Full Solution - Test No 3 (English)
Test No. 3 - Detailed Solution
Q1: Represent the following numbers on the number line.
(i) \( \sqrt{3} \)
Method: Calculate the approximate value using a calculator.
\( \sqrt{3} \approx 1.732 \)
This number lies between 1 and 2. We divide the space into 10 small parts. Since the value is approx 1.7, we mark the point at the 7th small mark.
1 2 √3
(ii) \( -2\frac{1}{7} \)
This is a negative number lying between -2 and -3.
We divide the segment between -2 and -3 into 7 equal parts. Starting from -2 (moving left), we select the 1st mark.
-3 -2 -2 1/7
(iii) \( 2\frac{3}{4} \)
This number lies between 2 and 3.
Divide the space between 2 and 3 into 4 equal parts. Mark the point at the 3rd mark.
2 3 2¾
Q2: Write the following in scientific notation.
Rule: Scientific notation is \( a \times 10^n \), where \( 1 \le a < 10 \).
1. Decimal moves Left = Positive Power (+).
2. Decimal moves Right = Negative Power (-).
(i) \( 48900 \)
The decimal is at the end: \( 48900.0 \). Move it 4 places to the left.
\( 48900.0 \xrightarrow{\text{4 left}} 4.89 \times 10^{+4} \)
Answer: \( 4.89 \times 10^4 \)
(ii) \( 0.0042 \)
Move the decimal 3 places to the right to get 4.2.
\( 0.0042 \xrightarrow{\text{3 right}} 4.2 \times 10^{-3} \)
Answer: \( 4.2 \times 10^{-3} \)
(iii) \( 0.65 \times 10^2 \)
This is not in standard form because \( 0.65 < 1 \).
Step 1: Convert 0.65 to scientific notation:
\( 0.65 = 6.5 \times 10^{-1} \)
Step 2: Substitute back into the expression:
\( (6.5 \times 10^{-1}) \times 10^2 \)
Step 3: Add the powers:
\( 6.5 \times 10^{-1 + 2} = 6.5 \times 10^1 \)
Answer: \( 6.5 \times 10^1 \)
Q3: Write two proper subsets of the following set.
Set: \( \{ x \mid x \in \mathbb{Q} \land 0 < x < 2 \} \)
This is the set of all rational numbers between 0 and 2. It contains infinite numbers. To write a proper subset, we can simply pick a finite set of numbers from this range.
Example 1: The set containing only the number 1.
\( \{ 1 \} \)
Example 2: The set containing 0.5 and 1.5.
\( \{ 0.5, 1.5 \} \)
Answer: \( \{1\} , \{1, 1.5\} \)
Q4: Factorize \( 3x^2 - 4x - 4 \).
Expression: \( 3x^2 - 4x - 4 \)
Logic: We need two numbers that:
1. Multiply to give (First coeff × Last constant): \( 3 \times -4 = \mathbf{-12} \).
2. Add to give -4 (Middle coeff).
Factors of -12:
\( -6 \times +2 = -12 \) (Correct)
\( -6 + 2 = -4 \) (Correct)
Step 1: Split the middle term \( -4x \) into \( -6x + 2x \).
\( = 3x^2 - 6x + 2x - 4 \)
Step 2: Factor by grouping.
\( = 3x(x - 2) + 2(x - 2) \)
Step 3: Take \( (x-2) \) as the common factor.
\( = (x - 2)(3x + 2) \)
Final Answer: \( (x - 2)(3x + 2) \)