IAS Prelims test Math for IAS Prelims Math Challenge

Solution

Correct: B

Let the two numbers be x and y. We are given that their sum is 50, so x + y = 50. We are also given that their product is 500, so xy = 500. We can solve these equations simultaneously to find the values of x and y. Rearranging the first equation gives y = 50 - x. Substituting this into the second equation gives x(50 - x) = 500. Expanding this equation gives 50x - x^2 = 500. Rearranging this equation gives x^2 - 50x + 500 = 0. Factoring this equation gives (x - 25)(x - 25) = 0, so x = 25. Therefore, y = 50 - x = 50 - 25 = 25. Hence, the two numbers are 25 and 25.

Solution

Correct: B

To solve for x, we need to isolate x on one side of the equation. Subtracting 5 from both sides gives 2x = 11 - 5, which simplifies to 2x = 6. Dividing both sides by 2 gives x = 6/2, which simplifies to x = 3.

Solution

Correct: A

To find the equation of the line, we need to find the slope and the y-intercept. The slope is given by (y2 - y1)/(x2 - x1) = (5 - 3)/(4 - 2) = 2/2 = 1. The equation of the line can be written in the form y = mx + b, where m is the slope and b is the y-intercept. We can substitute one of the points into this equation to solve for b. Using the point (2, 3), we get 3 = 1(2) + b, which simplifies to 3 = 2 + b. Subtracting 2 from both sides gives b = 1. Therefore, the equation of the line is y = x + 1.

Solution

Correct: A

To evaluate this expression, we need to follow the order of operations (PEMDAS). First, we evaluate the expressions inside the parentheses: 3 + 2 = 5 and 4 - 2 = 2. Then, we divide 5 by 2, which gives 2.5.

Solution

Correct: A

The general equation of a circle with center (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2. In this case, the center is (0, 0) and the radius is 4. Substituting these values into the equation gives (x - 0)^2 + (y - 0)^2 = 4^2, which simplifies to x^2 + y^2 = 16.

Solution

Correct: A

To evaluate this expression, we need to follow the order of operations (PEMDAS). First, we evaluate the exponents: 3^2 = 9 and 2^2 = 4. Then, we multiply 9 and 4, which gives 36.

Solution

Correct: A

To solve for x, we can multiply both sides of the equation by 4, which gives x = 9 * 4 = 36.

Solution

Correct: A

The equation of the line can be written in the form y = mx + b, where m is the slope and b is the y-intercept. We are given that the slope is 3, so m = 3. We are also given a point on the line, (1, 2), which we can substitute into the equation to solve for b. This gives 2 = 3(1) + b, which simplifies to 2 = 3 + b. Subtracting 3 from both sides gives b = -1. Therefore, the equation of the line is y = 3x - 1.

Solution

Correct: A

To evaluate this expression, we need to follow the order of operations (PEMDAS). First, we evaluate the expression inside the parentheses: 3 + 4 = 7. Then, we multiply 2 by 7, which gives 14.

Solution

Correct: A

The average speed is given by the total distance traveled divided by the total time. In this case, the total distance is 250 miles and the total time is 5 hours. Therefore, the average speed is 250/5 = 50 mph.

Solution

Correct: A

To evaluate this expression, we need to follow the order of operations (PEMDAS). First, we evaluate the expression inside the parentheses: 2 + 3 = 5. Then, we square 5, which gives 25.

Solution

Correct: B

To solve for x, we need to isolate x on one side of the equation. Subtracting 5 from both sides gives 2x = 11 - 5, which simplifies to 2x = 6. Dividing both sides by 2 gives x = 6/2, which simplifies to x = 3.

Solution

Correct: A

Since the pedal is pushing the crank at an angle of 30 degrees, we can find the distance traveled by the pedal using the formula for the arc length of a circle: s = r * theta, where s is the arc length, r is the radius of the circle, and theta is the angle in radians. In this case, r = 0.2 meters and theta = 30 degrees = pi/6 radians. Therefore, s = 0.2 * pi/6 = 0.1 * pi meters.

Solution

Correct: A

The average speed for the entire trip is given by the total distance divided by the total time. Let the distance from city A to city B be x. Then, the total distance is 2x. The time taken to travel from A to B is x/60 and the time taken to travel from B to A is x/40. Therefore, the total time is x/60 + x/40. The average speed is then given by 2x / (x/60 + x/40) = 2 / (1/60 + 1/40) = 2 / (4/240 + 6/240) = 2 / (10/240) = 2 * 240 / 10 = 48 km/h.

Solution

Correct: A

The general equation of a circle with center (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2. In this case, the center is (3, 4) and the radius is 5. Substituting these values into the equation gives (x - 3)^2 + (y - 4)^2 = 5^2, which simplifies to (x - 3)^2 + (y - 4)^2 = 25.

Solution

Correct: A

The volume of the water in the tank is given by the formula for the volume of a cylinder: V = pi * r^2 * h, where V is the volume, r is the radius, and h is the height of the water. In this case, r = 2 meters and h = 3 meters. Therefore, V = pi * 2^2 * 3 = 12 pi cubic meters.

Solution

Correct: B

The volume of a cube is given by the formula V = s^3, where V is the volume and s is the length of an edge. In this case, V = 64 cubic meters. Therefore, s^3 = 64. Taking the cube root of both sides gives s = 4 meters.

Solution

Correct: C

To solve for x, we can take the logarithm base 2 of both sides: x = log2(16) = log2(2^4) = 4.

Solution

Correct: B

The average speed is given by the total distance divided by the total time. In this case, the total distance is 180 miles and the total time is 3 hours. Therefore, the average speed is 180/3 = 60 mph.

Solution

Correct: A

To find the equation of the line, we need to find the slope and the y-intercept. The slope is given by (y2 - y1)/(x2 - x1) = (4 - 2)/(3 - 1) = 2/2 = 1. The equation of the line can be written in the form y = mx + b, where m is the slope and b is the y-intercept. We can substitute one of the points into this equation to solve for b. Using the point (1, 2), we get 2 = 1(1) + b, which simplifies to 2 = 1 + b. Subtracting 1 from both sides gives b = 1. Therefore, the equation of the line is y = x + 1.

Solution

Correct: B

To evaluate this expression, we need to follow the order of operations (PEMDAS). First, we evaluate the expressions inside the parentheses: 2 * 3 = 6 and 4 * 5 = 20. Then, we add 6 and 20, which gives 26.