HAT Quantitative Reasoning MCQs

Let tens digit be a and units digit b; then a + b = 11 and 9(b - a) = 27.
So b - a = 3, giving a = 4 and b = 7.

Set x - 3 equal to 5 and -5.
The resulting solutions are x = 8 and x = -2.

Square the equation: x² + 2 + 1/x² = 25.
Subtracting 2 gives x² + 1/x² = 23.

Use a² + b² = (a + b)² - 2ab.
Thus, 144 - 70 = 74.

Add the geometric terms directly.
Their total is 31.

The first term is 8 and common ratio is 1/2.
The sum is a/(1-r) = 8/(1/2) = 16.

For ax² + bx + c = 0, the sum of roots is -b/a.
Here, -(-9)/1 = 9.

For ax² + bx + c = 0, the product of roots is c/a.
Here, 20/1 = 20.

The digit sum of 568 is 19, which leaves remainder 1 when divided by 9.
Adding 8 makes the digit sum 27, divisible by 9.

The largest two-digit multiple of 7 is 7 × 14.
This equals 98.

The LCM uses prime powers 2³, 3, and 5.
Their product is 8 × 3 × 5 = 120.

The second number is product divided by the first number.
Thus, 216 ÷ 18 = 12.