A102416
B104976
C90576
D104776
Answer:
B. 104976
Read Explanation:
Methods to Calculate 324² (324² കണക്കാക്കാനുള്ള വഴികൾ)
1. Direct Multiplication (നേരിട്ടുള്ള ഗുണനം)
The most straightforward method is to multiply 324 by 324:
$324 \times 324 = 104976$
2. Using Algebraic Identity (ബീജഗണിത ഐഡൻ്റിറ്റി ഉപയോഗിച്ച്)
The identity $(a+b)^2 = a^2 + 2ab + b^2$ can be effectively used for mental calculations or breaking down larger numbers.
For $324^2$, we can consider it as $(300 + 24)^2$:
Here, $a = 300$ and $b = 24$.
$a^2 = 300^2 = 90000$
$2ab = 2 \times 300 \times 24 = 600 \times 24 = 14400$
$b^2 = 24^2 = 576$ (It is beneficial to memorize squares up to at least 30 or 50 for competitive exams).
Adding these values: $90000 + 14400 + 576 = 104976$.
Alternatively, you can also use $(325 - 1)^2 = 325^2 - 2(325)(1) + 1^2$. This demonstrates flexibility in choosing 'a' and 'b'.
Verification Techniques (പരിശോധനാ വിദ്യകൾ)
1. Unit Digit Check (യൂണിറ്റ് അക്കം പരിശോധിക്കുക)
The unit digit of $324^2$ must be the unit digit of $4^2$, which is $16$. Therefore, the unit digit of the final answer must be 6.
This technique helps in quickly eliminating incorrect options in multiple-choice questions.
2. Digital Sum (അക്കങ്ങളുടെ തുക)
The digital sum of a number is the sum of its digits, repeatedly summed until a single digit is obtained.
Digital sum of 324: $3 + 2 + 4 = 9$.
The digital sum of $324^2$ should be the digital sum of $9^2 = 81$, which is $8+1 = 9$.
Checking the answer 104976: $1+0+4+9+7+6 = 27$. The digital sum of 27 is $2+7 = 9$. This matches, providing a quick verification.
Note: While useful, the digital sum is not foolproof as different numbers can have the same digital sum. It's best used for quick elimination or verification alongside other methods.