If \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( x^{2} + \alpha x + \beta = 0 \), where \( \beta \neq 0 \), then what is the value of \( \alpha - \beta \)?
A shopkeeper marks the price of an article at ₹200. After allowing a discount of 10%, he still gains 20% on the cost price. What is the cost price of the article?
A person borrowed ₹9,000 at 7%, ₹12,000 at 8% and ₹15,000 at 9% simple interest per annum. He had to pay ₹50,700 at the end of n years. What is the value of n?
Consider the following statements:
1. The sum of the cubes of three consecutive natural numbers is divisible by 9.
2. Every even power of every odd number \( (> 1) \) when divided by 8 gives 1 as remainder.
Which of the above statements is/are correct?
A piece of cloth costs ₹10,000. If a 2 m longer piece of the same cloth is purchased for the same amount, it would cost ₹250 less per metre. What is the original length of the piece of cloth?
Two sides of a triangle forming a right angle are \( 6x^{2} \) and \( (2x^{2} - 1) \). If the area of the triangle is 84 square units, then what is the perimeter of the triangle?
A train X takes 2 hours less than a train Y to cover a distance of 192 km between two cities. Their average speeds differ by 16 km/hr. How long does the faster train take to cover the journey?
