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    Pre-RMO Test Series - Test 1: Pre-Regional Mathematical Olympiad

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    Ayush Kumar
    1. The document is a test paper for a pre-regional mathematical olympiad covering topics in mathematics like sets, number theory, and polynomials. 2. It contains 30 multiple choice questions with varying marks ranging from 2 to 5 marks per question. 3. All questions are compulsory and must be answered with integers from 00 to 99. There are no negative marks.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    Pre-RMO Test Series - Test 1: Pre-Regional Mathematical Olympiad

    Uploaded by

    Ayush Kumar
    616 views3 pages
    1. The document is a test paper for a pre-regional mathematical olympiad covering topics in mathematics like sets, number theory, and polynomials. 2. It contains 30 multiple choice questions with varying marks ranging from 2 to 5 marks per question. 3. All questions are compulsory and must be answered with integers from 00 to 99. There are no negative marks.

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    Original Title

    Aakash PRMO Test_1

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    1. The document is a test paper for a pre-regional mathematical olympiad covering topics in mathematics like sets, number theory, and polynomials. 2. It contains 30 multiple choice questions with varying marks ranging from 2 to 5 marks per question. 3. All questions are compulsory and must be answered with integers from 00 to 99. There are no negative marks.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    616 views3 pages

    Pre-RMO Test Series - Test 1: Pre-Regional Mathematical Olympiad

    Uploaded by

    Ayush Kumar
    1. The document is a test paper for a pre-regional mathematical olympiad covering topics in mathematics like sets, number theory, and polynomials. 2. It contains 30 multiple choice questions with varying marks ranging from 2 to 5 marks per question. 3. All questions are compulsory and must be answered with integers from 00 to 99. There are no negative marks.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 3

    Date : 24/06/2020 CODE - A

    Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005


    Ph.: 011-47623456

    MM : 102 Pre-RMO Test Series - Test 1 Time : 3 Hrs.

    Pre-Regional Mathematical Olympiad


    (For Class VIII to X Studying Students)

    Topics Covered :
    Mathematics : Sets, Number Theory, Polynomials

    INSTRUCTIONS :

    1. Questions 1 to 6 carry 2 marks each.

    2. Questions 7 to 21 carry 3 marks each.

    3. Questions 22 to 30 carry 5 marks each.

    4. All questions are compulsory.

    5. There are no negative marks.

    (ANSWER TO ALL QUESTIONS IN INTEGERS FROM 00 TO 99)

    1. The lengths of the sides of a right triangle are the integers a, b and c, and these integers have no common
    a+b+c
    factor. If a < b < c and (c – a) : b = 5 : 7, then find the value of  .
     3 

    3 x 3 − 2 x 2 + x + 1 3 x 3 − 2 x 2 + 5 x − 13
    2. If the distinct real number a, b, c satisfy the equation = , then find the
    3 x 3 − 2 x 2 − x − 1 3 x 3 − 2 x 2 − 5 x + 13
    value of 18(a + b + c).
    2
    3. The number of ordered pairs of real numbers (a, b) for which 2 x + 3 y + 5 and x = ay + b.
    =
    x + 3y
    4. Find the number of ordered pairs of integers (x, y) such that (7x + 2y) (4x + y) = 11.
     x   −12.5 
    5. f (x) 
    If [x] denotes the greatest integer not greater than x and= ×  , 0 < x < 90. If the range of
    12.5   x 
    f(x) consists of k elements then find the value of k.
    6. What is the largest integer that is a divisor of (p + 1) (p + 3) (p + 5) (p + 7) (p + 9) for all positive integers p?
    Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph.011-47623456
    (1)
    Class VIII to X Pre-RMO Test Series : Test 1A

    Let f ( x )=
    2
    7. x + 2 x − 1 − x − 2 x − 1, x > 1. Find the value of (f(9) + f(11) + f(15)) .

    8. Find the number of ordered pairs of positive integers (x, y) that satisfy x2 + y2 – xy = 37.
    9. Integers p, q, r satisfy p + q – r = 1 and p2 + q2 – r2 = –1. What is the sum of all possible values of p + q + r?

    10. If x = 1 1
    4
    (5 + 1)(5 + 1) (5 + 1)(5 1 1
    )
    , then find the value of ( x − 1− 5 ) .
    1 6
    16
    2 4 8 16
    +1
    11. N is the smallest natural number such that in decimal representation it ends with 6 and if we move last digit to
    the front of the number, we get a number 4 times larger than the given number. Find the sum of digits of N.
    12. If the ordered triplets of real numbers (x, y, z) satisfy x − y + z= x − y + z , x + y + z = 8 and x – y + z
    = 4, then find the value of xyz.
    13. Let a, b are integers and a + b is a root of x2 + ax + b = 0. What is the maximum possible value of a2?
    1
    14. In base x system, and 0.17 are numerals for the same number. Find x.
    5
    15. Let N = 6 + 66 + 666 + …+ 666…66, where there are hundred 6’s in the last term in the sum. How many times
    does the digit 4 occur in the number N?
    16. Find the number of ordered triples of real numbers (x, y, z) that satisfy x + yz = 6, y + xz = 6 and z + xy = 6.
     1  1
    17. Let f(x) = x2 + ax + b, where a, b > 0. If for all non-zero real x, f  x +  = f ( x ) + f   and the roots of f(x) = 0
     x x
    are integers, then find the value of 3
    ba .
    18. Find the integer x for which x5 = 164916224.
    19. b b 91 and a b +=
    Suppose a, b are positive real numbers such that a a += b a 84 . Find a + b.

    20. In base 8, the four-digit numeral bbcc is the square of the two-digits numeral aa . Find the value of
    (a + b + c).
    8
    1 p
    21. Let the sum ∑
    n =1 n(n + 1)(n + 2)
    written in its lowest form be . Find the value of p + q.
    q
    22. A pencil costs `11 and a sharpener cost `13. Find the number of ways in which a person can spend exactly
    `1000 to buy pencil and sharpeners.
    23. Let a, b and c be positive integers such that ‘a’ is the cube of an integer, c = b + 1, and a2 + b2 = c2. Find the
    sum of digits of least possible value of c.
    24. Five distinct two-digit numbers are in a geometric progression. Find the sum of second and fourth digit.
    25. abc and cba are, respectively, the base nine and base seven numerals for the same positive integer. Find
    the sum of digits of this integer when expressed in base ten.
    26. Suppose 2, 1, 4 are the roots of the equation x4 + ax2 + bx – c = 0. Find the value of c.
    27. The polynomials x3 + ax2 + 9x + 6 and x3 + bx2 + 6x + 3 have a common quadratic factor over the set of
    polynomials with integral coefficients. Find the value of (a + b).
    28. If the numbers of subsets of {1, 2, 3, 4, 5, 6, 7, 8, 9} that are subsets of neither {1, 2, 3, 4, 5} nor {4, 5, 6, 7, 8,
    9} is K, then find the sum of digits of K..
    29. Find the number of triplets of prime number (p, q, r) satisfying 3p4 – 5q4 – 4r2 = 26.
    30. The number of ordered pairs (x, y) satisfying 3x + 2y = 27 is a, then find the value of a, if x and y are non-
    negative integers.

      

    Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph.011-47623456
    (2)
    Edition: 2020-21

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