(a) Expand the following: [3] i. (a + 2b – 3c)2 ii.   2 4 5  x (b) Find the cube root of 74088. [3] (c) Let A = {factors of 24} and B = {factors of 30},

A couple more:

 

If two adjacent sides of a rectangle are (5x2 + 25xy + 4y2) and (2x2 – 2xy + 3y2), find its area

Area of a rectangle is just the product of its adjacent sides:

 

Area = (5x² + 25xy + 4y²)(2x² - 2xy + 3y²)

=  5x² *(2x² - 2xy + 3y²) + 25xy*(2x² - 2xy + 3y²) + 4y²*(2x² - 2xy + 3y²)

= 10x⁴ - 10x³y +15x²y²   +  50x³y -50 x²y² +75xy³   + 8x²y² - 8 xy³ + 12y⁴

 

I'll leave you to simplify this by collecting like terms together.

 

 

 

A two digit number is three times the sum of its digits. If 45 is added to the number; its digits are reversed. Find the number

 A two digit number can be represented as 10a + b  where a and b are single digits (a>0).

So:  10a + b = 3(a + b)         ...(1)

       10a + b + 45 = 10b + a  ...(2)

   

 From (1):  10a + b = 3a + 3b  or  7a - 2b = 0      ...(3)

From (2):    9a  - 9b = -45       or     a - b = -5       ...(4)

From (4) we have a = b - 5  ...(5) 

Use (5) in (3): 7(b-5) - 2b = 0  or 7b - 35 - 2b = 0 or 5b = 35  so b = 7.

Put this back into (5): a = 7 - 5  or a = 2.

So the original 2-digit number was 27

 

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Here are a few to be going on with:

 

(a) (i) Distribute the 2 over the other terms (i.e. just multiply the terms in brackets by 2): 2a + 4b - 6c

(ii) Doesn't show clearly in my browser.

 

(b) Just use the calculator.

 

(c) Factors of 24: A = {1, 2, 3, 4, 6, 8, 12, 24}   

     Factors of 30: B = {1, 2, 3, 5, 6, 10, 15, 30} 

    A ∪ B means the union of A and B:  A ∪ B = {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 24, 30}

    A ∩ B means the intersection of A and B (i.e. the items that are common to both): A ∩ B = {1, 2, 3, 6}

    A - B means the items that are left when you take away from A any items that are also in B: A - B = {4, 8, 12, 24}

   I assume n(A) means the number of items in A:

    n(A) = 8,   n(B) = 8,  n(A ∪ B) = 12, n(A ∩ B) = 4, n(A - B) = 4

    n(A) - n(A ∩ B) = 8 - 4 = 4 = n(A - B)

    n(A ∪ B) - n(B) = 12 - 8 = 4 = n(A - B)

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