The sum of all possible values of n where n ∈ N , x 0 and 10

Question

The sum of all possible values of n where n ∈ N , x >0 and 10< n ≤ 100 such that the equation [2 x 2]+ x - n =0 has a solution, is equal to [Note: [ x ] denotes largest integer less than or equal to x.] (A) 150 (B) 175 (C) 190 (D) 210

Tardigrade Admin · Accepted Answer

We have [2 x2]+x-n=0 ⇒ x text has to be an integer. ⇒ n =2 x 2+ x = x (2 x +1) ∴ n can be 21,36,55,78 corresponding to x=3,4,5,6. Hence, sum of all possible values of n is equal to 190. Ans. Note: If x is negative also then answer is 435 .]

Q. The sum of all possible values of $n$ where $n \in N, x > 0$ and $10 < n \leq 100$ such that the equation $[2 x^{2}] + x - n = 0$ has a solution, is equal to
[Note: $[x]$ denotes largest integer less than or equal to $x$ .]

150

175

190

210

We have $[2 x^{2}] + x - n = 0$
$\Rightarrow x has to be an integer.$
$\Rightarrow n = 2 x^{2} + x = x (2 x + 1)$
$∴$ n can be $21, 36, 55, 78$ corresponding to $x = 3, 4, 5, 6$ .
Hence, sum of all possible values of $n$ is equal to 190. Ans.
Note: If $x$ is negative also then answer is 435 .]

Q. The sum of all possible values of n where n∈N,x>0 and 10<n≤100 such that the equation [2x2]+x−n=0 has a solution, is equal to [Note: [x] denotes largest integer less than or equal to x.]

150

175

190

210

We have [2x2]+x−n=0 ⇒x has to be an integer. ⇒n=2x2+x=x(2x+1) ∴ n can be 21,36,55,78 corresponding to x=3,4,5,6. Hence, sum of all possible values of n is equal to 190. Ans. Note: If x is negative also then answer is 435 .]

Q. The sum of all possible values of $n$ where $n \in N, x > 0$ and $10 < n \leq 100$ such that the equation $[2 x^{2}] + x - n = 0$ has a solution, is equal to
[Note: $[x]$ denotes largest integer less than or equal to $x$ .]

We have $[2 x^{2}] + x - n = 0$
$\Rightarrow x has to be an integer.$
$\Rightarrow n = 2 x^{2} + x = x (2 x + 1)$
$∴$ n can be $21, 36, 55, 78$ corresponding to $x = 3, 4, 5, 6$ .
Hence, sum of all possible values of $n$ is equal to 190. Ans.
Note: If $x$ is negative also then answer is 435 .]