Study the following information and answer the questions that follow.
A number series is given as 20, a, b, c, d, 65
Where a, b, c and d are missing terms.
It is also given that:
I. a – 20 = (x2 + y)
II. The value of b is greater than a and the difference of b and a is equal to the
`[(x+1)^2+y]`
III. The value of c is `[(x+2)^2 + y]` more than b and the value of d is `[(x + 3)^2 +y]` more than c.
Note: x is equal to the HCF of 2 prime numbers and the value of y is equal to the smaller root of the quadratic equation `z^2 – z – 6 = 0`.
Which of the following is/are divisible by (y + 5)?

Options

A. A. Only d
B. B. Only a and b
C. C. Only b and d
D. D. Only a, b and d

Explanation

To solve the problem step by step, let's break down the information provided and find the values of \(a\), \(b\), \(c\), and \(d\).

### Step 1: Find \(x\) and \(y\)

1. **Finding \(y\)**:
- The quadratic equation given is \(z^2 - z - 6 = 0\).
- To find the roots, we can factor the equation:
\[ (z + 2)(z - 3) = 0 \]
- The roots are \(z = -2\) and \(z = 3\). The smaller root is \(y = -2\).

2. **Finding \(x\)**:
- \(x\) is defined as the HCF of two prime numbers. The two smallest prime numbers are 2 and 3, and their HCF is \(1\). Therefore, \(x = 1\).

### Step 2: Calculate \(y + 5\)

- Now that we have \(y = -2\):
\[ y + 5 = -2 + 5 = 3 \]

### Step 3: Calculate \(a\)

- The equation for \(a\) is given as:
\[ a - 20 = x^2 + y \]
- Substituting the values of \(x\) and \(y\):
\[ a - 20 = 1^2 + (-2) = 1 - 2 = -1 \]
- Therefore:
\[ a = 20 - 1 = 19 \]

### Step 4: Calculate \(b\)

- The value of \(b\) is greater than \(a\) and the difference between \(b\) and \(a\) is given by:
\[ b - a = (x + 1)^2 + y \]
- Substituting the values:
\[ b - 19 = (1 + 1)^2 + (-2) = 2^2 - 2 = 4 - 2 = 2 \]
- Therefore:
\[ b = 19 + 2 = 21 \]

### Step 5: Calculate \(c\)

- The value of \(c\) is given as:
\[ c = b + (x + 2)^2 + y \]
- Substituting the values:
\[ c = 21 + (1 + 2)^2 + (-2) = 21 + 3^2 - 2 = 21 + 9 - 2 = 28 \]

### Step 6: Calculate \(d\)

- The value of \(d\) is given as:
\[ d = c + (x + 3)^2 + y \]
- Substituting the values:
\[ d = 28 + (1 + 3)^2 + (-2) = 28 + 4^2 - 2 = 28 + 16 - 2 = 42 \]

### Step 7: Check divisibility by \(y + 5\)

- We need to check which of \(a\), \(b\), \(c\), and \(d\) are divisible by \(3\) (since \(y + 5 = 3\)):
- \(a = 19\) (not divisible by 3)
- \(b = 21\) (divisible by 3)
- \(c = 28\) (not divisible by 3)
- \(d = 42\) (divisible by 3)

### Conclusion

The values \(b\) and \(d\) are divisible by \(y + 5\).