Equation

Problem 1

Solve for $x$ in the equation: $3(x+3)-5=16$ .

See Solution

Problem 2

Find when balls A and B, rotating at different speeds, meet at the starting point again. A: 2 rotations in 26 min, B: 5 in 35 min.

See Solution

Problem 3

Find when balls A and B, rotating at different speeds, meet at the starting point again. A: 2 rotations in 26 min, B: 5 in 35 min.

See Solution

Problem 4

Calculate $8 \div 2(2+2)$ .

See Solution

Problem 5

Solve $3x - 5 = 16$ for $x$ and express your answer in set notation.

See Solution

Problem 6

Solve $3x - 5 = 16$ for $x$ and express your answer in set notation.

See Solution

Problem 7

Решите у в уравнении: $4y + 3 = 6y - 7$ .

See Solution

Problem 8

Evaluate the integral $\int_{-2}^{2} \frac{1+x^{2}}{1+2^{x}} d x$ .

See Solution

Problem 9

Find $x$ in the equation $\frac{2}{3}=\frac{5}{x}$ . Options: A $\frac{2}{15}$ , B 6, C $\frac{15}{2}$ , D 8.

See Solution

Problem 10

A boat travels 222 km on 74 liters. How much gas is needed for 39 km?

See Solution

Problem 11

A boat travels 129 km on 43 liters. How far can it go on 57 liters?

See Solution

Problem 12

Find $y$ if $\log _{y} \frac{1}{9}=3$ .

See Solution

Problem 13

Solve for $y$ in the equation $y^{2}+7 y-60=0$ .

See Solution

Problem 14

פתרו את המשוואה $3 x + 5 = 14$ ומצאו את ערך $x$ . תשובה: $x =$

See Solution

Problem 15

פתרו את המשוואה $3 x + 5 = 14$ ומצאו את $x =$

See Solution

Problem 16

Find the line equation through the origin with the same slope as the line between points $A(1,5)$ and $B(2,-3)$ .

See Solution

Problem 17

The probability of an athlete not winning 3 races is $\frac{1}{4}$ . Find the probability of winning: (i) only the 2nd race, (ii) all 3 races, (iii) exactly 2 races.

See Solution

Problem 18

Two discs with masses $m$ and $4m$ and radii $a$ and $2a$ roll without slipping. True statements? (A) Angular speed of center of mass is $\omega / 5$ (B) Angular momentum about $O$ is $81 m a^{2} \omega$ (C) Angular momentum about center of mass is $17 ma^{2} \omega / 2$ (D) $z$ -component of $\vec{L}$ is $55 m a^{2} \omega$

See Solution

Problem 19

A boat travels 222 km on 74 liters. How much gas for 39 km? Use the ratio: $\frac{74}{222} = \frac{x}{39}$ .

See Solution

Problem 20

A car travels 234 km on 39 liters. How many liters are needed for 414 km?

See Solution

Problem 21

Solve the equation $\frac{7x - 8}{5} = \frac{2x + 5}{4}$ .

See Solution

Problem 22

Solve the equation: $\frac{5(x+11)}{3}=\frac{3(1+x)}{2}$ for $x$ .

See Solution

Problem 23

Solve $3x - 5 = 16$ for $x$ and express your answer in set notation.

See Solution

Problem 24

Find when balls A and B, with rotation times of 26/2 and 35/5 minutes, meet at the starting point again.

See Solution

Problem 25

Find $k$ for the planes $x-2y+4z=10$ and $18x+17y+kz=50$ to be perpendicular. Options: (a) -4 (b) 4 (c) 2 (d) -2

See Solution

Problem 26

Find the time when balls A and B return to the starting point, given A rotates 2 times in 26 min and B 5 times in 35 min.

See Solution

Problem 27

Find $y$ if $\log_{y} \frac{1}{9} = 3$ .

See Solution

Problem 28

Find the $z$ -score for $x = 7$ given that the mean $\mu = 4$ and standard deviation $\sigma = 2$ .

See Solution

Problem 29

Find the $z$ -score for $x = 7$ given $\mu = 4$ and $\sigma = 2$ using $z = \frac{x - \mu}{\sigma}$ .

See Solution

Problem 30

Given a circle with center $O$ and diameter $MN$ , prove:
(i) $\angle MAN = 90^{\circ} + \angle PQR$ ,
(ii) $\angle QPR + 2 \times \angle MAN = 360^{\circ}$ .

See Solution

Problem 31

Convert the parabola equation $(x+6)^{2}=12(y-1)$ to standard form.

See Solution

Problem 32

Rewrite $y=2 x^{2}-4 x+7$ in focus-directrix form.

See Solution

Problem 33

Find $m \angle 12$ if $m \angle 3=2x+14$ and $m \angle 16=4x-16$ , with angles 3 and 12 as alternate exterior angles.

See Solution

Problem 34

Find $m \angle 16$ if $m \angle 1=5x+8$ and $m \angle 16=7x-20$ .

See Solution

Problem 35

Find $m \angle 13$ given $m \angle 7 = 4x + 6$ and $m \angle 15 = 13x - 6$ for parallel lines.

See Solution

Problem 36

Find $m \angle 14$ given $m \angle 1 = 67^{\circ}$ and $m \angle 18 = 42^{\circ}$ with parallel lines $l$ and $m$ . Options: $113^{\circ}$ , $96^{\circ}$ , $105^{\circ}$ , $109^{\circ}$ .

See Solution

Problem 37

Convert the parabola equation $y=-\frac{1}{8}(x-4)^{2}+7$ to standard form.

See Solution

Problem 38

Find the focus of the parabola $x = y^{2} + 4y + 10$ . Choose from: $(-6.25,2)$ , $(-2,6.25)$ , $(6.25,-2)$ , $(2,6.25)$ .

See Solution

Problem 39

Convert $y=-\frac{1}{8}(x-4)^{2}+7$ to standard form and choose the correct option from 1-4.

See Solution

Problem 40

Convert $y=x^{2}-4 x-8$ to vertex form. Which is correct? 1) $(x+2)^{2}=y-12$ 2) $(x+2)^{2}=y+12$ 3) $(x-2)^{2}=y-12$ 4) $(x-2)^{2}=y+12$

See Solution

Problem 41

Convert $y=3 x^{2}+12 x-1$ to vertex form and select the correct option from:
1. $y=3(x-4)^{2}+1$
2. $y=3(x+2)^{2}-13$
3. $y=3(x-2)^{2}+13$
4. $y=3(x+4)^{2}-1$

See Solution

Problem 42

Convert $y=x^{2}-4 x-8$ to vertex form. Which is correct?
1. $(x+2)^{2}=y+12$
2. $(x+2)^{2}=y-12$
3. $(x-2)^{2}=y+12$
4. $(x-2)^{2}=y-12$

See Solution

Problem 43

Convert $x=\frac{1}{4}(y-2)^{2}+3$ to standard form. Which option is correct?
1. $x=\frac{1}{4} y^{2}-y+4$
2. $y=\frac{1}{4} x^{2}-x+4$
3. $x=-\frac{1}{4} y^{2}-y-4$
4. $y=-\frac{1}{4} x^{2}-x-4$

See Solution

Problem 44

Find the sum of two alternate exterior angles: $6a - 33$ and $2x + 10$ .

See Solution

Problem 45

Prove that if $B P$ and $P Q$ are tangents, then (i) $A B \parallel P Q$ and (ii) $M P \times A M = B M \times M Q$ .

See Solution

Problem 46

Show that the roots of $m x^{2}+2 x+1=m$ are always real for any real constant $m$ .

See Solution

Problem 47

Show that the roots of the equation $m x^{2}+2 x+1=0$ are real for a real constant $m$ .

See Solution

Problem 48

Find the sum of two alternate exterior angles: $6x - 33$ and $2x + 10$ . What is the sum in degrees?

See Solution

Problem 49

Find the largest prime $k$ such that $x^{2} + 6x + 25 - 2kx > 0$ for all real $x$ .

See Solution

Problem 50

A passenger train travels at $29.2 \mathrm{~km/h}$ and a cattle train at $36.5 \mathrm{~km/h}$ . How long until the cattle train catches up?

See Solution

Problem 51

1. Prove $n^{7}-n$ is divisible by 42 for all positive integers $n$ . Show primes ≠ 2, 5 divide numbers like 1, 11, etc.
2. Prove if $p>3$ is prime, then $p^{2} \equiv 1(\bmod 24)$ .
3. Find the number of trailing zeros in $1000!$ .
4. If $p$ and $p^{2}+2$ are primes, prove $p^{3}+2$ is prime.
5. Prove $\operatorname{gcd}(2^{a}-1,2^{b}-1)=2^{\operatorname{gcd}(a, b)}-1$ for positive integers $a, b$ .

See Solution

Problem 52

Solve the equation $-3 h^{2} / 2 - 24 h - 45 / 2 = 0$ .

See Solution

Problem 53

Prove that $n^{7}-n$ is divisible by 42 for all positive integers $n$ and that primes other than 2 or 5 divide infinitely many of $1, 11, 111, 1111,$ etc.

See Solution

Problem 54

Find the values of $k$ such that the curve $y=x^{2}-4x+kx+3$ is above the line $y+3x=2$ .

See Solution

Problem 55

Prove that for any positive integer $n$ , $n^{7}-n$ is divisible by 42. Also, show $p^{2} \equiv 1(\bmod 24)$ for primes $p>3$ .

See Solution

Problem 56

Prove that $n^{7}-n$ is divisible by 42 for all positive integers $n$ and $p>3$ prime, $p^{2} \equiv 1(\bmod 24)$ .

See Solution

Problem 57

If $\overline{F B} \| \overline{E C}$ and $m \angle E F B=103^{\circ}$ , find $m \angle C E F$ .

See Solution

Problem 58

Find the sum of the angle measures of two alternate exterior angles: $5x - 26$ and $2x + 10$ .

See Solution

Problem 59

Find $a$ and $b$ if the line $y=x+h$ is tangent to $y=\frac{k}{1-x}$ and $k=\frac{(h+a)^{2}}{b}$ .

See Solution

Problem 60

絲带的长度是繩子长度的幾分之幾？繩子 70 厘米，絲带 21 厘米。

See Solution

Problem 61

一盒雞蛋有 $\frac{3}{4}$ 打，每隻雞蛋值 $1 \frac{4}{5}$ 元，求兩盒的總價。

See Solution

Problem 62

把一條麻繩分成 5 段，每段長 $3 \frac{1}{5}$ 米，剩下 $2 \frac{3}{8}$ 米，求原長多少米？

See Solution

Problem 63

Find the sum of two alternate exterior angles: $6x - 33$ and $3x + 13$ . What is their total measure in degrees?

See Solution

Problem 64

Find the integral of $x^{2}$ with respect to $x$ .

See Solution

Problem 65

Rewrite $y=-x^{2}-6 x+1$ as $y=a-(x+b)^{2}$ . Find the max value of $y$ and its $x$ value. Sketch the curve.

See Solution

Problem 66

Find the length of line segment $AB$ where $A$ and $B$ are intersections of $2x + 3y = 6$ and $x^2 - 2y^2 - xy = 0$ .

See Solution

Problem 67

Initially, there are 84 white and 57 black pebbles. After adding equal pebbles, the ratio is $11:8$ . Find the final white pebbles.

See Solution

Problem 68

22) Given $\int_{4}^{6} f(x) dx=5$ and $\int_{10}^{4} f(x) dx=8$ , find $\int_{6}^{10}(4 f(x)+10) dx$ .
23) Find the tangent line equation to $y=e^{2x}$ at $x=1$ .

See Solution

Problem 69

26) Find the integral for the volume of a solid with base $R$ and height 5 times the base: (a) $25 \int_{a}^{b}(g(x)-f(x))^{2} dx$ (b) $5 \int_{a}^{b}(g(x)-f(x)) dx$ (c) $5 \int_{a}^{b}(f(x)-g(x)) dx$ (d) $5 \int_{a}^{b}(f(x)-g(x))^{2} dx$
27) Solve $\frac{dy}{dx}=\frac{x}{y}$ with $y=4$ at $x=2$ : (a) $\sqrt{\frac{1}{2} x^{2}+14}$ (b) $\sqrt{2 x^{2}+8}$ (c) $\sqrt{x^{2}+6}$ (d) $\sqrt{x^{2}+12}$

See Solution

Problem 70

Find the side lengths of $\triangle ABC$ with ratio $2:3:4$ and perimeter 108.

See Solution

Problem 71

已知 $z=1-2 i$ 且 $z+a \cdot \bar{z}+b=0$ ，求 $a$ 和 $b$ 的值。选项为 A. $a=1, b=-2$ B. $a=-1, b=2$ C. $a=1, b=2$ D. $a=-1, b=-2$ 。

See Solution

Problem 72

Find the area of a circular tray with a diameter of 28 cm. Use $\pi=\frac{22}{7}$ .

See Solution

Problem 73

Find the dimensions of a rectangle with area $21 \mathrm{yd}^{2}$ and length $1 \mathrm{yd}$ less than twice the width.

See Solution

Problem 74

Find the area between the curves $y=3 x^{2}$ , $y=0$ , and $x=3$ . Choose integration with respect to $x$ or $y$ .

See Solution

Problem 75

已知半径为1的球体，求四棱锥体积最大时的高度选项：A. $\frac{1}{3}$ B. $\frac{1}{2}$ C. $\frac{\sqrt{3}}{3}$ D. $\frac{\sqrt{2}}{2}$

See Solution

Problem 76

A box has 24 fruits: 7 cherries, 8 oranges, 9 lemons. How many lemons must be sold for cherry pick probability to be 50%? A. 0 B. 1 C. 2 D. 3

See Solution

Problem 77

If $\sum_{n=0}^{\infty} \cos ^{2 n} \theta=5$ , find $\cos 2 \theta$ .

See Solution

Problem 78

Solve for $x$ in the equation $2 x + 9 = 16$ .

See Solution

Problem 79

Solve for $x$ in the equation: $1 + 2 = x$ .

See Solution

Problem 80

Find the apparent distance from the top of a 4.20 cm benzene layer to the bottom of a 6.20 cm water layer at normal incidence.

See Solution

Problem 81

Find the slope of the line given by $y-8=-\frac{1}{2}(x-2)$ . Answer as an integer or fraction.

See Solution

Problem 82

Three forces $F_{1}, F_{2}, F_{3}$ at point $O$ are in equilibrium. Given $F_{1} = 10.0 \mathrm{~N}$ , $F_{2} = 5.0 \mathrm{~N}$ , angle = 60°. Find $F_{3}$ .

See Solution

Problem 83

An object creates a focused image $30.9 \mathrm{~cm}$ right of a converging lens. A diverging lens $14.2 \mathrm{m}$ away shifts the screen $18.5 \mathrm{~cm}$ right. Find the diverging lens' focal length in cm.

See Solution

Problem 84

Identify the property shown: $E K + K F = K E + F K$ - multiplication, transitive, subtraction, reflexive, or none?

See Solution

Problem 85

Identify the property shown: $BK + KF = KE + FK$ (options: multiplication, transitive, subtraction, reflexive, none).

See Solution

Problem 86

A mud house has a cone roof over a cylinder. Given dimensions, find $A B$ , volume, surface area, and house choice.

See Solution

Problem 87

A lens creates a real image $249 \mathrm{~cm}$ from the object and $1 \frac{2}{3}$ times its size. Find the focal length.

See Solution

Problem 88

Find the focal length of a lens given an object height of 3.25 cm, distance to image 6 cm, and distance to lens 16 cm.

See Solution

Problem 89

Calculate the sum: $500 + 300 + 200$ .

See Solution

Problem 90

Solve the quadratic equation $12x^{2} - 25x = 0$ .

See Solution

Problem 91

Solve the equation: $x^{2}-16 x+61=2 x-20$ .

See Solution

Problem 92

Find the volume of a cuboid with face areas $35 \mathrm{~cm}^{2}$ , $42 \mathrm{~cm}^{2}$ , and $14 \mathrm{~cm}^{2}$ .

See Solution

Problem 93

Solve the equation $x^{4}-2 x^{3}-3 x^{2}=0$ .

See Solution

Problem 94

A car takes 4 s to stop over 20 m. Can we find its speed before braking without knowing acceleration? Options: (A) Yes, $20 \mathrm{~m} / 4 \mathrm{~s}$ . (B) Yes, double average speed. (C) No, need acceleration for $\Delta x=v_{0} t+\frac{1}{2} a t^{2}$ . (D) No, velocity definition includes acceleration.

See Solution

Problem 95

Solve for $x$ in the equation $2 + 3 = x$ .

See Solution

Problem 96

Solve the equation: $\frac{4 z}{z+1}+\frac{5}{z}=\frac{6 z+5}{z^{2}+2}$ .

See Solution

Problem 97

Karan borrowed \$3,650 for 5 months at 10% interest. What is her monthly payment amount?

See Solution

Problem 98

How many cans of Coke are left after selling 862,456 from 2,564,546? Round to the nearest ten thousand.

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Problem 99

Identify the property shown in $3 x(y+2)=(y+2) 3 x$ : distributive, identity, commutative, or associative?

See Solution

Problem 100

Medina dan Adi bermain catur. Medina menang 3 kali dan draw 2 kali. Apakah pernyataan berikut benar atau salah?
1. Skor Adi = 21
2. Skor Medina = 44
3. Total skor = 52

See Solution

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Equation

Problem 1

Solve for x x x in the equation: 3(x+3)−5=16 3(x+3)-5=16 3(x+3)−5=16.

Problem 2

Find when balls A and B, rotating at different speeds, meet at the starting point again. A: 2 rotations in 26 min, B: 5 in 35 min.

Problem 3

Find when balls A and B, rotating at different speeds, meet at the starting point again. A: 2 rotations in 26 min, B: 5 in 35 min.

Problem 4

Calculate 8÷2(2+2) 8 \div 2(2+2) 8÷2(2+2).

Problem 5

Solve 3x−5=16 3x - 5 = 16 3x−5=16 for x x x and express your answer in set notation.

Problem 6

Solve 3x−5=16 3x - 5 = 16 3x−5=16 for x x x and express your answer in set notation.

Problem 7

Решите у в уравнении: 4y+3=6y−74y + 3 = 6y - 74y+3=6y−7.

Problem 8

Evaluate the integral ∫−221+x21+2xdx \int_{-2}^{2} \frac{1+x^{2}}{1+2^{x}} d x ∫−22​1+2x1+x2​dx.

Problem 9

Find x x x in the equation 23=5x \frac{2}{3}=\frac{5}{x} 32​=x5​. Options: A 215 \frac{2}{15} 152​, B 6, C 152 \frac{15}{2} 215​, D 8.

Problem 10

A boat travels 222 km on 74 liters. How much gas is needed for 39 km?

Problem 11

A boat travels 129 km on 43 liters. How far can it go on 57 liters?

Problem 12

Find y y y if log⁡y19=3 \log _{y} \frac{1}{9}=3 logy​91​=3.

Problem 13

Solve for y y y in the equation y2+7y−60=0 y^{2}+7 y-60=0 y2+7y−60=0.

Problem 14

פתרו את המשוואה 3x+5=14 3 x + 5 = 14 3x+5=14 ומצאו את ערך x x x. תשובה: x= x = x=

Problem 15

פתרו את המשוואה 3x+5=14 3 x + 5 = 14 3x+5=14 ומצאו את x= x = x=

Problem 16

Find the line equation through the origin with the same slope as the line between points A(1,5) A(1,5) A(1,5) and B(2,−3) B(2,-3) B(2,−3).

Problem 17

The probability of an athlete not winning 3 races is 14 \frac{1}{4} 41​. Find the probability of winning: (i) only the 2nd race, (ii) all 3 races, (iii) exactly 2 races.

Problem 18

Problem 19

A boat travels 222 km on 74 liters. How much gas for 39 km? Use the ratio: 74222=x39 \frac{74}{222} = \frac{x}{39} 22274​=39x​.

Problem 20

A car travels 234 km on 39 liters. How many liters are needed for 414 km?

Problem 21

Solve the equation 7x−85=2x+54 \frac{7x - 8}{5} = \frac{2x + 5}{4} 57x−8​=42x+5​.

Problem 22

Solve the equation: 5(x+11)3=3(1+x)2 \frac{5(x+11)}{3}=\frac{3(1+x)}{2} 35(x+11)​=23(1+x)​ for x x x.

Problem 23

Solve 3x−5=16 3x - 5 = 16 3x−5=16 for x x x and express your answer in set notation.

Problem 24

Find when balls A and B, with rotation times of 26/2 and 35/5 minutes, meet at the starting point again.

Problem 25

Find k k k for the planes x−2y+4z=10 x-2y+4z=10 x−2y+4z=10 and 18x+17y+kz=50 18x+17y+kz=50 18x+17y+kz=50 to be perpendicular. Options: (a) -4 (b) 4 (c) 2 (d) -2

Problem 26

Find the time when balls A and B return to the starting point, given A rotates 2 times in 26 min and B 5 times in 35 min.

Problem 27

Find y y y if log⁡y19=3 \log_{y} \frac{1}{9} = 3 logy​91​=3.

Problem 28

Find the z z z-score for x=7 x = 7 x=7 given that the mean μ=4 \mu = 4 μ=4 and standard deviation σ=2 \sigma = 2 σ=2.

Problem 29

Find the z z z-score for x=7 x = 7 x=7 given μ=4 \mu = 4 μ=4 and σ=2 \sigma = 2 σ=2 using z=x−μσ z = \frac{x - \mu}{\sigma} z=σx−μ​.

Problem 30

Given a circle with center OOO and diameter MNMNMN, prove:(i) ∠MAN=90∘+∠PQR\angle MAN = 90^{\circ} + \angle PQR∠MAN=90∘+∠PQR,(ii) ∠QPR+2×∠MAN=360∘\angle QPR + 2 \times \angle MAN = 360^{\circ}∠QPR+2×∠MAN=360∘.

Problem 31

Convert the parabola equation (x+6)2=12(y−1)(x+6)^{2}=12(y-1)(x+6)2=12(y−1) to standard form.

Problem 32

Rewrite y=2x2−4x+7 y=2 x^{2}-4 x+7 y=2x2−4x+7 in focus-directrix form.

Problem 33

Find m∠12 m \angle 12 m∠12 if m∠3=2x+14 m \angle 3=2x+14 m∠3=2x+14 and m∠16=4x−16 m \angle 16=4x-16 m∠16=4x−16, with angles 3 and 12 as alternate exterior angles.

Problem 34

Find m∠16 m \angle 16 m∠16 if m∠1=5x+8 m \angle 1=5x+8 m∠1=5x+8 and m∠16=7x−20 m \angle 16=7x-20 m∠16=7x−20.

Problem 35

Find m∠13 m \angle 13 m∠13 given m∠7=4x+6 m \angle 7 = 4x + 6 m∠7=4x+6 and m∠15=13x−6 m \angle 15 = 13x - 6 m∠15=13x−6 for parallel lines.

Problem 36

Find m∠14 m \angle 14 m∠14 given m∠1=67∘ m \angle 1 = 67^{\circ} m∠1=67∘ and m∠18=42∘ m \angle 18 = 42^{\circ} m∠18=42∘ with parallel lines l l l and m m m. Options: 113∘ 113^{\circ} 113∘, 96∘ 96^{\circ} 96∘, 105∘ 105^{\circ} 105∘, 109∘ 109^{\circ} 109∘.

Problem 37

Convert the parabola equation y=−18(x−4)2+7 y=-\frac{1}{8}(x-4)^{2}+7 y=−81​(x−4)2+7 to standard form.

Problem 38

Find the focus of the parabola x=y2+4y+10 x = y^{2} + 4y + 10 x=y2+4y+10. Choose from: (−6.25,2) (-6.25,2) (−6.25,2), (−2,6.25) (-2,6.25) (−2,6.25), (6.25,−2) (6.25,-2) (6.25,−2), (2,6.25) (2,6.25) (2,6.25).

Problem 39

Convert y=−18(x−4)2+7 y=-\frac{1}{8}(x-4)^{2}+7 y=−81​(x−4)2+7 to standard form and choose the correct option from 1-4.

Problem 40

Convert y=x2−4x−8 y=x^{2}-4 x-8 y=x2−4x−8 to vertex form. Which is correct? 1) (x+2)2=y−12(x+2)^{2}=y-12(x+2)2=y−12 2) (x+2)2=y+12(x+2)^{2}=y+12(x+2)2=y+12 3) (x−2)2=y−12(x-2)^{2}=y-12(x−2)2=y−12 4) (x−2)2=y+12(x-2)^{2}=y+12(x−2)2=y+12

Solve for $x$ in the equation: $3(x+3)-5=16$ .

Calculate $8 \div 2(2+2)$ .

Solve $3x - 5 = 16$ for $x$ and express your answer in set notation.

Solve $3x - 5 = 16$ for $x$ and express your answer in set notation.

Решите у в уравнении: $4y + 3 = 6y - 7$ .

Evaluate the integral $\int_{-2}^{2} \frac{1+x^{2}}{1+2^{x}} d x$ .

Find $x$ in the equation $\frac{2}{3}=\frac{5}{x}$ . Options: A $\frac{2}{15}$ , B 6, C $\frac{15}{2}$ , D 8.

Find $y$ if $\log _{y} \frac{1}{9}=3$ .

Solve for $y$ in the equation $y^{2}+7 y-60=0$ .

פתרו את המשוואה $3 x + 5 = 14$ ומצאו את ערך $x$ . תשובה: $x =$

פתרו את המשוואה $3 x + 5 = 14$ ומצאו את $x =$

Find the line equation through the origin with the same slope as the line between points $A(1,5)$ and $B(2,-3)$ .

The probability of an athlete not winning 3 races is $\frac{1}{4}$ . Find the probability of winning: (i) only the 2nd race, (ii) all 3 races, (iii) exactly 2 races.

A boat travels 222 km on 74 liters. How much gas for 39 km? Use the ratio: $\frac{74}{222} = \frac{x}{39}$ .

Solve the equation $\frac{7x - 8}{5} = \frac{2x + 5}{4}$ .

Solve the equation: $\frac{5(x+11)}{3}=\frac{3(1+x)}{2}$ for $x$ .

Solve $3x - 5 = 16$ for $x$ and express your answer in set notation.

Find $k$ for the planes $x-2y+4z=10$ and $18x+17y+kz=50$ to be perpendicular. Options: (a) -4 (b) 4 (c) 2 (d) -2

Find $y$ if $\log_{y} \frac{1}{9} = 3$ .

Find the $z$ -score for $x = 7$ given that the mean $\mu = 4$ and standard deviation $\sigma = 2$ .

Find the $z$ -score for $x = 7$ given $\mu = 4$ and $\sigma = 2$ using $z = \frac{x - \mu}{\sigma}$ .

Given a circle with center $O$ and diameter $MN$ , prove:
(i) $\angle MAN = 90^{\circ} + \angle PQR$ ,
(ii) $\angle QPR + 2 \times \angle MAN = 360^{\circ}$ .

Convert the parabola equation $(x+6)^{2}=12(y-1)$ to standard form.

Rewrite $y=2 x^{2}-4 x+7$ in focus-directrix form.

Find $m \angle 12$ if $m \angle 3=2x+14$ and $m \angle 16=4x-16$ , with angles 3 and 12 as alternate exterior angles.

Find $m \angle 16$ if $m \angle 1=5x+8$ and $m \angle 16=7x-20$ .

Find $m \angle 13$ given $m \angle 7 = 4x + 6$ and $m \angle 15 = 13x - 6$ for parallel lines.

Find $m \angle 14$ given $m \angle 1 = 67^{\circ}$ and $m \angle 18 = 42^{\circ}$ with parallel lines $l$ and $m$ . Options: $113^{\circ}$ , $96^{\circ}$ , $105^{\circ}$ , $109^{\circ}$ .

Convert the parabola equation $y=-\frac{1}{8}(x-4)^{2}+7$ to standard form.

Find the focus of the parabola $x = y^{2} + 4y + 10$ . Choose from: $(-6.25,2)$ , $(-2,6.25)$ , $(6.25,-2)$ , $(2,6.25)$ .

Convert $y=-\frac{1}{8}(x-4)^{2}+7$ to standard form and choose the correct option from 1-4.

Convert $y=x^{2}-4 x-8$ to vertex form. Which is correct? 1) $(x+2)^{2}=y-12$ 2) $(x+2)^{2}=y+12$ 3) $(x-2)^{2}=y-12$ 4) $(x-2)^{2}=y+12$

Convert $y=3 x^{2}+12 x-1$ to vertex form and select the correct option from:
1. $y=3(x-4)^{2}+1$
2. $y=3(x+2)^{2}-13$
3. $y=3(x-2)^{2}+13$
4. $y=3(x+4)^{2}-1$

Convert $y=x^{2}-4 x-8$ to vertex form. Which is correct?
1. $(x+2)^{2}=y+12$
2. $(x+2)^{2}=y-12$
3. $(x-2)^{2}=y+12$
4. $(x-2)^{2}=y-12$

Find the sum of two alternate exterior angles: $6a - 33$ and $2x + 10$ .

Prove that if $B P$ and $P Q$ are tangents, then (i) $A B \parallel P Q$ and (ii) $M P \times A M = B M \times M Q$ .

Show that the roots of $m x^{2}+2 x+1=m$ are always real for any real constant $m$ .

Show that the roots of the equation $m x^{2}+2 x+1=0$ are real for a real constant $m$ .

Find the sum of two alternate exterior angles: $6x - 33$ and $2x + 10$ . What is the sum in degrees?

Find the largest prime $k$ such that $x^{2} + 6x + 25 - 2kx > 0$ for all real $x$ .

A passenger train travels at $29.2 \mathrm{~km/h}$ and a cattle train at $36.5 \mathrm{~km/h}$ . How long until the cattle train catches up?

Solve the equation $-3 h^{2} / 2 - 24 h - 45 / 2 = 0$ .

Prove that $n^{7}-n$ is divisible by 42 for all positive integers $n$ and that primes other than 2 or 5 divide infinitely many of $1, 11, 111, 1111,$ etc.

Find the values of $k$ such that the curve $y=x^{2}-4x+kx+3$ is above the line $y+3x=2$ .

Prove that for any positive integer $n$ , $n^{7}-n$ is divisible by 42. Also, show $p^{2} \equiv 1(\bmod 24)$ for primes $p>3$ .

Prove that $n^{7}-n$ is divisible by 42 for all positive integers $n$ and $p>3$ prime, $p^{2} \equiv 1(\bmod 24)$ .

If $\overline{F B} \| \overline{E C}$ and $m \angle E F B=103^{\circ}$ , find $m \angle C E F$ .

Find the sum of the angle measures of two alternate exterior angles: $5x - 26$ and $2x + 10$ .

Find $a$ and $b$ if the line $y=x+h$ is tangent to $y=\frac{k}{1-x}$ and $k=\frac{(h+a)^{2}}{b}$ .

一盒雞蛋有 $\frac{3}{4}$ 打，每隻雞蛋值 $1 \frac{4}{5}$ 元，求兩盒的總價。

把一條麻繩分成 5 段，每段長 $3 \frac{1}{5}$ 米，剩下 $2 \frac{3}{8}$ 米，求原長多少米？

Find the sum of two alternate exterior angles: $6x - 33$ and $3x + 13$ . What is their total measure in degrees?

Find the integral of $x^{2}$ with respect to $x$ .

Rewrite $y=-x^{2}-6 x+1$ as $y=a-(x+b)^{2}$ . Find the max value of $y$ and its $x$ value. Sketch the curve.

Find the length of line segment $AB$ where $A$ and $B$ are intersections of $2x + 3y = 6$ and $x^2 - 2y^2 - xy = 0$ .

Initially, there are 84 white and 57 black pebbles. After adding equal pebbles, the ratio is $11:8$ . Find the final white pebbles.

22) Given $\int_{4}^{6} f(x) dx=5$ and $\int_{10}^{4} f(x) dx=8$ , find $\int_{6}^{10}(4 f(x)+10) dx$ .
23) Find the tangent line equation to $y=e^{2x}$ at $x=1$ .

Find the side lengths of $\triangle ABC$ with ratio $2:3:4$ and perimeter 108.

已知 $z=1-2 i$ 且 $z+a \cdot \bar{z}+b=0$ ，求 $a$ 和 $b$ 的值。选项为 A. $a=1, b=-2$ B. $a=-1, b=2$ C. $a=1, b=2$ D. $a=-1, b=-2$ 。

Find the area of a circular tray with a diameter of 28 cm. Use $\pi=\frac{22}{7}$ .

Find the dimensions of a rectangle with area $21 \mathrm{yd}^{2}$ and length $1 \mathrm{yd}$ less than twice the width.

Find the area between the curves $y=3 x^{2}$ , $y=0$ , and $x=3$ . Choose integration with respect to $x$ or $y$ .