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?

Select the answer using the code given below:

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 \(4321012345 \times 98766789\)?

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

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

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 \(Y\)?

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 \( (1+\tan \alpha \tan \beta)^{2}+(\tan \alpha-\tan \beta)^{2} \) 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 the 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 \( \tan \alpha \cdot \cot 2 \beta \) equal to?

What is the value of \( \sin^{2}\theta\,\cos^{2}\theta\bigl(\sec^{2}\theta+\cosec^{2}\theta\bigr) \) equal to?

If \(64^{\sin^{2}\theta}+64^{\cos^{2}\theta}=16\) where \(0\leqslant\theta\leqslant\frac{\pi}{2}\), then 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 across a river, the angles of depression of two points \( P \) and \( Q \) on the banks on opposite side 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 a triangle \(A B C, \angle A B C=60^{\circ}\) and \(A D\) is the altitude. If \(A B=6\ \mathrm{cm}\) and \(B C=8\ \mathrm{cm}\), then what is the area of the triangle?