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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 satisfy this?

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

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

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 \( N \) ?

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 \(432101 2345 \times 98766789\ )?

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

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

\(p\) 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 \(x=-1, y=1, z=5\)?

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 ?

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?

What is the value of \( \displaystyle \frac{1}{ \sqrt{10}+\ sqrt{9}}+\ frac{1}{\sqrt {11}+\sqrt {10}}+\ldots +\frac{1}{ \sqrt{196} +\sqrt{195 }} \)?

A train \( X \) takes 24 seconds to pass a person standing on the platform , and a 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 ?