Multiple Choice

1. In a group of 30 students, 18 like soccer, 15 like basketball, and 7 like both. How many like neither?

   33

   26

   4

   23

2. Let $P=\{x\mid x\text{ is a natural number and }x<10\}$ and $Q=\{x\mid x\text{ is even and }0

   $\{1,2,3,4,5,6,7,8,9\}$

   $\{2,4,6\}$

   $\{2,4,6,8\}$

   $\{2,4,6,8,10,12\}$

3. A set $S$ has exactly $32$ subsets. How many elements are in $S$?

   5

   6

   16

   4

4. With $U=\{1,2,\dots,10\}$, $A=\{\text{odd numbers}\}$, $B=\{\text{prime numbers}\}$. Find $A'\cap B$.

   $\{2\}$

   $\{1,3,5,7,9\}$

   $\{2,4,6,8,10\}$

   $\{3,5,7\}$

5. In a survey of 100 students: $M=40$, $S=35$, $E=30$; $M\cap S=15$, $S\cap E=12$, $M\cap E=10$, $M\cap S\cap E=5$. How many liked only Math?

   15

   35

   40

   20

6. Which statement is false for $A=\{x,y\}$, $B=\{y,x\}$?

   $A\subset B$

   $A=B$

   $A\subseteq A$

   $\emptyset\subseteq A$

7. Let $A=\{x\in\mathbb{R}\mid 2\le x<7\}$ and $B=\{x\in\mathbb{R}\mid 5

   $\{x\in\mathbb{R}\mid 5\le x\le 7\}$

   $\{x\in\mathbb{R}\mid 5

   $\{x\in\mathbb{R}\mid 2\le x\le 10\}$

   $\{x\in\mathbb{R}\mid 2\le x<5\}$

8. With $U=\{1,2,\dots,10\}$, $A=\{1,2,3,4,5\}$, $B=\{4,5,6,7\}$. Find $(A\setminus B)'$.

   $\{1,2,3\}$

   $\{1,2,3,4,5,6,7\}$

   $\{8,9,10\}$

   $\{4,5,6,7,8,9,10\}$

9. Give an equivalent form of $(X\cup Y)'$.

   $X'\cup Y'$

   $X'\cap Y'$

   $(X\cap Y)'$

   $X\cup Y$

10. If $A=\{\text{red, blue}\}$, $B=\{\text{small, medium, large}\}$, find $|A\times B|$.

    9

    8

    5

    6

11. A tank starts at $0$ L and fills at $5$ L/min. How many liters after $4$ min?

    1.25 L

    20 L

    1 L

    9 L

12. A bike rental has a fixed cost (y-intercept) of $5$ € and costs $15$ € at $2$ h. What is the fixed cost?

    10 €

    5 €

    20 €

    15 €

13. A pitcher has $1000$ mL at $t=0$ and loses $50$ mL each minute. How much after $3$ min?

    1150 mL

    850 mL

    997 mL

    950 mL

14. Robot A: $10$ m in $2$ s; Robot B: $12$ m in $3$ s. Which is faster?

    Impossible to determine

    Both are equally fast

    Robot B

    Robot A

15. A baby weighs $4$ kg at $1$ mo and $6$ kg at $3$ mo. Assuming steady gain, what at $2$ mo?

    4.5 kg

    5.5 kg

    6 kg

    5 kg

16. Apples cost $2$ €/kg (linear). Which point does \emph{not} lie on the line?

    $(4,8)$

    $(2,4)$

    $(0,0)$

    $(5,9)$

17. A car starts with $50$ L fuel and uses $5$ L per $100$ km. Fuel left after $300$ km?

    45 L

    25 L

    35 L

    65 L

18. The postage table is: $1$ kg $\to 3$ €, $2$ kg $\to 5$ €, $3$ kg $\to 7$ €. What is the cost for $4$ kg?

    10 €

    11 €

    8 €

    9 €

19. Water heats to $100^\circ$C at $t=5$ min and stays at $100^\circ$C until $t=15$ min. Temperature at $t=12$ min?

    $100^\circ$C

    $105^\circ$C

    $95^\circ$C

    $112^\circ$C

20. Points $(1,2)$, $(3,4)$, $(5,6)$ lie on a line. Next point following the same pattern?

    $(7,8)$

    $(8,9)$

    $(6,7)$

    $(7,9)$

21. A snail travels from $(3,5)$ to $(10,5)$. How many units?

    13 units

    7 units

    10 units

    5 units

22. Midpoint between $8$ and $16$ (blocks east). Where is the treasure?

    24 blocks

    12 blocks

    4 blocks

    8 blocks

23. Three ramps have the following rise/run (same horizontal run $8$ m): Ramp A rises $2$ m, Ramp B rises $3$ m, Ramp C rises $4$ m. Which is steepest?

    Ramp A

    Ramp B

    Ramp C

    All are equally steep

24. Road 1 goes from $(1,1)$ to $(5,3)$. Road 2 starts at $(1,4)$ and is parallel to Road 1. Which point could Road 2 pass through?

    $(5,6)$

    $(5,5)$

    $(5,4)$

    $(3,7)$

25. A horizontal ground is the line segment from $(0,3)$ to $(6,3)$. A vertical flagpole stands at base $(4,3)$. Which could be the top?

    $(4,7)$

    $(5,3)$

    $(3,4)$

    $(4,3)$

26. A beanstalk grows steadily: Mon $4$ ft, Tue $6$ ft, Wed $8$ ft. How tall on Saturday?

    16 ft

    10 ft

    12 ft

    14 ft

27. From $(2,2)$ to $(5,6)$ along grid lines (only right/up). Describe a valid path and total distance.

    3 right and 4 up; total 12 units

    3 right and 4 up; total 5 units

    3 right and 4 up; total 7 units

    4 right and 3 up; total 7 units

28. A car starts at $(2,3)$ and repeats the move “$1$ right, $2$ up” three times. Final coordinates?

    $(5,5)$

    $(3,9)$

    $(3,5)$

    $(5,9)$

29. Line $P$ from $(1,2)$ to $(5,4)$. Line $Q$ starts at $(2,5)$, parallel to $P$ and same length. Other end of $Q$?

    $(6,7)$

    $(7,9)$

    $(3,7)$

    $(4,9)$

30. Which is farther: $5\times 10^3$ miles or $3\times 10^4$ miles?

    $5\times 10^3$ miles

    They are the same

    $3\times 10^4$ miles

    Impossible to determine

31. A farmer has $8\times 10^2$ apples and sells $3\times 10^2$. How many left?

    $11\times 10^2$ apples

    $24\times 10^4$ apples

    $5\times 10^0$ apples

    $5\times 10^2$ apples

32. Simplify $\displaystyle \frac{(3x^{-2}y^3)^2}{(9x^4y^{-1})^{-1}}$ for $x,y\ne 0$.

    $81x^{-8}y^7$

    $81x^8y^7$

    $y^5/81$

    $81y^5$

33. Evaluate $({-2})^3-4^{-2}+(-1)^0\cdot 2^1$.

    $-\dfrac{33}{16}$

    $-22$

    $-\dfrac{97}{16}$

    $-\dfrac{129}{16}$

34. Compute $\displaystyle \frac{(4.5\times 10^7)(3.0\times 10^{-3})}{9.0\times 10^2}$ in scientific notation.

    $15\times 10^1$

    $1.5\times 10^3$

    $1.5\times 10^4$

    $1.5\times 10^2$

35. Find $(2.7\times 10^5)+(3.4\times 10^3)$ in scientific notation.

    $2.734\times 10^3$

    $2.7034\times 10^5$

    $6.1\times 10^8$

    $2.734\times 10^5$

36. Simplify $\left(\dfrac{a^3 b^{-2}}{a^{-1} b^4}\right)^{-2}(a^2 b^3)^3$, $a,b\ne 0$.

    $\dfrac{a^2}{b^{21}}$

    $\dfrac{b^{21}}{a^2}$

    $a^{-14} b^{15}$

    $a^2 b^{21}$

37. If $x=3^k$ and $y=3^{k+2}$, compute $\dfrac{y}{x^2}$.

    $3^{2-k}$

    $3^{2+k}$

    $3^{2-2k}$

    $3^{k-2}$

38. A field is $3.2\times 10^3$ m by $1.5\times 10^2$ m. A tractor cultivates $6.0\times 10^4$ m$^2$/h. Time to finish?

    $8.0\times 10^0$ hours

    $0.8\times 10^1$ hours

    $80$ hours

    $8.0\times 10^1$ hours

39. Simplify completely:

    $x^{11}y^4$

    $x^2 y^8$

    $\dfrac{y^4}{x^{11}}$

    $\dfrac{x^{11}}{y^4}$

40. Solve $5^{2x-1}=125^{x-3}$.

    $x=-8$

    $x=8$

    $x=9$

    $x=2$

41. Which is largest: $3.05\times 10^4,\ 2.9\times 10^4,\ 3\times 10^4,\ 3.1\times 10^3$?

    $2.9\times 10^4$

    $3\times 10^4$

    $3.1\times 10^3$

    $3.05\times 10^4$

42. Factorise completely: $3xy-24x+5y-40$.

    $(3x+8)(y-5)$

    $(3x-5)(y+8)$

    $(3x+5)(y-8)$

    $(xy-8)(3+5)$

43. Factorise $6x^2-19x+10$.

    $(x-2)(6x-5)$

    $(2x-5)(3x-2)$

    $(2x+5)(3x+2)$

    $(6x-1)(x-10)$

44. Express $2x^2+12x-7$ in the form $a(x+h)^2+k$.

    $2(x+3)^2-25$

    $2(x+3)^2-7$

    $(x+3)^2-25$

    $2(x+6)^2-79$

45. Solve $3x^2=5x+2$.

    $x=-2,\ -1/3$

    $x=2,\ 1/3$

    $x=2,\ -\dfrac{1}{3}$

    $x=1/3,\ -2$

46. Solve $x^2-8x+5=0$ by completing the square.

    $x=-4\pm\sqrt{11}$

    $x=4\pm\sqrt{11}$

    $x=8\pm\sqrt{11}$

    $x=4\pm\sqrt{21}$

47. Solve $2x^2-7x-3=0$ using the quadratic formula.

    $x=\dfrac{-7\pm\sqrt{73}}{4}$

    $x=\dfrac{7\pm\sqrt{73}}{4}$

    $x=\dfrac{7\pm\sqrt{73}}{2}$

    $x=\dfrac{7\pm\sqrt{25}}{4}$

48. A rectangle has width $w$ m and length $w+5$ m with area $84$ m$^2$. Find $w$.

    $w=6$ m

    $w=84/5$ m

    $w=7$ m

    $w=12$ m

49. A projectile: $h=-5t^2+20t+15$ (meters). When does it hit the ground?

    $t=2+\sqrt{7}$ s

    $t=2-\sqrt{7}$ s

    $t=2+\sqrt{19}$ s

    $t=3$ s

50. Factorise completely: $2x^2y+6x-xy-3$.

    $(2xy+1)(x-3)$

    $(xy+3)(2x-1)$

    $(xy-3)(2x+1)$

    $(2x-y)(xy+3)$

51. For which $k$ does $x^2-6x+k=0$ have exactly one real solution?

    $k=36$

    $k=-9$

    $k=9$

    $k=0$