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Equations, SM-Bank 067

Find the value of  \(\large r\) in the formula \(A=\pi r^2\) given \(A=10\). Assume  \(\large r\) is positive and give your answer correct to 1 decimal place.  (2 marks)

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\(r\approx 1.8\)

Show Worked Solution
\(A\) \(=\pi r^2\)
\(\pi r^2\) \(=10\)
\(r^2\) \(=\dfrac{10}{\pi}\)
\(\sqrt{r^2}\) \(=\sqrt{\dfrac{10}{\pi}}\)
\(r\) \(=1.784\ldots\)
  \(\approx 1.8\ (1\text{ d.p.})\)

Equations, SM-Bank 066

Solve the quadratic equation  \(x^2+3=147\)  for  \(x<0\).  (2 marks)

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\(x=-12\)

Show Worked Solution
\(x^2+3\) \(=147\)
\(x^2\) \(=147-3\)
\(x^2\) \(=144\)
\(\sqrt{x^2}\) \(=\sqrt{144}\)
\(x\) \(=\pm12\)

 
\(\therefore\ \text{Since}\ x<0\ x=-12\)

Equations, SM-Bank 065

Solve the quadratic equation  \(x^2-1=48\)  for  \(x>0\).  (2 marks)

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\(x=7\)

Show Worked Solution
\(x^2-1\) \(=48\)
\(x^2\) \(=48+1\)
\(x^2\) \(=49\)
\(\sqrt{x^2}\) \(=\sqrt{49}\)
\(x\) \(=\pm7\)

 
\(\therefore\ \text{Since}\ x>0\ \ x=7\)

Equations, SM-Bank 064

A square has an area of \(121\ \text{cm}^2\).

  1. Find the side length of the square.  (2 marks)

  2. Find the perimeter of the square.  (2 marks)

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a.    \(11\ \text{cm}\)

b.    \(44\ \text{cm}\)

Show Worked Solution
a.    \(A\) \(=s^2\)
  \(A\) \(=121\ \ \text{(given)}\)
  \(\therefore\ s^2\) \(=121\)
  \(\sqrt{s^2}\) \(=\sqrt{121}\)
  \(s\) \(=11\)

 
\(\text{Side length}=11\ \text{cm}\)

b.    \(\text{Perimeter}\) \(=4\times s\)
    \(=4\times 11\)
    \(=44\)

 
\(\text{Perimeter}=44\ \text{cm}\)

Equations, SM-Bank 063

A square has an area of \(36\ \text{m}^2\).

  1. Find the side length of the square.  (2 marks)

  2. Find the perimeter of the square.  (2 marks)

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a.    \(6\ \text{m}\)

b.    \(24\ \text{m}\)

Show Worked Solution
a.    \(A\) \(=s^2\)
  \(A\) \(=36\ \ \text{(given)}\)
  \(\therefore\ s^2\) \(=36\)
  \(\sqrt{s^2}\) \(=\sqrt{36}\)
  \(s\) \(=6\)

 
\(\text{Side length}=6\ \text{m}\)

b.    \(\text{Perimeter}\) \(=4\times s\)
    \(=4\times 6\)
    \(=24\)

 
\(\text{Perimeter}=24\ \text{m}\)

Equations, SM-Bank 062

Solve, giving all solutions correct to 1 decimal place.

  1. \(x^2=8\)  (2 marks)

  2. \(x^2=323\)  (2 marks)

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a.    \(x=\pm 2.8\)

b.    \(x=\pm 18.0\)

Show Worked Solution
a.    \(x^2\) \(=8\)
  \(\sqrt{x^2}\) \(=\pm\sqrt{8}\)
  \(x\) \(=\pm 2.828…\ =\pm 2.8\)

 

b.    \(x^2\) \(=323\)
  \(\sqrt{x^2}\) \(=\pm\sqrt{323}\)
  \(x\) \(=\pm 17.972…\ =\pm 18.0\)

Equations, SM-Bank 061

Solve, giving all solutions.

  1. \(x^2=25\)  (2 marks)

  2. \(x^2=289\)  (2 marks)

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a.    \(x=\pm 5\)

b.    \(x=\pm 17\)

Show Worked Solution
a.    \(x^2\) \(=25\)
  \(\sqrt{x^2}\) \(=\pm\sqrt{25}\)
  \(x\) \(=\pm 5\)

 

b.    \(x^2\) \(=289\)
  \(\sqrt{x^2}\) \(=\pm\sqrt{289}\)
  \(x\) \(=\pm 17\)

Equations, SM-Bank 060

Show that \(x^2=64\) has 2 solutions.  (2 marks)

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\(\text{See worked solutions}\)

Show Worked Solution
\(8^2=8\times 8=64\ \ \therefore \ x=8\ \text{is a solution.}\)
\((-8)^2=-8\times -8=64\ \ \therefore \ x=-8\ \text{is a solution.}\)

 
\(\therefore x=8\ \text{and }x=-8\ \text{are both solutions of }x^2=64\)

Equations, SM-Bank 059

Show that \(x^2=100\) has 2 solutions.  (2 marks)

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\(\text{See worked solutions}\)

Show Worked Solution
\(10^2=10\times 10=100\ \ \therefore \ x=10\ \text{is a solution.}\)
\((-10)^2=-10\times -10=100\ \ \therefore \ x=-10\ \text{is a solution.}\)

 
\(\therefore x=10\ \text{and }x=-10\ \text{are both solutions of }x^2=100\)

Equations, SM-Bank 058

Show that \(x^2=9\) has 2 solutions.  (2 marks)

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\(\text{See worked solutions}\)

Show Worked Solution
\(3^2=3\times 3=9\ \ \therefore \ x=3\ \text{is a solution.}\)
\((-3)^2=-3\times -3=9\ \ \therefore \ x=-3\ \text{is a solution.}\)

 
\(\therefore x=3\ \text{and }x=-3\ \text{are both solutions of }x^2=9\)

Equations, SM-Bank 057 MC

Which of the following is not equal to \(16\)?

  1. \(8^2\)
  2. \((-4)^2\)
  3. \(4\times 4\)
  4. \(2^4\)
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\(A\)

Show Worked Solution
\(\text{Option A: }\ \ \) \(8^2=8\times 8=64\ne 16\)
\(\text{Option B: }\ \ \) \((-4)^2=-4\times -4=16\ \ \checkmark\)
\(\text{Option C: }\ \ \) \(4\times 4=16\ \ \checkmark\)
\(\text{Option D: }\ \ \) \(2^4=2\times 2\times 2\times 2=16\ \ \checkmark\)

 
\(\Rightarrow A\)

Equations, SM-Bank 056 MC

Which of the following is not equal to  \(9\)?

  1. \(-3\times -3\)
  2. \(4.5^2\)
  3. \(\sqrt{81}\)
  4. \(3^2\)
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\(B\)

Show Worked Solution
\(\text{Option A: }\ \ \) \(-3\times -3=9\ \ \checkmark\)
\(\text{Option B: }\ \ \) \(4.5^2=20.25\ne 9\)
\(\text{Option C: }\ \ \) \(\sqrt{81}=9\ \ \checkmark\)
\(\text{Option D: }\ \ \) \(3^2=9\ \ \checkmark\)

\(\Rightarrow B\)

Special Properties, SMB-013 MC

Which statement is always true?

  1. Scalene triangles have two angles that are equal.
  2. All angles in a parallelogram are equal.
  3. The opposite sides of a trapezium are equal in length.
  4. The diagonals of a rhombus are perpendicular to each other.
Show Answers Only

`D`

Show Worked Solution

`text{Consider each option:}`

`A:\ \text{Isosceles (not scalene) have two equal angles.}`

`B:\ \text{Only opposite angles in a parallelogram are equal.}`

`C:\ \text{At least one pair of opposite sides of a trapezium are not equal.}`

`D:\ \text{Rhombuses have perpendicular diagonals.}`

`=>D`

Equations, SM-Bank 056

The formula for converting degrees Celsius to Fahrenheit is  \(F=\dfrac{9C}{5}+32\).

  1. Find \(F\) is \(C=35\).  (2 marks)

  2. Find \(C\) is \(F=68\).  (2 marks)

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a.   \(F=95\)

b.   \(C=20\)

Show Worked Solution
a.    \(F\) \(=\dfrac{9C}{5}+32\)
  \(F\) \(=\dfrac{9\times 35}{5}+32\)
  \(F\) \(=\dfrac{315}{5}+32\)
  \(F\) \(=63+32=95\)

 

b.    \(F\) \(=\dfrac{9C}{5}+32\)
  \(68\) \(=\dfrac{9C}{5}+32\)
  \(\dfrac{9C}{5}\) \(=68-32\)
  \(\dfrac{9C}{5}\) \(=36\)
  \(9C\) \(=36\times 5\)
  \(9C\) \(=180\)
  \(C\) \(=\dfrac{180}{9}=20\)

Equations, SM-Bank 055

Solve  \(2(x+3)=15\).  (2 marks)

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\(x=4.5\)

Show Worked Solution
\(2(x+3)\) \(=15\)
\(2\times x+2\times 3\) \(=15\)
\(2x+6\) \(=15\)
\(2x\) \(=9\)
\(x\) \(=\dfrac{9}{2}\)
\(x\) \(=4.5\)

Equations, SM-Bank 054

Solve  \(3(x-1)=24\).  (2 marks)

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\(x=9\)

Show Worked Solution
\(3(x-1)\) \(=24\)
\(3\times x+3\times -1\) \(=24\)
\(3x-3\) \(=24\)
\(3x\) \(=27\)
\(x\) \(=\dfrac{27}{3}\)
\(x\) \(=9\)

Equations, SM-Bank 053

Write an algebraic equation for the perimeter of the rectangle below and use it to calculate the value of \(x\) given the perimeter is \(62\).  (2 marks)

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\(x=3\)

Show Worked Solution

\(P=2l+2w\)

\(62\) \(=2(7x+1)+2\times 3x\)
\(62\) \(=2\times 7x+2\times 1 +6x\)
\(62\) \(=14x+2 +6x\)
\(20x+2\) \(=62\)
\(20x\) \(=60\)
\(x\) \(=3\)

Equations, SM-Bank 052

An isosceles triangle has a perimeter of  \(46\)  and its base is 12. Find the length of its equal sides.  (2 marks)

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\(l=17\)

Show Worked Solution
\(P\) \(=2l+b\)
\(46\) \(=2l+12\)
\(2l\) \(=34\)
\(l\) \(=\dfrac{34}{2}\)
\(l\) \(=17\)

Equations, SM-Bank 050

Find the value of  \(c\)  in the formula  \(c=\sqrt{a^2+b^2}\)  if  \(a=12\)   and  \(b=5\).  (2 marks)

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\(c=13\)

Show Worked Solution
\(c\) \(=\sqrt{a^2+b^2}\)
\(c\) \(=\sqrt{12^2+5^2}\)
\(c\) \(=\sqrt{144+25}\)
\(c\) \(=\sqrt{169}\)
\(c\) \(=13\)

Equations, SM-Bank 048

Solve  \(\dfrac{2n+3}{2}=-1\).  (2 marks)

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\(n=-2\dfrac{1}{2}\)

Show Worked Solution
\(\dfrac{2n+3}{2}\) \(=-1\)
\(2n+3\) \(=-1\times 2\)
\(2n+3\) \(=-2\)
\(2n\) \(=-5\)
\(n\) \(=\dfrac{-5}{2}\)
\(n\) \(=-2\dfrac{1}{2}\)

Equations, SM-Bank 047

Solve  \(\dfrac{3x}{5}-8=1\).  (2 marks)

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\(x=15\)

Show Worked Solution
\(\dfrac{3x}{5}-8\) \(=1\)
\(\dfrac{3x}{5}\) \(=1+8\)
\(\dfrac{3x}{5}\) \(=9\)
\(3x\) \(=9\times 5\)
\(3x\) \(=45\)
\(x\) \(=15\)

Equations, SM-Bank 044

Find the value of  \(A\)  in the formula  \(A=\dfrac{h}{2}(a+b)\)  if  \(h=8\),  \(a=7\)   and  \(b=3\).  (2 marks)

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\(A=40\)

Show Worked Solution
\(A\) \(=\dfrac{h}{2}(a+b)\)
\(A\) \(=\dfrac{8}{2}(7+3)\)
\(A\) \(=4\times 10\)
\(A\) \(=40\)

Equations, SM-Bank 035

Two consecutive integers have a sum of 147.

  1. Given the first number is \(x\), write an algebraic equation to represent this information.  (2 marks)

  2. Solve your equation algebraically to find the two numbers.  (2 marks)

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a.    \(2x+1=147\)

b.    \(73 , 74\)

Show Worked Solution

a.    \(\text{Given the 1st number is}\ x\text{, the 2nd number is }x+1.\)

\(\therefore\ \text{Equation is:}\)

\(x+(x+1)\) \(=147\)
\(2x+1\) \(=147\)

 

b.    \(2x+1\) \(=147\)
  \(2x\) \(=146\)
  \(x\) \(=73\)

 
\(\therefore\ \text{Numbers are }73\ \text{and }74.\)

Equations, SM-Bank 034

Two consecutive integers have a sum of 25.

  1. Given the first number is \(x\), write an algebraic equation to represent this information.  (2 marks)

  2. Solve your equation algebraically to find the two numbers.  (2 marks)

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a.    \(2x+1=25\)

a.    \(12 , 13\)

Show Worked Solution

a.    \(\text{Given the 1st number is}\ x\text{, the 2nd number is }x+1.\)

\(\therefore\ \text{Equation is:}\)

\(x+(x+1)\) \(=25\)
\(2x+1\) \(=25\)
b.    \(2x+1\) \(=25\)
  \(2x\) \(=24\)
  \(x\) \(=12\)

 
\(\therefore\ \text{Numbers are }12\ \text{and }13.\)

Special Properties, SMB-010 MC

Which of these are always equal in length?

  1. the diagonals of a rhombus
  2. the diagonals of a parallelogram
  3. the opposite sides of a parallelogram
  4. the opposite sides of a trapezium
Show Answers Only

`C`

Show Worked Solution

`text{Consider each option:}`

`A:\ \text{rhombus diagonals are perpendicular but not always equal}`

`B:\ \text{parallelogram diagonals not always equal (see below)}`

`C:\ \text{always true (see above)}`

`D:\ \text{at least 1 pair of opposite sides of a trapezium are not equal}`
\(\Rightarrow C\)

Special Properties, SMB-009 MC

`PQRS` is a parallelogram.

Which of these must be a property of `PQRS`?

  1. Line `PS` is perpendicular to line `PQ`.
  2. Line `PQ` is parallel to line `PS`.
  3. Diagonals `PR` and `SQ` are perpendicular.
  4. Line `PS` is parallel to line `QR`.
Show Answers Only

`D`

Show Worked Solution

`text{By elimination:}`

`A\ \text{and}\ B\ \text{clearly incorrect.}`

`C\ \text{true if all sides are equal (rhombus) but not true for all parallelograms.}`

`text(Line)\ PS\ text(must be parallel to line)\ QR.`

`=>D`

Special Properties, SMB-004 MC

Which one of the following triangles is impossible to draw?

  1. a right angled triangle with two acute angles
  2. an isosceles triangle with one right angle
  3. a scalene triangle with three acute angles
  4. a right angled triangle with one obtuse angle
Show Answers Only

`D`

Show Worked Solution

`text(A right angle = 90°.)`

`text{Since an obtuse angle is greater than 90°, it is impossible for}`

`text(a triangle, with an angle sum less than 180°, to have both.)`

`=>D`

Special Properties, SMB-002 MC

A triangle has two acute angles.

What type of angle couldn't the third angle be?

  1. an acute angle
  2. an obtuse angle
  3. a right-angle
  4. a reflex angle
Show Answers Only

`D`

Show Worked Solution

`text(A triangle’s angles add up to 180°, and a reflex angle is)`

`text(greater than 180°.)`

`:.\ text(The third angle cannot be reflex.)`

`=>D`

Equations, SM-Bank 032

Verify that \(x=0.3\) is a solution of the equation \(8x+0.5=2.9\).  (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution

\(\text{When}\ x=7\)

\(LHS\) \(=8x+0.5\)
  \(=8\times 0.3+0.5\)
  \(=2.4+0.5=2.9\)
  \(=RHS\)

  
\(\therefore\ x=0.3\ \text{ is a solution}\)

Equations, SM-Bank 031

Verify that \(x=7\) is a solution of the equation \(\dfrac{x-3}{2}=2\).  (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution

\(\text{When}\ x=7\)

\(LHS\) \(=\dfrac{x-3}{2}\)
  \(=\dfrac{7-3}{2}\)
  \(=\dfrac{4}{2}=2\)
  \(=RHS\)

  
\(\therefore\ x=7\ \text{ is a solution}\)

Equations, SM-Bank 030

Verify that \(x=\dfrac{3}{4}\) is a solution of the equation \(4x-5=-2\).  (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution

\(\text{When}\ x=\dfrac{3}{4}\)

\(LHS\) \(=4x-5\)
  \(=4\times \dfrac{3}{4}-5\)
  \(=3-5=-2\)
  \(=RHS\)

  
\(\therefore\ x=\dfrac{3}{4}\ \text{ is a solution}\)

Equations, SM-Bank 029

Verify that \(x=1.5\) is a solution of the equation \(\dfrac{2x}{3}+3=4\).  (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution

\(\text{When}\ x=1.5\)

\(LHS\) \(=\dfrac{2x}{3}+3\)
  \(=\dfrac{2\times 1.5}{3}+3\)
  \(=1+3=4\)
  \(=RHS\)

  
\(\therefore\ x=1.5\ \text{ is a solution}\)

Equations, SM-Bank 028

Verify that \(x=2\) is a solution of the equation \(2x+3=7\).  (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution

\(\text{When}\ x=2\)

\(LHS\) \(=2x+3\)
  \(=2\times 2 +3\)
  \(=7\)
  \(=RHS\)

  
\(\therefore\ x=2\ \text{ is a solution}\)

Equations, SM-Bank 025

\(3\) is subtracted from a quarter of \(x\) and the result is \(-\dfrac{1}{2}\).

Write an equation and solve it algebraically to find the value of \(x\).  (3 marks)

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\(x=10\)

Show Worked Solution
\(\dfrac{x}{4}-3\) \(=-\dfrac{1}{2}\)
\(\dfrac{x}{4}\) \(=-\dfrac{1}{2}+3\)
\(\dfrac{x}{4}\) \(=2\dfrac{1}{2}\)
\(x\) \(=4\times 2\dfrac{1}{2}\)
\(x\) \(=10\)

 

Equations, SM-Bank 020

Solve  \(8-\dfrac{x}{9}=-1\)  (2 marks)

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\(x=81\)

Show Worked Solution
\(8-\dfrac{x}{9}\) \(=-1\)
\(\dfrac{-x}{9}\) \(=-1-8\)
\(\dfrac{-x}{9}\) \(=-9\)
\(-x\) \(=-9\times 9\)
\(-x\) \(=-81\)
\(x\) \(=81\)