Test

Assume \( N \) 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 \( N \)?

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?

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?

What is the remainder when \( 111^{222}+222^{333}+333^{444} \) is divided by 5?

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

What are the last three digits in the multiplication of \( 4321012345 \times 98766789 \)?

A train \( X \) passes a person standing on a platform in 24 seconds, and train \( Y \) passes a person standing on a platform in 18 seconds. They pass each other in opposite directions in 20 seconds. What is the ratio of the speed of \( X \) to the speed of \( Y \)?

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

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

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?

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. \(p + q = 1\)

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

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

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

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

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

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

A person purchases one kg of tea powder from each of the four places \( A, B, C, D \) at the rate of \( ₹ 1000 \) per \( 1 \mathrm{~kg}, 2 \mathrm{~kg}, 4 \mathrm{~kg}, 5 \mathrm{~kg} \). If on an average he purchased \( x \mathrm{~kg} \) of tea powder per \( ₹ 1000 \), then what is the approximate value of \( x \)?

Which of the following sequences has a geometric mean of 16?

If the geometric mean of two numbers is 10 and one of the numbers is 5, what is the other number?

Consider the sequence: 2, 4, 8, 16, 32, 64. What is the geometric mean of this sequence?