1 Angles of Parallelogram
The adjacent angles of a parallelogram are \((2x + 12)^\circ\) and \((3x - 62)^\circ\), then the value of \(x\) is:
Concept:
1
Properties of Parallelograms: In any parallelogram, consecutive (adjacent) angles are supplementary. This means their sum is always equal to \(180^\circ\).
Hint:
1
Set up the equation: \((2x + 12) + (3x - 62) = 180\). Combine the \(x\) terms and the constant numbers, then solve for \(x\).
Mistakes to Avoid:
!
Equating the Angles: Opposite angles in a parallelogram are equal, but these are adjacent angles! Do not set \(2x + 12 = 3x - 62\). They must add up to 180.
👉 Scroll horizontally to read the full handwritten solution.
Let the adjacent angles be \( \angle A = (2x + 12)^\circ \) and \( \angle B = (3x - 62)^\circ \).
Since adjacent angles of a parallelogram are supplementary:
\( \angle A + \angle B = 180^\circ \)
\( (2x + 12) + (3x - 62) = 180 \)
\( 5x - 50 = 180 \) [½ M]
\( 5x = 180 + 50 \)
\( 5x = 230 \)
\( x = \frac{230}{5} = 46 \) [½ M]
Correct Option: (C)
2 Fractional Exponents
\( \frac{(144)^{1/2} + (256)^{1/2}}{3^2 - 2} = \)
Concept:
1
Laws of Exponents: A fractional exponent of \(1/2\) is exactly the same as finding the square root of the number. Therefore, \(x^{1/2} = \sqrt{x}\).
Hint:
1
Calculate the numerator first: find the square roots of 144 and 256, then add them together. Then calculate the denominator: square 3 and subtract 2. Finally, divide the results.
Mistakes to Avoid:
!
Halving Instead of Rooting: Students often mistakenly think \( (144)^{1/2} \) means \(144 \div 2 = 72\). This is incorrect. It means \(\sqrt{144}\), which is 12!
👉 Scroll horizontally to read the full handwritten solution.
Given expression: \( \frac{(144)^{1/2} + (256)^{1/2}}{3^2 - 2} \)
We know that \( a^{1/2} = \sqrt{a} \).
\( (144)^{1/2} = \sqrt{144} = 12 \)
\( (256)^{1/2} = \sqrt{256} = 16 \)
Substitute these values into the numerator:
Numerator = \( 12 + 16 = 28 \) [½ M]
Now solve the denominator:
Denominator = \( 3^2 - 2 = 9 - 2 = 7 \)
Final Calculation:
\( = \frac{28}{7} = 4 \) [½ M]
Correct Option: (B)
3 Sum of Odd Numbers
The value of \(1 + 3 + 5 + 7 + 9 + ........ + 199\) is:
Concept:
1
Sum of Odd Natural Numbers: The sum of the first \(n\) consecutive odd natural numbers starting from 1 is always equal to \(n^2\). For example, \(1+3 = 2^2\), \(1+3+5 = 3^2\), etc.
Hint:
1
Find how many terms (\(n\)) are in the sequence. The general formula for the \(n\)-th odd number is \((2n - 1)\). Set \(2n - 1 = 199\) and solve for \(n\). Then square your answer!
Mistakes to Avoid:
!
Assuming \(n = 199\): Many students assume that because the last number is 199, there are 199 terms. Since we are skipping all even numbers, there are only half that amount!
👉 Scroll horizontally to read the full handwritten solution.
The given sequence is the sum of odd numbers:
\( 1 + 3 + 5 + ... + 199 \)
The sum of the first \(n\) odd numbers is \(n^2\).
To find \(n\), we use the formula for the \(n\)-th odd term:
\( 2n - 1 = 199 \)
\( 2n = 199 + 1 \)
\( 2n = 200 \)
\( n = 100 \) [½ M]
Therefore, the sum is \( n^2 \):
Sum = \( 100^2 \) [½ M]
Correct Option: (B)
4 Perfect Cubes
The least number by which 1536 should be divided so that the quotient becomes a perfect cube, is:
Concept:
1
Prime Factorisation: To make a number a perfect cube, the exponent of every prime factor must be a multiple of 3 (e.g., \(2^3, 5^6, 7^9\)). Any factor that doesn't form a triplet must be divided out.
Hint:
1
Break 1536 down into its prime factors. You will get a lot of 2's and one other number. Group the 2's into triplets (groups of 3). Which number is left over without a triplet? That is your answer.
Mistakes to Avoid:
!
Guess and Check: Dividing 1536 by 8 gives 192. Is 192 a perfect cube? No. Dividing by 6 gives 256. Perfect cube? No. Dividing by 4 gives 384. No. Don't guess; always use prime factorization for guaranteed accuracy.
👉 Scroll horizontally to read the full handwritten solution.
Step 1: Find the prime factorisation of 1536.
\( 1536 = 2 \times 768 \)
\( 1536 = 2 \times 2 \times 384 \)
\( 1536 = 2 \times 2 \times 2 \times 192 \)
... continuing this process:
\( 1536 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \)
\( 1536 = 2^9 \times 3^1 \) [½ M]
Group the prime factors into triplets (powers of 3):
\( 1536 = (2^3) \times (2^3) \times (2^3) \times 3 \)
The factor 3 does not form a triplet.
To make the quotient a perfect cube, we must divide by 3. [½ M]
Correct Option: (A)
5 Number Systems
The base-60 system is also called as:
Concept:
1
Ancient Number Systems: The ancient Mesopotamians (Babylonians) used a base-60 numeral system known as the Sexagesimal system. It is the reason we still have 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a circle today!
Hint:
1
"Deci" means 10. "Bi" means 2. Look for the prefix that relates to 60.
Mistakes to Avoid:
!
Confusing with our current system: Our modern system is the Decimal system (Base-10) derived from Hindu-Arabic numerals. Computers use Binary (Base-2).
👉 Scroll horizontally to read the full handwritten solution.
Let's review the number systems based on their base:
- Base 2: Binary System
- Base 10: Decimal System
- Base 60: Sexagesimal System [1 M]
Therefore, the base-60 system is called the Sexagesimal system.
Correct Option: (B)
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