Full Solution - Test No 4 (English)
Test No. 4 - Detailed Solution
Q1: Write the following in ordinary notation.
Rule:
1. Positive Power (+): Move decimal to the RIGHT.
2. Negative Power (-): Move decimal to the LEFT.
(i) \( 1.77 \times 10^7 \)
The power is +7. Move the decimal point 7 places to the right.
\( 1.77 \xrightarrow{\text{2 jumps}} 177 \)
We still need 5 more jumps, so add 5 zeros.
\( 17700000 \)
Answer: \( 17,700,000 \)
(ii) \( 5.5 \times 10^{-6} \)
The power is -6. Move the decimal point 6 places to the left.
Start: 5.5
1 jump: .55
2 jumps: .055
6 jumps: .0000055
Answer: \( 0.0000055 \)
Q2: Express the following recurring decimals in the form \( \frac{p}{q} \).
(i) \( 0.\overline{4} \)
Let \( x = 0.4444... \)    --- (i)
Since 1 digit is repeating, multiply both sides by 10:
\( 10x = 4.4444... \quad \text{--- (ii)} \)
Subtract equation (i) from (ii):
\( 10x - x = 4.4444... - 0.4444... \)
\( 9x = 4 \)
\( x = \frac{4}{9} \)
Answer: \( \frac{4}{9} \)
(ii) \( 0.\overline{37} \)
Let \( x = 0.373737... \)    --- (i)
Since 2 digits (37) are repeating, multiply both sides by 100:
\( 100x = 37.373737... \quad \text{--- (ii)} \)
Subtract equation (i) from (ii):
\( 100x - x = 37.3737... - 0.3737... \)
\( 99x = 37 \)
\( x = \frac{37}{99} \)
Answer: \( \frac{37}{99} \)
Q3: Is there any set which has no proper subset? If yes, name it.
Answer: Yes, the Empty Set (or Null Set).
Symbol: \( \phi \) or \( \{ \} \)
Reasoning: A proper subset must contain fewer elements than the original set. Since the empty set contains 0 elements, it is impossible to form a subset with fewer than 0 elements.
Q4: Factorize \( x^2 + x - 12 \).
Expression: \( x^2 + x - 12 \)
Logic (Mid-Term Break): We need two numbers that:
1. Multiply to give -12.
2. Add to give +1 (Coefficient of middle term).
Finding Factors:
The numbers are +4 and -3 because:
\( 4 \times (-3) = -12 \) (Check)
\( 4 + (-3) = +1 \) (Check)
Step 1: Split the middle term \( +x \) into \( +4x \) and \( -3x \).
\( = x^2 + 4x - 3x - 12 \)
Step 2: Factor by grouping.
Take \( x \) common from first pair, and \( -3 \) from second pair.
\( = x(x + 4) - 3(x + 4) \)
(Note: Taking -3 common changes -12 to +4)
Step 3: Take \( (x+4) \) as the common factor.
\( = (x + 4)(x - 3) \)
Final Answer: \( (x + 4)(x - 3) \)