Product Of Two Vectors

We are given the equation , which has two equal roots. Let the two equal roots be , and the other two roots be and . Thus, the polynomial can be factored as: By expanding the factored form: and multiplying this with , we get: Expanding this product gives the equation: By comparing the coefficients with the original equation , we obtain the system of equations: 1. , 2. , 3. , 4. .
Step 1: Solving the system of equations From equation 1, we have .
Substitute this into the second equation: Now, substitute into the third equation: Now solve this system of equations. After solving, we find . Thus, the value of is .

To find a unit vector perpendicular to the vectors and , we proceed as follows: Step 1: Compute the cross product The cross product of and is:

Expand the determinant: Simplify: Step 2: Normalize the cross product To find a unit vector perpendicular to and , normalize : First, compute the magnitude of : Thus, the unit vector is: Step 3: Compare with the options The unit vector matches option (3): Final Answer:

Step 1: Define the events:
- : Product is even.
- : Product is divisible by 4.
- : Product is not divisible by 4 (i.e., divisible by 2 but not 4). We need to find , the probability that the product is even but not divisible by 4.
Step 2: Calculate the total number of possible outcomes. - There are 100 numbers to choose from, and we are choosing 2 numbers with replacement, so the total number of possible outcomes is .
Step 3: Compute , the probability that the product is even.
- A product is even if at least one number is even. Since half the numbers from 1 to 100 are even (i.e., 50 even numbers), the probability that a randomly chosen number is even is .
- The probability that both numbers are odd (and thus the product is odd) is .
- Therefore, the probability that the product is even (at least one number is even) is: Step 4: Compute , the probability that the product is divisible by 4.
- For the product to be divisible by 4, at least one of the numbers must be divisible by 4. Numbers divisible by 4 between 1 and 100 are 4, 8, 12, ..., 100. There are 25 such numbers.
- The probability that both numbers are divisible by 4 is .
- Therefore, the probability that the product is divisible by 4 is:
Step 5: Compute , the probability that the product is even but not divisible by 4.
- This occurs when one number is divisible by 2 (i.e., even), but neither number is divisible by 4.
The probability that a number is divisible by 2 but not 4 is .
- The probability that both numbers are divisible by 2 but not by 4 is .
- Thus, .