(A ∩ B)'.

Venn diagram: Sets canalso be represented bygeometrical diagrams suchas circles, rectangles,squares or ovals. Theelements of a set arewritten inside the enclosedshape. This type ofrepresentation is called avenn diagram.

CONCEPT MAP

The complement of the unionofsets A and B is the set of elemntsnot in A ∪ B and is written as(A ∪ B)'

INTERSECTION: Intersectionof set A and set B is a set whichconsists of common elementsthat belong to both set A andset B. It is denoted by A ∩ B.

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BASIC PRACTICE

1. It is given that the sets P = {letters in the word SCIENCE) and Q = {factors of 15}.(i) List all the elements of:

(a) set P (b) set Q(ii) Write down the value of:

(a) n(P) (b) n(Q)(iii) Illustrate each of the following sets with a Venn diagram.

(a) Set P (b) Set Q

(iv) Fill in each blank with the symbol ∈ or ∈ .

(a) 3 Q (b) 10 Q

2. Fill in each blank with the symbol = or ≠ .(i) Set R = {x : prime numbers and 7 < x < 10}

∴ R φ

(ii) Set S = {multiples of 5 which are divisible by 4}

∴ S φ

3. Fill in each blank with the symbol ⊂ or ⊂ .

(i) {2, 5} {1, 2, 3, 4, 5} (ii) {1, 2, 3} {odd numbers}

(iii) {a, e} {vowels} (iv) {2, 3, 5, 7} {prime numbers < 10}

(v) {squares} {rectangles} (vi) {parallelograms} {rhombuses}

4. Illustrate the relationship between the sets in a Venn diagram, in each of the following.Then shade the region representing the complement of a set and list all its elements.

(i) ξ = {x : 1 ≤ x ≤ 9, x is an integer}

M = {x : x is a prime number}

(ii) ξ = {factors of 12}

N = {multiples of 6}

5. Given that the universal set, ξ = {x : 1 ≤ x ≤ 9, x is an integer}, set A = {1, 2, 3, 4, 5} andset B = {2, 4, 6}, complete the following.(i) Elements of A ∩ B(ii) Elements A ∪ B

(iii) (a) List all the elements of sets ξ , A and B in the Venn diagram.

(b) Elements of (A ∩ B)'(c) Elements of (A ∪ B)'

6. In the Venn diagrams in the answer space, ξ = F ∪ G. On the diagrams, shade

(i) the set F',

(ii) the set H where H ⊂ G and H ∩ F = φ .

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7. The universal set,

ξ = {x : 2 ≤ x ≤ 9, x is an integer},set F = {x : x is a prime number} andset G = {x : x is a multiple of 3}.(i) On the Venn diagram in the answer space, write down all the elements of the sets.(ii) Find (a) n(G')' (b) n(F ∪ G')

8. In the given Venn diagram, find

42

31

5

98

76

Aξ

1011

B

(i) n(A) (ii) n(B') (iii) n(A ∩ B) (iv) n(A ∪ B)'

9. Given ξ = {a, b, c, d, e, f, g, h, i, j, k}, C = {b, e, a, d}, D = {j, a, d, e} and E = {b, a, d, g, e}.Determine(i) C ∩ D ∩ E (ii) C ∪ D ∪ E (iii) n[(C ∪ D) ∩ E'] (iv) n[(C ∪ E)' ∩ D]

10. In the Venn diagram, given that n(F) = 12, n(G) = 15 and n(F ∩ G) = 6. If n( ξ ) = 25, find(i) n(F') (ii) n(G') (iii) n(F ∪ G)'

ξF G

11. In the Venn diagram, given that n(H) = 32,

n(K) = 25, n(H ∪ K) = 50 and n( ξ ) = 53. Find(i) n(H ∩ K) (ii) n(H ∪ K)'

ξH K

12. Determine A ∩ B.(i) A = {K, U, C, H, I, N, G}, B = {C, H, I, N, A}(ii) A = {odd numbers less than 10}, B = {prime numbers less that 10}

(iii) A = 2 8

2 x3 3

− ≤ ≤ , B =

3 32, , ,3

2 4 − −

(iv) A = {factors of 15}, B = {factors of 20}

13. Given ξ = {p, q, r, s, t, u, v, w, x, y, z}, G = {q, r, u, v, w}, H = {p, q, r, s, t, u} and K = {q,t, u, y}. List the elements for(i) G ∩ H, (ii) G ∩ K, (iii) H ∩ K, (iv) G ∩ H ∩ K

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FURTHER PRACTICE

1.ξ

A B

I II III

IV

Given A = {multiples of 5}and B = {square numbers}.The number 25 should be in the region:(A) I (B) II (C) III (D) IV

2. Given ξ = {x : 10 ≤ x ≤ 20, x ∈ integers},

M = {numbers whose sum of digits is an even number},and N = {prime numbers}.n(M ∪ N) = ________.(A) 4 (B) 5 (C) 6 (D) 7

3.P

Q

In the Venn diagram above, ξ = P ∪ Q. Which of the following statements is correct?

(A) P ∩ Q = P (B) P ⊂ Q (C) P ∩ Q' = ∅ (D) P ∪ Q = P

4. In the Venn diagram, the universal set, ξ = {students in class 7}, set M = {students whopassed Mathematics} and set N = {students who passed English}.

ξM N

It is given that n(M) = 50, n(N) = 60 and n(M ∩ N) = 12. If 2 students failed in bothsubjects, find the total number of students who sat for the examination.(A) 98 (B) 100 (C) 112 (D) 120

5. The Venn diagram shows sets ξ , P and Q.

ξP

Q

The shaded region in the Venn diagram represents set(A) P ∩ Q (B) P' ∩ Q (C) P ∩ Q' (D) P' ∩ Q'

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6. In the Venn diagram, ξ is the universal set. List all the elements of set Q'.

ξ P Q

2

31 4

5

6

7

(A) {2, 3} (B) {1, 2, 3} (C) {5, 6, 7} (D) {2, 3, 5, 6, 7}

7. The Venn diagram shows all the number of elements in sets ξ , F and G. Find n(F ∩ G').

ξ F G

2 1 3

4

(A) 2 (B) 3 (C) 5 (D) 1

8. The Venn diagram shows all the elements in sets ξ , H and K. Find n(H' ∪ K).

ξ H K

1

23

45

6

7 8 9

(A) 3 (B) 6 (C) 7 (D) 8

9. The Venn diagram shows all the elements in sets ξ , P and Q. List all the elements ofP' ∩ Q'.

ξ PQ

1

26 4

3 5

7

8

(A) {1, 2, 6} (B) {1, 2, 4, 6} (C) {3, 5, 7, 8} (D) {3, 4, 5, 7, 8}

10. The universal set,

ξ = {x : 2 ≤ x ≤ 9 and x is an integer},

set P = {x : x is a prime number} and

set Q = {x : x is an odd number}.

Find n(P ∩ Q}'.

(A) 5 (B) 6 (C) 7 (D) 8

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BRAIN WORKS

1.

Q

P

The Venn diagram shows the relationship between sets ξ , P and Q.

(a) Fill in the blanks in the answer space using the set notation ∪ , ∩ or ⊂ to showthe relationship between sets P and Q.(i) P Q (ii) P Q P=d d

(b) Express in set notation, as simply as possible,(i) P ∩ Q' (ii) the subset shaded in the Venn diagram.

2. The Venn diagram shows sets ξ , D and F. Separate regionsare labelled I, II, III and IV.(a) State the region representing:

(i) (D ∪ F)' (ii) D ∩ F'(b) The universal set,

ξ = {x : x is an integer},

setD = {x : x is a multiple of 3}and set F = {x : x is a multiple of 5}.State the region in which each of the following elements belongs.(i) 15 (ii) 51

3. Given ξ = {x : 20 ≤ x ≤ 40, x is an integer},

P = {the sum of digits of the number are even numbers},Q = {perfect squares},

and R = {product of the digits is less than 10}.Find(a) P' (b) n(Q') (c) n(R')

4. The Venn diagram on the right shows sets S, T and V.

15

23

5

9

8

7

6

ξ

12

T

V

S13

Determine(a) S' (b) n(T') (c) n(V')

ξD F

I

II III IV

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(b) ξ = {3, 4, 5, 7, 9, 10, 12, 15, 18}

X = {even numbers}Y = {multiples of 3}

(c) ξ = {D, R, O, C, I, H, G, A, I, K, Y, J, S}

X = {O, R, C, H, I, D}Y = {D, A, I, S, Y}

11. A survey of 30 youths shows that 20 like to read and 15 like outdoor games. Find howmany of them who like(a) both reading and outdoor games, (b) reading only, (c) outdoor games only.

12. A class consists of 35 students. There are 20 girls and 9 librarians in the class. If one-third of the boys are librarians,(a) find the number of girls who are librarians,(b) draw a Venn diagram to show the intersection of sets.

13. In a survey involving 60 children, it was found that 45 of them read comics, 18 readdetective stories and 5 do not read at all. Use a Venn diagram to calculate the number ofchildren that read both comics and detective stories.

14. If ξ = {x : 1 ≤ x ≤ 50, x is an integer}, S = {numbers with two same digits} andT = {multiples of 5}, (a) find S ∪ T (b) determine whether (i) S ⊂ (S ∪ T)

(ii) T ⊂ (S ∪ T).

15. Given ξ = {k, l, m, n, o, p, q, r, s, t}, F = {l, n, p, q, r}, G = {k, n, q, s} andH = {l, m, n, r, s, t}. Determine (a) n(F ∪ G)' (b) n(G ∪ H)'

16. If ξ = {x : 10 ≤ x ≤ 30}, K = {multiples of 5}, L = {numbers whose sum of digits aredivisible by 3}, find (a) K ∩ L (b) K ∪ L (c) n(K ∪ L)'

17. In each case below, determine the complement of the union of sets and represent themby using Venn diagrams. Shade the region that represents the complement of the union.

(a) ξ = {days in a week}, A = {weekend}, B = {days that begin with the letter �T�}

(b) ξ = {1, 2, 3, ..., 9, 10}, C = {prime numbers}, D = {factors of 12}

(c) ξ = {e, f, g, h, i, j, k, l}, E = {g, h, k, l}, F = {g, h, k}

18. The Venn diagram shows the number of elements in set Hand set K. Given n( ξ ) = 30 and n(K) = 13, find(a) n(H ∪ K)(b) n(H ∪ K)'.

MULTIPLE ANSWER QUESTIONS

1. Describe the shaded region.(A) A � B (B) (A � B)'(C) A ∩ B' (D) A ∪ B'

ξH K

3x x 8

A B

U

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2. Which of the following collections is a set?(A) The collection of natural numbers between 2 and 20(B) The collection of numbers which satisfy the equation x2 � 5x + 6 = 0(C) The collection of prime numbers between 1 and 100(D) The collection of all beautiful women in Jalandhar

3. Which of the following is equivalent to A ⊂ B?

(A) A � B = φ (B) A ∩ B = A (C) A ∪ B = B (D) A ∩ B

4. If A = {1, 2, (3, 4), 5}, then which of the following statements is incorrect?(A) {3, 4} ∈A (B) {(3, 4)}, ⊂ A (C) {3, 4} ⊂ A (D) {(2, 3)} ∈ A

5. If A and B are any two sets, then A � B = ______.(A) A ∩ B' (B) A � (A ∩ B) (C) (A' ∪ B') (D) B ∩ A'

6. If A and B are any two sets, then which of the following is true?(A) A ∩ B ⊂ A (B) A ⊂ A ∪ B (C) (A � B) ⊂ A (D) A ⊂ (A � B)

PARAGRAPH QUESTIONS

Passage - I

The universal set,

ξ = {x : 10 ≤ x ≤ 30, x is an integer},

set L = {x : x is a multiple of 3},set M = {x : x is a number when divided by 5 leaves 3 as the remainder} andset N = {x : x is a number such that the difference of its digits = 1}.Based on this information answer the questions given below.(i) List all the elements of set M and set N.(ii) Find n(L ∪ N)'(iii) Find n(M ∪ L).

In a certain examination, the candidates can offer papers in English or Hindi or both thesubjects. The number of candidates who appeared in the examination is 1000 of whom650 appeared in English and 200 appeared in both English and Hindi.

Based on this information answer the questions given below.

(i) Find the number of candidates who offered paper in Hindi.

(ii) Find the number of candidates who offered paper in English only.

(iii) Find the number of cadidates who offered paper in Hindi only.

Passage - II

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Basic PracticeANSWERS

1) (i) (a) {S, C, I, E, N} (b) {1, 3, 5, 15} (ii) (a) 5 (b) 4

(iii) (a) S

I

CNE (b)

15

15

3 (iv) (a) 3 ∈ Q (b) 10 ∉ Q

2) (i) R = φ (ii) S ≠ φ3) (i) ⊂ (ii) ⊂ (iii) ⊂ (iv) ⊂ (v) ⊂ (vi) ⊂

4) (i) 2

57

31

46

89(ii)

6

12

12

34

5) (i) (2, 4) (ii) {1, 2, 3, 4, 5, 6}

(iii) (a) 2

7A

4 6 8

9

B

1

35

(b) {1, 3, 5, 7, 8, 9} (c) {7, 8, 9}

6) (i)

F G

(ii)

F G

H

7) (i) 2

F

4

6

8

9

G

57

3 (ii) (a) 3 (b) 6

8) (i) 5 (ii) 5 (iii) 2 (iv) 2 9) (i) {a, d, e} (ii) {a, b, d, e, g, j} (iii) 1 (iv) 110)(i) 13 (ii) 10 (iii) 4 11) (i) 7 (ii) 3

12) (i) {C, H, I, N} (ii) {3, 5, 7} (iii) 3 3

2, ,2 4

− −

(iv) {1, 5}

13) (i) {q, r, u} (ii) {q, u} (iii) {q, t, u} (iv) {q, u}14)(i) {4, 16} (ii) {16} (iii) {16} (iv) {16}

15)(i) ag

A

cfd

e

B

h

b (ii) o

s

C

u

m

a

D

v

r

el

(iii)

I RE

16)

P

Q (ii) P

Q

17) (i) {2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16} (ii) {2, 5, 6, 8, 10, 12, 13, 14, 16}(iii) {2, 4, 6, 7, 8, 9, 10, 12, 14, 16} (iv) 12

18) (i) {E, F, I, C, N, T, M, A, G} (ii) 919) (i) {8, 16, 24, 25, 32, 36, 40, 48, 49} (ii) {8, 12, 16, 22, 24, 32, 40, 42, 48}

(iii) 8 (iv) 11

20) (i) A

C

M

D

H

G

N

E

B

F I

(ii) 52

M

53

54

55

57

N

56

58

51

(iii)

N

M

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1) (a) (i) P ⊂ Q (ii) P ∩ Q = P (b) (i) φ (ii) P' ∩ Q2) (a) (i) I (ii) II (b) (i) III (ii) II3) (a) P' {21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 40} (b) n(Q') = 19 (c) n(R') = 114) (a) {6, 7, 13, 15} (b) 7 (c) 9 5) (a) 14 (b) 14 (c) 76) (a) x = 5 (b) 18 (c) 19

7) (a) P

(b) QP

QP

(c) (d)

Q

P

R

8) (a) P (b) Q (c) P9) (a) H = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50} K = {1, 2, 4, 5, 10, 20}

(b) 3 (c) (i) (H ∩ K) ⊂ H (ii) (H ∩ K) ⊂ K10) (a) 18 (b) 7 (c) 11 11) (a) 5 (b) 15 (c) 10

12) (a) 4 (b) 5

G

1016 4

L B

13) 8

14) (a) {5, 10, 11, 15, 20, 22, 25, 30, 33, 35, 40, 44, 45, 50} (b) (i) yes (ii) yes15) (a) 3 (b) 2 16) (a) {15, 30} (b) {10, 12, 15, 18, 20, 21, 24, 25, 27, 30} (c) 1117) (a) (A ∪ B)' = {Mon, Wed, Fri} (b) (C ∪ D)' = {8, 9, 10} (c) (E ∪ F)' {e, f, i, j}

A B

C D

E G

F

18) (a) 28 (b) 2

1) A, C 2) A, B, C 3) A, B, C 4) C, D 5) A, B 6) A, B, C

Passage - I:

(i) M = {13, 18, 23, 28} N = {10, 12, 21, 23}

(ii) 12

(iii) 10

Passage - II:

(i) 550 (ii) 450 (iii) 350

Brain works

Further Practice

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

21. 22.

B C D B B D D C C A

B A B C A A C B D D

B C

Multiple answer Questions

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CLASS - VII

IIT F

oundatio

n &

Olym

pia

d E

xplo

rer - M

ath

em

atic

s Cla

ss - VII

FOUNDATION OLYMPIAD&

IntegratedSyllabus

UNIQUE ATTRACTIONS●

● Cross word Puzzles

● Graded Exercise

Basic Practice■

Further Practice■

Brain Works■

● Multiple Answer Questions

● Paragraph Questions

Detailed solutionsfor all problems

of IIT Foundation &Olympiad Explorer

are available in this book

CLASS - X

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� Simple, clear and systematic presentation

� Concept maps provided for every chapter

� Set of objective and subjective questions at the

end of each chapter

� Previous contest questions at the end of each

chapter

� Designed to fulfill the preparation needs for

international/national talent exams, olympiads

and all competitive exams

YOUR

COACH

India’s FIRST scientifically designed portalfor Olympiad preparation• Olympiad & Talent Exams preparation packages

Analysis Reports Previous question papers• •Free Demo Packages Free Android Mobile App• •

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