11 Quadrilaterals
If the sum of the opposite angles of a quadrilateral is $240^\circ$ and the other two angles are equal. The measure of equal angles is:
Concept:
1
Angle Sum Property of Quadrilaterals: The sum of all four interior angles of any quadrilateral is always exactly $360^\circ$.
Hint:
1
Let the two equal angles be $x$. Set up the equation: $240^\circ + x + x = 360^\circ$. Now solve for $x$!
Mistakes to Avoid:
!
Assuming Parallelogram Properties: Do not assume opposite angles are equal (e.g., $120$ each). The question says it's a general quadrilateral, so just rely on the total sum of $360^\circ$.
👉 Scroll horizontally to read the full handwritten solution.
Let the quadrilateral be $ABCD$.
Sum of two opposite angles = $240^\circ$
Let the other two equal angles be $x$.
By angle sum property of quadrilateral:
Sum of all four angles = $360^\circ$
$240^\circ + x + x = 360^\circ$ [½ M]
$240^\circ + 2x = 360^\circ$
$2x = 360^\circ - 240^\circ$
$2x = 120^\circ$
$x = 60^\circ$ [½ M]
The measure of the equal angles is $60^\circ$.
Correct Option: (A)
12 Squares & Square Roots
How many non-perfect squares are there between $50^2$ and $51^2$?
Concept:
1
Numbers between Squares: The exact number of non-perfect square numbers lying strictly between the squares of two consecutive natural numbers $n$ and $(n+1)$ is always exactly $2n$.
Hint:
1
Identify $n$. Here, our consecutive numbers are $50$ and $51$. Therefore, $n = 50$. Simply use the formula $2n$.
Mistakes to Avoid:
!
Manual Subtraction: If you try to calculate $51^2 - 50^2$, you get $2601 - 2500 = 101$. But you must subtract $1$ more because "between" excludes the endpoints! It's much safer to just use the $2n$ rule.
👉 Scroll horizontally to read the full handwritten solution.
The numbers are $50^2$ and $51^2$.
These are consecutive numbers $n$ and $(n+1)$, where $n = 50$.
We know that there are exactly $2n$ non-perfect square numbers
between the squares of two consecutive numbers $n$ and $(n+1)$. [½ M]
Therefore, the number of non-perfect squares = $2 \times 50$
$= 100$ [½ M]
Correct Option: (B)
13 Exponents & Powers
The number $4,70,394$ in standard form is written as:
Concept:
1
Standard Form (Scientific Notation): A number is in standard form when written as $k \times 10^n$, where $1 \le k < 10$ and $n$ is an integer.
Hint:
1
Place the decimal point immediately after the very first digit (the $4$). Count how many places you had to jump to the left from the end of the number to place it there. That count is your exponent for the $10$.
Mistakes to Avoid:
!
Wrong Decimal Placement: Options like $47.0394$ are incorrect because the first number ($47$) is greater than $10$. Standard form requires the first number to be between $1$ and $9.99$.
👉 Scroll horizontally to read the full handwritten solution.
The given number is $470394$.
To write this in standard form ($k \times 10^n$ where $1 \le k < 10$):
Move the decimal point to the left until only one non-zero digit
remains in front of it.
$470394.0 \rightarrow$ shift decimal $5$ places to the left. [½ M]
This gives us $4.70394$.
Since we shifted $5$ places to the left, we multiply by $10^5$.
Standard form = $4.70394 \times 10^5$ [½ M]
Correct Option: (A)
14 Number Systems
The Roman numeral for 2713 is:
Concept:
1
Roman Numerals Expansion: Break the number down into its place values (Thousands, Hundreds, Tens, Ones) and convert each part individually. $M = 1000$, $D = 500$, $C = 100$, $X = 10$, $I = 1$.
Hint:
1
Break $2713$ into $2000 + 700 + 10 + 3$. For $2000$, use two $M$'s. For $700$, use $D$ ($500$) plus two $C$'s ($200$).
Mistakes to Avoid:
!
Repeating D: Option C has $MMDD$. In Roman Numerals, symbols $V$, $L$, and $D$ are NEVER repeated! Only $I$, $X$, $C$, and $M$ can be repeated (up to three times).
👉 Scroll horizontally to read the full handwritten solution.
Let's expand the number $2713$ based on place value:
$2713 = 2000 + 700 + 10 + 3$ [½ M]
Now convert each part to Roman numerals:
- $2000 = MM$
- $700 = DCC$ (Since $500 = D$ and $100+100 = CC$)
- $10 = X$
- $3 = III$
Combining them all together:
$= MMDCCXIII$ [½ M]
Correct Option: (B)
15 Quadrilaterals
The diagonals of a kite:
Concept:
1
Properties of a Kite: A kite is a quadrilateral with two distinct pairs of equal-length sides that are adjacent to each other. Its diagonals intersect at exactly $90^\circ$ (they are perpendicular).
Hint:
1
Think of a physical kite flying in the sky. The two wooden sticks forming the cross inside it cross exactly at right angles.
Mistakes to Avoid:
!
Bisection Confusion: Many students select Option A. However, in a kite, only the longer main diagonal bisects the shorter cross diagonal. The shorter one does NOT bisect the longer one! So they do not mutually "bisect each other".
👉 Scroll horizontally to read the full handwritten solution.
Let's recall the diagonal properties of a Kite:
1. The diagonals intersect at $90^\circ$ (they are perpendicular). [½ M]
2. Only ONE diagonal bisects the other.
(Therefore, they do not "mutually" bisect each other).
Checking the options:
(A) is incorrect because they don't both bisect each other.
(C) is misleading because one DOES bisect the other.
(B) is perfectly true: They are perpendicular. [½ M]
Correct Option: (B)
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