16
RAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION-2018 / STD - VIII / QP + SOLUTIONS 1 1 N.B. For Q. No. 2 to 5 proper procedure and explanation is necessary. Q.1 Choose the correct alternative answer in each of the following. [10 M] 1. 5 0.1 ...... (A) 0.1 (B) 1 (C) 0.001 (D) 0.00001 Sol: (D) 5 5 1 5 1 0.1 10 10 0.00001 100000 Topic:Integers, Decimals & Fractions / Sub-topic:Decimals/ L-Easy / Ganit Prabhutwa Examination 2. Find the hypotenuse of a right angled triangle having sides 20 and 21 (A) 29 (B) 28 (C) 41 (D) 25 Sol: (A) 2 2 2 c a b 2 2 2 20 21 c 400 441 C a=20 b=21 841 2 841 c 841 29 c Topic:Geometry / Sub-topic:Triangles / L-Easy / Ganit Prabhutwa Examination 3. Find the value of 2 2 a b if 12 a b and 10 ab (A) 124 (B) 104 (C) 164 (D) 144 Sol: (A) 12 a b ; 10 ab JEE | MEDICAL-UG | BOARDS | KVPY | NTSE | OLYMPIADS GANIT PRABHUTWA EXAMINATION-2018 Time: 3 hrs. Date: 09.12.2018 Std. VIII Max.Marks : 100

GANIT PRABHUTWA EXAMINATION-2018 Date: 09.12 ... - Rao …...RAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION-2018 / STD - VIII / QP + SOLUTIONS 8 8 10 minutes + 55 seconds They will

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RAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION-2018 / STD - VIII / QP + SOLUTIONS

1 1

N.B. For Q. No. 2 to 5 proper procedure and explanation is necessary.

Q.1 Choose the correct alternative answer in each of the following. [10 M]

1. 50.1 ......

(A) 0.1 (B) 1 (C) 0.001 (D) 0.00001

Sol: (D)

55 1 5 10.1 10 10 0.00001100000

Topic:Integers, Decimals & Fractions / Sub-topic:Decimals/ L-Easy / Ganit Prabhutwa Examination

2. Find the hypotenuse of a right angled triangle having sides 20 and 21

(A) 29 (B) 28 (C) 41 (D) 25

Sol: (A)2 2 2c a b

2 2 220 21c

400 441

C

a=20

b=21

841

2 841c

841 29c

Topic:Geometry / Sub-topic:Triangles / L-Easy / Ganit Prabhutwa Examination

3. Find the value of 2 2a b if 12a b and 10ab (A) 124 (B) 104 (C) 164 (D) 144

Sol: (A)

12a b ; 10ab

JEE | MEDICAL-UG | BOARDS | KVPY | NTSE | OLYMPIADS

GANIT PRABHUTWA EXAMINATION-2018Time: 3 hrs.

Date: 09.12.2018 Std. VIII Max.Marks : 100

RAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION-2018 / STD - VIII / QP + SOLUTIONS

2 2

2 2 2 2a b a b ab

2 2 212 2 10a b

2 2144 20 a b

2 2 124a b

Topic:Algebra / Sub-topic:_/ L-Easy / Ganit Prabhutwa Examination

4. Which member of the set 2,3,5,7,9,11,13 is not a prime number ?

(A) 2 (B) 11 (C) 9 (D) 5

Sol: (C)

9 is not a prime number.

9 3 3

Topic:Number System / Sub-topic:Prime Numbers / L-Easy / Ganit Prabhutwa Examination

5. Which of the following is the product of 8,937 and 125 ?

(A) 93700 (B) 937000 (C) 7496500 (D) 4685000

Sol: (B)

The given three numbers are 8; 937; 125

8 937 125

937 125 8

937 1000

937000Topic:Integers, Decimals & Fractions / Sub-topic: Integers/ L-Easy / Ganit Prabhutwa Examination

6. Find : :x y z if 2 3 4x y z

(A) 6 : 4 : 3 (B) 2 :3: 4 (C) 4: 3:8 (D) 3: 2 :3Sol: (A)

2 3 4x y z k

2kx ; 3

ky ; 4kz

: : : :2 3 4k k kx y z

1 1 1: :2 3 4

6 : 4 : 3

Topic: Ratios and Proportions / Sub-topic:Proportions / L-Easy / Ganit Prabhutwa Examination

RAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION-2018 / STD - VIII / QP + SOLUTIONS

3 3

7. Find the difference between the simple interest and the compound interest on Rs. 1600 at the rate 5 p.c.p.a.for 2 years.

(A) Rs. 160 (B) Rs. 80 (C) Rs. 4 (D) Rs. 5

Sol: (C)

S.I. 1600 2 5 160

100 100PTR

. . 1100

trC I P P

251600 1 1600100

4411600 1400

411600 164400

. . . . 164 160C I S I

Rs. 4

Topic:Simple Interest and Compound Interest/ Sub-topic:_/ L-Medium / Ganit Prabhutwa Examination

8. Find the middle term (geometric mean) of 0.25 and 400.

(A) 10 (B) 1 (C) 100 (D) 0.1

Sol: (A)2b ac , ,a b c geometric progression

2 400 0.25b

2 4 25b

4 25b

100 10b

Topic:Sequence and Series/ Sub-topic: Geometric Progressions/ L-Easy / Ganit Prabhutwa Examina-tion

9. Mohanlal got a profit 14

th

of the cost price when he sold an article for Rs. 400. Find the profit percent.

(A) 40 (B) 10 (C) 25 (D) 20

Sol: (C)

Let cost price be x

RAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION-2018 / STD - VIII / QP + SOLUTIONS

4 4

profit 4x

Selling price 400

Profit percentage P ro fi t 1 0 0

.C P

/ 4 100xx

25%

Topic:Profit and Loss/ Sub-topic:_/ L-Medium / Ganit Prabhutwa Examination

10. Whole numbers m,n are such that 121nm . Find the value of 11 nm

(A) 1024 (B) 10000 (C) 1 (D) 1000

Sol: (D)2121 11 121nm

11, 2m n

1 2 1 31 11 1 10 1000nm

Topic:Number System / Sub-topic:Exponents / L-Medium / Ganit Prabhutwa Examination

(B) Write only answer of each of the following subquestions. [10 M]

1. The sides of a rectangle are 24 cm and 18 cm. Find the perimeter of the square drawn on its diagonal.

Sol. In :BCD

2 2 2BD BC CD

2 224 18

A B

CDE

F

24 cm

18 cm

576 324

2 900BD cm

30 , perimeter 4 30 120BD cm cm

Topic:Mensuration/ Sub-topic:Rectangles_/ L-Medium / Ganit Prabhutwa Examination

2. Find the values of x and a, from the information given in the adjoining figure.

105o65o

Sol. 180ACP ACB

105 180a

RAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION-2018 / STD - VIII / QP + SOLUTIONS

5 5

180 105o oa

75a

Also, ABC ACB x {external angle of a triangle}

A x

aB C P

105o

65o

65o a x

65 75o ox

140x

140 , 75ox a

Topic:Lines and Angles / Sub-topic:_/ L-Medium / Ganit Prabhutwa Examination

3. Rakhi completes a task in 10 days, while Rama takes 5 days to finish the same task. If they work

together, in how many days will they finish the task ?

Sol. Rakhi completes a task in 10 days.

In one day, Rakhi completes 1

10 of the task.

Similarly, in one day, Rama completes 15 of the task.

If Rakhi and Rama completes the task in ‘x’ days working together.

In one day, they will do 1x of the task.

1 1 110 5 x

1 1 210x

10 days3

x

They will finish the task in 103 or 3.33 days.

Topic:Time and Work / Sub-topic:_/ L-Easy / Ganit Prabhutwa Examination

4. If 183 = 5832, then find the value of 23 5.832 5 .

Sol. 318 5832

32 3 2 13 5.832 5 5832 10 5 18 10 25 45

Topic:Integers, Decimals & Fractions / Sub-topic:Decimals_/ L-Easy / Ganit Prabhutwa Examination

RAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION-2018 / STD - VIII / QP + SOLUTIONS

6 6

5. The bisectors of and B C of ABC meet at incentre I. 54m A , find the measure of BIC .

Sol.

A

B C

I

In , 180ABC A B C

54 180B C

126B C ...(i)

In , 180BIC IBC ICB BIC ...(ii)

1But 2

IBC B ...(iii) { the angles are bisected}

1and 2

ICB C ...(iv)

from (ii), (iii), (iv)

1 1 1802 2

B C BIC

1 1802

B C BIC

1 126 1802

O BIC

180 63BIC

117BIC

Topic:Geometry / Sub-topic:Triangles/ L-Medium / Ganit Prabhutwa Examination

Q.2 Solve the following : [15 M]

1. PQRS is a rhombus : 40m MQR . Find the measures of ; , ,QSR PQR QPS .

RAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION-2018 / STD - VIII / QP + SOLUTIONS

7 7

Sol. Given : 40oMQR

2PQR MQR { QM is angle bisector of PQR }

40°

2 40 80o o

180oQRS PQR

180 80o oQRS

100oQRS

In :QRS

180oQRS QSR SQR

100 40 180o o oQSR { SQR MQR }

40oQSR

Also, QPS QRS

100oQPS

40 , 80 and 100o o oQSR PQR QPS

Topic:Quadrilaterals and Polygons/ Sub-topic:Quadrilaterals/ L-Medium / Ganit Prabhutwa Examina-tion

2. Exactly at what time between 2 and 3 will the hour hand overlap the minute hand ?

Sol. At 2 o’ clock, hour hand will be at 2 and minute hand will be at 12.

In one hour, hour hand covers 360 3012

oo per hour..

In one minute, hour hand cover 030 1

60 min 2

o

per minute.

Also, in one minute, minute hand covers 360 6 per minute60

oo

12 12

3

4567

8910

11

Let they meet after x minutes :

16 602

ox x

6 602

oxx

11 602

ox

120 minutes11

x

They will after 12011 or 10.909 minutes or

10 minutes 10 60 seconds11

RAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION-2018 / STD - VIII / QP + SOLUTIONS

8 8

10 minutes + 55 seconds They will overlap at 2 hours 10 minutes and 55 seconds

Topic:Geometry/ Sub-topic:_/ L-Medium / Ganit Prabhutwa Examination

3. Find the average of the first 20 consecutive odd numbers.

Sol. First 20 consecutive odd numbers are 1, 3, 5, ----

Sum 2 12n a n d

20 2 1 20 1 22

10 2 19 2

10 2 38

10 40 400

400Average= 2020

Topic:Sequence and Series / Sub-topic:Arithmetic Progression/ L-Easy / Ganit Prabhutwa Examination

4. By simple interest, a certain amount triples in 12 years at a certain rate of interest.Find the rate of interest.

Sol. Let rate of interest be R

12

3100

P RP P

{ T = 12 years}

123 1100

R

12 2100

R

2 10012

R

50 % 16.67% p.a.3

R

Topic:Simple Interest and Compound Interest/ Sub-topic:Simple Interest_/ L-Medium / Ganit PrabhutwaExamination

5. Construct a rectangle having diagonal 6.0 cm and the measure of an angle between the diagonals 120°.

Sol.

A D

B C0

3cm 3cm

3cm 3cm

Clearly CD = 3 cmOD = 3 cm

RAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION-2018 / STD - VIII / QP + SOLUTIONS

9 9

i) With he help of ruler draw a line segment AC of length 6 cm and mark its mid point O on it.

A O C3 3ii) With the help of compass draw an arc of radius 3 cm from O and from C on one side of line segment

AC to get point D.

A O C3 3iii) Similarly draw an arc from O and from A on other side of line segment AC to get point B.

Join ABCD

A O C3 3

Topic:Constructions / Sub-topic:_/ L-Medium / Ganit Prabhutwa Examination

Q.3 Solve the following subquestions. [20 M]

1. The points A, B, C, D, E, F lie on a circle. Find m BAF m FED m DCB .

Sol.

A

B

C D

E

F

Join F and C

Now, FEDC is a cyclic quadrilateral

also, FABC is a cyclic quadrilateral

Now, 0180 .......FED FCD i (opposite angles of a cyclic quadrilateral)

also 0180 .......BAF FCB ii (opposite angles of a cyclic quadrilateral)

Now adding (i) and (ii)0 0180 180FED FCD BAF FCB

RAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION-2018 / STD - VIII / QP + SOLUTIONS

1010

0360

0360BAF FED DCB DCB FCB FCD

Topic:Quadrilaterals and Polygons/ Sub-topic:Polygons/ L-Medium / Ganit Prabhutwa Examination2. GCD of two numbers is 23 and their product is 19044. Find all possible pairs of such numbers.

Sol. Let the numbers be x, y

GCD of , 23x y

19044xy

19044 828

23LCM

of , 23GCD x y , 828 23 36LCM 2 2of , 23 2LCM x y

Numbers can be 2 223,828 , 23 2 , 23 3

23,828 and 92,207

Topic:Number System/ Sub-topic:HCF and LCM_/ L-Difficult / Ganit Prabhutwa Examination

3) Simplify 2

2

12 277 18

x xx x

Sol. 2

2

3 912 277 18

x xx xx x

9x 2x

3 .2

xx

x + 2 - 5 5or 1x + 2 2x

Topic:Algebra/ Sub-topic:Quadratic equation/ L-Easy / Ganit Prabhutwa Examination

4) Solve :1 2 1 7 12

6 3 4 6x x x

Sol.1 2 1 7 2 6 1

6 3 4 6x x x

1 8 3 21 136 12 6

x x x

2 2 29 3 26x x

RAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION-2018 / STD - VIII / QP + SOLUTIONS

1111

27 27x

1x

Topic:Algebra/ Sub-topic:_/ L-Medium / Ganit Prabhutwa Examination

5) In the adjoining figure, line || line m. From the information given in the figure find the measure of .ABC

Sol.

A

BD C

0110ABC

Topic:Lines and Angles/ Sub-topic:_/ L-Easy / Ganit Prabhutwa ExaminationQ.4) Solve the following subquestions. [20 M]

1. Divide 4 24 by 2 2 .x x x Find quotient and remainder. Write answer as

dividend = divisor × quotient + remainderSol.

2 2 2x x

2 2 2x x 4 4x 4 3 22 2x x x ( )

( )

3 22 2 4x x 3 22 4 4xx x

( )

22 4 4x x 22 4 4x x

0

2Quotient 2 2x x

Remainder = 0

RAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION-2018 / STD - VIII / QP + SOLUTIONS

1212

4 2 24 2 2 2 2 0x x x x x

Topic:Algebra/ Sub-topic:_/ L-Medium / Ganit Prabhutwa Examination2. The speed of a boat in still water is 25 km/hr. The time required for the boat to travel 120 km against the

stream is 32 times the time required to travel the same distance along the same stream. Find the speed of the

stream.

Sol. Let speed of stream /x km hr

Speed against stream 25 /x km hr

Speed along stream 25 /x km hr

distanceTime =speed

Time against stream 32

(Time along stream)

120 3 12025 2 25x x

2 25 3 25x x

5 25x 5 /x km hr

Topic:Time, Speed and Distance/ Sub-topic:_/ L-Medium / Ganit Prabhutwa Examination

3. Solve : 2 326 0.63

xo

Sol. 2 326 0.63

xo

2 32 613 10

x

2 35 33 5

x

1 3 225 5

3 3

x

13 22

x

RAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION-2018 / STD - VIII / QP + SOLUTIONS

1313

13 22

x

56

x

Topic:Algebra/ Sub-topic:Exponents_/ L-Easy / Ganit Prabhutwa Examination4. If a polygon has 90 diagonals, then find the number of sides of the polygon.

Sol. Number of diagonals 3

2n n

2 3 902

n n

2 3 180n n 2 3 180 0n n

12 15 0n n

12 Rejectedn

15n

Topic:Geometry/ Sub-topic:Polygons/ L-Medium / Ganit Prabhutwa ExaminationQ.5) Solve the following subquestions. [25 M]

1. Factorise : 2 28 8 5 14x x x x

Sol. 2 28 8 5 14x x x x

2Let 8x x y

5 14y y

2 5 14y y

2 7y y

2 28 2 8 7x x x x ( 2 8x x y )

2 8 2 1 7x x x x

Consider : 2 8 2x x

2 4Roots2

b b aca

8 64 82

RAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION-2018 / STD - VIII / QP + SOLUTIONS

1414

8 722

8 6 22

4 3 2

4 3 2 4 3 2 1 7x x x x

Topic:Algebra/ Sub-topic:Factorisation_/ L-Medium / Ganit Prabhutwa Examination

2. Find 33

1 3if 3 4x xx x

Sol.33 4xx

1 43

xx

On cubing,

31 6427

xx

33

1 1 1 64327

x x xx x x

33

1 4 643 13 27

xx

33

1 64 427

xx

33

1 17227

xx

Topic:Algebra/ Sub-topic:_/ L-Medium / Ganit Prabhutwa Examination

3. In the adjoining figure, 90 , , , , , 12, 15.oQPS QPR PSR PR QS PS y RS x PR PQ

then - (i) Prove QPS is similar to QRP

(ii) Find the values of , .x y

P

Q R S

15 12

RAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION-2018 / STD - VIII / QP + SOLUTIONS

1515

Sol. In &QPS QRP

90oQPS QRP

same anglesPQS RQP

Thus andQPS QRP are similar by A A test

P

Q R S

15 12

QP QS PSQR QP RP

Also in PQR

2 2 2PQ QR PR

2 2 215 12QR

9QR

159 15 12

QS PS

5 93 15 12

x y

16 and 20x y

Topic:Geometry/ Sub-topic:Triangles/ L-Medium / Ganit Prabhutwa Examination

4. In the adjoining figure ABC and PQR are congruent. All small triangles are congruent, each side ofsmall triangle is 4 find the area of the figure.

P

Q R

A B

C

4

4

4 4 4

4

44

Sol.

4 44 4

44

4 4

44 4

4

4 4

44

4A B

RQ

C

P

Area of the figure = Area of PQR (equilateral with side length = 12)

+ 3 (area of smaller equilateral triangles with side length = 4)

RAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION-2018 / STD - VIII / QP + SOLUTIONS

1616

2 23 312 3 44 4

36 3 12 3 48 3 sq.units

Topic:Geometry/ Sub-topic:Triangles and Polygons_/ L-Medium / Ganit Prabhutwa Examination5. Some kg of tea at the rate of Rs. 320 per kg is mixed with 5 kg of tea at the rate of Rs. 280 per kg. Now

the rate of the mixture so formed is Rs. 300 per kg. How much tea of the rate of Rs. 320 per kg was taken?Sol. Let the amount of tea of Rs. 320 per kg taken be x kg.

Mixture contains x kg. of Rs. 320 per kg tea. + 5 kg. of Rs. 280 per kg tea.

Rate of mixture 320 5 280

3005

xx

320 280 5 300 5x x

32 140 30 150x x

2 10x

5 .x kg

Topic:Mixtures/ Sub-topic:_/ L-Medium / Ganit Prabhutwa Examination