Untitled

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

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

Use the code given below to choose the answer :

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

p varies directly as \(\left(x^{2}+y^{2}+z^{2}\right)\). 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?

What is the value of \ (\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 }}\)?

A train \( X \) passes a person standing on a platform in 24 seconds , and train \( Y \) passes a person standing on a 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 will be the maximum value for the expression \(8 \sin \ theta - 4 \ sin^{2} \theta \)?

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 ?