If $\oplus \div \odot = 2$, $\oplus \div \triangle = 3$, $\odot + \triangle = 5$, and $\Delta \times \otimes = 10$,
then the value of $(\otimes - \oplus)^2$ is:
Solving Algebraic Equations System
We are given a system of equations with symbols:
- Equation 1: $\oplus \div \odot = 2$
- Equation 2: $\oplus \div \triangle = 3$
- Equation 3: $\odot + \triangle = 5$
- Equation 4: $\Delta \times \otimes = 10$
We need to find the value of $(\otimes - \oplus)^2$.
Deriving Variable Values
First, let's express variables in terms of others:
- From Equation 1: $\oplus = 2 \times \odot$
- From Equation 2: $\oplus = 3 \times \triangle$
- Equating the expressions for $\oplus$: $2 \times \odot = 3 \times \triangle \implies \odot = \frac{3}{2} \times \triangle$
Now, substitute $\odot$ into Equation 3:
- $(\frac{3}{2} \times \triangle) + \triangle = 5$
- $\frac{5}{2} \times \triangle = 5$
- Solving for $\triangle$: $\triangle = 5 \times \frac{2}{5} = 2$
Find the values of the other variables:
- $\odot = 5 - \triangle = 5 - 2 = 3$
- $\oplus = 3 \times \triangle = 3 \times 2 = 6$
- From Equation 4: $\otimes = 10 \div \triangle = 10 \div 2 = 5$
Calculating the Final Expression
We need to calculate $(\otimes - \oplus)^2$ using the derived values:
- $\otimes = 5$
- $\oplus = 6$
- $(\otimes - \oplus)^2 = (5 - 6)^2$
- $= (-1)^2$
- $= 1$
The value of $(\otimes - \oplus)^2$ is 1.
Important Questions from Numerical Reasoning
- $P, Q, R, S, X$, and $Y$ are distinct single-digit whole numbers taking values from 0 to 9.
$PQ$ is a two-digit number with $Q$ being in the units place and $P$ in the tens place. Similarly, $RS$ is a two-digit number.
It is known that $PQ$ and $RS$ are consecutive numbers and
$(PQ)^2 + (RS)^2 = XYP$, with $XYP$ being a three-digit number.
The value of $Y$ is __________ - Let $p_1$ and $p_2$ denote two arbitrary prime numbers. Which one of the following statements is correct for all values of $p_1$ and $p_2$?
- The remainder when $98!$ is divided by $101$ is equal to ________
- A 'frabjous' number is defined as a 3 digit number with all digits odd, and no two adjacent digits being the same. For example, 137 is a frabjous number, while 133 is not. How many such frabjous numbers exist?
Ankita has to climb 5 stairs starting at the ground, while respecting the following rules:
1. At any stage, Ankita can move either one or two stairs up.
2. At any stage, Ankita cannot move to a lower step.
Let $F(N)$ denote the number of possible ways in which Ankita can reach the $N^{th}$ stair. For example, $F(1) = 1$, $F(2) = 2$, $F(3) = 3$. The value of $F(5)$ is ________.
