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A real number \( x \) is such that the sum of the number and four times its square is minimum. What is this number?

The difference of the squares of two positive integers \( m \) and \( n (m > n) \) is 72. How many pairs of positive integers will satisfy this?

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Assume \( N \) is a 5-digit number. When \( N \) is divided by \( 6, 12, 15, 24 \), the remainders are respectively \( 2, 8, 11, 20 \). What is the greatest value of \( N \)?

What is the remainder when \(111^{222} + 222^{333} + 333^{444}\) is divided by 5?

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

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

Assume \( N, 11 \) is the least positive multiple which, when divided by \( 6, 12, 15, 18 \), leaves a remainder of 5. Which of the following is correct?

\(\frac{1}{\sqrt{10}+\sqrt{9}} + \frac{1}{\sqrt{11}+\sqrt{10}} + \frac{1}{\sqrt{12}+\sqrt{11}} + \ldots + \frac{1}{\sqrt{196}+\sqrt{195}}\) is equal to?

A train \( X \) passes a person standing on platform in 24 seconds and train \( Y \) passes a person standing on platform in 18 seconds. They pass each other in 20 seconds while moving in opposite directions. What is the ratio of the speed of \( X \) to the speed of \( Y \)?

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

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

What is \( (1+\tan \alpha \tan \beta)^{2}+(\tan \alpha-\tan \beta)^{2} \) 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 these statements is/are correct?

If \(0 \leqslant (\alpha-\beta) \leqslant (\alpha+\beta) \leqslant \frac{\pi}{2}\), \(\tan(\alpha+\beta)=\sqrt{3}\) and \(\tan(\alpha-\beta)=\frac{1}{\sqrt{3}}\), then what is the value of \(\tan\alpha \cdot \cot 2\beta\)?

What is the value of \( \sin^{2} \theta \cos^{2} \theta(\sec^{2} \theta+\operatorname{cosec}^{2} \theta) \)?

If \(64^{\sin^{2}\theta}+64^{\cos^{2}\theta}=16\) where \(0\leqslant\theta\leqslant\frac{\pi}{2}\), what is the value of \(\tan\theta+\cot\theta\)?

If \( \operatorname{cosec} \theta - \cot \theta = m \) and \( \sec \theta - \tan \theta = n \), then what is \( \operatorname{cosec} \theta + \sec \theta \) equal to?

From a point X on a bridge over a river, the angles of depression to two points P and Q on the opposite bank of the river are α and β respectively. If the point X is at a height h above the surface of the river, what is the width of the river if α and β are complementary?

In triangle \( A B C \), \( \angle A B C=60^{\circ} \) and \( A D \) is the altitude. If \( A B=6 \mathrm{~cm} \) and \( B C=8 \mathrm{~cm} \), what is the area of the triangle?

If \( p \) and \( q \) are the roots of the equation \( x^{2} - \sin^{2}\theta\,x - \cos^{2}\theta = 0 \), what is the minimum value of \( p^{2} + q^{2} \)?

The arithmetic mean of n numbers is M. If the sum of the first (n-1) terms is k, what is the nth number?

What is the geometric mean of \(3,9,27,81,243,729,2187\)?

A person buys 1 kg of tea powder from each of four places \(A, B, C, D\) at the rates of 1000 rupees per \(1 \mathrm{~kg}, 2 \mathrm{~kg}, 4 \mathrm{~kg}, 5 \mathrm{~kg}\) respectively. If he buys \(x \mathrm{~kg}\) of tea powder on average for 1000 rupees, what is the approximate value of \(x\)?

What is the sum of the largest and smallest 4-digit numbers formed using single-digit prime numbers (without repetition)?

What is the remainder when \( 3^{255} \) is divided by 28?

What is the value of \( x(0 \leqslant x \leqslant 8) \) if the remainder is 0 when \(100^{97} + 100^{54} + x + 1\) is divided by 9?