Permutations and Combinations Test 47

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The area of a triangle whose vertices are $$\displaystyle \left( { x }_{ 1 },{ y }_{ 1 } \right) ,\left( { x }_{ 2 },{ y }_{ 2 } \right) $$ and $$\displaystyle \left( { x }_{ 3 },{ y }_{ 3 } \right) $$ is given by $$\displaystyle \Delta =\frac { 1 }{ 2 } \left| { x }_{ 1 }\left( { y }_{ 2 },{ y }_{ 3 } \right) +{ x }_{ 2 }\left( { y }_{ 3 },{ y }_{ 1 } \right) +{ x }_{ 3 }\left( { y }_{ 1 },{ y }_{ 2 } \right) \right| $$. The points $$\displaystyle \left( { x }_{ 1 },{ y }_{ 1 } \right) ,\left( { x }_{ 2 },{ y }_{ 2 } \right) $$ and $$\displaystyle \left( { x }_{ 3 },{ y }_{ 3 } \right) $$ are collinear of $$\displaystyle \Delta =0$$

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Determine the area of the triangle whose vertices are $$\displaystyle \left( \frac { 1 }{ 2 } ,\frac { -1 }{ 2 } \right) ,\left( 2,\frac { -1 }{ 2 } \right) $$ and $$\displaystyle \left( 2,\frac { \sqrt { 3 } -1 }{ 2 } \right) $$.

Permutations an...

Question 1

$$\displaystyle 3\sqrt { 3 } $$

$$\displaystyle \frac { 3\sqrt { 3 } }{ 8 } $$

$$\displaystyle 12\sqrt { 3 } $$

$$\displaystyle \frac { 3\sqrt { 3 } }{ 4 } $$

Question 2

$$11!$$

$$12!$$

$$\dfrac{12!}{2}$$

$$1$$

Question 3

$$19!$$

$$\dfrac{19!}{10!}$$

$$\dfrac{19!}{10!^2}$$

$$20!$$

Question 4

$$(-1)^x = 1; (-1)^y = 1$$

$$(-1)^x = 1; (-1)^y = -1$$

$$(-1)^x = -1; (-1)^y = 1$$

$$(-1)^x = -1; (-1)^y = -1$$

Question 5

$$7$$ or $$14$$

$$7$$

$$14$$

$$14$$ or $$21$$

Question 6

$$4!\times4!$$

$$7!$$

$$3!\times4!$$

$$8!$$

Question 7

$$1825$$

$$3801$$

$$4845$$

$$300$$

Question 8

91

81

71

61

Question 9

0

1

2

3

4

Question 10

85 $$\displaystyle { units }^{ 2 }$$

87 $$\displaystyle { units }^{ 2 }$$

90 $$\displaystyle { units }^{ 2 }$$

91 $$\displaystyle { units }^{ 2 }$$

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