Untitled

If \( (a^{3}+b^{3}) \) is proportional to \( (a^{2}-b^{2}) \), then \( (a^{2}-a b+b^{2}) \) is proportional to

\( x^{2}-20=\sqrt{20+\sqrt{20+\sqrt{20+\sqrt{20+\ldots \text{ infinite terms }}}}} \) then what is \( x \) equal to?

If (\( x+k \)) is the HCF of \( x^{2}+p x+q \) and \( x^{2}+q x+p \), where \( p \neq q \), then what is the value of \( k \)?

If \( \frac{a+b}{b+c} = \frac{c+d}{d+a} \) where \( a \neq c \), then which one of the following is correct?

If the sum of the squares of the roots of the equation \( \mathrm{x}^{2}-14 \mathrm{x}+\mathrm{k}=0 \) is 100, then what is the value of \( k \)?

If \( \frac{\sqrt{x+20}+\sqrt{x-1}}{\sqrt{x+20}-\sqrt{x-1}}=\frac{7}{3} \), then what is the value of \( \sqrt{(x+20)(x-1)} \)?

What is the HCF of \(x^{8}-y^{8}\) and \(x^{7}-y^{7}+x^{5}y^{2}-x^{2}y^{5}\)?

Consider the following statements:
1. \( x^{4}(y-z)+y^{4}(z-x)+z^{4}(x-y) \) is positive if \( x>y>z \).
2. \( x^{4}(y-z)+y^{4}(z-x)+z^{4}(x-y) \) is negative if \( x
Which of the above statements is/are correct?

What is \( \frac{x^{6} \cdot-24 x^{4}+144 x^{2}}{\left(x^{2}+4 \sqrt{3} x+12\right)(x-2 \sqrt{3})^{2}} \) equal to?

Consider the following statements :
1. \((ab+bc+ca)\) is a factor of
\[a^{2}(b-c)^{3}+b^{2}(c-a)^{3}+c^{2}(a-b)^{3}\]
2. \((a+b+c)\) is a factor of
\[a^{2}(b-c)^{3}+b^{2}(c-a)^{3}+c^{2}(a-b)^{3}\]
Which of the statements given above is/are correct?

What is \(\frac{1}{x(x-y)(x-z)} + \frac{1}{y(y-z)(y-x)} + \frac{1}{z(z-x)(z-y)}\) equal to?

If three times the greater of two numbers is divided by the smaller number, the quotient will be 6 and the remainder will be 6. If five times the smaller number is divided by the greater number, the quotient will be 2 and the remainder will be 3. What is the difference between the numbers?

X and Y can do a piece of work in 45 days and 40 days respectively. They begin to work together, but X leaves after n days and then Y completes the remaining work in 23 days. What is n equal to?

There are two stations X and Y, 1320 km apart. A train starts from station X at 6 a.m. and moves at an average speed of \( 60 \mathrm{~km} / \mathrm{hr} \). At 2 p.m. another train starts from Y towards X and moves at an average speed of \( 80 \mathrm{~km} / \mathrm{hr} \). When do they meet?

Two glasses of equal volume are filled with a mixture of alcohol and water in the ratio \(3: 2\) and \(4: 1\), respectively. These glasses are emptied into a third glass. What is the ratio of alcohol and water in the third glass?

480 persons working 10 hours per day complete one-fourth of a work in 10 days. How many additional persons are to be employed in order to complete the remaining work in 20 days, working 8 hours per day?

12 women and 16 men can do a piece of work in 5 days. 13 women and 24 men can do it in 4 days. How long will 25 women and 50 men take to do it?

If \( x \) is the HCF and \( y \) is the LCM of \( \frac{3}{5}, \frac{6}{25}, \frac{9}{20}, \frac{27}{50} \), then which one of the following is correct?

There are two natural numbers \( x \) and \( y \), where \( x > y \). When \( x \) is divided by 6, it leaves the remainder 2; and when \( y \) is divided by 6, it leaves the remainder 3. What is the remainder when \( (x - y) \) is divided by 6?

Consider the number \( \mathrm{N}=12^{6} \times 3^{8} \times 5^{3} \). Which of the following statements is/are correct? 1. The number of odd factors of N is 60. 2. The number of even factors of N is 720. Select the correct answer using the code given below:

If \( \frac{\log_{10} a}{b-c}=\frac{\log_{10} b}{c-a}=\frac{\log_{10} c}{a-b},\,(a \neq b \neq c) \), then what is the value of abc?

The value of a 2-digit number is 5 times the sum of the digits. What is the product of the digits?

What is the angle between the hour hand and the minute hand of a clock when the clock shows 4 hours 40 minutes?

At what time between 3:00 and 4:00 will the hands of a clock be at right angles?

If the angle between the hour and minute hand of a clock is 150 degrees, what time could it be?

A clock shows 6:20. What is the angle between the hour and minute hands?