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Basics of Mathematics — JEE Main & Advanced IOQM PYQs

88 previous year questions from Basics of Mathematics with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

88 questionsIOQMSolutions on every page
1

Find the sum of all positive integers n for which |2^n+5^n-65 | is a perfect square.

Olympiad workbook Solution View → 2

A total fixed amount of N thousand rupees is given to three persons A, B, C , every year, each being given an amount proportional to her age. In the first year, A got half the tota

Olympiad workbook Solution View → 3

A five digit number n= a b c d e is such that when divided respectively by 2,3,4,5,6 the remainders are a, b, c, d, e . What is the remainder when n is divided by 100 ?

Olympiad workbook Solution View → 4

Find the largest 2-digit number N which is divisible by 4 , such that all integral powers of N end with N .

Olympiad workbook Solution View → 5

For any real number t , let [ t ] denote the largest integer t . Suppose that N is the greatest integer such that [ [ [ N ] ] =4 . Find the sum of digit of N .

Olympiad workbook Solution View → 6

Find the number of integer solutions to ||x|-2020| < 5 .

Olympiad workbook Solution View → 7

For a natural number n , let n^ denote the number obtained by deleting zero digits, if any. (For example, if n=260, n^ =26 ; if n=2020 , n^ =22 .) Find the number of 3 -digit numbe

Olympiad workbook Solution View → 8

Find the number of pairs (a, b) of natural numbers such that b is a 3 -digit number, a+1 divides b-1 and b divides a^2+a+2 .

Olympiad workbook Solution View → 9

Positive integers a, b, c satisfy a b a-b =c . What is the largest possible value of a+b+c not exceeding 99?

Olympiad workbook Solution View → 10

Let X be the set of all even positive integers n such that the measure of the angle of some regular polygon is n degrees. Find the number of elements in X .

Olympiad workbook Solution View → 11

Let x, y be positive integers such that x^4=(x-1) (y^3-23 )-1 . Find the maximum possible value of x+y .

Olympiad workbook Solution View → 12

Find the number of triples of real numbers (a, b, c) such that a²⁰+b²⁰+c²⁰=a²⁴+b²⁴+c²⁴=1 .

Olympiad workbook Solution View → 13

Determine the number of positive integral value of p for which there exists a triangle with side a, b and c which satisfy a^2+ (p^2+9 ) b^2+9 c^2-6 a b-6 p b c=0

Olympiad workbook Solution View → 14

Find the number of elements in the set (a, b) N: 2 a, b 2023, _a(b)+6 _b(a)=5 .

Olympiad workbook Solution View → 15

Find the number of positive integers n such that n + n+1 < 11 .

Olympiad workbook Solution View → 16

What is the least positive integer by which 2^5 3^6 4^3 5^3 6^7 should be multiplied so that, the product is a perfect square?

Olympiad workbook Solution View → 17

The prime numbers a, b and c are such that a+b^2 =4 c^2 . Determine the sum of all possible values of a+b+c .

Olympiad workbook Solution View → 18

Find the number of ordered pairs (a, b) such that a, b 10,11, , 29,30 and GCD (a, b)+ LCM (a, b)=a+b .

Olympiad workbook Solution View → 19

If x and y are positive integers such that (x-4) (x-10)=2^y , find the maximum possible value of x+y .

Olympiad workbook Solution View → 20

For a positive integer n , let n denote the perfect square integer closest to n . For example, 74 =81 , 18 =16 . If N is the smallest positive integer such that 91 120 143 180 N =9

Olympiad workbook Solution View → 21

The positive real numbers a, b, c satisfy: aligned & a 2 b+1 + 2 b 3 c+1 + 3 c a+1 =1 & 1 a+1 + 1 2 b+1 + 1 3 c+1 =2 aligned What is the value of 1 a + 1 b + 1 c ?

Olympiad workbook Solution View → 22

A contractor has two teams of workers : team A and team B. Team A can complete a job in 12 days and team B can do the same job in 36 days. Team A starts working on the job and team

Olympiad workbook Solution View → 23

A pen costs ₹ 11 and a notebook costs ₹ 13 . Find the number of ways in which a person can spend exactly ₹ 1000 to buy pens and notebooks.

Olympiad workbook Solution View → 24

Let m, n be natural numbers such that m+3 n-5=5 LCM (m, n)-11 GCD (m, n) Find the maximum possible value of m+n .

Olympiad workbook Solution View → 25

In a class, the total numbers of boys and girls are in the ratio 4: 3 . On one day it was found that 8 boys and 14 girls were absent from the class and that the number of boys was

Olympiad workbook Solution View → 26

How many positive integers less than 1000 have the property that the sum of the digits of each such number is divisible by 7 and the number itself is divisible by 3 ?

Olympiad workbook Solution View → 27

Find the smallest positive integer n 10 such that n+6 is a prime and 9 n+7 is a perfect square.

Olympiad workbook Solution View → 28

An integer n is such that [ n 9 ] is a three digit number with equal digits, and [ n-172 4 ] is a 4 digit number with the digits 2,0,2,4 in some order. What is the remainder when n

Olympiad workbook Solution View → 29

For each positive integer n , consider the highest common factor h _ n of the two numbers n !+1 and (n+1)! . For n < 100 , find the largest value of h_n .

Olympiad workbook Solution View → 30

Let p, q be two-digit numbers neither of which are divisible by 10 . Let r be the four-digit number by putting the digits of p followed by the digits of q (in order). As p, q vary,

Olympiad workbook Solution View → 31

Let and be positive integers such that 16 37 < < 7 16 . Find the smallest possible value of .

Olympiad workbook Solution View → 32

Let X be the set consisting of twenty positive integers n, n+2, , n+38 . The smallest value of n for which any three numbers a, b, c X , not necessarily distinct, form the sides of

Olympiad workbook Solution View → 33

Let the rational number p / q be closest to but not equal to 22 / 7 among all rational numbers with denominator < 100 . What is the value of p-3 q ?

Olympiad workbook Solution View → 34

Find the number of positive integers less than 101 that can not be written as the difference of two squares of integers.

Olympiad workbook Solution View → 35

Let T be the smallest positive integers which, when divided by 11,13,15 leaves remainders in the sets 7,8,9 , 1,2,3 , 4,5,6 respectively. What is the sum of the square of the digit

Olympiad workbook Solution View → 36

Let a and b natural numbers such that 2 a-b, a-2 b and a+b are all distinct squares. What is the smallest possible value of b ?

Olympiad workbook Solution View → 37

Let N denote the number of all natural numbers n such that n is divisible by a prime p> n and p < 20 . What is the value of N ?

Olympiad workbook Solution View → 38

Let E denote the set of all natural numbers n such that 3 < n < 100 and the set 1,2,3, , n can be partitioned in to 3 subsets with equal sums. Find the number of elements of E .

Olympiad workbook Solution View → 39

Let N be the smallest positive integer such that N +2 ~N +3 ~N + +9 ~N is a number all whose digits are equal. What is the sum of the digits of N ?

Olympiad workbook Solution View → 40

A pen costs Rs. 13 and a note book costs Rs. 17. A school spends exactly Rs. 10000 in the year 2017-18 to buy x pens and y note books such that x and y are as close as possible (i.

Olympiad workbook Solution View → 41

Let a b c be a three digit number with nonzero digits such that a ^2+ b ^2= c ^2 . What is the largest possible prime factor of a b c ?

Olympiad workbook Solution View → 42

Consider all 6-digit numbers of the form abccba where b is odd. Determine the number of all such 6-digit numbers that are divisible by 7 .

Olympiad workbook Solution View → 43

A book is published in three volumes, the pages being numbered from 1 onwards. The page numbers are continued from the first volume to the second volume to the third. The number of

Olympiad workbook Solution View → 44

If a, b, c 4 are integers, not all equal, and 4 a b c=(a+3)(b+3)(c+3) , then what is the value of a+b+c ?

Olympiad workbook Solution View → 45

Find the number of integer solutions to ||x|-2020| < 5 .

Olympiad workbook Solution View → 46

Find the largest 2-digit number N which is divisible by 4 , such that all integral powers of N end with N .

Olympiad workbook Solution View → 47

The prime numbers a, b and c are such that a+b^2 =4 c^2 . Determine the sum of all possible values of a+b+c .

Olympiad workbook Solution View → 48

If x and y are positive integers such that (x-4) (x-10)=2^y , find the maximum possible value of x+y .

Olympiad workbook Solution View → 49

A five digit number n= a b c d e is such that when divided respectively by 2,3,4,5,6 the remainders are a, b, c, d, e . What is the remainder when n is divided by 100 ?

Olympiad workbook Solution View → 50

For a natural number n , let n^ denote the number obtained by deleting zero digits, if any. (For example, if n=260, n^ =26 ; if n=2020 , n^ =22 .) Find the number of 3 -digit numbe

Olympiad workbook Solution View → 51

What is the least positive integer by which 2^5 3^6 4^3 5^3 6^7 should be multiplied so that, the product is a perfect square?

Olympiad workbook Solution View → 52

Find the sum of all positive integers n for which |2^n+5^n-65 | is a perfect square.

Olympiad workbook Solution View → 53

A total fixed amount of N thousand rupees is given to three persons A, B, C , every year, each being given an amount proportional to her age. In the first year, A got half the tota

Olympiad workbook Solution View → 54

For any real number t , let [ t ] denote the largest integer t . Suppose that N is the greatest integer such that [ [ [ N ] ] =4 . Find the sum of digit of N .

Olympiad workbook Solution View → 55

Find the number of ordered pairs (a, b) such that a, b 10,11, , 29,30 and GCD (a, b)+ LCM (a, b)=a+b .

Olympiad workbook Solution View → 56

For a positive integer n , let n denote the perfect square integer closest to n . For example, 74 =81 , 18 =16 . If N is the smallest positive integer such that 91 120 143 180 N =9

Olympiad workbook Solution View → 57

Find the number of pairs (a, b) of natural numbers such that b is a 3 -digit number, a+1 divides b-1 and b divides a^2+a+2 .

Olympiad workbook Solution View → 58

Positive integers a, b, c satisfy a b a-b =c . What is the largest possible value of a+b+c not exceeding 99?

Olympiad workbook Solution View → 59

Let m, n be natural numbers such that m+3 n-5=5 LCM (m, n)-11 GCD (m, n) Find the maximum possible value of m+n .

Olympiad workbook Solution View → 60

Let X be the set of all even positive integers n such that the measure of the angle of some regular polygon is n degrees. Find the number of elements in X .

Olympiad workbook Solution View → 61

Let x, y be positive integers such that x^4=(x-1) (y^3-23 )-1 . Find the maximum possible value of x+y .

Olympiad workbook Solution View → 62

The positive real numbers a, b, c satisfy: aligned & a 2 b+1 + 2 b 3 c+1 + 3 c a+1 =1 & 1 a+1 + 1 2 b+1 + 1 3 c+1 =2 aligned What is the value of 1 a + 1 b + 1 c ?

Olympiad workbook Solution View → 63

Determine the number of positive integral value of p for which there exists a triangle with side a, b and c which satisfy a^2+ (p^2+9 ) b^2+9 c^2-6 a b-6 p b c=0

Olympiad workbook Solution View → 64

Find the number of triples of real numbers (a, b, c) such that a²⁰+b²⁰+c²⁰=a²⁴+b²⁴+c²⁴=1 .

Olympiad workbook Solution View → 65

Find the number of elements in the set (a, b) N: 2 a, b 2023, _a(b)+6 _b(a)=5 .

Olympiad workbook Solution View → 66

Let and be positive integers such that 16 37 < < 7 16 . Find the smallest possible value of .

Olympiad workbook Solution View → 67

A contractor has two teams of workers : team A and team B. Team A can complete a job in 12 days and team B can do the same job in 36 days. Team A starts working on the job and team

Olympiad workbook Solution View → 68

Find the number of positive integers n such that n + n+1 < 11 .

Olympiad workbook Solution View → 69

A pen costs ₹ 11 and a notebook costs ₹ 13 . Find the number of ways in which a person can spend exactly ₹ 1000 to buy pens and notebooks.

Olympiad workbook Solution View → 70

In a class, the total numbers of boys and girls are in the ratio 4: 3 . On one day it was found that 8 boys and 14 girls were absent from the class and that the number of boys was

Olympiad workbook Solution View → 71

How many positive integers less than 1000 have the property that the sum of the digits of each such number is divisible by 7 and the number itself is divisible by 3 ?

Olympiad workbook Solution View → 72

An integer n is such that [ n 9 ] is a three digit number with equal digits, and [ n-172 4 ] is a 4 digit number with the digits 2,0,2,4 in some order. What is the remainder when n

Olympiad workbook Solution View → 73

Let X be the set consisting of twenty positive integers n, n+2, , n+38 . The smallest value of n for which any three numbers a, b, c X , not necessarily distinct, form the sides of

Olympiad workbook Solution View → 74

Let p, q be two-digit numbers neither of which are divisible by 10 . Let r be the four-digit number by putting the digits of p followed by the digits of q (in order). As p, q vary,

Olympiad workbook Solution View → 75

For each positive integer n , consider the highest common factor h _ n of the two numbers n !+1 and (n+1)! . For n < 100 , find the largest value of h_n .

Olympiad workbook Solution View → 76

Find the smallest positive integer n 10 such that n+6 is a prime and 9 n+7 is a perfect square.

Olympiad workbook Solution View → 77

Let E denote the set of all natural numbers n such that 3 < n < 100 and the set 1,2,3, , n can be partitioned in to 3 subsets with equal sums. Find the number of elements of E .

Olympiad workbook Solution View → 78

Let a b c be a three digit number with nonzero digits such that a ^2+ b ^2= c ^2 . What is the largest possible prime factor of a b c ?

Olympiad workbook Solution View → 79

A pen costs Rs. 13 and a note book costs Rs. 17. A school spends exactly Rs. 10000 in the year 2017-18 to buy x pens and y note books such that x and y are as close as possible (i.

Olympiad workbook Solution View → 80

Let the rational number p / q be closest to but not equal to 22 / 7 among all rational numbers with denominator < 100 . What is the value of p-3 q ?

Olympiad workbook Solution View → 81

Consider all 6-digit numbers of the form abccba where b is odd. Determine the number of all such 6-digit numbers that are divisible by 7 .

Olympiad workbook Solution View → 82

Let N be the smallest positive integer such that N +2 ~N +3 ~N + +9 ~N is a number all whose digits are equal. What is the sum of the digits of N ?

Olympiad workbook Solution View → 83

A book is published in three volumes, the pages being numbered from 1 onwards. The page numbers are continued from the first volume to the second volume to the third. The number of

Olympiad workbook Solution View → 84

Find the number of positive integers less than 101 that can not be written as the difference of two squares of integers.

Olympiad workbook Solution View → 85

Let N denote the number of all natural numbers n such that n is divisible by a prime p> n and p < 20 . What is the value of N ?

Olympiad workbook Solution View → 86

Let a and b natural numbers such that 2 a-b, a-2 b and a+b are all distinct squares. What is the smallest possible value of b ?

Olympiad workbook Solution View → 87

Let T be the smallest positive integers which, when divided by 11,13,15 leaves remainders in the sets 7,8,9 , 1,2,3 , 4,5,6 respectively. What is the sum of the square of the digit

Olympiad workbook Solution View → 88

If a, b, c 4 are integers, not all equal, and 4 a b c=(a+3)(b+3)(c+3) , then what is the value of a+b+c ?

Olympiad workbook Solution View →
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