Ramanujan Olympiad - Aatmdeep

NAME OF THE CANDIDATE

Time : 2 Hours.Maximum Marks : 120Total Number of Questions : 40

HIGHER FOR CLASS-IX & X

AATMDEEP VIDYALAYA Siktaur Chauraha, Deoria Bypass Gorakhpur-273010,Contact no. : 9935703444, 8853211001, 8172835555

Website : www.aatmdeepvidyalaya.com, Email ID : [email protected]

GENERAL INSTRUCTIONSPlease read the instructions carefully. You are allotted 5 minutes specifically for this purpose.

1. The question paper consists of 40 Objective Questions, for each question you will be awarded 3 marks ifyou have darkened only the bubble corresponding to the correct answer and zero mark if no bubble isdarkened. In case of bubbling of incorrect answer, minus one (–1) marks will be awarded.

2. It is mandatory to fill the required informations on the OMR Sheet.3. Blank papers, clip boards, log tables, slide rule, calculators, mobile or any other electronic gadgets in any

form are not allowed.

4. All rough work should be done only in the extra blank papers provided by the invigilator with the questionpaper.

5. Use Black or Blue Ball Pen to fill the bubbles on OMR Sheet.

Ramanujan Olympiad

by

Class-IX & X

AATMDEEP VIDYALAYA Siktaur Chauraha, Deoria Bypass Gorakhpur-273010,Contact no. : 9935703444, 8853211001, 8172835555Website : www.aatmdeepvidyalaya.com, Email ID : [email protected] VIDYALAYA 2

1. The value of m3n1m

1n3nmn2m3m

151066532

is equal to :

(A) 0 (B) 1 (C) m2 (D) none of these

2. The top of a broken tree has its toptouching the ground (as shown in thefigure) at a distance of 10 m from thebottom. If the angle made by thebroken part with ground is 300, then thelength of the broken part is

(A) m310 (B) m3

20

(C) 20 m (D) m320

3. If ,31

22

22

babababa

then ba

(A) 1 (B) 2 (C) 3 (D) 4

4. A quantity Q is obtained by adding three quantities. The first is a constant (k), the second varies directly

with the square root by y and the third varies directly with the cube root of y . If ,60,1 Qy when

230,64 Qy and ,660,729 Qy then the value of constant (k) is :(A) 10 (B) 20 (C) 30 (D) 40

5. If O is the centre of the circle shown below and ,BCABOA then the value of x is :

(A) 15º (B) 30º (C) 60º (D) 45º

6. A three-digit number is eleven times the two-digit number formed by using the hundred’s and the unit’s digitof the three-digit number respectively (ten’s place of two digit number is same as hundred’s place of threedigit number). If the digit in ten’s place is one greater than the digit in hundred’s place, then what is the digitin the unit’s place ?(A) 2 (B) 3 (C) 4 (D) 1

7. A three digit number which on being subtracted from another three-digit number consisting of the samedigits in reverse order gives 594. The minimum possible sum of all the three digits of this number is(A) 6 (B) 7 (C) 8 (D) Cannot be determined

Class-IX & X


Directions : (Q.8-9) : Answer the questions based on the following informations.A cow is tied at point A by arope. Neither the rope nor thecow is allowed to enter thetriangle ABC. º60BACand mACAB 10 .

A

B

C

8. What is the area that can be grazed by the cow, if the length of the rope is 8 m ?

(A) msq31128 (B) msq64 (C) msq

3160

(D) msq3

320

9. What is the area that can be grazed by the cow, if the length of the rope is 12 m ?

(A) msq31364 (B) msq144 (C) msq

3368

(D) msq)120(

10. The adjoining figures shows a set of concentric squares. If the diagonalof the innermost square is 2 unit and if the distance between thecorresponding corners of any two successive squares is 1 unit, find thedifference between the areas of the third and the fifth squares, countingthe innermost suqare.

(A) unitsq210 (B) unitsq32 (C) unitsq10 (D) unitsq20

11. The figure shows the rectangle ABCDwith a semicircle and a circle inscribedinside in it as shown. What is the ratioof the radii of the circle to that of thesemicircle ?

A

DO C

B

(A) 1:)12( 2 (B) 1:)12(2 2(C) 2:)12( 2 (D) 1:)12( 2

12. Two circle, both of radii 1 cm, intersect such that the circumference of each one pases through thecentre of the other. What is area (in sq cm) of the common region ?

(A) 43

3

(B)

23

32

(C) 23

34

(D) 23

32

Directions (Q. 13-14) : Answer the questions based on the following information.In the adjoining figure, I and II are circles with centresP and Q respectively. The two circles touch each otherand have a common tangent that touches them at pointsR and S respectively. This common tangent meets theline joining P and Q at O. The diameters of I and II arein the ratio 4 : 3. It is also known that the length of POis 28 cm.

RS

QP

III

O

13. What is the ratio of the length of PQ to that of QO ?(A) 1 : 4 (B) 1 : 3 (C) 3 : 8 (D) 3 : 4

Class-IX & X


14. What is the radius of the circle II ?(A) 2 cm (B) 3 cm (C) 4 cm (D) 5 cm

15. In the adjoining figure.AB = 3 cm, BC = 4 cm

º90ABC Calculatethe radius of inscribedcircle.

A

B C(A) 1 cm (B) 2 cm(C) 0.5 cm (D) 1.5 cm

16. In the figure given below (not drawn to scale), A, B andC are three points on a circle with centre O. The chordBA is extended to a point T such that CT becomes atangent to the circle at point C. If º30ATC and

º50ACT , then the angle BOA is : B A

C

O50º

30º T(A) 100º (B) 150º(C) 80º (D) not possible to determine

17. In the given diagram, ABCD is arectangle with AE = EF = FB. What isthe ratio of the area of the triangle CEFand that of the rectangle ?

A E F B

CD

(A) 1 : 4 (B) 1 : 6(C) 2 : 5 (D) 2 : 3

18. The line AB is 6 m in length and is a tangent to theinner one of the two concentric circles at point C. It isknown that the radii of the two circles are integers.The radius of the outer circle is :

BC

A

(A) 5 m (B) 4 m (C) 6 m (D) 3 m

19. AB is the diameter of the given circle,while points C and D l ie on thecircumference as shown. If AB is 15cm, AC is 12 cm and BD is 9 cm, findthe area of quadrilateral ACBD.

A B

C

D(A) 54p sq cm (B) 216p sq cm(C) 162p sq cm (D) 108 sq cm.

20. The side of a square is6 cm. Find the ratio ofthe areas of circlesinscribed and circumscribed about it.

a

r

R

aD C

BA

(A) 1 : 2(B) 1 : 5(C) 2 : 5(D) 1 : 4

Class-IX & X


21. PQRS is a square.SR is a tangent (atpoints S) to the circle with centre O andTR = OS.Then, the ratio of area of thecircle to the area of the square is.

(A) 3

(B) 711

(C) 3

(D) 117

22. The sum of the areas of two circles, which touch each other extremally, is 153 . If the sum of their radii is15, find the ratio of the larger to the smaller radius.(A) 4 : 1 (B) 2 : 1 (C) 3 : 1 (D) 1 : 1

23. In the given figure, two concentric circlesare shown with centre O . PQRS is asquare inscribed in the outer circle. It alsocircumscribes the inner circle, touching

it at points DCB ,, and .A What is theratio of the perimeter of the outer circle tothat of quadrilateral ABCD :

DS

cOA

QBP

R

(A) 4

(B) 23

(C) 2

(D)

24. In the figure below, P, Q and R are points on a circle with centre O. The tangent to the circle at R intersectssecant PQ at T. If 55ºQRT and 25ºQTR , find POQ.

P

Q

R

O

T550

250

(A) 110º (B) 100º (C) 90º (D) 50º

25. ,)( 2 MPBBE where BE is a two digit number and MPB is a three digit number such that B,E,M & P

are distinct digits, then ?M(A) 2 (B) 3 (C) 9 (D) None of these

26. When you reverse the digits of the number 13, the number increases by 18. How many other two digitnumbers increase by 18 when their digits are reversed?(A) 5 (B) 6 (C) 7 (D) 8

Class-IX & X


27. In the figure below, MON MPO 90NQO and OQ is the bisector

for MON and ,10QN .7/40ORFind .OP(A) 4.8 (B) 4.5(C) 4 (D) 5

28. In the adjoining figure, ADABAC 5and 8 ADAC . Then, the area of

the rectangle ABCD isA B

CD

(A) 36 (B) 50(C) 60 (D) Cannot be answered

Directions : (Q.29-30) : Answer questions on the basis of the information given below:An airline has a certain free luggage allowance and charges for excess luggage at a fixed rate per kg. Twopassengers, Raja and Praja have 60 kg of luggage between them, and are charged Rs. 1200 and Rs.2400respectively for excess luggage. Had the entire luggage belonged to one of them, the excess luggagecharge would have been Rs.5400.

29. What is the weight of Praja’s luggage?(A) 20 kg (B) 25 kg (C) 30 kg (D) 35 kg

30. What is the free luggage allowance?(A) 10 kg (B) 15 kg (C) 20 kg (D) 25 kg

31. A semi-circle is drawn with AB as its diameter. From C, a point on AB, a line perpendicular to AB is drawnmeeting the circumference of the semi-circle at D. Given that AC = 2 cm and CD = 6 cm, the area of thesemi-circle (in sq. cm) will be :(A) 32 (B) 50 (C) 5.40 (D) 81

32. It is possible to construct a triangle PQR with :

(A) 00 25PQR,110PRQ3cm,RQPR

(B) 00 301PQR,20QPR5cm,QR4cm,PQ

(C) 00 30PRQ,60PQR6cm,QR3cm,PQ

(D) 6cmPR3cm,QR2cm,PQ

33. If ,3 cbax then the value of )()()(3)()()( 333 cxbxaxcxbxax is :

(A) cba (B) )()()( accbba (C) 0 (D) none of these

Class-IX & X


Consider a number N = 2 1 P 5 3 Q 4 (Qu. no. 34 to 35)

34. Number of ordered pairs (P, Q) so that the number ‘N' is divisible by 9, is(A) 11 (B) 12 (C) 10 (D) 8

35. Number of values of Q so that the number ‘N’ is divisible by 8, is(A) 4 (B) 3 (C) 2 (D) 6

36. If a two-digit number is divided by the number having same digits written in reverse order, we get 4 as quotientand 3 as remainder and if the number is divided by the sum of the digits then 8 as a quotient and 7 as aremainder is obtained. Find the number.(A) 71 (B) 51 (C) 17 (D) none

37. If (a2 + b2)3 = (a3 + b3)2 and ab 0 then the numerical value of ab

ba

is equal to ______

(A) 2/3 (B) 1/3 (C) 4/3 (D) 5/3

38. In the following figure, the diameter ofthe circle is 3 cm. AB and MN are twodiameters such that MN isperpendicular to AB. In addition, CG isperpendicular to AB such that AE:EB= 1:2, and DF is perpendicular to MNsuch that NL:LM = 1:2. The length ofDH in cm is

(A) 122 (B) 2/))122((

(C) 2/))123(( (D) 3/))122((

39. If ba is 6 more than dc and ba is 3 less than dc , then ca is :(A) 0.5 (B) 1 (C) 1.5 (D) None of these

40. Find the number of values of N (N is Integer) such that NN 2 becomes a prime number..(A) 2 (B) 3 (C) 4 (D) 1

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Ramanujan Olympiad - Aatmdeep