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Equations, SM-Bank 069

Find the value of  \(\large c\)  in the formula \(c^2=a^2+b^2\) given \(a=12\) and \(b=14\). Assume  \(\large c\)  is positive and give your answer correct to 2 decimal places.  (2 marks)

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\(c\approx 18.44\)

Show Worked Solution
\(c^2\) \(=a^2+b^2\)
\(c^2\) \(=12^2+14^2\)
\(c^2\) \(=144+196\)
\(\sqrt{c^2}\) \(=\sqrt{340}\)
\(c\) \(=18.4390\ldots\)
  \(\approx 18.44\)

Equations, SM-Bank 068

Find the value of  \(\large c\)  in the formula \(c^2=a^2+b^2\), given \(a=7\) and \(b=24\). Assume  \(\large c\)  is positive.  (2 marks)

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\(c=25\)

Show Worked Solution
\(c^2\) \(=a^2+b^2\)
\(c^2\) \(=7^2+24^2\)
\(c^2\) \(=49+576\)
\(\sqrt{c^2}\) \(=\sqrt{625}\)
\(c\) \(=25\)

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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\(x=7\)

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

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

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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\(\text{See worked solutions}\)

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

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

Equations, SM-Bank 059

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

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\(\text{See worked solutions}\)

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

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

Equations, SM-Bank 058

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

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\(\text{See worked solutions}\)

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

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

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

\(\Rightarrow B\)