Let x, y, z be variables such that (x + y + z) = k, where k is a constant. If \( (x + z - y) \times (x - z + y) \) is proportional to yz, then (y + z - x) is proportional to:
N \( = 2^{n} + 1 \) is the smallest 5-digit number which when divided by 2, \( 2^{2} \), \( 2^{3} \), \( 2^{4} \), ..., \( 2^{n} \) leaves a remainder 1. What is the value of n?
If \( \alpha \) and \( \beta \) are the roots of the equation \( x + a + b = \frac{abx}{ab + ax + bx} \), then what is \( (\alpha \beta + \alpha + \beta) \) equal to?
Consider a 2-digit number N. Let P be the product of the digits of the number. If P is added to square of the digit in the tens place of N, we get 84. If P is added to the square of the digit in the unit place of N, we get 60. What is the value of P + N?
A mixture of 100 L contains kerosene and turpentine oil in the ratio 3 : 2. What is the minimum quantity of kerosene in litres (whole number) that should be mixed in the mixture so that the resulting mixture has 20% of kerosene?
