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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?

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

Using the code given below , choose the answer:

Select the answer using the code given below :

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 × 987 66789?

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

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 \( N \) is the least positive multiple of 11 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?

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

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

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?

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?

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

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?

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 ?

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?