FSc Mathematics Number Systems | Short Question Answer. The following questions and answers are very important from Chapter No.1 of FSc. ICS Part-I | 1st Year Mathematics, for the final examinations of all Boards of Punjab as well as Pakistan:
Chapter No.1: Number Systems
Short Questions and Answers
Q.1: Define a rational number.
Answer:
A rational number is a number which can be written in the form `\frac{p}{q}`, where p and q are relatively prime integers and `q\ne 0`.
Answer:
A rational number is a number which can be written in the form `\frac{p}{q}`, where p and q are relatively prime integers and `q\ne 0`.
Q.2: Whether an integer is a rational number?
Answer:
An integer m can be written as `m=\frac{m}{1}` `(\frac{Int}{Int})`, so an integer is a rational number.
Answer:
An integer m can be written as `m=\frac{m}{1}` `(\frac{Int}{Int})`, so an integer is a rational number.
Q.3: Whether a natural number is a rational
number?
Answer: A natural number n can be written as `n=\frac{n}{1}``(\frac{Int}{Int})`, so a natural number is a rational number.
Q.4: Whether an odd integer is a rational
number?
Answer: An odd integer `2n-1` can be written as `2n-1=\frac{2n-1}{1}``(\frac{Int}{Int})`, so an odd integer is a rational number.
Q.5: Whether an even integer is a rational
number?
Answer: An even integer `2n` can be written
as `2n=\frac{2n}{1}``(\frac{Int}{Int})`, so an even integer is a
rational number.
Q.6: Prove that for any real number ‘a’; a.0
= 0.
Answer:
a.0 = a[1+(-1)] (additive inverse law)
= a[1-1] (definition of subtraction)
= a.1 - a.1 (distributive law)
= a-a (multiplicative identity law)
= a + (-a) (definition of subtraction)
= 0 (additive inverse law)
a.0 = a[1+(-1)] (additive inverse law)
= a[1-1] (definition of subtraction)
= a.1 - a.1 (distributive law)
= a-a (multiplicative identity law)
= a + (-a) (definition of subtraction)
= 0 (additive inverse law)
Q.7: For ` a,b,c\in R, c\ne 0`, prove that `
\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}`. (Gujranwala Board, Lahore Board)
Answer:
R.H.S = `\frac{a+b}{c}=(a+b).\frac{1}{c}=a.\frac{1}{c}+b.\frac{1}{c}` (right distributive law)
= `\frac{a}{c}+\frac{b}{c}`
Answer:
R.H.S = `\frac{a+b}{c}=(a+b).\frac{1}{c}=a.\frac{1}{c}+b.\frac{1}{c}` (right distributive law)
= `\frac{a}{c}+\frac{b}{c}`
Q.8: For ` a,b,c,d\in R, b\ne 0, d\ne 0`,
prove that ` \frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}`. (Lahore Board)
Answer:
R.H.S = `\frac{ad+bc}{bd}=(ad+bc).\frac{1}{bd}`
= `ad.\frac{1}{bd}+bc.\frac{1}{bd}` (right distributive law)
= `\frac{a}{b}.(d.\frac{1}{d})+\frac{c}{d}.(b.\frac{1}{b})=\frac{a}{b}.1+\frac{c}{d}.1` (inverse law)
= `\frac{a}{b}+\frac{c}{d}=` L.H.S. (identity law)
Answer:
R.H.S = `\frac{ad+bc}{bd}=(ad+bc).\frac{1}{bd}`
= `ad.\frac{1}{bd}+bc.\frac{1}{bd}` (right distributive law)
= `\frac{a}{b}.(d.\frac{1}{d})+\frac{c}{d}.(b.\frac{1}{b})=\frac{a}{b}.1+\frac{c}{d}.1` (inverse law)
= `\frac{a}{b}+\frac{c}{d}=` L.H.S. (identity law)
Q.9: Define an irrational number.
Answer:
An irrational number is a number which cannot be written in the form `\frac{p}{q}`, where p and q are relatively prime integers and `q\ne0`.
Answer:
An irrational number is a number which cannot be written in the form `\frac{p}{q}`, where p and q are relatively prime integers and `q\ne0`.
Q.10: Differentiate between rational and irrational numbers.
Answer:
Rational Number: A rational number is a number which can be written in the form `\frac{p}{q}`, where p and q are relatively prime integers and `q\ne0`.
Irrational Number: An irrational number is a number which cannot be written in the form `\frac{p}{q}`, where p and q are relatively prime integers and `q\ne0`.
Answer:
Rational Number: A rational number is a number which can be written in the form `\frac{p}{q}`, where p and q are relatively prime integers and `q\ne0`.
Irrational Number: An irrational number is a number which cannot be written in the form `\frac{p}{q}`, where p and q are relatively prime integers and `q\ne0`.
Q.11: Show that `\sqrt{3}` is an irrational number. (Sargodha
and Lahore Boards)
Answer:
If `\sqrt{3}` is a rational number, then there exist relatively prime integers p and q, such that:
Answer:
If `\sqrt{3}` is a rational number, then there exist relatively prime integers p and q, such that:
`\sqrt{3}=\frac{p}{q}`
`\Rightarrow3=\frac{p^2}{q^2}`
`\Rightarrow3q=\frac{p^2}{q}` (1)
Which is not possible, because L.H.S of (1) is an integer
while its R.H.S. is not an integer. Hence, `\sqrt{3}` is an irrational number.Q.12: Show that `\sqrt{5}` is an irrational number. (Lahore
Boards)
Answer:
If `\sqrt{5}` is a rational number, then there exist relatively prime integers p and q, such that:
`\sqrt{5}=\frac{p}{q}`
`\Rightarrow5=\frac{p^2}{q^2}`
`\Rightarrow5q=\frac{p^2}{q}` (1)
Which is not possible, because L.H.S of (1) is an integer while its R.H.S. is not an integer. Hence, `\sqrt{5}` is an irrational number.
Answer:
If `\sqrt{5}` is a rational number, then there exist relatively prime integers p and q, such that:
`\sqrt{5}=\frac{p}{q}`
`\Rightarrow5=\frac{p^2}{q^2}`
`\Rightarrow5q=\frac{p^2}{q}` (1)
Which is not possible, because L.H.S of (1) is an integer while its R.H.S. is not an integer. Hence, `\sqrt{5}` is an irrational number.
Q.13: Whether ` \pi` is rational or irrational? Explain by
defining ` \pi`. (Federal Board)
Answer:
`\pi` is an irrational number. It is the ratio of the circumference of a circle to its diameter.
Answer:
`\pi` is an irrational number. It is the ratio of the circumference of a circle to its diameter.
Q.14: What is the decimal representation of a rational
number?
Answer:
Rational numbers are represented by repeating decimals or terminating decimals. For example,
`\frac{4}{3}=1.333...,` 3 repeats
`\frac{3}{11}=0.272727...`, 27 repeats
`\frac{5}{7}=0.714285714285714285…` 714285 repeats
Answer:
Rational numbers are represented by repeating decimals or terminating decimals. For example,
`\frac{4}{3}=1.333...,` 3 repeats
`\frac{3}{11}=0.272727...`, 27 repeats
`\frac{5}{7}=0.714285714285714285…` 714285 repeats
Repeating decimals that consist of zeros from some point on
are called terminating decimals. Some examples are:
`\frac{1}{2}=0.5000...,\frac{12}{4}=3.000...,\frac{8}{25}=0.32000...`
`\frac{1}{2}=0.5000...,\frac{12}{4}=3.000...,\frac{8}{25}=0.32000...`
It is usual to omit the repetitive zeros in terminating
decimals.
For example,
`\frac{1}{2}=0.5, \frac{12}{4}=3, \frac{8}{25}=0.32`
For example,
`\frac{1}{2}=0.5, \frac{12}{4}=3, \frac{8}{25}=0.32`
Q.15: What is the decimal representation of an irrational
number?
Answer:
The irrational numbers are represented by non-repeating decimals. For example, the decimal 0.101001000100001000001… does not repeat because the number of zeros between the ones keeps growing, so it is an irrational number. Similarly, `\sqrt{2}`=1.414213562… is also an irrational number.
Answer:
The irrational numbers are represented by non-repeating decimals. For example, the decimal 0.101001000100001000001… does not repeat because the number of zeros between the ones keeps growing, so it is an irrational number. Similarly, `\sqrt{2}`=1.414213562… is also an irrational number.
Q.16: Define a binary operation in a set.
Answer:
If `a*b\in A` for every a, `b\in A`, then ‘*’ is called the binary operation in a set A.
Answer:
If `a*b\in A` for every a, `b\in A`, then ‘*’ is called the binary operation in a set A.
Q.17: Whether `'\div'` is a binary operation in the set of
rational numbers Q?
Answer:
Since the division of any rational number by 0 is not a rational number, i.e., `\frac{r}{0}\notin Q` for all `r\in Q`, so `'\div'` is not a binary operation in the set of rational numbers Q.
Answer:
Since the division of any rational number by 0 is not a rational number, i.e., `\frac{r}{0}\notin Q` for all `r\in Q`, so `'\div'` is not a binary operation in the set of rational numbers Q.
Q.18: Is '+' is a binary operation in the set of irrational
numbers ` Q^{c}`?
Answer:
Since the sum of two irrational numbers may not be an irrational number, for example, `(1+\sqrt{3})+(1-\sqrt{3})=2\notin Q^c`, So '+' is not a binary operation in the set of irrational numbers `Q^c`.
Answer:
Since the sum of two irrational numbers may not be an irrational number, for example, `(1+\sqrt{3})+(1-\sqrt{3})=2\notin Q^c`, So '+' is not a binary operation in the set of irrational numbers `Q^c`.
Q.19: Is '.' is a binary operation in the set of irrational
numbers ` Q^{c}`?
Answer:
Since the multiplication of two irrational numbers may not be an irrational number, for example, `(\sqrt{2})(\sqrt{2})=2\notin Q^c`, So '.' is not a binary operation in the set of irrational numbers `Q^c`.
Answer:
Since the multiplication of two irrational numbers may not be an irrational number, for example, `(\sqrt{2})(\sqrt{2})=2\notin Q^c`, So '.' is not a binary operation in the set of irrational numbers `Q^c`.
Q.20: Is '-' is a binary operation in the set of irrational
numbers ` Q^{c}`?
Answer:
Since the subtraction of two irrational numbers may not be an irrational number, for example, `(5-\sqrt{3})-(2-\sqrt{3})=3\notin Q^c`, So '-' is not a binary operation is the set of irrational numbers `Q^c`.
Answer:
Since the subtraction of two irrational numbers may not be an irrational number, for example, `(5-\sqrt{3})-(2-\sqrt{3})=3\notin Q^c`, So '-' is not a binary operation is the set of irrational numbers `Q^c`.
NOTE:
The remaining short question and answers of this chapter (Number Systems) will be published soon Insha Allah.
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