21 Squares & Square Roots
Find the square root of $11025$ using the prime factorization method.
Concept:
1
Prime Factorization Method: To find the square root, break the number down into its basic prime factors. Group identical factors into pairs. From every pair, take exactly one factor, and multiply them together to get the square root.
Hint:
1
The number $11025$ ends in 5, so it is divisible by 5. Also, $1+1+0+2+5 = 9$, which means it is divisible by 3. Start dividing by the smallest primes (3, 5, 7) until you reach 1!
Mistakes to Avoid:
!
Forgetting to Pair: A common mistake is writing down all the factors and multiplying them. Remember: you must take only ONE number from each PAIR of identical primes!
👉 Scroll horizontally to read the full handwritten solution.
Let's find the prime factors of $11025$:
$11025 \div 3 = 3675$
$3675 \div 3 = 1225$
$1225 \div 5 = 245$
$245 \div 5 = 49$
$49 \div 7 = 7$
$7 \div 7 = 1$
Prime Factorization:
$11025 = 3 \times 3 \times 5 \times 5 \times 7 \times 7$ [1 M]
Grouping them into pairs:
$11025 = (3 \times 3) \times (5 \times 5) \times (7 \times 7)$
Taking one number from each pair for the square root:
$\sqrt{11025} = 3 \times 5 \times 7$
$\sqrt{11025} = 105$ [1 M]
Final Answer: 105
22 Quadrilaterals
If $ABCD$ is a rectangle with $\angle BAC = 32^\circ$. Determine $\angle DBC$.
Concept:
1
Rectangle Diagonal Properties: In a rectangle, all corner angles are exactly $90^\circ$. Additionally, the diagonals are equal in length and bisect each other, which divides the rectangle into four isosceles triangles!
Hint:
1
Look at corner $B$. The total angle $\angle ABC = 90^\circ$. If you find $\angle DBA$, you can subtract it from 90 to find $\angle DBC$. Remember that the diagonals form an isosceles triangle with the top side $AB$.
Mistakes to Avoid:
!
Diagonals Bisecting Corners: A very common error is assuming the diagonal of a rectangle cuts the $90^\circ$ corner perfectly in half ($45^\circ$ each). This ONLY happens in squares, not standard rectangles!
👉 Scroll horizontally to read the full handwritten solution.
Let the diagonals $AC$ and $BD$ intersect at point $O$.
In a rectangle, diagonals are equal and bisect each other.
Therefore, $OA = OB$.
In $\triangle OAB$, since two sides are equal, it is an isosceles triangle.
Angles opposite to equal sides are equal:
$\angle OBA = \angle OAB$
We are given $\angle BAC = 32^\circ$ (which is the same as $\angle OAB$).
So, $\angle OBA = 32^\circ$. [1 M]
We know the corner angle of a rectangle is $90^\circ$:
$\angle ABC = 90^\circ$
$\angle OBA + \angle OBC = 90^\circ$
$32^\circ + \angle DBC = 90^\circ$ (since $\angle OBC$ is the same as $\angle DBC$)
$\angle DBC = 90^\circ - 32^\circ$
$\angle DBC = 58^\circ$ [1 M]
Final Answer: $58^\circ$
23 Ancient Number Systems
Represent the following numbers in the Mesopotamian number system:
(A) $7530$
(B) $3605$
(A) $7530$
(B) $3605$
Concept:
1
Base-60 (Sexagesimal): The Mesopotamian system uses powers of 60 instead of powers of 10. You must repeatedly divide the number by 60 to find the values for the $60^2$, $60^1$, and $60^0$ place columns.
Hint:
1
For 7530: Divide by 60. The remainder is your "ones" column. Take the quotient and divide by 60 again. That remainder is your "sixties" column!
Mistakes to Avoid:
!
Zero Placeholders: Mesopotamians eventually used a blank space to represent zero. If a division leaves a zero remainder (like in 3605), you must leave a blank space in that column!
👉 Scroll horizontally to read the full handwritten solution.
(A) Representing 7530:
We divide by 60 to find the place values:
$7530 \div 60 = 125$ with a remainder of $\mathbf{30}$.
$125 \div 60 = \mathbf{2}$ with a remainder of $\mathbf{5}$.
The values from highest to lowest place are: $2, 5, 30$.
$7530 = (2 \times 60^2) + (5 \times 60^1) + (30 \times 60^0)$
In cuneiform representation:
2 vertical wedges | 5 vertical wedges | 3 corner wedges [1 M]
(B) Representing 3605:
$3605 \div 60 = 60$ with a remainder of $\mathbf{5}$.
$60 \div 60 = \mathbf{1}$ with a remainder of $\mathbf{0}$.
The values from highest to lowest place are: $1, 0, 5$.
$3605 = (1 \times 60^2) + (0 \times 60^1) + (5 \times 60^0)$
In cuneiform representation:
1 vertical wedge | [Blank Space for Zero] | 5 vertical wedges [1 M]
24 Algebraic Expansion
Multiply:
(i) $(3x + 2y)$ and $(5x + 3y)$
(ii) $(2x + 3y)$ and $(4x - 5y)$
(i) $(3x + 2y)$ and $(5x + 3y)$
(ii) $(2x + 3y)$ and $(4x - 5y)$
Concept:
1
Binomial Multiplication: Use the Distributive Property (often remembered as FOIL: First, Outer, Inner, Last). Multiply every term in the first bracket by every term in the second bracket, then combine the like terms.
Hint:
1
For (i): $3x(5x) + 3x(3y) + 2y(5x) + 2y(3y)$. Calculate each piece and add them together!
Mistakes to Avoid:
!
Sign Errors: In part (ii), be extremely careful with the negative sign! A positive times a negative is a negative. $3y \times -5y = -15y^2$.
👉 Scroll horizontally to read the full handwritten solution.
Part (i):
$(3x + 2y)(5x + 3y)$
Multiply $3x$ with both terms, then $2y$ with both terms:
$= 3x(5x + 3y) + 2y(5x + 3y)$
$= 15x^2 + 9xy + 10xy + 6y^2$
Combine the middle "like terms" ($9xy + 10xy$):
$= 15x^2 + 19xy + 6y^2$ [1 M]
Part (ii):
$(2x + 3y)(4x - 5y)$
$= 2x(4x - 5y) + 3y(4x - 5y)$
$= 8x^2 - 10xy + 12xy - 15y^2$
Combine the middle "like terms" ($-10xy + 12xy$):
$= 8x^2 + 2xy - 15y^2$ [1 M]
25 Divisibility
Without actual division find the remainder when $379843$ is divided by $3$.
Concept:
1
Remainder Rule for 3: The remainder when any number is divided by 3 is exactly equal to the remainder when the *sum of its digits* is divided by 3.
Hint:
1
Add the digits: $3 + 7 + 9 + 8 + 4 + 3$. Take that total sum and divide it by 3. The remainder you get is your final answer!
Mistakes to Avoid:
!
Long Division: Do not set up a long division bracket for $379843 \div 3$. The question explicitly forbids "actual division" of the large number, and doing it will cost you the marks!
👉 Scroll horizontally to read the full handwritten solution.
To find the remainder without division, we use the divisibility rule for 3.
Step 1: Find the sum of all digits in the number $379843$.
Sum = $3 + 7 + 9 + 8 + 4 + 3$
Sum = $34$ [1 M]
Step 2: Divide the sum by 3 to find the remainder.
$34 \div 3 = 11$ with a remainder of $1$.
Therefore, when $379843$ is divided by 3, the remainder is $1$. [1 M]
Final Answer: 1
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