Prime Number Properties Analysis
This solution examines the properties of statements involving two arbitrary prime numbers, $p_1$ and $p_2$. A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself.
Evaluating Prime Number Statements
- Statement 1: $p_1 + p_2$ is not a prime number.
- Test Case: Let $p_1 = 2$ and $p_2 = 3$. Both are prime.
- Sum: $p_1 + p_2 = 2 + 3 = 5$.
- Result: 5 is a prime number. Therefore, this statement is false as it's not true for all primes.
- Statement 2: $p_1p_2$ is not a prime number.
- Definition: A prime number has only 1 and itself as divisors.
- Consider the product $p_1p_2$. Since $p_1$ and $p_2$ are primes, they are both greater than 1.
- The divisors of $p_1p_2$ include $1$, $p_1$, $p_2$, and $p_1p_2$.
- As $p_1 > 1$ and $p_2 > 1$, $p_1$ is a divisor other than 1 and $p_1p_2$. Similarly for $p_2$.
- Therefore, $p_1p_2$ always has more than two divisors, making it a composite number.
- Result: This statement is always true.
- Statement 3: $p_1 + p_2 + 1$ is a prime number.
- Test Case: Let $p_1 = 2$ and $p_2 = 3$.
- Sum + 1: $p_1 + p_2 + 1 = 2 + 3 + 1 = 6$.
- Result: 6 is not a prime number. Therefore, this statement is false.
- Statement 4: $p_1p_2 + 1$ is a prime number.
- Test Case: Let $p_1 = 3$ and $p_2 = 5$. Both are prime.
- Product + 1: $p_1p_2 + 1 = (3)(5) + 1 = 15 + 1 = 16$.
- Result: 16 is not a prime number. Therefore, this statement is false.
Correct Statement Identification
The analysis confirms that only the second statement, "$p_1p_2$ is not a prime number", is universally true for any two prime numbers $p_1$ and $p_2$. This is because the product of two primes will always have $p_1$ and $p_2$ as factors, in addition to 1 and the product itself, disqualifying it from being prime.
Important Questions from Numerical Reasoning
- 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 ________.- In a zoo, three lions and four tigers eat 390 kg of food every week. In another zoo, four lions and five tigers eat 500 kg of food every week. Lions and tigers eat different amounts of food, but all individuals of the same species eat the same amount. The amount of food a single lion eats per week is ________ kg.
(Answer in integer) Consider a spherical globe rotating about an axis passing through its poles. There are three points P, Q, and R situated respectively on the equator, the north pole, and midway between the equator and the north pole in the northern hemisphere. Let P, Q, and R move with speeds $v_P$, $v_Q$, and $v_R$, respectively.
Which one of the following options is CORRECT?
A dozer pushes up a 100 kg spool of cable along a $20^ \circ$ incline road at a constant velocity as shown in the figure. The figure shows a dozer pushing a spool (diameter 400 mm) up a $20^ \circ$ incline. Point A is the contact point between the spool and the road, and Point B is the contact point between the dozer bucket and the spool. The coefficient of static friction between the dozer bucket and the spool (Point B) is 0.45, and coefficient of kinetic friction between road and the spool (Point A) is 0.15.
Consider the spool only slides up the incline. The maximum normal force in N acting at Point B, is ___________ [rounded off to 1 decimal place]
