1. ## questions about binomial expansions, coordinate geometry

1.
Given that the expansion of (a+x)(1-2x)^n in ascending powers of x is 3-41x+bx^2+.....Find the values of the constants a, n and b.

2.
Given that the coefficient of x^2 in the expansion of (k+x)[2-(x/2)]^6 is 84, find the value of the constant k.

3.
Solutions to this question by accurate drawing will not be accepted. A parallelogram ABCD in which A(8,2) and B(2,6). The equation of BC is y=(1/2)x+5 and X is the point on BC such that AX is perpendicular to BC. Equation of AX is y=-2x+18. Coordinate of X is (5.2,7.6). Given also that BC = 5 BX, find the coordinates of C and D.

2. Originally Posted by wintersoltice
1.
Given that the expansion of (a+x)(1-2x)^n in ascending powers of x is 3-41x+bx^2+.....Find the values of the constants a, n and b.
$\displaystyle (1 - 2x)^n \approx 1^n + n \cdot 1^{n - 1} \cdot (-2x)^1 + \frac{n(n - 1)}{2} \cdot 1^{n - 2} \cdot (-2x)^2$

So
$\displaystyle (a + x)(1 - 2x)^n \approx (a + x)(1 - 2nx + 2n(n - 1)x^2)$

$\displaystyle = a + (1 - 2na)x + (2n(n - 1)a - 2n)x^2$ (Keeping only terms to second order in x.)

-Dan

3. 3.
Solutions to this question by accurate drawing will not be accepted. A parallelogram ABCD in which A(8,2) and B(2,6). The equation of BC is y=(1/2)x+5 and X is the point on BC such that AX is perpendicular to BC. Equation of AX is y=-2x+18. Coordinate of X is (5.2,7.6). Given also that BC = 5 BX, find the coordinates of C and D.

This is good, you have been given coordinates of X, and you know that the magnitude of vector BC is 5 times that of vector BX

First find BX:

BX= ((5.2-2),(7.6-6))=(3.2,1.6)

now BC = 5BX= 5(3.2,1.6) = (16,8)

So now you know that point C has position vector (16,8) relative to B,

so that C=((16+2),(8+6)) = (18,14)

Now to find D, note that vector CD is the same as vector BA,

and BA= ((8-2),(2-6)) = (6,-4)

so CD = (6,-4)

and you find that D is at ((18+6),(14-4))= (24,10)

Hope this helps?

4. Hello, wintersoltice!

3. A parallelogram $\displaystyle ABCD$ in which $\displaystyle A(8,2)$ and $\displaystyle B(2,6).$
The equation of $\displaystyle BC$ is: .$\displaystyle y\:=\:\frac{1}{2}x+5$
$\displaystyle X$ is the point on $\displaystyle BC$ such that $\displaystyle AX \perp BC$
Equation of $\displaystyle AX$ is: .$\displaystyle y\:=\:-2x+18$ .
. . . not needed
Coordinates of $\displaystyle X$ are $\displaystyle (5.2,7.6)$
Given that $\displaystyle BC = 5 BX$, find the coordinates of $\displaystyle C\text{ and }D.$
Code:
      |
|                             o C
|                        *
|                   *
|          X   *
|          * (5.2,7.6)
|   B o     *
|   (2,6)    *
|             *
|              *
|               o(8,2)
|               A
- + - - - - - - - - - - - - - - - - -
|
$\displaystyle BC$ is five times $\displaystyle BX.$
Moving from B to X, we move 3.2 right and 1.6 up.

Moving from B to C, we will move 5 times as much.
. . $\displaystyle \begin{array}{ccccc}x &=& 2 + 5(3.2) &=& 18 \\ y &=& 6 + 5(1.6) &=& 14\end{array}\quad\Rightarrow\quad C(18,14)$

Moving from B to A, we move 6 right and 4 down.
Moving from C to D, we will move the same.
. . $\displaystyle \begin{array}{ccccc}x &=& 18 + 6 &=& 24 \\y &=& 14 - 4 &=& 10\end{array} \quad\Rightarrow\quad D(24,10)$