If \( \lambda \) is an integer and \( \alpha \), \( \beta \) are the roots of \( 4x^{2} - 16x + \lambda = 0 \) such that \( 1 < \alpha < 2 \) and \( 2 < \beta < 3 \), then how many values can \( \lambda \) take?
If \( x^{2} = y + z \), \( y^{2} = z + x \) and \( z^{2} = x + y \), then what is the value of \( \frac{1}{x + 1} + \frac{1}{y + 1} + \frac{1}{z + 1} \)?
A is a set of positive integers such that when divided by 2, 3, 4, 5 and 6 leaves the remainder 1, 2, 3, 4 and 5 respectively. How many integers between 0 and 100 belong to the set A?
In an examination, a student was asked to divide a certain number by 8. By mistake he multiplied it by 8 and got the answer 2016 more than the correct answer. What was the number?
In the quadratic equation \( x^{2} + ax + b = 0 \), \( a \) and \( b \) can take any value from the set {1, 2, 3, 4}. How many pairs of values of \( a \) and \( b \) are possible in order that the quadratic equation has real roots?
The sum of the squares of two positive integers is 208. If the square of the larger number is 18 times the smaller number, then what is the difference of the larger and smaller numbers?
