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BINOMIAL THEOREM — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSBINOMIAL THEOREM1 Mark
MCQ
The number of terms with integral coefficients in the expansion of $\Big(17^{\frac{1}{3}}+35^{\frac{1}{2}}\text{x}\Big)^{600}$ is:
  • A
    100
  • B
    50
  • C
    150
  • ✓
    101
✓

Answer

Correct option: D.
101
  1. 101
Solution:
The general term $T_{r+1}$ in the given expansion is given by ${^\text{600}}\text{C}_{\text{r}}(17^{\frac{1}{3}})^{600-\text{r}}\Big(35^{\frac{1}{2}}\text{x}\Big)^{\text{r}}$
$={^\text{600}}\text{C}_{\text{r}}(17^{\frac{1}{3}})^{200-\frac{\text{r}}{3}}\times\Big(35^{\frac{\text{r}}{2}}\text{x}\Big)^{\text{r}}$
Now, $T_{r+1}$​​​​​​​ is an integer if $\frac{\text{r}}{2}$ and $\frac{\text{r}}{2}$ are integers for all $0\leq\text{r}\leq600$
Thus, we have
r = 0, 6, 12, ....600
Since, It is an A.P
So, $600 = 0 + (\text{n} - 1)6$
$\Rightarrow \text{n}=101$
Hence, there are 101 terms with integral coefficients.

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