Test 01

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?

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?

Three persons \( \mathrm{A}, \mathrm{B} \), 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 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 the work?

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

Suppose the LCM of two given numbers is L and their 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?

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

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

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

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

If \( x^{n}-p y^{n}+q z^{n},\; x^{2}+a b y z-b z x-a x y \) is divisible by, 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 \( 2^{x-1} + 2^{3-x} \)?

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

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 \( \operatorname{HCF}(c, d)=1,\;c\neq d \)?

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

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

If the polynomial \( x^{n} - p y^{n} + q z^{n} \) is divided by \( x^{2} + a b y z - b z x - a x y \), what is the relationship between \( p \) and \( q \)?

If the polynomial \( x^{n} - p y^{n} + q z^{n} \) is divided by \( x^{2} + a b y z - b z x - a x y \), what is the relationship between \( a^{n} \) and \( b^{n} \)?

If \( x^{n}-p y^{n}+q z^{n} \) is divided by \( x^{2}+a b y z - b z x - a x y \), what could be the value of \( n \)?