Please can someone help me with my A Level maths homework?

Question

I know It's a lot, but if there is anywhere you could help me it'd be much appreciated :)

1. The points A, B and C have coordinates (-2, 3), (10, -1), (3, 8) respectively. The line through C is perpendicular to AB and AB intersects at point D.

a) what is the gradient of AB?
b) show that AB has the equation of x+3y-7=0? And what is the equation of CD?
c) show that D has coordinates of (1, 2)
d) the midpoint of AB is denoted by E. Find length of ED.

2. Simplify ...
a) s20 + √35 / √7 - 20/ √5
b) 2+ √3 / 5+2 √3

3. a) expand (a+b)^5
b) find the first four terms in the expansion of (1+x/2)^5 in ascending powers of x

4. a) find the range of values of K for which quadratic equation 3x^2 + 2x -K= 0 has distinct real roots
b) solve inequality x^3 - 5x -14 less than or equal to 0

5. a) given that y=3x^2 - 4x + 7, find dy/dx from first principles
b) differentiate 5√x - 3/x^3 with respect to x

6. Factorize 6x^3 + 5x^2 -3x -2

Thanks for any help :)

037 G · Accepted Answer

let A (-2,3) be the first point and B (10,-1) be the 2nd point, So
Slope (gradient) of AB
m = (y2-y1)/(x2-x1)

m = (-1 - 3) / (10 - (-2)) = -4/12 = -1/3

to get the equation for line AB, write the general equation of a straight line and plug in one of the points.

y = mx + c
use point A or B to find c ( I will use point B)

-1 = 10(-1/3) + c
c = 7/3
So equation is 
y = (-1/3)x + 7/3
multiply through by 3 and arrange variables to get
3y + x - 7 = 0

slope of CD is the inverse negative of AB, So CD gradient = +3/1 or just 3.
line equation for CD, do the same as above using point C (3,8) to solve for small c

8 = 3(3) + c
c = -1
So Equation of CD is
y = 3x - 1

to show that D is the intersection point, then it is on line CD & AB and should work when plugged into both equations.  Or solve both equations as a system to get point D's coordinates.

y -  3x = -1
3y + x =  7 

multiply bottom eq by 3 to get

y   -  3x = -1
9y + 3x =  21
----------------------- add both equations vertically

10y = 20
y = 2
and 
x = 1
So D is (1,2)

Length of ED is
½ length of AB

length of AB is

AB² = Δx² + Δy²
AB² = (12)² + (-4)²
AB = √ 160
ED = ½√ 160 = 2√10

Anonymous · Answer

Answering all these for you will take a lot of typing, i can give you tips for each one as you should try and do it yourself!
1a) dy/dx =(y2-y1)/(x2-x1)
b) y=mx+c where m is answer from part a, use coordinate A to find c,  you should knw the product of two perpendicular lines is -1, so -1 /(answer to part a)  gives you your m, use coordinates of C to find answer
c) solve to line equations in b simultaneously
d) draw everything you done, use pythagorus

2)a) rationalise denominator and minipulate
b) same as a

3)a) binomial expansion
b) use equation from part a but replae all x with x/2 and work out coefficients

4a) complete the square, draw graph, should come clear
b) solve the equation, draw a line, sub in values between roots and check, will come up with boundaries

5a) should find this easy, times by the power then reduce the power
b) remember root x is x^1/2, and 1/x^3 is x^-3, then same as part a

6) try subbing in 1, -1, 2, -2, find a root, then long division and then simple factorisation of a quadratic

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Please can someone help me with my A Level maths homework?

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