Senior Maths Challenge 2024 Answers

Senior Maths Challenge 2024: Answers and Detailed Explanations

The Senior Mathematical Challenge (SMC) is a prestigious competition designed to stimulate mathematical interest and problem-solving skills in senior secondary school students. This article provides detailed solutions and explanations for the 2024 Senior Maths Challenge questions. We'll delve into each problem, breaking down the reasoning and showcasing different approaches where applicable. Remember, understanding the process is just as important as getting the right answer. This resource aims to help you not just understand the solutions, but to improve your overall mathematical problem-solving abilities. Please note: Since the actual 2024 Senior Maths Challenge questions are not yet available at the time of writing, this article will provide example problems and solutions mirroring the style and difficulty of the actual challenge. These examples will cover a range of topics commonly tested.

Example Problems and Solutions:

Problem 1: Number Theory

Question: Find the smallest positive integer n such that n² + 1 is divisible by 10.

Solution: A number is divisible by 10 if its last digit is 0. Therefore, we need n² + 1 to end in 0, which means n² must end in 9. Let's consider the last digits of perfect squares:

• 1² = 1
• 2² = 4
• 3² = 9
• 4² = 16
• 5² = 25
• 6² = 36
• 7² = 49
• 8² = 64
• 9² = 81
• 10² = 100

The last digits repeat in a cycle of 20 (0,1,4,9,6,5,6,9,4,1,0...). We see that the last digit is 9 when n ends in 3 or 7. The smallest positive integer n satisfying this condition is 3. Let's check: 3² + 1 = 10, which is divisible by 10.

Therefore, the smallest positive integer n is 3.

Problem 2: Algebra

Question: If 3x + 2y = 7 and x - y = 1, find the value of x + y.

Solution: We have a system of two linear equations. We can solve for x and y using either substitution or elimination. Let's use elimination:

Multiply the second equation by 2: 2x - 2y = 2

Now add this equation to the first equation:

(3x + 2y) + (2x - 2y) = 7 + 2 5x = 9 x = 9/5

Substitute x = 9/5 into the second equation:

(9/5) - y = 1 y = 9/5 - 1 y = 4/5

Therefore, x + y = (9/5) + (4/5) = 13/5

Therefore, x + y = 13/5

Problem 3: Geometry

Question: A right-angled triangle has hypotenuse of length 10 and one leg of length 6. What is the area of the triangle?

Solution: Let the legs of the right-angled triangle be a and b, and the hypotenuse be c. We are given that c = 10 and a = 6. By the Pythagorean theorem, a² + b² = c².

6² + b² = 10² 36 + b² = 100 b² = 64 b = 8 (since b must be positive)

The area of a triangle is given by (1/2) * base * height. In a right-angled triangle, the legs are the base and height.

Area = (1/2) * 6 * 8 = 24

Therefore, the area of the triangle is 24 square units.

Problem 4: Probability

Question: A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn at random without replacement. What is the probability that both marbles are red?

Solution: The probability of drawing a red marble on the first draw is 5/8 (5 red marbles out of 8 total marbles). After drawing one red marble, there are 4 red marbles left and 7 total marbles. The probability of drawing a second red marble is 4/7.

The probability of both events happening is the product of their individual probabilities:

P(both red) = (5/8) * (4/7) = 20/56 = 5/14

Therefore, the probability that both marbles are red is 5/14.

Problem 5: Sequences and Series

Question: Find the sum of the arithmetic series: 2 + 5 + 8 + ... + 29.

Solution: This is an arithmetic series with first term a = 2 and common difference d = 3. To find the number of terms, we use the formula for the nth term: a_n = a + (n-1)*d.

29 = 2 + (n-1)3 27 = (n-1)3 9 = n-1 n = 10

There are 10 terms in the series. The sum of an arithmetic series is given by:

S_n = (n/2)(a + a_n)

S₁₀ = (10/2)(2 + 29) = 5 * 31 = 155

Therefore, the sum of the arithmetic series is 155.

Further Challenge Problems and Advanced Concepts

The examples above represent a range of difficulty levels typically found in the SMC. To further enhance your preparation, consider practicing problems involving:

• Trigonometry: Problems involving trigonometric identities, solving triangles, and applications to geometry.
• Calculus: Basic differentiation and integration problems, particularly those related to rates of change and areas.
• Coordinate Geometry: Finding equations of lines and circles, calculating distances and areas using coordinates.
• Number Theory: Problems involving prime numbers, divisibility rules, and modular arithmetic.
• Combinatorics: Counting problems involving permutations and combinations.

Remember, consistent practice is key. Work through past papers, focus on understanding the underlying concepts, and don't be afraid to explore different approaches to problem-solving. The more you practice, the more confident and proficient you will become in tackling challenging mathematical problems.

Frequently Asked Questions (FAQ)

• Q: What resources are available to help me prepare for the SMC?

  • A: Past papers are invaluable resources. Many websites and textbooks offer practice problems mirroring the SMC's style and difficulty.

• Q: What topics should I focus on when preparing?

  • A: Focus on algebra, geometry, number theory, and basic calculus. A strong foundation in these areas will equip you to handle most SMC problems.

• Q: Is there a time limit for the SMC?

  • A: Yes, the SMC has a strict time limit. Practice working under time constraints to improve your speed and efficiency.

• Q: What should I do if I get stuck on a problem?

  • A: Don't spend too much time on a single problem. Move on to other questions and return to the difficult ones later if time permits. Consider different approaches or try simplifying the problem.

Conclusion

The Senior Maths Challenge is a valuable opportunity to test and improve your mathematical skills. While achieving a high score requires significant effort and dedication, this article aims to equip you with the necessary tools and insights to succeed. Remember to focus not just on memorizing formulas but on truly understanding the underlying concepts and applying them creatively to solve a diverse range of problems. Through persistent practice and a methodical approach, you can confidently tackle the challenges presented and enhance your mathematical abilities. Good luck!