Question

Assuming that for integers $ a,b,c $ , the function is $ f(x) = a{x^2} + bx + c $ . Let the conditions be: $
  f(1) = 0\;, \\
  40 < f(6) < 50, \\
  60 < f(7) < 70,\; \\
  1000t < f(50) < 1000(t + 1) \\
  $
; for some $ t $ .
The task is to find out the value of $ t $ from the following;
(a) $ 2 $
(b) $ 3 $
(c) $ 4 $
(d) $ 5\;or\;more $

Answer 1

Hint: When we look at what is given to us we see a number of relations to the function $ f(x) = a{x^2} + bx + c $ . We need to observe the relations carefully and establish connections between each relation. Find patterns and use it to solve the inequalities among the given relations to find the value of $ t $ . Keep in mind the properties of $ a,b,c $ since they are integers, it is a crucial step in solving this problem.

Complete step-by-step answer:
Integers are given as: $ a,b,c $ , for the function $ f(x) = a{x^2} + bx + c $
Note down what is given to us in the question above;
 $
  f(1) = 0\;, \\
  40 < f(6) < 50, \\
  60 < f(7) < 70,\; \\
  1000t < f(50) < 1000(t + 1) \;
  $
We need to find the value of $ t $ that can satisfy the inequality;
Let us start solving with the first inequality:
If it is said that;
 $
  f(1) = 0 \\
   \Rightarrow a + b + c = 0\; \to (1) \\
  $
It is also said that;
 $
  40 < f(6) < 50 \\
   \Rightarrow 40 < 36a + 6b + c < 50 \;
  $
But we know from $ (1) $ ; that $ (a + b + c) = 0 $
So separating: $ 36a + 6b + c $ as its equivalent to;
 $
   \Rightarrow 35a + 5b + (a + b + c) \\
   \Rightarrow 35a + 5b \;
  $
Further we can bring back the previous inequality and write it as;
 $ \Rightarrow 40 < 35a + 5b < 50 $
Simplifying the above equation by dividing it throughout by $ 5 $
 $ \Rightarrow 8 < 7a + b < 10 $
Remember that it is said that $ a $ and $ b $ are integers, so the only possible integers that will satisfy the inequality would imply that;
 $ \Rightarrow 7a + b = 9\; \to (2),\, $ only integer between $ 8 $ and $ 10 $
Similarly proceed with solving the following inequalities;
 $
  60 < f(7) < 70 \\
   \Rightarrow 60 < 49a + 7b + c < 70 \\
   \Rightarrow 60 < 48a + 6b < 70,\;\because \;(a + b + c) = 0 \;
  $
Now dividing throughout by $ 6 $ we get,
 $ \Rightarrow 10 < 8a + b < \dfrac{{35}}{3} $
Now similar to the solution of previous inequality, we eliminate all possibilities which are not integers and conclude that $ 8a + b = 11 \to (3) $
Taking the equalities $ (2) $ and $ (3) $ and solving them, we know;
 $
   \Rightarrow 8a + b - (7a + b) = 11 - 9 \\
   \Rightarrow a = 2 \;
  $
If $ a = 2 $ , we can calculate value of $ b $ to be ; $ b = - 5 $
Solving $ (1) $ by substituting; $ a = 2 $ and $ b = - 5 $
 $ \Rightarrow \therefore c = 3 $
Substituting the found values of $ a,b,c $ in the original function $ f(x) $
 $ \Rightarrow f(x) = 2{x^2} - 5x + 3 $
Now find $ f(50) $
 $
   \Rightarrow f(50) = 2 \times {(50)^2} + 5 \times (50) + 3 \\
   \Rightarrow f(50) = 5000 - 250 + 3 \\
   \Rightarrow f(50) = 4753 \;
  $
But in the question it is said that:
 $
  \because 1000t < f(50) < 1000(t + 1) \\
   \Rightarrow t = 4 \;
  $
Therefore the correct value of $ t $ is $ 4 $ .
Right option is option (c) $ 4 $
So, the correct answer is “OPTION C”.

Note: There are different types of equations, depending on the degree of each equation. If the degree is $ 1 $ the equation is said to be linear, if the degree is $ 2 $ then it is a quadratic equation and so on. The names have come about because depending on what the degree of the equation is the number of roots or solutions for that equation is the degree itself. Solving an equation can be done manually or using matrices or any other method.