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If \( a^2 - bc = \alpha, b^2 - ac = \beta, c^2 - ab = \gamma \), then what is the value of \( \frac{a \alpha + b \beta + c \gamma}{(a+b+c)(\alpha+\beta+\gamma)} \)?

If A and B can finish a work in 10 days, B and C can finish the same work in 12 days, and C and A can finish the same work in 15 days; then in how many days can A, B, and C together finish half of this work?

If \( \mathrm{x}^{4}+\alpha \mathrm{x}^{3}+\beta \mathrm{x}^{2}+\gamma \mathrm{x}-1 \) has a factor \( (\mathrm{x}-1)^{3} \), what will be its other factor?

A two-digit number is such that the sum of this number and the number formed by reversing its digits is 55. Additionally, the difference between this number and the number formed by reversing its digits is 45. What is the product of the digits?

For what relationship between a and b is the equation \( \sin \theta = \frac{a + b}{2 \sqrt{ab}} \) possible?

Three persons A, B, and C together can complete a work in 36 days. A and B together can do five times the work done by C alone; B and C together can do as much work as A alone. If A and C together can do n times the work done by B alone, what is the value of n?

If \( \frac{2a}{3} = \frac{4b}{5} = \frac{3c}{4} \), what is the value of \( \frac{18}{a} \sqrt{a^{2} + c^{2} - b^{2}} \)?

The sum of deviations of n numbers from 10 and 20 are a and b respectively. If \( \frac{\mathrm{b}}{\mathrm{a}} = -4 \), what is the mean of these n numbers?

If the median \(M\) of the observations \[12,1,8,54,61,28,45,35,21,17\] is given, what is the value of \(2M+5\)?

What is the number of real roots of the equation \( \sqrt{\mathrm{x}+9}=\mathrm{x}-3 \)?

If \( \mathrm{x}=97+56 \sqrt{3} \), then what is the value of \( \sqrt[4]{\mathrm{x}}+\frac{1}{\sqrt[4]{\mathrm{x}}} \)?

Suppose the LCM of two given numbers is L and the HCF is H. L and H are in the ratio 3:2. If the sum of the two numbers is 45, what is the product of these numbers?

A person walks from his house at an average speed of \( 3 \mathrm{~km} / \mathrm{hr} \) and reaches the office 40 minutes early. If he walks at an average speed of \( 2 \mathrm{~km} / \mathrm{hr} \), he will reach the office 40 minutes late. What is the distance between his house and the office?

A person uses two different speeds to cover the distance from his home to the office. If he walks at a speed of 3 km/h, he arrives 40 minutes early. If he walks at a speed of 2 km/h, he arrives 40 minutes late. What is the distance between the home and the office?

If a person reaches the office 40 minutes early by walking at a speed of 3 km/h and reaches 40 minutes late by walking at a speed of 2 km/h, what will be his average speed?

If a person walks at a speed of 3 km/h and reaches the office 40 minutes early, what distance does he cover?

Consider the following statements:
1. If \((\mathrm{a}+\mathrm{b})\) is directly proportional to \((\mathrm{a}-\mathrm{b})\), then \((\mathrm{a}^{2}+\mathrm{b}^{2})\) is directly proportional to ab.
2. If a is directly proportional to b, then \((a^{2}-b^{2})\) is directly proportional to ab.
Which of the statements given above is/are correct?

Consider the following statements:
1. If \((a+b)\) is directly proportional to \((a-b)\), then \((a^2+b^2)\) is directly proportional to ab.
2. If a is directly proportional to b, then \((a^2-b^2)\) is directly proportional to ab.
Which of the statements given above is/are correct?

Consider the following statements:
1. If \((a+b)\) is directly proportional to \((a-b)\), then \((a^2+b^2)\) is directly proportional to \(ab\).
2. If \(a\) is directly proportional to \(b\), then \((a^2-b^2)\) is directly proportional to \(ab\).
Which of the statements given above is/are correct?

Consider the following statements:
1. If \((a+b)\) is directly proportional to \((a-b)\), then \((a^2+b^2)\) is directly proportional to ab.
2. If a is directly proportional to b, then \((a^2-b^2)\) is directly proportional to ab.
Which of the statements given above is/are correct?

Consider the following statements:
1. If \((a+b)\) is directly proportional to \((a-b)\), then \((a^2+b^2)\) is directly proportional to \(ab\).
2. If \(a\) is directly proportional to \(b\), then \((a^2-b^2)\) is directly proportional to \(ab\).
Which of the statements given above is/are correct?

Consider the following statements :
1. If \( (\mathrm{a}+\mathrm{b}) \) is directly proportional to \( (\mathrm{a}-\mathrm{b}) \), then \( \left(\mathrm{a}^{2}+\mathrm{b}^{2}\right) \) is directly proportional to ab.
2. If a is directly proportional to b, then \( \left(a^{2}-b^{2}\right) \) is directly proportional to \( a b \).
Which of the statements given above is/are correct?

Consider the following statements:
1. If \( (a+b) \) is directly proportional to \( (a-b) \), then \( (a^{2}+b^{2}) \) is directly proportional to ab.
2. If a is directly proportional to b, then \( (a^{2}-b^{2}) \) is directly proportional to \( ab \).
Which of the statements given above is/are correct?

Consider the following statements:
1. If (a+b) is directly proportional to (a−b), then (a²+b²) is directly proportional to ab.
2. If a is directly proportional to b, then (a²−b²) is directly proportional to ab.
Which of the statements given above is/are correct?

Consider the following statements :
1. If \((a+b)\) is directly proportional to \((a-b)\), then \((a^2 + b^2)\) is directly proportional to ab.
2. If a is directly proportional to b, then \((a^2 - b^2)\) is directly proportional to \(ab\).
Which of the statements given above is/are correct?

Consider the following statements:
1. If \( (a+b) \) is directly proportional to \( (a-b) \), then \( (a^2+b^2) \) is directly proportional to ab.
2. If a is directly proportional to b, then \( (a^2-b^2) \) is directly proportional to \( ab \).
Which of the statements given above is/are correct?

Consider the following statements :
1. If \( (\mathrm{a}+\mathrm{b}) \) is directly proportional to \( (\mathrm{a}-\mathrm{b}) \), then \( \left(\mathrm{a}^{2}+\mathrm{b}^{2}\right) \) is directly proportional to ab.
2. If a is directly proportional to b, then \( (a^{2}-b^{2}) \) is directly proportional to \( a b \).
Which of the statements given above is/are correct?

Consider the following statements :
1. If \( (a+b) \) is directly proportional to \( (a - b) \), then \( (a^2 + b^2) \) is directly proportional to ab.
2. If a is directly proportional to b, then \( (a^2 - b^2) \) is directly proportional to ab.
Which of the statements given above is/are correct?

Consider the following statements:
1. If \((a+b)\) is directly proportional to \((a-b)\), then \((a^2+b^2)\) is directly proportional to \(ab\).
2. If \(a\) is directly proportional to \(b\), then \((a^2-b^2)\) is directly proportional to \(ab\).
Which of the statements given above is/are correct?

Consider the following statements:
1. If \((a+b)\) is directly proportional to \((a-b)\), then \((a^2+b^2)\) is directly proportional to ab.
2. If a is directly proportional to b, then \((a^2-b^2)\) is directly proportional to ab.
Which of the statements given above is/are correct?

Consider the following statements:
1. If \((\mathrm{a}+\mathrm{b})\) is directly proportional to \((\mathrm{a}-\mathrm{b})\), then \((\mathrm{a}^{2}+\mathrm{b}^{2})\) is directly proportional to \(ab\).
2. If \(a\) is directly proportional to \(b\), then \((a^{2}-b^{2})\) is directly proportional to \(ab\).
Which of the statements given above is/are correct?

Consider the following statements:
1. If \( (a+b) \) is directly proportional to \( (a-b) \), then \( (a^2+b^2) \) is directly proportional to \( ab \).
2. If \( a \) is directly proportional to \( b \), then \( (a^2-b^2) \) is directly proportional to \( ab \).
Which of the statements given above is/are correct?

The combined age of a man and his wife is 6 times the combined age of their children. Two years ago their combined age was 10 times the combined age of their children; and six years later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 years old?

The combined age of a man and his wife is 6 times the combined age of their children. Two years ago their combined age was 10 times the combined age of their children; and six years later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 years old?

The combined age of a man and his wife is 6 times the combined age of their children. Two years ago their combined age was 10 times the combined age of their children; and six years later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 years old?

The combined age of a man and his wife is 6 times the combined age of their children. Two years ago their combined age was 10 times the combined age of their children; and six years later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 years old?

The combined age of a man and his wife is 6 times the combined age of their children. Two years ago their combined age was 10 times the combined age of their children; and six years later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 years old?

The combined age of a man and his wife is 6 times the combined age of their children. Two years ago their combined age was 10 times the combined age of their children; and six years later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 years old ?

The combined age of a man and his wife is 6 times the combined age of their children. Two years ago their combined age was 10 times the combined age of their children; and six years later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 years old ?

The combined age of a man and his wife is 6 times the combined age of their children. Two years ago their combined age was 10 times the combined age of their children; and six years later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 years old?

The combined age of a man and his wife is 6 times the combined age of their children. Two years ago their combined age was 10 times the combined age of their children; and six years later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 years old ?

The combined age of a man and his wife is 6 times the combined age of their children. Two years ago their combined age was 10 times the combined age of their children; and six years later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 years old?

The combined age of a man and his wife is 6 times the combined age of their children. Two years ago their combined age was 10 times the combined age of their children; and six years later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 years old?

The combined age of a man and his wife is 6 times the combined age of their children. Two years ago their combined age was 10 times the combined age of their children; and six years later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 years old?

The combined age of a man and his wife is 6 times the combined age of their children. Two years ago their combined age was 10 times the combined age of their children; and six years later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 years old?

The combined age of a man and his wife is 6 times the combined age of their children. Two years ago their combined age was 10 times the combined age of their children; and six years later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 years old?

The combined age of a man and his wife is 6 times the combined age of their children. Two years ago their combined age was 10 times the combined age of their children; and six years later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 years old?

The combined age of a man and his wife is 6 times the combined age of their children. Two years ago their combined age was 10 times the combined age of their children; and six years later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 years old?

What is
\[
\begin{array}{r}
3(\sin x-\cos x)^{4}+6(\sin x+\cos x)^{2}+ \\
4(\sin x)^{6}+4(\cos x)^{6}
\end{array}
\]
equal to?

What is \(3(\sin x - \cos x)^{4} + 6(\sin x + \cos x)^{2} + 4(\sin x)^{6} + 4(\cos x)^{6}\) equal to?

What is \(3(\sin x-\cos x)^{4} + 6(\sin x+\cos x)^{2} + 4(\sin x)^{6} + 4(\cos x)^{6}\) equal to?

What is
\[
\begin{array}{r}
3(\sin x-\cos x)^{4}+6(\sin x+\cos x)^{2}+ \\
4(\sin x)^{6}+4(\cos x)^{6}
\end{array}
\]
equal to?

What is
\[
\begin{array}{r}
3(\sin x-\cos x)^{4}+6(\sin x+\cos x)^{2}+ \\
4(\sin x)^{6}+4(\cos x)^{6}
\end{array}
\]
equal to?

What is \(3(\sin x-\cos x)^4 + 6(\sin x+\cos x)^2 + 4(\sin x)^6 + 4(\cos x)^6\) equal to?

What is \
\[
\begin{array}{r}
3(\sin x-\cos x)^{4}+6(\sin x+\cos x)^{2}+ \\
4(\sin x)^{6}+4(\cos x)^{6}
\end{array}
\]
equal to?

What is \[3(\sin x-\cos x)^{4}+6(\sin x+\cos x)^{2}+4(\sin x)^{6}+4(\cos x)^{6}\] equal to?

What is
\[
\begin{array}{r}
3(\sin x-\cos x)^{4}+6(\sin x+\cos x)^{2}+ \\
4(\sin x)^{6}+4(\cos x)^{6}
\end{array}
\]
equal to?

What is \(3(\sin x - \cos x)^{4} + 6(\sin x + \cos x)^{2} + 4(\sin x)^{6} + 4(\cos x)^{6}\) equal to?

What is
\[
\begin{array}{r}
3(\sin x-\cos x)^{4}+6(\sin x+\cos x)^{2}+ \\
4(\sin x)^{6}+4(\cos x)^{6}
\end{array}
\]
equal to?

What is
\[
\begin{array}{r}
3(\sin x-\cos x)^{4}+6(\sin x+\cos x)^{2}+ \\
4(\sin x)^{6}+4(\cos x)^{6}
\end{array}
\]
equal to?

What is \( 3(\sin x - \cos x)^{4} + 6(\sin x + \cos x)^{2} + 4(\sin x)^{6} + 4(\cos x)^{6} \) equal to?

What is \(3(\sin x - \cos x)^{4} + 6(\sin x + \cos x)^{2} + 4(\sin x)^{6} + 4(\cos x)^{6}\) equal to?

What is
\[
\begin{array}{r}
3(\sin x-\cos x)^{4} + 6(\sin x+\cos x)^{2} + \\
4(\sin x)^{6} + 4(\cos x)^{6}
\end{array}
\]
equal to?

What is \(3(\sin x-\cos x)^{4} + 6(\sin x+\cos x)^{2} + 4(\sin x)^{6} + 4(\cos x)^{6}\) equal to?

What is the value of \( \sin \theta+\cos \theta \), if \( \theta \) satisfies the equation \( \cot^{2} \theta - (\sqrt{3}+1)\cot \theta + \sqrt{3} = 0 \); \( 0 < \theta < \frac{\pi}{4} \)?

What is the value of \( \sin \theta+\cos \theta \), if \( \theta \) satisfies the equation \( \cot^{2}\theta-(\sqrt{3}+1)\cot\theta+\sqrt{3}=0 \); \( 0<\theta<\frac{\pi}{4} \)?

What is the value of \( \sin \theta+\cos \theta \), if \( \theta \) satisfies the equation \( \cot^{2} \theta-(\sqrt{3}+1)\cot \theta+\sqrt{3}=0 \); \(0<\theta<\frac{\pi}{4}\)?

What is the value of \( \sin \theta+\cos \theta \), if \( \theta \) satisfies the equation \( \cot^{2}\theta - (\sqrt{3}+1)\cot\theta + \sqrt{3} = 0 \); \( 0 < \theta < \tfrac{\pi}{4} \)?

What is the value of \( \sin \theta+\cos \theta \), if \( \theta \) satisfies the equation \( \cot^{2} \theta-(\sqrt{3}+1) \cot \theta+\sqrt{3}=0 \); \( 0<\theta<\frac{\pi}{4} \) ?

What is the value of \( \sin \theta+\cos \theta \), if \( \theta \) satisfies the equation \( \cot^{2}\theta - (\sqrt{3}+1)\cot\theta + \sqrt{3} = 0 \); \( 0 < \theta < \tfrac{\pi}{4} \)?

What is the value of \( \sin \theta+\cos \theta \), if \( \theta \) satisfies the equation \( \cot^{2} \theta - (\sqrt{3}+1) \cot \theta + \sqrt{3} = 0 \); \( 0 < \theta < \frac{\pi}{4} \)?

What is the value of \( \sin \theta+\cos \theta \), if \( \theta \) satisfies the equation \( \cot^{2} \theta-(\sqrt{3}+1) \cot \theta+\sqrt{3}=0 \); \( 0<\theta<\frac{\pi}{4} \)?

What is the value of \( \sin \theta + \cos \theta \), if \( \theta \) satisfies the equation \( \cot^{2} \theta - (\sqrt{3} + 1) \cot \theta + \sqrt{3} = 0 \); \( 0 < \theta < \frac{\pi}{4} \)?

What is the value of \( \sin \theta + \cos \theta \), if \( \theta \) satisfies the equation \( \cot^{2} \theta - (\sqrt{3} + 1) \cot \theta + \sqrt{3} = 0 \); \( 0 < \theta < \frac{\pi}{4} \)?

What is the value of \(\sin \theta + \cos \theta\), if \(\theta\) satisfies the equation \(\cot^{2} \theta - (\sqrt{3}+1) \cot \theta + \sqrt{3} = 0\); \(0 < \theta < \frac{\pi}{4}\)?

What is the value of \( \sin \theta + \cos \theta \), if \( \theta \) satisfies the equation \( \cot^{2} \theta - (\sqrt{3}+1)\cot \theta + \sqrt{3} = 0 \); \(0 < \theta < \frac{\pi}{4}\)?

What is the value of \( \sin \theta + \cos \theta \), if \( \theta \) satisfies the equation \( \cot^{2} \theta - (\sqrt{3} + 1) \cot \theta + \sqrt{3} = 0 \); \( 0 < \theta < \frac{\pi}{4} \)?

What is the value of \( \sin \theta + \cos \theta \), if \( \theta \) satisfies the equation \( \cot^{2} \theta - (\sqrt{3} + 1) \cot \theta + \sqrt{3} = 0 \); \( 0 < \theta < \frac{\pi}{4} \)?

What is the value of \( \sin \theta+\cos \theta \), if \( \theta \) satisfies the equation \( \cot^{2} \theta-(\sqrt{3}+1)\cot \theta+\sqrt{3}=0 \); \(0<\theta<\tfrac{\pi}{4}\)?

What is the value of \( \sin \theta+\cos \theta \), if \( \theta \) satisfies the equation \( \cot^{2} \theta - (\sqrt{3}+1) \cot \theta + \sqrt{3} = 0 \); \( 0<\theta<\frac{\pi}{4} \)?

Which one of the following is a value of \( \theta \), if \( \theta \) satisfies the equation \( \tan 2 \theta \tan 4 \theta - 1 = 0 \); \( 0 < \theta < \frac{\pi}{2} \) ?

Which one of the following is a value of \( \theta \), if \( \theta \) satisfies the equation \( \tan 2 \theta \tan 4 \theta - 1 = 0 \); \( 0 < \theta < \frac{\pi}{2} \)?

Which one of the following is a value of \( \theta \), if \( \theta \) satisfies the equation \( \tan 2 \theta \tan 4 \theta - 1 = 0 \); \( 0<\theta<\frac{\pi}{2} \) ?

Which one of the following is a value of \( \theta \), if \( \theta \) satisfies the equation \( \tan 2 \theta \tan 4 \theta-1=0 \); \( 0<\theta<\frac{\pi}{2} \) ?

Which one of the following is a value of \( \theta \), if \( \theta \) satisfies the equation \( \tan 2 \theta \tan 4 \theta - 1 = 0 \); \( 0 < \theta < \frac{\pi}{2} \)?

Which one of the following is a value of \( \theta \), if \( \theta \) satisfies the equation \( \tan 2 \theta \tan 4 \theta - 1 = 0 \); \( 0 < \theta < \tfrac{\pi}{2} \)?

Which one of the following is a value of \( \theta \), if \( \theta \) satisfies the equation \( \tan 2 \theta \tan 4 \theta - 1 = 0 \); \( 0 < \theta < \frac{\pi}{2} \)?

Which one of the following is a value of \( \theta \), if \( \theta \) satisfies the equation \( \tan 2 \theta \tan 4 \theta - 1 = 0 \); \( 0<\theta<\frac{\pi}{2} \)?

Which one of the following is a value of \( \theta \), if \( \theta \) satisfies the equation \( \tan 2 \theta \tan 4 \theta-1=0 \); \( 0<\theta<\frac{\pi}{2} \) ?

Which one of the following is a value of \( \theta \), if \( \theta \) satisfies the equation \( \tan 2 \theta \tan 4 \theta - 1 = 0 \); \( 0 < \theta < \frac{\pi}{2} \)?

Which one of the following is a value of \( \theta \), if \( \theta \) satisfies the equation \( \tan 2 \theta \tan 4 \theta - 1 = 0 \); \( 0<\theta<\frac{\pi}{2} \)?

Which one of the following is a value of \( \theta \), if \( \theta \) satisfies the equation \( \tan 2 \theta \tan 4 \theta-1=0 \); \( 0<\theta<\frac{\pi}{2} \) ?

Which one of the following is a value of \(\theta\), if \(\theta\) satisfies the equation \(\tan 2\theta\,\tan 4\theta - 1 = 0\); \(0 < \theta < \frac{\pi}{2}\)?

Which one of the following is a value of \( \theta \), if \( \theta \) satisfies the equation \( \tan 2 \theta \tan 4 \theta -1 = 0 \); \( 0<\theta<\frac{\pi}{2} \) ?

Which one of the following is a value of \( \theta \), if \( \theta \) satisfies the equation \( \tan 2 \theta \tan 4 \theta - 1 = 0 \); \( 0 < \theta < \frac{\pi}{2} \)?

Which one of the following is a value of \( \theta \), if \( \theta \) satisfies the equation \( \tan 2 \theta \tan 4 \theta - 1 = 0 \); \( 0<\theta<\frac{\pi}{2} \)?