Read the following quadric equation carefully and answer the questions given below. Equation 1: ax² + bx + c = 0 Equation 2: dy² + ey + c = 0 Note: (i) c is a single digit prime number greater than 2 (ii) d and b are two digits prime number less than 20 (iii) d is greater than 11 (iv) b is greater than d (v) Smallest roots of both the equation are same (vi) No root is irrational (vii) e = b + 1 (viii) 3c is greater than b

Find the value of $a$.

Question

Read the following quadric equation carefully and answer the questions given below. Equation 1: ax² + bx + c = 0 Equation 2: dy² + ey + c = 0 Note: (i) c is a single digit prime number greater than 2 (ii) d and b are two digits prime number less than 20 (iii) d is greater than 11 (iv) b is greater than d (v) Smallest roots of both the equation are same (vi) No root is irrational (vii) e = b + 1 (viii) 3c is greater than b

Find the value of $a$.

Accepted Answer

**Solution:**

Here is the extracted text from the image:
From the conditions given in question 
possible values of c c = 3, 5, 7 
Possible values of d d = 13, 17, 19 
Possible values of b b = 17, 19 
here we can't take b = 13 because b is always greater than d As 3c is greater than d
 if c = 3 3c = 9 (invalid) if c = 5 3c = 15 (invalid)
 if c = 7 3c = 21 (valid) d cannot be 19 because if d = 19 then there will be no value of b such that b will be greater to d 
now, c = 7 
d = 13, 17 b = 17, 19 
Possible value of c, d, b and e respectively
 (i) 7, 13, 17, 18.........(1) 
(ii) 7, 13, 19, 20 .........(2) 
(iii) 7, 17, 19, 20.........(3)
For the roots to be rational d(discriminant) will be perfect square 
For equation 2 
e² – 4dc must be perfect square Using value from (1) 324 – 4 × 13 × 7 = –40 d < 0 
Roots are not real hence, Values are invalid Here is the extracted text from the image:
Using value from (2) 400 - 4 × 13 × 7 = 36 d is a perfect square Roots are rational number hence, values are valid
Using value from (3) 400 - 4 × 17 × 7 = -76 d < 0 
roots are not real hence, values are invalid
So, c = 7 d = 13 b = 19 e = 20
For Equation 2 
13y² + 20y + 7 = 0 
13y² + 13y + 7y + 7 = 0 
y = -7/13, -1 
Smallest root = -1 
Now putting in equation 1 
a(-1)² + 19(-1) + 7 = 0 a = 19 - 7 a = 12

answer :(c) 12

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