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

1 Website : www.raoiit.comRao IIT Academy 1

Q.1 A) Choose the correct alternative and write it against the correctsub – question number. (10)

1) Ray PQ and ray PS are opposite rays. How many points are common to the two rays ?(A) 0 (B) 1 (C) 2 (D) 3

Sol. (B)Common point is ‘P’

SQP

Topic : Lines / Sub Topic : Rays / Level : Easy / Ganit Prabhutwa Examination

2) What is the decimal form of 361 ?90

(A) 4.011 (B) 4.01 (C) 4.0001 (D) 4.01Sol. (D)

361 360 1 1490 90 90 90

4 0.0111..... 4.01

Topic : Number System / Sub Topic : Decimals / Level : Moderate / Ganit Prabhutwa Examination3) The centre of the circle passing through all the three vertices of PQR lies on the side PQ.

What type of angle is P ?

(A) acute angle (B) obtuse angle (C) right angle (D) can not be decidedSol. (A)

Since angle R must be right angle, P must be acute angleP

R Q

C

Topic :Circles, Triangles / Sub Topic : Circles / Level :Moderate / Ganit Prabhutwa Examination

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

GANIT PRABHUTWA EXAMINATION

Std - VIIIDate : 10 - 12 - 2017Total Marks : 100Time : 3 Hours

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4) A money lender charges Rs. 63 as interest for 6 months on Rs. 420. What is the rate of interestp.c.p.a ?

(A) 33 % (B) 13 % (C) 30 % (D) 10 %

Sol. (C)

Time = 6 months = 6 1

12 2yr yr

63100

P R TSI

420 1 63100 2

R

300 30%10

R

Topic :Percentages / Sub Topic : Simple Interest / Level : Moderate / Ganit Prabhutwa Examina-tion

5) A shopkeeper sells an article for Rs. 240, and suffers a loss of 4 %. What should be the sellingprice of that article to gain a profit of 12 % ?

(A) 252 (B) 236 (C) 262 (D) 280

Sol. (D)

Let 100CP x

After loss of 4 %

SP = 96 x = 240

240 10 2.596 4

x

so, C.P. 100 2.5

= 250

S.P. after 12 % profit

= 112 % of 250

112 250100

56 5

= 280

Topic :Profit & Loss / Sub Topic : Profit and loss / Level : Moderate / Ganit Prabhutwa Examina-tion




6) What is the averages of the numbers 2 1 17 ,3 ,0,1, ?3 4 12

(A) 4 (B) 2.4 (C) 3 (D) 4.2Sol. (B)

2 1 17 3 0 13 4 12

5

1 2 1 1115 3 4 12

1 8 3 1 1 1211 11 15 12 5 5

= 2.4

Topic : Data Handling / Sub Topic :Averages / Level :Moderate / Ganit Prabhutwa Examination

7) What should be added to 2 2 11 x x so that the sum will be divisible by 3x as well as by

5x ?

(A) 4 (B) – 4 (C) 3 (D) –3Sol. (B)

2 2 11 3 5x x k a x x

2 22 11 2 15x x k ax ax a

1, 11 15a k

4k

Topic : Quadratic Equations / Sub Topic :Factorization / Level : Moderate / Ganit PrabhutwaExamination

8) If 2 3x y and 5 6y z find the ratio :x z(A) 1 : 3 (B) 9 : 5 (C) 5 : 9 (D) 3 : 5

Sol. (B)

2 3x y 5 6y z

32

xy

65

yz

Multiply the two ratios to get so, 3 6 92 5 5

x y xy z z

Topic : Ratios & Proportions / Sub Topic : Ratios / Level : Easy / Ganit Prabhutwa Examination




9) Area of a circle is 1386 sq.cm. Find its diameter. 227

(A) 42 cm (B) 21 cm (C) 132 cm (D) 84 cm

Sol. (A)

2 1386r

222 13867

r

2 1386 722

r

2 63 7 r

21 r

Diameter = 2r

2 21

42cm

Topic :Mensuration / Sub Topic :Area of circles / Level :Easy / Ganit Prabhutwa Examination

10) If 1813

x then find x.

(A) 4 (B) –4 (C) 14 (D)

14

Sol. (D)

1813

x

4 133

x

4 13 3 x

4 1 x

14

x

Topic : Exponents & Powers/ Sub Topic : Exponents / Level : Easy / Ganit Prabhutwa Examina-tion




Q.1 B) Write only answers of each of the following sub-questions. (10)

1) Area of a trapezium is 84 cm2. Length of one of its parallel sides is 10 cm and its height is 6 cm.What is the length of the other parallel side ?

Sol. A = 84 cm2

a

b

h = 6

1842

a b h

184 10 62

b

10 28b

18b cm

Topic : Mensuration / Sub Topic : Trapezium area / Level : easy / Ganit Prabhutwa Examination

2) Two numbers differ by 20. One third of the greater number is half of the smaller number.Find the numbers.

Sol. 20a b ...(1)

1 13 2

a b

2 3 0a b ...(2)

2 2 40– 2 3 0

40

a ba b

b

40b

40 20a

60a

Topic : Linear Equation in one/two variable / Sub Topic : Linear Equation in one/two variable/Level : Easy / Ganit Prabhutwa Examination




3) In the adjoining figure, line l | | line m. Line n is a transversal. Find values of x and y.

x y + 10

2y – 30

n

l

m

Sol. 10 180x y [Linear pair]

170x y ...(1)

10 2 30y y [Corresponding angles]

x y + 10

2y – 30

n

l

m

40 y

40 170 130x x

Topic : Parallel lines / Sub Topic : Crossponding angles, supplementary length of chord / Level :Easy / Ganit Prabhutwa Examination

4) In a circle of diameter 10 cm, a chord is at a distance of 6 cm from the centre. Find the lengthof the chord.

Sol. Not possible to solve using diameter as 10 cm.

Hence we assume 10 cm is given as the radius and not diameter.

A B6 10O

In ,OAB

2 2 2OB OA AB

2 2 210 6 AB

2 100 36AB

2 64AB

8AB cm

Hence length of chord = 2 16AB cm

Topic : Circles / Sub Topic :Length of chord / Level :Moderate / Ganit Prabhutwa Examination




5) Find the condition on the values of x, so that the triangle of sides 40 cm, 37 cm, and x cmcan be constructed.

Sol. (1) 40 37 x

77x

(2) 40 37x

40 37

x3x

(3) 37 40x

3x Using (1), (2) & (3)

3 77x

Topic : Triangles / Sub Topic :Triangle inequality / Level :Moderate / Ganit Prabhutwa Examina-tion

Q.2 Solve the following sub-questions. (15)

1) Factorize : 3 3 2 28 4 8x y x y xy

Sol. 3 3 2 28 4 8x y x y xy

3 32 4 2x y xy x y

32 3 2 2 4 2x y x y x y xy x y

22 2 3 2 4x y x y x y xy

2 22 2 4x y x xy y

Topic : Factorization / Sub Topic : (a+b)3 / Level : Moderate / Ganit Prabhutwa Examination2) Gopal leaves home and by walking at two third of his usual speed reaches his school 10

minutes late. How much time does he take usually to reach the school from home ?

Sol. DST

.... 1DxT

2 .... 23 10

DxT

From (1) & (2)

2103

Tx T x

3 2 20Tx Tx x

20 minsT

Topic :Linear Equation in one variable / Sub Topic : Linear Equation in one variable / Level :Moderate / Ganit Prabhutwa Examination




3) Divide the polynomial 3 26 11 6x x x by (x – 2) and write the remainder..

Sol.2 4 3x x 3 26 11 6x x x 2x 3 22x x

24 11 6x x 24 8x x

3 6x 3 6x

0

Remainder = 0

Alternately if 3 26 11 6p x x x x , then the remainder when divided by (x – 2) is

3 22 2 6 2 11 2 6 0p

Topic : Polynomials / Sub Topic : Long division / Remainder theorem / Level : Moderate / GanitPrabhutwa Examination

4) Seema purchased 8 chocolates for Rs. 10. How many chocolates should she sell for Rs. 10 togain a profit of 60 % ?

Sol. 8chocolates .10Rs

Required profit = 60%

1608 10 16100

SP for chocolates

Rs.

Hence SP of one chocolate = 2 Rs.

Hence for 10 Rs. number of chocolates to be sold 10 52

chocolates.

Topic : Profit and Loss / Sub Topic : Profit and Loss / Level : Moderate / Ganit Prabhutwa Exami-nation

5) In the adjoining figure, ABCE is a square with side 14 cm. If B, C, D are collinear andl(ED) = 50 cm, find l(BD).

A

B C

E

D




Sol.

14 50

D

EA

14

B14 C

1 BD,In AECD

2 2 2ED EC CD

2 2 250 14 CD

2 2 250 14 50 14 50 14CD

36 64 6 8 48CD cm

Hence 14 48 62BD BC CD cm

Topic : Pythagorus theorem / Sub Topic : Pythagorus theorem / Level : Moderate / Ganit PrabhutwaExamination

Q.3 Solve, the following sub-questions. (20)1) If 6,a b c and 2 2 2 29,a b c find the value of .ab bc ac Sol. 6a b c ...(i)

2 2 2 29a b c ...(ii)

2 26a b c

2 2 2 2 36a b c ab ac bc Using equation (ii)

29 2 36ab ac bc

7 3.52

ab ac bc

Topic : Algebraic Identities / Sub Topic : (x+y+z)2 / Level :Moderate / Ganit Prabhutwa Examina-tion2) Two years before, a machine was purchased for Rs. 62500. If its price reduces by 10% every

year, what is the price of that machine today?Sol. C.P = Rs. 62500

Price reduces 10% per year.

So, it becomes 9

10

th

after every year..

After two years,

Price will be 9 962500

10 10

625 81

50625 .Rs

Topic : Percentages / Sub Topic : Percentages / Level : Moderate / Ganit Prabhutwa Examination




3) Mohan bought a bicycle for Rs. 1040. He paid the amount in the form of notes of Rs. 100,Rs. 50 and Rs. 20. Number of 100 rupee notes was half of 50 rupee notes and number of 20rupee notes was three times the number of 100 rupee notes. Find the number of notes of eachdenomination.

Sol. Let there be x number of 100 rupees notes

Hence number of 50 rupees notes = 2x

Hence number of 25 rupees notes = 3x

100 50 2 20 3 1040x x x

10 10 6 104x x x

26 104x

4x

Rs.100 4 notes

Rs.50 2 4 8 notes

Rs.20 3 4 12 notes

Topic : Linear Equation in one variable / Sub Topic : Linear Equation in one variable / Level :Moderate / Ganit Prabhutwa Examination

4) In the adjoining figure, ABCD is a trapezium.D C

BA

5

3

P

x-5

x-3

Seg. AB || seg. DC

Find the value of x.

Sol. APB CPD

5 33 5

AP PB xCP PD x

A B

CDx–5 3

x–35

P

5 25 3 9x x

2 16x

8x

Topic : Geometry / Sub Topic : Geometry of s / Level : Moderate / Ganit Prabhutwa Examination




5) 1/ 4 1216 1296 6x x . Find the value of x.

Sol. 1/ 4 1216 1296 6x x

3 11/ 4

16 61296

x x

3 1

1/ 44

16 66

x x

3 116 66

x x

3 1 16 6 6x x 3 26 6x x

3 2x x

2 2x

1x

Topic : Exponents & Powers / Sub Topic : Exponents & Powers / Level : Moderate / Ganit PrabhutwaExamination

Q.4 Solve, the following sub-questions. (20)

1) Construct LMN such that ( ) 5.2 , ( ) 5.7 ,l LM cm l LN cm height LT = 4.5 cm.

Sol. Step (1) : First draw any line & draw length LT perpendicular to it of length 4.5 cm. (Drawing perpendicularto a line is a standard procedure)

L

4.5 cm

TStep (2) : Take distance L M = 5.2 in compass & draw an arc with L as centre of cutting the line in M.Similarly take distance LN = 5.7 cm in compass & draw an arc with L as centre & cutting line in N.Note that there will be two possibilities for points M & N as shown.

L

NTMM

N




Hence total 2 non-congruent triangles possible as shown.

L

NTM

5.74.55.2

L

N TM

4.55.25.7

Topic : Construction / Sub Topic : Construction of s / Level :Moderate / Ganit Prabhutwa

Examination

2) The L.C.M. of two number is fourteen times their G.C.D. The sum of the G.C.D and the L.C.Mis 600. Find two pairs of such numbers.

Sol. LCM = 14 GCD and LCM + GCD = 600

14 (GCD) + GCD = 600

GCD = 40 and LCM = 560

As GCD = 23 5 and LCM = 42 5 7

One of the numbers must have 23 in the factorization and the other should have 24 as a factor..

Both numbers should have 5 in the factorization but only one number should have 7 in the factorization.

This gives two pairs of numbers as 3 42 5 7, 2 5 and 3 42 5, 2 5 7

i.e. 280;80 and 40;560

Topic : LCM & GCD / Sub Topic : LCM & GCD / Level : Moderate_ / Ganit Prabhutwa Examina-tion

3) Perimeters of a rectangle and a square are equal. The ratio of the length and breadth of therectangle is 5 : 4. Find the ratio of the length of the rectangle to the side of the square.

Sol. Take length and breadth of rectangle as 5x and 4x

perimeter of rectangle = perimeter of square = 18 x

side length of square 92x

length of rectangle 5 10 :9side of square 9 / 2

xx

Topic : Ratio & Proportion / Sub Topic : Rectangles & squares / Level : Moderate / Ganit PrabhutwaExamination




4) Solve, 1 21 3

2 2 3x x

.

Sol.1 21 3

2 2 3x x

Multiply both side by 6

1 26 1 6 3

2 2 3x x

3 6 9 2x x

6 2 9 3x x

4 6x

3x

Topic : Linear Equation in one variable / Sub Topic : Linear Equation in one variable / Level :Easy / Ganit Prabhutwa Examination

5) A train running at a speed 0f 90 km/hr, crosses a bridge in 36 seconds. Another train havinglength 100 meters less, is running at a speed of 45 km/hr. In how much time will the train crossthe same bridge?

Sol. Total distance needed to cross a bridge

= length of bridge + length of train

= speed time

for first train, it is 3690 0.93600

km = 900 m

for the second train, it will be 100 m less = 800 m

= speed time

time required 800 0.8 845 / 45 450

m hrskm hr

8 3600 seconds 64 seconds450

Topic : Linear Equation in one variable / Sub Topic : Linear Equation in one variable / Level :Moderate / Ganit Prabhutwa Examination




Q.5 Solve the following sub-question. (25)

1) ABCD is a rectangle. Prove that seg. AC seg. BD

Sol.

D

C

A

B

ABD CDB by RHS congruency test as hypotenuse BD is common and 90oBAD BCD

Hence AB=CD

Now,

ABC DCB by SAS congruency test.

as AB = CD, BC is the common side and 90oABC DCB

AC DB

Topic : Geometry / Sub Topic : Rectangles / Level : Easy / Ganit Prabhutwa Examination

2) The price of sugar is increased by 20%. By what percent the consumption of sugar be reducedso that expenditure on sugar will remain the same?

Sol. If the base price of sugar = 100 x Rs/kg

then the final increased price 120100 . 120 /100

x x Rs kg

Let initial consumption = y kg

initial expenditure = 100 xy Rs.

Since final expenditure = initial expenditure

= 100 xy Rs.

final consumption final epxenditure 100

final price 120xyx

56y kg

reduction in consumption 56 6y yy kg

% reduction / 6 100 16.67%yy

Topic : Percentages / Sub Topic : Percentages / Level :Moderate / Ganit Prabhutwa Examination




3) Factorize : 2 2 21 2 2x yz x y z .

Sol. 2 2 21 2 2x yz x y z

2 2 21 2 2x x y yz z

= 2 21x y z

1 1x y z x y z ..... using 2 2a b a b a b

1 1x z y x z y

Topic : Factorization / Sub Topic : (a+b)2 / Level : Moderate / Ganit Prabhutwa Examination

4) In a terminal examination Arun scored 82 marks in mathematics. Ajit scored 5 marks less than

Amit and Amit scored 4 marks more than the average marks of the class. If the average ofmarks of Arun, Ajit and Amit is 2 more than the average of marks of the class, find the average

of marks of the class.

Sol. Let average marks of the class = x

Amit’s marks = x + 4

Ajit’s marks = 4 5 1x x

Average of Arun, Ajit and Amit’s marks

82 1 4 85 2

3 3x x x

= 2 more than class average

= x + 2

85 2 3 2x x

85 6 3 2x x

79x

Topic : Linear Equation in one variable / Sub Topic :Linear Equation in one variable / Level :Moderate / Ganit Prabhutwa Examination




5) A copper wire is bent to form a circle whose area is 616 sq. cm. If the same wire is bent to form

a square, what will be the area of that square? 227

.

Sol.r

Let the perimeter of the wire 2p r

2pr

Area 2 2

2 22 616

4 4p pr cm

2 222616 4 616 4 88 4 22 88 887

p cm

88p as 0 88p p cm

Hence perimeter of the square = 88 cm

side 224p cm

Area of square 2

2 222 4844p cm

Topic :Mensuration / Sub Topic : Area of circle / square / Level : Easy / Ganit Prabhutwa Exami-nation


std - viii time : 3 hours - rao iit€¦ · rao iit academy / ganit prabhutwa examination / std -...

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