IOQM 2021 previous year paper · official paper and PYQ-pattern practice

IOQM 2021 PYQ-pattern questions

Showing 7 of the 7 PYQ-pattern questions in this set — freshly worded in that year's pattern, not the original questions. Answers and solutions open under each question.

1. Q1 · Number Theory

   What is the remainder when 7¹⁰⁰ is divided by 48?

   • A. 1
   • B. 7
   • C. 25
   • D. 49
   Answer and solution

   Answer: A

   7² = 49 ≡ 1 (mod 48). Thus 7¹⁰⁰ = (7²)⁵⁰ ≡ 1⁵⁰ = 1 (mod 48), giving remainder 1.

2. Q2 · Combinatorics

   In how many ways can 5 identical red balls and 3 identical blue balls be arranged in a row?

   • A. 28
   • B. 35
   • C. 56
   • D. 70
   Answer and solution

   Answer: C

   The number of arrangements = 8!/(5!×3!) = (8×7×6)/(3×2×1) = 56.

3. Q3 · Trigonometry

   If sin θ + cos θ = √2, find the value of sin θ cos θ.

   • A. 1/4
   • B. 1/3
   • C. 1/2
   • D. 2/3
   Answer and solution

   Answer: C

   Squaring both sides: (sin θ + cos θ)² = 2, which gives sin² θ + 2 sin θ cos θ + cos² θ = 2. Since sin² θ + cos² θ = 1, we get 1 + 2 sin θ cos θ = 2, therefore sin θ cos θ = 1/2.

4. Q4 · Combinatorics

   In how many ways can the letters of the word OLYMPIAD be arranged such that the vowels O, I, A occupy even positions?

   • A. 576
   • B. 720
   • C. 1440
   • D. 2880
   Answer and solution

   Answer: D

   8 distinct letters, 4 even positions. Place 3 vowels in even positions: P(4,3)=24 ways. Remaining 5 letters fill remaining 5 positions: 5!=120. Total = 24×120 = 2880.

5. Q5 · Trigonometry

   If tan A = 3/4 where A is an acute angle, find the value of 100 × sin A.

   • A. 60
   • B. 75
   • C. 80
   • D. 48
   Answer and solution

   Answer: A

   If tan A = 3/4, we can form a right triangle with opposite = 3, adjacent = 4, hypotenuse = 5. Thus sin A = 3/5, so 100 × sin A = 100 × 3/5 = 60.

6. Q6 · Geometry

   In triangle ABC, the angle bisector from A meets BC at D. If AB = 8, AC = 12, and BC = 15, what is the length of BD?

   • A. 5
   • B. 6
   • C. 7
   • D. 8
   Answer and solution

   Answer: B

   By the angle bisector theorem, BD/DC = AB/AC = 8/12 = 2/3. Since BD + DC = 15, we have BD = (2/5) × 15 = 6.

7. Q7 · Algebra

   Find the number of integer solutions (x, y) to the equation x² - 4y² = 17, where both x and y are integers.

   • A. 0
   • B. 4
   • C. 8
   • D. 12
   Answer and solution

   Answer: B

   Factor as (x-2y)(x+2y)=17, prime. The four factorizations 1·17, 17·1, (-1)(-17), (-17)(-1) each give integer x,y (checking sum/difference parity works since 17 is odd). This yields (9,±4) and (-9,±4), totaling 4 integer solutions.