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The difference of the square of two natural numbers m and n (m > n) is 72. How many pairs of natural numbers will satisfy?

Let N be a 5-digit number. When N is divided by 6, 12, 15, 24 it leaves respectively 2, 8, 11, 20 as remainders. What is the greatest value of N?

p varies directly as \(x^{2}+y^{2}+z^{2}\). When \(x=1, y=2, z=3\), then \(p=70\). What is the value of \(p\) when \(x=-1, y=1, z=5\)?

Let N be the least positive multiple of 11 that leaves a remainder of 5 when divided by 6, 12, 15, 18. Which one of the following is correct?

A real number x is such that the sum of the number and four times its square is the least. What is that number?

What are the last three digits in the multiplication of 4321012345 \times 98766789?

Train X crosses a man standing on the platform in 24 seconds and train Y crosses a man standing on the platform in 18 seconds. They cross each other while running in opposite directions in 20 seconds. What is the ratio of speed of X to speed of Y?

What is \(\frac{1}{\sqrt{10}+\sqrt{9}}+\frac{1}{\sqrt{11}+\sqrt{10}}+\ldots+\frac{1}{\sqrt{196}+\sqrt{195}}\) equal to?

Consider the following statements :
I. \( \tan 50^{\circ}-\cot 50^{\circ} \) is positive
II. \( \cot 25^{\circ}-\tan 25^{\circ} \) is negative

Which of the statements is/are correct?

Let \(p, q\) be the roots of the equation \(x^{2}+m x - n = 0\) and \(m, n\) be the roots of the equation \(x^{2}+p x - q = 0\) (\(m, n, p, q\) are non-zero numbers). Which of the following statements is/are correct?

What is \( (1+\tan \alpha \tan \beta)^{2}+(\tan \alpha-\tan \beta)^{2} \) equal to?

What is the maximum value of \( 8 \sin \theta - 4 \sin^{2} \theta \)?