Unlock the complete solution to Review Exercise 1 from the new 9th Class Math syllabus by Punjab Curriculum and Textbook Board (PCTB), Lahore. This step-by-step guide is designed to help students master the latest concepts with ease and confidence.
Questions with Correct Options and Explanations:
1. \(\sqrt{7}\) is:
(a) integer
(b) rational number
(c) irrational number (Correct)
(d) natural number
Explanation: The square root of 7 is not a perfect square and cannot be expressed as a fraction, making it irrational.
2. \(\pi\) and \(e\) are:
(a) natural numbers
(b) integers
(c) rational numbers
(d) irrational numbers (Correct)
Explanation: Both \(\pi\) and \(e\) are non-repeating, non-terminating decimals, classifying them as irrational numbers.
3. If \(n\) is not a perfect square, then \(\sqrt{n}\) is:
(a) rational number
(b) natural number
(c) integer
(d) irrational number (Correct)
Explanation: If \(n\) is not a perfect square, its square root cannot be expressed as a fraction, making it irrational.
4. \(\sqrt{3} + \sqrt{5}\) is:
(a) whole number
(b) integer
(c) rational number
(d) irrational number (Correct)
Explanation: The sum of two irrational numbers \(\sqrt{3}\) and \(\sqrt{5}\) is also irrational.
5. For all \(x \in \mathbb{R}, x = x\) is called:
(a) reflexive property (Correct)
(b) transitive number
(c) symmetric property
(d) trichotomy property
Explanation: Reflexive property states that any element is equal to itself.
6. Let \(a, b, c \in \mathbb{R}\), then \(a > b\) and \(b > c \Rightarrow a > c\) is called ________ property.
(a) trichotomy
(b) transitive (Correct)
(c) additive
(d) multiplicative
Explanation: The transitive property applies when the relation holds across three elements in this manner.
7. \(2^x \times 8^x = 64\), then \(x =\):
(a) \(\frac{3}{2}\) (Correct)
(b) \(\frac{3}{4}\)
(c) \(\frac{5}{6}\)
(d) \(\frac{2}{3}\)
Explanation: Solving the equation \(2^x \times (2^3)^x = 64 \Rightarrow 2^{4x} = 2^6 \Rightarrow x = \frac{3}{2}\).
8. Let \(a, b \in \mathbb{R}\), then \(a = b\) and \(b = a\) is called ________ property.
(a) reflexive
(b) symmetric (Correct)
(c) transitive
(d) additive
Explanation: Symmetric property states that if \(a = b\), then \(b = a\).
9. \(\sqrt{75} + \sqrt{27} =\):
(a) \(\sqrt{102}\)
(b) \(9\sqrt{3}\) (Correct)
(c) \(5\sqrt{3}\)
(d) \(8\sqrt{3}\)
Explanation: Simplify \(\sqrt{75} = 5\sqrt{3}\) and \(\sqrt{27} = 3\sqrt{3}\), so \(\sqrt{75} + \sqrt{27} = 5\sqrt{3} + 3\sqrt{3} = 9\sqrt{3}\).
10. The product of \((3 + \sqrt{5})(3 – \sqrt{5})\) is:
(a) prime number
(b) odd number
(c) irrational number
(d) rational number (Correct)
Explanation: Using the difference of squares formula, \((3 + \sqrt{5})(3 – \sqrt{5}) = 3^2 – (\sqrt{5})^2 = 9 – 5 = 4\), which is a rational number.
2. If \(a = \frac{3}{2}\), \(b = \frac{5}{3}\), and \(c = \frac{7}{5}\), then verify that:
(i) \(a(b + c) = ab + ac\)
\[b + c = \frac{5}{3} + \frac{7}{5} = \frac{25 + 21}{15} = \frac{46}{15}\]
\[a(b + c) = \frac{3}{2} \cdot \frac{46}{15} = \frac{138}{30} = \frac{23}{5}\]
\[ab = \frac{3}{2} \cdot \frac{5}{3} = \frac{5}{2}, \quad ac = \frac{3}{2} \cdot \frac{7}{5} = \frac{21}{10}\]
\[ab + ac = \frac{5}{2} + \frac{21}{10} = \frac{25 + 21}{10} = \frac{46}{10} = \frac{23}{5}\]
Verified
\[a(b + c) = ab + ac = \frac{23}{5}\]
(ii) \((a + b)c = ac + bc\)
\[a + b = \frac{3}{2} + \frac{5}{3} = \frac{9 + 10}{6} = \frac{19}{6}\]
\[(a + b)c = \frac{19}{6} \cdot \frac{7}{5} = \frac{133}{30}\]
\[ac = \frac{3}{2} \cdot \frac{7}{5} = \frac{21}{10}, \quad bc = \frac{5}{3} \cdot \frac{7}{5} = \frac{7}{3}\]
\[ac + bc = \frac{21}{10} + \frac{7}{3} = \frac{63 + 70}{30} = \frac{133}{30}\]
Verified
\[(a + b)c = ac + bc = \frac{133}{30}\]
3. If \(a = \frac{4}{3}\), \(b = \frac{5}{2}\), \(c = \frac{7}{4}\), then verify the associative property of real numbers with respect to addition and multiplication.
Addition
\((a + b) + c = a + (b + c)\)
\[a + b = \frac{4}{3} + \frac{5}{2} = \frac{8 + 15}{6} = \frac{23}{6}\]
\[(a + b) + c = \frac{23}{6} + \frac{7}{4} = \frac{92 + 42}{24} = \frac{134}{24} = \frac{67}{12}\]
\[b + c = \frac{5}{2} + \frac{7}{4} = \frac{10 + 7}{4} = \frac{17}{4}\]
\[a + (b + c) = \frac{4}{3} + \frac{17}{4} = \frac{16 + 51}{12} = \frac{67}{12}\]
Verified
\[(a + b) + c = a + (b + c) = \frac{67}{12}\]
Multiplication
\((a \cdot b) \cdot c = a \cdot (b \cdot c)\)
\[a \cdot b = \frac{4}{3} \cdot \frac{5}{2} = \frac{20}{6} = \frac{10}{3}\]
\[(a \cdot b) \cdot c = \frac{10}{3} \cdot \frac{7}{4} = \frac{70}{12} = \frac{35}{6}\]
\[b \cdot c = \frac{5}{2} \cdot \frac{7}{4} = \frac{35}{8}\]
\[a \cdot (b \cdot c) = \frac{4}{3} \cdot \frac{35}{8} = \frac{140}{24} = \frac{35}{6}\]
Verified
\[(a \cdot b) \cdot c = a \cdot (b \cdot c) = \frac{35}{6}\]
4. Is \(0\) a rational number?
Yes, \(0\) is a rational number because it can be expressed as \(\frac{0}{1}\) where \(1 \neq 0\).
5. State the trichotomy property of real numbers.
For any real number \(a\), exactly one of the following is true:
\[a > 0, \quad a = 0, \quad a < 0\]
6. Find two rational numbers between \(4\) and \(5\).
Two rational numbers between \(4\) and \(5\) are:
\[4.1 = \frac{41}{10}, \quad 4.5 = \frac{9}{2}\]
7. Simplify the following
(i) \[\sqrt[5]{\frac{x^{15}y^{35}}{z^{20}}}\]
\[= \frac{x^{15/5}y^{35/5}}{z^{20/5}} = \frac{x^3y^7}{z^4}\]
(ii) \[\sqrt[3]{(27)^{2x}}\]
\[= (27)^{2x/3} = (3^3)^{2x/3} = 3^{2x}\]
(iii) \[\frac{6(3)^{n+2}}{3^{n+1} – 3^n}\]
Simplify the denominator:
\[3^{n+1} – 3^n = 3^n(3 – 1) = 2 \cdot 3^n\]
Substitute:
\[\frac{6(3)^{n+2}}{2 \cdot 3^n} = \frac{6 \cdot 3^n \cdot 3^2}{2 \cdot 3^n} = \frac{6 \cdot 9}{2} = 27\]
Answer
\[27\]
8. The sum of three consecutive odd integers is \(51\). Find the three integers.
Let the integers be \(x – 2\), \(x\), \(x + 2\):
\[(x – 2) + x + (x + 2) = 51\]
\[3x = 51 \]
\[ x = 17\]
The integers are:
\[15, 17, 19\]
9.Abdullah picked up \(96\) balls and placed them into two buckets. One bucket has \(28\) more balls than the other bucket. How many balls were in each bucket?
Let the balls in the first bucket be \(x\) and in the second bucket be \(x + 28\):
\[x + (x + 28) = 96\]
\[2x + 28 = 96 \]
\[ 2x = 68 \]
\[ x = 34\]
The buckets contain:
\[34, \quad 34 + 28 = 62\]
Answer
\[34, 62\]
10. Salma invested Rs. \(3,50,000\) in a bank, which paid simple profit at the rate of \(7\frac{1}{4}\%\) per annum. After \(2\) years, the rate was increased to \(8\%\) per annum. Find the amount she had at the end of \(7\) years.
Convert \(7\frac{1}{4}\%\) to a decimal:
\[7\frac{1}{4}\% = \frac{29}{4}\% = \frac{29}{4 \cdot 100} = \frac{29}{400}\]
For the first \(2\) years:
\[\text{Interest} = \text{Principal} \cdot \text{Rate} \cdot \text{Time}\]
\[I_1 = 3,50,000 \cdot \frac{29}{400} \cdot 2 = 3,50,000 \cdot \frac{58}{400} = 50,750\]
For the next \(5\) years at \(8\%\):
\[I_2 = 3,50,000 \cdot \frac{8}{100} \cdot 5 = 3,50,000 \cdot \frac{40}{100} = 1,40,000\]
Total interest:
\[I = 50,750 + 1,40,000 = 1,90,750\]
Total amount:
\[\text{Amount} = \text{Principal} + \text{Interest} = 3,50,000 + 1,90,750 = 5,40,750\]
Answer
\[\text{Amount} = 5,40,750\]
