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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?

None

Use the code given below to choose the answer:

None

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 will be 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 leaves a remainder of 5 when divided by \(6,12,15,18\). Which of the following is correct?

\[\frac{1}{\sqrt{10}+\sqrt{9}}+\frac{1}{\sqrt{11}+\sqrt{10}}+\frac{1}{\sqrt{12}+\sqrt{11}}+\ldots+\frac{1}{\sqrt{196}+\sqrt{195}}\]
What is it equal to?

A train \( X \) takes 24 seconds to pass a person standing on the platform, and train \( Y \) takes 18 seconds to pass a person standing on the platform. 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 \(0 \leq \theta \leq \frac{\pi}{2}\), 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 \( \alpha \) and \( \beta \) respectively. If the point \( X \) is at a height \( h \) above the surface of the river, what is the width of the river if \( \alpha \) and \( \beta \) 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 \( n \)th 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 will be the remainder when \(3^{255}\) is divided by 28?

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