Untitled

For which value of x (0 ≤ x ≤ 8) will the remainder be 0 when \(100^{97} + 100^{54} + x + 1\) is divided by 9?

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?

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

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

p is proportional to the follower of \(x^2 + y^2 + z^2\). When \(x=1, y=2, z=3\), \(p=70\). What is the value of \(p\) when \(x=-1, y=1, z=5\)?

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

Assume \( p, q \) are the roots of the equation \( x^{2} + m x - n = 0 \) and \( m, n \) are the roots of the equation \( x^{2} + p x - q = 0 \), where \( (m, n, p, q) \) are non-zero numbers. Which of the following statement(s) is/are correct?

``\[
\begin{array}{l}
\frac{1}{\sqrt{10}+\sqrt{9}}+\frac{1}{\sqrt{11}+\sqrt{10}}+\\
\frac{1}{\sqrt{12}+\sqrt{11}}+\ldots+\frac{1}{\sqrt{196}+\sqrt{195}}
\end{array}
\]
What is it equal to?``

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

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 \( Y \)?

What is \((1+\tan\alpha\tan\beta)^{2}+(\tan\alpha-\tan\beta)^{2}\) equal to?

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 true? I. \( m(m+n) = -1 \) II. \( p + q = 1 \) Use the code given below to choose the answer:

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?

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

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

What is the value of \( \sin^{2}\theta\,\cos^{2}\theta\left(\sec^{2}\theta+\operatorname{cosec}^{2}\theta\right) \) 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 \( 64^{\sin^{2} \theta} + 64^{\cos^{2} \theta} = 16 \) where \( 0 \leqslant \theta \leqslant \frac{\pi}{2} \), then what is the value of \( \tan \theta + \cot \theta \)

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

If \(\cosec \theta - \cot \theta = m\) and \(\sec \theta - \tan \theta = n\), then what is \(\cosec \theta + \sec \theta\) equal to?

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 over a river, the angles of depression to two points P and Q on the opposite bank of the river are α and β respectively. If the point X is at a height h above the river surface, what is the width of the river if α and β are complementary?

In triangle \( A B C \), \( \angle A B C=60^{\circ} \) and \( A D \) is the altitude. If \( A B=6\mathrm{~cm} \) and \( B C=8\mathrm{~cm} \), 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 nth number?

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

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

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

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 \( \tan\alpha \cdot \cot2\beta \)?

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

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

Use the code given below to choose the answer: