# Question 1:

## Binomial Expansion of $(2-3x)^5$

**Working/Answer:**

$$\left[(2-3x)^5\right] = \ldots + \binom{5}{1}2^4(-3x) + \binom{5}{2}2^3(-3x)^2 + \ldots$$

$$= 32, \quad -240x, \quad +720x^2$$

| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt at binomial with correct coefficient combined with correct power of $x$ | M1 | Need correct binomial coefficient with correct power of $x$; ignore errors in powers of 2 or 3, sign or bracket errors |
| $32$ | B1 | Must be simplified to $32$; must be the only constant term in final answer |
| $-240x$ | A1 | cao; $x$ is required; not $\pm 240x$ |
| $+720x^2$ | A1 | cao; can follow omission of negative sign in working |

**Total: 4 marks**

**Special Case:** Descending powers of $x$:
$$(-3x)^5 + 2\times5\times(-3x)^4 + 2^2\times\binom{5}{3}\times(-3x)^3 + \ldots = -243x^5 + 810x^4 - 1080x^3 + \ldots$$
Award as s.c. M1B1A0A0 if completely correct, or M1B0A0A0 for correct binomial coefficient with correct power of $x$

**Alternative Method 1:**
$$\left[(2-3x)^5\right] = 2^5\left(1 + \binom{5}{1}\left(\frac{-3x}{2}\right) + \binom{5}{2}\left(\frac{-3x}{2}\right)^2 + \ldots\right)$$
**M1B0A0A0** — M1 for correct binomial coefficient combined with correct power of $x$ in the bracket; answers must simplify to $32, -240x, +720x^2$ for full marks

**Method 2 (Multiplying out):** B1 for $32$; M1A1A1 for other terms with M1 awarded if $x$ or $x^2$ term correct. Completely correct is 4/4.