IOQM MOCK TEST - 1

Indian Olympiad Qualifier in Mathematics (IOQM) – 2020-21

Time: 3 hours Maximum Marks: 30

INSTRUCTIONS

Please read the instructions carefully

A: General:
1. Immediately fill in the particulars on this page of the Test Booklet with Blue/Black Ball point
pen.
2. Use Black Ball Point Pen only for writing particulars on Answer Sheet. Use of pencil is
strictly prohibited.
3. Blank papers, clipboards, log tables, slide rules, calculators, cellular phones, pagers and
electronic gadgets in any form are not allowed.
4. The answer sheet, a machine-gradable Objective Response Sheet (ORS) is provided separately.
5. Do not Tamper/mutilate the ORS or this booklet.
6. No additional sheets will be provided for rough work.
7. On completion of this test, the candidate must hand over the Answer Sheet to the Invigilator
on duty in the Room/Hall. However, the candidates are allowed to take away this Test
Booklet with them.

B: Questions paper format & Marking Schema:

1. The question paper consists of 30 questions of Mathematics.
2. Answers will be whole numbers from 0 to 99, where students must fill OMR corresponding to
correct answer.
3. If your answer is 9 then you will have to fill in as 09.
4. For each question you will be awarded 1 marks if you darken the bubble corresponding to the
correct answer ONLY and zero (0) marks if no bubbles are darkened. There is no negative
Marking.

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Mathematics

1. K is the number of positive integers less than 20002001 and not containing digits other than 0 and 2 . Find sum
of digits of K.

2. K is the largest real number having the following the following property: for any positive real numbers a, b, c,
a  b  c  K and abc  K . Find K  . (  x  is the greatest integer less than or equal to x)

3. Triangle ABC is isosceles with AC  BC and C  1200 . Points D and E are chosen on segment AB so that
AD  DE  EB . Find CDE (in degress)

1
4. The geometric progression an  has first term a1  1536 and common ratio  . What is the value of n for which
2
the product of the first n terms is maximum?

Let an be the coefficient of x in expansion of  3  x  . Find

n 32 33 34
5.    ......
a2 a3 a4

6. Each digit of a four digit number is one of 1 , 2 , 3 , 4 . Every two adjacent digits are different. The first and the
last digits are also different. Moreover, the first digit is not greater than any other digit. How many such four digit
numbers are there?

9 cot C
7. In triangle ABC, if 9BC2  9CA2  19AB2  0 , what is the value of ?
cot A  cot C

8. If all positive integers relatively prime to 105 (including 1 ) are arranged in ascending order, then in this
arrangement the kth term is 178 . Find k.

What is the number of real roots of the equation, log x   log x   2  0 ;   denotes floor function?
2
9.

10. Two non-intersecting circles, not lying inside each other, are drawn in a plane. Two lines pass through a point P
which lies outside each circle. The first line intersects the first circle at A and A I and the second circle at B and
BI ; here A and B are closer to P than A I and BI and P lies on segment AB. Analogously, the second line
intersects the first circle at C and CI and the second circle at D and DI . If A, B, C, D are concyclic then find
PA . PB . PAI . PBI
PC . PD . PCI . PDI

11. How many subsets a, b, c of 3,  2,  1, 0, 1, 2, 3 are there such that ax  by  c  0 makes an acute angle
with positive x axis.

12. 2a is the number of pairs of positive integers x, y with x  y such that gcd  x, y   5! and cm  x, y   50! Find a.

13. A natural number N has exactly 12 divisors (including 1 and N), which are numbered in increasing order
d1  d2  d3  ...  d12 . It is known that the divisor dd4 1 is equal to the product  d1  d2  d4  d8 . Find d4 .

14. In an acute angled triangle ABC, point H is the intersection point of altitude CE to AB and altitude BD to AC. A
circle with DE as its diameter intersects AB and AC at points F and G. FG and AH intersect at point K. If BC  25 ,
BD  20 , BE  7 and AK  p , find 10p .

15. A new sequence is obtained from the sequences of positive integers 1 , 2 , 3 , 4 , 5 , 6 .... , be deleting all
perfect squares. Find the sum of the digits of 2018th term of this new sequences.

16. Find the least value of the expression  x  y  y  z  , given that x, y, z are positive numbers satisfying the
equation xyz  x  y  z   676 .

17. Find the value of the smallest positive integer m such that the equation x2  2 m  5 x  100m  9  0 has only
integer solutions.

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BD 2 AE 3
18. In ABC , D, E are on BC, AC such that  ,  . AD and BE intersect at F. Fin the value of 12 .
DC 3 EC 4
AF BF
. .
FD FE

19. Find number of all 5 digit natural numbers such that in each of them, the sum of digits is 10 , the digits 1 , 2 , 3
occurs atleast once, and 3 or more of the digits are identical.

n  n n  n 
20.            divides 2
2018
; n  3 and n  N . Find sum of all possible values of n.
 0  1   2   3 

21. Find p such that the three roots of 5x3  5 p  1 x2   71p  1 x   66p  1  0 are positive integers.

 1093 
22. What is the remainder when  31  is divided by 100 ;   is floor function.
10  3 

23. Let ABCD be a quadrilateral with AB parallel to DC. Let E, F be points on AB and CD. The segments AF and DE
Area  ABCD 
intersect at G, while the segments BF and CE intersects at H. If  k , then find the least value of k.
Area EGFH

3 5
24. Let a, b, c be positive integers and loga b  , logc d  . if a  c  9 , then find b  d .
2 4

25. In triangle ABC , C  900 , D is a point on BC such that AD bisects BAC internally, CD  1.5 cm ,
BD  2.5 cm . Find AC (in cm).

x  y 13 4  y
26. Real numbers x and y satisfy the equation   . Find xy .
6 xy x

27. N is 50 digit number (in the decimal scale). All digit except the 26th digit (from the left) are 1 . If N is divisible by
13 , find the 26th digit.

28. Suppose the real numbers  and  satisfy the equations 3  32  5  1 and 3  32  5  5 . Find    .

x  x   x   x 
29. x is an integer satisfying         ....    1001 . Find the largest prime divisor of x (   is floor
1!   2!   3!  10! 
function).

30. For real numbers x, y, u, v satisfying 2x2  3y2  10 , 3u2  8v 2  6 , 4xv  3yu  2 15 , the maximum value of
x  y  z is k, then find 3k 2  4 15 .
________________________________________________________________________________________________
*** Best of Luck ***

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FIITJEE – The Fundamental Approach………. IOQM Mock Test-1 [5]

IOQM MOCK TEST - 1

Indian Olympiad Qualifier in Mathematics (IOQM) – 2020-21

ANSWERS KEY
Q.No. ANS
1. 10
2. 05
3. 60
4. 12
5. 18
6. 28
7. 05
8. 82
9. 03
10. 01
11. 43
12. 14
13. 13
14. 86
15. 11
16. 52
17. 90
18. 35
19. 20
20. 30
21. 76
22. 08
23. 04
24. 93
25. 03
26. 06
27. 03
28. 02
29. 73
30. 31

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