C D S Test

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 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?