CDS 1

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 will satisfy this?

Use the code given below to choose the answer:

Assume \( N \) is a 5-digit number. When \( N \) is divided by \( 6, 12, 15, 24 \), the respective remainders are \( 2, 8, 11, 20 \). 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 \(4321012345 \times 98766789\)?

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

Suppose \(N, 11\) is the least positive common multiple which, when divided by \(6, 12, 15, 18\), leaves a remainder of 5. Which 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 \( X \) 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 \( Y \)?

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?

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 the value of \( \tan \alpha \cdot \cot 2 \beta \)?

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

If \(64^{\sin^{2}\theta} + 64^{\cos^{2}\theta} = 16\) where \(0 \leq \theta \leq \frac{\pi}{2}\), what is the value of \(\tan\theta + \cot\theta\)?

From a point \(X\) on a bridge over a river, the angles of depression to two points \(P\) and \(Q\) on the opposite bank 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 triangle \(ABC\), \(\angle ABC = 60^{\circ}\) and \(AD\) is the altitude. If \(AB = 6\ \mathrm{cm}\) and \(BC = 8\ \mathrm{cm}\), what is the area of the triangle?

If \( p \) and \( q \) are the roots of the equation \( x^{2} - \sin^{2} \theta\, x - \cos^{2} \theta = 0 \), what is the minimum value of \( p^{2} + q^{2} \)?

The arithmetic mean of \( n \) numbers is \( M \). If the sum of the first \( (n-1) \) terms is \( k \), what is the \( n \)th number?

What is the geometric mean of \(3,9,27,81,243,729,2187\)?

A person buys tea powder from four places \(A, B, C, D\) at the rates of \(1\,\mathrm{kg}\), \(2\,\mathrm{kg}\), \(4\,\mathrm{kg}\), and \(5\,\mathrm{kg}\) respectively for 1000 rupees each. If he buys on average \(x\,\mathrm{kg}\) of tea powder for 1000 rupees, what is the approximate value of \(x\)?

What is the sum of the largest and smallest 4-digit numbers formed using single-digit prime numbers (without repetition)?

What will be the remainder when \(3^{255}\) is divided by 28?

What is the value of \( x \) (where \( 0 \le x \le 8 \)) if the remainder is 0 when \((100^{97} + 100^{54} + x + 1)\) is divided by 9?