JEE Main 2016 (2)

Algebra Level 4

Given that $f(x)=x^{2}-px+q$ has has two prime numbers as roots and $p+q=11$ where $p$ is an odd positive integer.

$f(1)+f(2)+f(3)+f(4)+\cdots+f(48)$

If the value of the above expression is $A$ , then find $\dfrac{A}{16}-11$ .

Try my set JEE main 2016 .

2017 None of these choices 2015 2018 2016.5

3 solutions

Akshat Sharda
Mar 20, 2016

Let $a$ and $b$ be the roots of $f(x)=x^2-px+q$ , then the sum of roots of $f(x)$ is $p$ and multiplication of roots of $f(x)$ is equal to $q$ .

$p+q=11 \Rightarrow a+b+ab=11$

As $p$ is odd, one of the roots have to be $2$ . Let, $a=2$ , then the value of $b=3$ .

Therefore, $f(x)=x^2-5x+6$ .

$\begin{aligned} & = f(1)+f(2)+f(3)+f(4)+\ldots +f(48) \\ & = \displaystyle \sum_{1≤x≤48} (x^2-5x+6) \\ & = \frac{48\cdot 49 \cdot 97}{6}-5\cdot \frac{48\cdot 49}{2}+6\cdot 48 = 32432 \\ & \Rightarrow \frac{32432}{16}-11=2027-11=\boxed{2016}\end{aligned}$

If $a$ = 3 and $b$ = 5 then $a + b + ab = 3 + 5 + 3\cdot5 = 23 \neq$ an even number. Rather we can say that $p + q = 11$ and $p$ is odd implying $q$ must be even under given conditions. As $q$ is product of two primes and is even, one of the primes must be $2$ .

Rohit Hill - 5 years, 2 months ago

Sorry, that was a silly error.

Akshat Sharda - 5 years, 2 months ago

You can add one to that equation and get (a+1)(b+1)=12.

Kushagra Sahni - 5 years, 2 months ago

one root is 2 because P is odd not because of p+q=11 since for primes >2 odd(a) +odd(b) + odd(ab) = odd

Somesh Patil - 5 years, 2 months ago

What's that?!

Rohit Hill - 5 years, 2 months ago

p denotes sum of roots of the quadratic equation and since for primes > 2 sum cannot be odd one root must be 2

Somesh Patil - 5 years, 2 months ago

I've corrected my mistake. Thanks!

Akshat Sharda - 5 years, 2 months ago

To minimise calculation note that from $x \geq 4$ , the expression can be rewritten as $\displaystyle \sum_{x=1}^{45} (x+1)(x) = \frac{45\times 46 \times 91}{6} + \frac{45\times 46}{2} = 32430$

and also, $f(2)=f(3)=0 , f(1)=2 \Rightarrow A = 2+ 32430 = 32432$

Vishnu Bhagyanath - 5 years, 2 months ago

Same way !!!

abc xyz - 5 years, 2 months ago

Shourya Pandey
Mar 21, 2016

Yippee! I guessed the solution because I assumed that the answer was going to be $2016$ :P

For you these type of questions are anyway a piece of cake aren't they?

Kushagra Sahni - 5 years, 2 months ago

Umair Siddiqui
Mar 21, 2016

Nice problem...i hope this comes in Jee 2016😂😂

JEE Main 2016 (2)

Try my set JEE main 2016 .

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