Welcome to the Fractions for IIT JEE practice quiz. This set of 25 moderate-level questions is designed to test your understanding of rational numbers, partial fractions, and recurring decimals as they apply to competitive mathematics.
Key concepts covered:
- Summation of telescoping series and infinite geometric progressions.
- Properties of the fractional part function
{x}. - Partial fraction decomposition techniques.
- Operations with recurring decimals and continued fractions.
- Rationalizing complex denominators and comparing rational exponents.
Note: All mathematical expressions are provided in standard notation. Ensure you understand the properties of LCM and HCF for fractions before starting.
Q1. Find the sum of the series ##\sum_{r=1}^{n} \frac{1}{r(r+1)}##.
The series is a telescoping sum: ##\sum (\frac{1}{r} - \frac{1}{r+1}) = (1 - 1/2) + (1/2 - 1/3) + \dots + (1/n - 1/(n+1)) = 1 - 1/(n+1) = n/(n+1)##.
Q2. Express the recurring decimal ##0.1\bar{2}## as a fraction in its simplest form.
Let ##x = 0.1222...## then ##10x = 1.222...## and ##100x = 12.222...##. Subtracting gives ##90x = 11##, so ##x = 11/90##.
Q3. Solve the inequality ##\frac{x-1}{x+1} \geq 0##.
The critical points are 1 and -1. Testing intervals shows the expression is positive for ##x > 1## and ##x < -1##. Since the denominator cannot be zero, ##x \neq -1##, but ##x=1## is included.
Q4. Solve for ##x## if the fractional part ##\{x\} = 0.7## and ##x \in [0, 2]##.
The fractional part function is periodic with period 1. If ##\{x\} = 0.7##, then ##x = n + 0.7## where ##n## is an integer. For the range ##[0, 2]##, ##n## can be 0 or 1, giving ##0.7## and ##1.7##.
Q5. What is the Least Common Multiple (LCM) of the fractions ##\frac{2}{3}##, ##\frac{4}{9}##, and ##\frac{8}{27}##?
LCM of fractions = (LCM of numerators) / (HCF of denominators). LCM(2, 4, 8) = 8. HCF(3, 9, 27) = 3. Result = 8/3.
Q6. What is the Highest Common Factor (HCF) of the fractions ##\frac{3}{5}##, ##\frac{6}{25}##, and ##\frac{9}{20}##?
HCF of fractions = (HCF of numerators) / (LCM of denominators). HCF(3, 6, 9) = 3. LCM(5, 25, 20) = 100. Result = 3/100.
Q7. If ##\frac{1}{a}##, ##\frac{1}{b}##, and ##\frac{1}{c}## are in Arithmetic Progression (AP), then ##a, b, c## are in:
By definition, if the reciprocals of a sequence form an Arithmetic Progression, the original sequence is in Harmonic Progression.
Q8. In the partial fraction decomposition ##\frac{3x+2}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}##, find the value of ##A##.
Multiply by ##(x-1)## and set ##x=1##: ##A = \frac{3(1)+2}{1+2} = 5/3##.
Q9. Simplify the sum: ##\frac{1}{1+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{4}}##.
Rationalizing each term: ##(\sqrt{2}-1) + (\sqrt{3}-\sqrt{2}) + (\sqrt{4}-\sqrt{3}) = \sqrt{4} - 1 = 2 - 1 = 1##.
Q10. If ##x = 1 + \frac{1}{1 + \frac{1}{1 + \dots}}## (infinite continued fraction), find the value of ##x## for ##x > 0##.
The expression can be written as ##x = 1 + 1/x##, which leads to ##x^2 - x - 1 = 0##. Solving for ##x > 0## gives the golden ratio ##(1+\sqrt{5})/2##.
Q11. How many positive proper fractions ##\frac{p}{q}## in simplest form exist such that ##q = 12##?
The number of irreducible fractions with denominator ##q## is given by Euler's totient function ##\phi(12)##. ##\phi(12) = 12(1 - 1/2)(1 - 1/3) = 12(1/2)(2/3) = 4##. The fractions are 1/12, 5/12, 7/12, and 11/12.
Q12. What is the length of the period of the decimal expansion of ##\frac{1}{13}##?
##1/13 = 0.\overline{076923}##. The repeating block contains 6 digits.
Q13. The value of the recurring decimal ##0.999...## is exactly equal to:
Let ##x = 0.999...## then ##10x = 9.999...##. Subtracting gives ##9x = 9##, so ##x = 1##.
Q14. Find the sum of the infinite geometric series ##\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots##.
Sum ##S = \frac{a}{1-r}## where ##a = 1/3## and ##r = 1/3##. ##S = (1/3) / (1 - 1/3) = (1/3) / (2/3) = 1/2##.
Q15. Which of the following is true regarding the comparison of ##2^{1/2}## and ##3^{1/3}##?
Raising both to the 6th power: ##(2^{1/2})^6 = 2^3 = 8## and ##(3^{1/3})^6 = 3^2 = 9##. Since ##9 > 8##, ##3^{1/3} > 2^{1/2}##.
Q16. Find the value of ##x## that satisfies the equation ##\frac{2}{x} - \frac{3}{2x} = \frac{1}{4}##.
Simplify the left side: ##\frac{4-3}{2x} = \frac{1}{2x}##. Then ##\frac{1}{2x} = \frac{1}{4} \Rightarrow 2x = 4 \Rightarrow x = 2##.
Q17. The function ##f(x) = \frac{x^2-1}{x-1}## is defined for which set of values?
The function is undefined when the denominator is zero, i.e., ##x - 1 = 0 \Rightarrow x = 1##. For all other real values, it is defined.
Q18. Rationalize the denominator of ##\frac{1}{\sqrt[3]{2}-1}##.
Use the identity ##a^3 - b^3 = (a-b)(a^2 + ab + b^2)##. Let ##a = 2^{1/3}## and ##b = 1##. Multiplying numerator and denominator by ##(2^{1/3})^2 + 2^{1/3}(1) + 1^2## gives the result.
Q19. Find the sum of the first 5 terms of the sequence ##\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots##.
This is a GP with ##a = 1/2, r = 1/2, n = 5##. Sum ##S = \frac{1/2(1-(1/2)^5)}{1-1/2} = 1 - 1/32 = 31/32##.
Q20. The partial fraction decomposition of ##\frac{1}{x^2-1}## is:
##\frac{1}{(x-1)(x+1)} = \frac{A}{x-1} + \frac{B}{x+1}##. Using the cover-up method, ##A = 1/(1+1) = 1/2## and ##B = 1/(-1-1) = -1/2##.
Q21. How many rational numbers of the form ##\frac{k}{12}## (where ##k## is an integer) lie strictly between ##\frac{1}{3}## and ##\frac{1}{2}##?
Convert to denominator 12: ##1/3 = 4/12## and ##1/2 = 6/12##. The only integer ##k## such that ##4 < k < 6## is ##k = 5##. Thus, there is 1 such number.
Q22. If ##\frac{a}{b} = \frac{c}{d}##, then according to the property of Componendo and Dividendo, ##\frac{a+b}{a-b}## is equal to:
The property states that if ##a/b = c/d##, then ##(a+b)/(a-b) = (c+d)/(c-d)## provided the denominators are non-zero.
Q23. Find the fractional part of ##-\pi##, denoted by ##\{-\pi\}##.
The fractional part ##\{x\} = x - \lfloor x \rfloor##. Since ##\pi \approx 3.14159##, ##-\pi \approx -3.14159##. Thus ##\lfloor -\pi \rfloor = -4##. Therefore, ##\{-\pi\} = -\pi - (-4) = 4 - \pi##.
Q24. Calculate the sum of the infinite Arithmetico-Geometric Progression: ##\sum_{k=1}^{\infty} \frac{k}{3^k}##.
Using the formula for ##\sum n r^n = \frac{r}{(1-r)^2}## with ##r = 1/3##, we get ##(1/3) / (1 - 1/3)^2 = (1/3) / (4/9) = 3/4##.
Q25. Simplify the fraction ##\frac{0.5}{0.125}## to its simplest integer or fractional form.
##0.5 = 1/2## and ##0.125 = 1/8##. Thus, ##(1/2) / (1/8) = 8/2 = 4##.