Mathematiq'19 03 Qualifying Round

Explanation
The expression can be simplified as follows:
1- [1- {1- (1-1-1)}]
1- [1- {1- (0)}]
1- [1- {1}]
1- [1- 1]
1- 0
The final result is 0. Therefore, the answer should be 0.

Explanation
Ten years ago, the age ratio of A to B was 1:2. Let's assume their ages ten years ago were x and 2x respectively. Now, their present ages can be represented as 3x and 4x. The total of their present ages will be 3x + 4x = 7x. Since the question does not provide any specific values for x, we cannot determine the exact sum of their present ages. Therefore, the answer cannot be determined based on the given information.

Explanation
The quadratic equation x^2 + 5x + 7 = 0 has 0 real roots because the discriminant (b^2 - 4ac) is negative. In this case, the discriminant is (5^2 - 4(1)(7)) = 25 - 28 = -3, which is less than 0. Therefore, the equation does not intersect the x-axis and has no real solutions.

Explanation
Let's assume the smaller integer is x. The next consecutive integer would be x+1. According to the given information, the product of these two integers is 12 more than 12 times their sum. This can be written as x(x+1) = 12 + 12(x + (x+1)). Simplifying this equation, we get x^2 + x = 12 + 24x + 12. Rearranging the terms, we have x^2 - 23x - 24 = 0. Factoring this quadratic equation, we find (x-24)(x+1) = 0. Therefore, the smaller integer is 24.

Explanation
If x = 3 is the solution of the given equation, then we can substitute x = 3 into the equation and solve for k. Plugging in x = 3, we get 3(3)^2 + (k-1)(3) + 9 = 0. Simplifying this equation, we have 27 + 3k - 3 + 9 = 0. Combining like terms, we get 3k + 33 = 0. Subtracting 33 from both sides, we have 3k = -33. Dividing both sides by 3, we find that k = -11. Therefore, the value of k is -11.

Explanation
If the average of 7 consecutive numbers is 33, then the sum of these numbers is 33 multiplied by 7, which equals 231. Since the numbers are consecutive, the largest number will be the last number in the sequence. Therefore, the largest number is 36.

Explanation
The given quadratic function y = (x - 5)2 - 3 is in the form of y = a(x - h)2 + k, where (h, k) represents the coordinates of the vertex. In this case, the vertex is (5, -3). The equation of the line of symmetry is given by x = h, where h is the x-coordinate of the vertex. Therefore, the equation of the line of symmetry is x = 5.

Explanation
The average of a set of numbers is found by adding all the numbers together and then dividing the sum by the total number of values. In this case, the first five prime numbers are 2, 3, 5, 7, and 11. Adding these numbers together gives a sum of 28. Dividing 28 by 5 (the total number of values) gives an average of 5.6.

Explanation
The question states that 1/4 of a number subtracted from 1/3 of the number gives 12. To solve this, we can set up an equation: (1/3)x - (1/4)x = 12. To simplify, we can find a common denominator of 12, which gives us (4/12)x - (3/12)x = 12. Combining like terms, we get (1/12)x = 12. To solve for x, we multiply both sides of the equation by 12, giving us x = 12 * 12 = 144. Therefore, the number is 144.

Explanation
To find 20% of 30% of 800kg, we first calculate 30% of 800kg. This is done by multiplying 800 by 0.30, which equals 240kg. Then, we find 20% of 240kg by multiplying 240 by 0.20, which equals 48kg. Therefore, the correct answer is 48kg.

Explanation
If 15% of the cost of the holiday was spent on airfares, and the airfares cost £630, we can set up the equation: 15% of x = £630. To solve for x, we can divide both sides of the equation by 0.15 (or multiply by 100/15). This gives us x = £630 / 0.15 = £4200. Therefore, the total cost of the holiday was £4200.

Explanation
After 3 people take their glasses off, the number of people wearing glasses decreases by 3. Since the total number of people in the room is 60, 30% of them were originally wearing glasses, which is 18 people. After 3 people take their glasses off, the number of people wearing glasses becomes 18 - 3 = 15. To find the percentage of people wearing glasses now, we divide the number of people wearing glasses (15) by the total number of people in the room (60) and multiply by 100. This gives us (15/60) * 100 = 25%. Therefore, the correct answer is 25%.

Explanation
The given statement, 2x2 + 4r – 1 ≥ 5, is a quadratic inequality because it involves a quadratic term (2x2) and a linear term (4r – 1) with a relational operator (≥). Quadratic inequalities are mathematical statements that involve a quadratic expression and an inequality symbol, such as ≥, ≤, >, or

Explanation
The graph in the question shows a downward-opening parabola with its vertex at (-2, 3). The equation that represents this graph is y = -(x + 2) (x - 3).