Finding an unknown questions with answers

Questions

1. Question 1Find a from the x term

   In the expansion of $(1 + ax)^{5}$, the coefficient of $x$ is $10$. Find the value of $a$.

   Show the worked solution

   1. Step 1: Write the general term Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ for this expansion$T_{r+1} = {}^{5}C_{r}(1)^{5 - r}(ax)^{r}$
   2. Step 2: Choose r for the x term, substitute and simplify$T_{2} = {}^{5}C_{1}(1)^{4}(ax)^{1}$$= 5ax$
   3. Step 3: Its coefficient is 10: set up the equation and solve$5a = 10$$a = 2$
2. Question 2Find a from the x term

   In the expansion of $(1 + ax)^{6}$, the coefficient of $x$ is $-24$. Find the value of $a$.

   Show the worked solution

   1. Step 1: Write the general term Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ for this expansion$T_{r+1} = {}^{6}C_{r}(1)^{6 - r}(ax)^{r}$
   2. Step 2: Choose r for the x term, substitute and simplify$T_{2} = {}^{6}C_{1}(1)^{5}(ax)^{1}$$= 6ax$
   3. Step 3: Its coefficient is -24: set up the equation and solve$6a = -24$$a = -4$
3. Question 3Find a from the x² term

   In the expansion of $(1 + ax)^{4}$, the coefficient of $x^2$ is $150$. Given that $a > 0$, find the value of $a$.

   Show the worked solution

   1. Step 1: Write the general term Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ for this expansion$T_{r+1} = {}^{4}C_{r}(1)^{4 - r}(ax)^{r}$
   2. Step 2: Choose r for the x² term, substitute and simplify$T_{3} = {}^{4}C_{2}(1)^{2}(ax)^{2}$$= 6a^2x^2$
   3. Step 3: Its coefficient is 150: set up the equation and solve$6a^2 = 150$$a^2 = 25$$a = 5$
4. Question 4Find a from the x² term

   In the expansion of $(1 + ax)^{6}$, the coefficient of $x^2$ is $135$. Given that $a > 0$, find the value of $a$.

   Show the worked solution

   1. Step 1: Write the general term Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ for this expansion$T_{r+1} = {}^{6}C_{r}(1)^{6 - r}(ax)^{r}$
   2. Step 2: Choose r for the x² term, substitute and simplify$T_{3} = {}^{6}C_{2}(1)^{4}(ax)^{2}$$= 15a^2x^2$
   3. Step 3: Its coefficient is 135: set up the equation and solve$15a^2 = 135$$a^2 = 9$$a = 3$
5. Question 5Find n

   In the expansion of $(1 + 3x)^{n}$, the coefficient of $x^2$ is $252$. Find the value of $n$.

   Show the worked solution

   1. Step 1: Write the x² term, then write ⁿC₂ in terms of n$T_{3} = {}^{n}C_{2}(3x)^{2}$$= \dfrac{n(n - 1)}{2} \times 9x^2$
   2. Step 2: Its coefficient is 252: set up the equation$\dfrac{n(n - 1)}{2} \times 9 = 252$$n(n - 1) = 56$$n^2 - n - 56 = 0$
   3. Step 3: Factorise and solve: n is a positive whole number$(n - 8)(n + 7) = 0$$n = 8$($n = -7$ is rejected: n can't be negative.)
6. Question 6Find n

   In the expansion of $(1 + x)^{n}$, the coefficient of $x^2$ is $36$. Find the value of $n$.

   Show the worked solution

   1. Step 1: Write the x² term, then write ⁿC₂ in terms of n$T_{3} = {}^{n}C_{2}(x)^{2}$$= \dfrac{n(n - 1)}{2}x^2$
   2. Step 2: Its coefficient is 36: set up the equation$\dfrac{n(n - 1)}{2} = 36$$n(n - 1) = 72$$n^2 - n - 72 = 0$
   3. Step 3: Factorise and solve: n is a positive whole number$(n - 9)(n + 8) = 0$$n = 9$($n = -8$ is rejected: n can't be negative.)