Question

If \(\mathop {\lim }\limits_{n \to \infty }\)\(\sum\limits_{k = 1}^n {\left( {{{\left( {\frac{{3k}}{n}} \right)}^2} + 2} \right)\frac{3}{n}} \) = \(\int\limits_0^6 {f(x)}\) dx, then
1. f(0) = 1
2. f(x) = \(\frac{1}{3}{x^2}-\frac{1}{2}x + 1\)
3. \(\mathop {\lim }\limits_{n \to \infty }\) \(\sum\limits_{k = 1}^n {\left( {{{\left( {\frac{{3k}}{n}} \right)}^2} + 2} \right)\frac{3}{n}} \) = 15
4. \(\mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {\left( {{{\left( {\frac{{3k}}{n}} \right)}^2} + 2} \right)\frac{3}{n}} \) = 5

Answer 1

Correct Answer - Option 3 : \(\mathop {\lim }\limits_{n \to \infty }\) \(\sum\limits_{k = 1}^n {\left( {{{\left( {\frac{{3k}}{n}} \right)}^2} + 2} \right)\frac{3}{n}} \) = 15

Let \(\frac{k}{n}\) = x, \(\frac{1}{n}\) = dx

3\(\int\limits_0^1 {(9{x^2} + 2)} \)dx = \(\int\limits_0^3 {({x^2} + 2)}\)dx

3\(\left[ {\frac{{9{x^3}}}{3} + 2x} \right]_0^1\) = 15