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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?

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

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?

A person borrowed ₹10,000 for 9 months at an annual interest rate of 12% compounded quarterly. How much interest will he have to pay after 9 months?

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

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

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

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

Assume the \( \mathrm{LCM} \) and \( \mathrm{HCF} \) of two given numbers are \( \mathrm{L} \) and \( \mathrm{H} \) respectively. 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?

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

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

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

If \( x\left(a-b+\frac{a b}{a-b}\right)=y\left(a+b-\frac{a b}{a+b}\right) \) and \( x+y=2a^{3} \), then what is the value of \( x-y \)?

How many digits are there in the expansion of \(125^{100}\)? (Given \( \log_{10}2=0.301 \))

Which of the following is a factor of \( 3 \sqrt{3} x^{3}+2 \sqrt{2} y^{3}-18 x y+6 \sqrt{6} \)?

If \( x^{n}-p y^{n}+q z^{n} \) is divisible by \( x^{2}+a b y z - b z x - a x y \), then what is \( \frac{p}{a^{n}} - \frac{q}{b^{n}} \) equal to?

If \(3^{x-1}+3^{3-x}=6\), then what is the value of \(2^{x-1}+2^{3-x}\)?

What is the HCF of \(a c x^{3}+b c x^{2}+a d x^{2}+a c d x+b d x+b c d\) and \(a d x^{3}+a c x^{2}+b d x^{2}+b c x+a c d x+b c d\) if \(\mathrm{HCF}(c,d)=1, c \neq d\)?

A person borrowed ₹15,000 for 6 months at an annual interest rate of 10% compounded quarterly. How much interest will he have to pay after 6 months?

If ₹ 20,000 is lent for 1 year at an 8% annual interest rate compounded quarterly, what will be the total interest?

₹ 25,000 is lent for 9 months at a quarterly compounded annual interest rate of 6%. What will be the total interest after 9 months?