Natural Numbers

Problem 1101

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 1102

Differentiate $x^{e^{x}}$ with respect to $x$ .

See Solution

Problem 1103

What is the value of $6 \div 2(1+2)$ ?

See Solution

Problem 1104

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 1105

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 1106

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 1107

Solve $60 \div 2(3+7)$ .

See Solution

Problem 1108

Prove that in a given diagram, (i) $A B$ is parallel to $P Q$ and (ii) $M P \times A M = B M \times M Q$ .

See Solution

Problem 1109

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

See Solution

Problem 1110

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 1111

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

See Solution

Problem 1112

Find $k$ such that $x^{2}+k(x+2)+3(x+1)>0$ for all $x$ . What is the range of $k$ ?

See Solution

Problem 1113

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

See Solution

Problem 1114

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

See Solution

Problem 1115

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

See Solution

Problem 1116

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

See Solution

Problem 1117

Calculate $(1^{4} \times 2^{2} + 3^{3}) - 2^{5} \div 4$ .

See Solution

Problem 1118

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

See Solution

Problem 1119

Calculate $31 - 32 \div 4$ .

See Solution

Problem 1120

Solve the system of equations: $5x - 14y = -23$ and $-6x + 7y = 8$ .

See Solution

Problem 1121

In triangle $D E F$ , find the longest side given $m \angle D=59^{\circ}, m \angle E=76^{\circ}, m \angle F=45^{\circ}$ .

See Solution

Problem 1122

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 1123

Calculate $(4+27)-32 \div 4$ .

See Solution

Problem 1124

Multiply and simplify $(2 x-5)(3 x-3)$ .

See Solution

Problem 1125

Find the ball's instantaneous velocity at $t=10.0$ s given $x(t)=0.000015 t^{5}-0.004 t^{3}+0.4 t$ .

See Solution

Problem 1126

Determine if the series converges or diverges: $\sum_{n=1}^{\infty}\left[\left(\frac{6}{7}\right)^{n}-\frac{3}{2^{n}}\right]$ . If it converges, find the sum.

See Solution

Problem 1127

Find the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n(n+2)}$ .

See Solution

Problem 1128

Find the limit of the sequence $a_{n}=\sqrt{n^{2}+3 n}-n$ . Options: A) 3 B) 2 C) $1 / 2$ D) $3 / 2$ E) 0 F) 1 G) $\infty$ H) does not exist.

See Solution

Problem 1129

Determine if the series are convergent or divergent:
(I) $\sum_{n=1}^{\infty}\left(\frac{n+3}{2 n-5}\right)^{n}$
(II) $\sum_{n=1}^{\infty} \sqrt[4]{\frac{2}{n^{3}}}$
(III) $\sum_{n=1}^{\infty} \frac{2^{n}}{n ! \sqrt{n+1}}$
Explain your reasoning.

See Solution

Problem 1130

Determine if these series converge or diverge, explaining your reasoning: (I) $\sum_{n=1}^{\infty}\left(\frac{n+3}{2 n-5}\right)^{n}$ , (II) $\sum_{n=1}^{\infty} \sqrt[4]{\frac{2}{n^{3}}}$ , (III) $\sum_{n=1}^{\infty} \frac{2^{n}}{n ! \sqrt{n+1}}$ .

See Solution

Problem 1131

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 1132

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

See Solution

Problem 1133

Determine if these series are absolutely convergent, conditionally convergent, or divergent:
(I) $\sum_{n=1}^{\infty}\left(\frac{\pi}{2}\right)^{n}$
(II) $\sum_{n=1}^{\infty} \frac{(-1)^{n-1} e}{\sqrt{n}}$
(III) $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}(\sqrt{n}+1)^{3}}$

See Solution

Problem 1134

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 1135

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 1136

What time is nearest to midnight: A. 11:55 a.m., B. 12:06 a.m., C. 11:50 a.m., D. 12:03 a.m.?

See Solution

Problem 1137

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 1138

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 1139

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

See Solution

Problem 1140

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

See Solution

Problem 1141

Find the range of values of $k$ for which the curve $y=x^{2}-4 x+k x+3$ is above the line $y+3 x=2$ .

See Solution

Problem 1142

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 1143

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 1144

甲、乙兩地相距 $7 \frac{1}{2}$ 公里，爸爸駕車走了 $1 \frac{1}{2}$ 公里，求他走了全程的幾分之幾？

See Solution

Problem 1145

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 1146

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 1147

Any two parallel lines lie in the same plane. True or False?

See Solution

Problem 1148

A ball's position is given by $x(t)=0.000015 t^5 - 0.004 t^3 + 0.4 t$ . Find its velocity at $t=10.0$ s.

See Solution

Problem 1149

Is it true or false that 'If two rays are parallel, then the lines containing them must be coplanar'?

See Solution

Problem 1150

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

See Solution

Problem 1151

Find values of $k$ so that $2 x^{2}+k^{2}+2-2(k+2) x > 0$ for all $x$ .

See Solution

Problem 1152

A ball's position is given by $x(t)=0.000015 t^5 - 0.004 t^3 + 0.4 t$ . Find its velocity at $t=10.0$ s.

See Solution

Problem 1153

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

See Solution

Problem 1154

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 1155

Prove $x^{2}-x+2 \geq \frac{7}{4}$ for all values of $x$ .

See Solution

Problem 1156

Find $x$ such that $1+x^{2} \geq-\frac{7 x-x^{2}}{3}$ and determine the minimum of $y=1+x^{2}+\frac{7 x-x^{2}}{3}$ .

See Solution

Problem 1157

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 1158

Find all solutions for the function $f(x)=2 x^{3}-6 x^{2}+9 x-27$ .

See Solution

Problem 1159

Find the integral $\int \sin (3 x) d x$ . Choose from: (a) $3 \cos (3 x)+C$ , (b) $\frac{1}{3} \cos (3 x)+C$ , (c) $-3 \cos (3 x)+C$ , (d) $-\frac{1}{3} \cos (3 x)+C$ .

See Solution

Problem 1160

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 1161

Calculate the integral $\int_{1}^{5} \frac{x-1}{x} dx$ and choose the correct answer: (a) $5-\ln 5$ , (b) $4-\ln 5$ , (c) $2-\ln 5$ , (d) $1-\ln 5$ .

See Solution

Problem 1162

Find the limit: $\lim _{x \rightarrow 1} \frac{2 \cdot \ln (x)}{\mathrm{e}^{x}-1}$ . Options: (a) $\frac{2}{e}$ (b) 1 (c) 0 (d) nonexistent.

See Solution

Problem 1163

Find $f(1)$ given $f^{\prime}(x)=3 x^{2}$ and $f(2)=3$ . Choices: (a) -7 (b) -4 (c) 7 (d) 10

See Solution

Problem 1164

17) Find the area of the region between $y=2x$ and $y=x^2$ . 18) If a cube's side length $S=2$ in decreases at 5 in³/min, find the rate of change of $S$ in in/min.

See Solution

Problem 1165

Find the derivative $\frac{d}{d x}(\tan (\ln (x)))$ . What is $\frac{d y}{d x}$ for $\sqrt{x}+y^{2}=x y+2$ at (4,0)?

See Solution

Problem 1166

8) Find the limit: $\lim _{h \rightarrow 0} \frac{\cos \left((x+h)^{2}\right)-\cos \left(x^{2}\right)}{h}$ .
9) Determine $x$ for the maximum of $y=\frac{4}{3} x^{3}-8 x^{2}+15 x$ on $[0,4]$ .

See Solution

Problem 1167

A particle's velocity is $v(t)=e^{t-1}-3 \sin (t-1)$ . What is its motion at $t=1$ ? (a) speeding up, (b) slowing down, (c) neither, (d) at rest.

See Solution

Problem 1168

At $t=2$ , what does $R^{\prime}(2)=4$ mean? (a) Depth is 4 inches. (b) Rate is 4 in/hr. (c) Rate of change is 4 in/hr². (d) Depth increased by 4 inches.

See Solution

Problem 1169

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 1170

Find the limit: $\lim _{x \rightarrow \infty} \frac{4 \cdot \ln (x)+4}{3 x}$ . Choose (a) 2, (b) -2, (c) 0, or (d) nonexistent.

See Solution

Problem 1171

Find the $x$ -coordinate of the inflection point of $f(x)=\int_{1}^{x}(3t-6t^{2})dt$ . Options: (a) $-\frac{1}{4}$ (b) $\frac{1}{4}$ (c) 0 (d) $\frac{1}{2}$ . Evaluate $\int \sin(3x)dx$ . Options: (a) $3\cos(3x)+C$ (b) $\frac{1}{3}\cos(3x)+C$ (c) $-3\cos(3x)+C$ (d) $-\frac{1}{3}\cos(3x)+C$ . Count removable discontinuities of $y=\frac{x+2}{x^{4}+16}$ . Options: (a) one (b) two (c) three (d) four.

See Solution

Problem 1172

Which type of triangle cannot be both obtuse and one of the following: equilateral, acute, isosceles, or scalene?

See Solution

Problem 1173

Find $f'(x)$ if $f(x)=\frac{x^{2}+1}{3 x}$ and evaluate $\int(2 x+3)(x^{2}+3 x)^{4} dx$ .

See Solution

Problem 1174

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 1175

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

See Solution

Problem 1176

Which type of triangle cannot be both scalene and acute, right and acute, equilateral and acute, or isosceles and acute?

See Solution

Problem 1177

Find the integral of $f(x) = \sin 2x$ . Choose from: $(-1/2) \cos 2x + C$ or others.

See Solution

Problem 1178

已知向量 $\vec{a}, \vec{b}$ 满足 $|\vec{a}|=1,|\vec{b}|=\sqrt{3},|\vec{a}-2 \vec{b}|=3$ ，求 $\vec{a} \cdot \vec{b}$ 。

See Solution

Problem 1179

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 1180

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

See Solution

Problem 1181

Identify the point that does not lie on the line formed by 3 of the following points: $(-2,-3)$ , $(3,2)$ , $(-1,0)$ , $(-3,-4)$ .

See Solution

Problem 1182

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 1183

Simplify $\frac{36 a^{3} b^{2} x}{27 a b^{3} y}$ .

See Solution

Problem 1184

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 1185

Tom has 4 different marbles and 3 identical yellow ones. How many unique pairs of marbles can he select?

See Solution

Problem 1186

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 1187

Express $\tan 35^{\circ}+\tan 55^{\circ}$ using $k$ if $\cos 35^{\circ}=k$ .

See Solution

Problem 1188

Tom has 4 unique marbles and 3 identical yellow ones. How many ways can he choose 2 marbles?

See Solution

Problem 1189

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 1190

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

See Solution

Problem 1191

Simplify $(3 x^{2}-16 x+21) \div (x-3)$ .

See Solution

Problem 1192

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

See Solution

Problem 1193

Calculate the total fuel cost from the transaction: 35.270 L at \$1.619/L, with HST included of \$6.57. Total: \$57.10.

See Solution

Problem 1194

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

See Solution

Problem 1195

Evaluate the integral: $\int \frac{3 a x}{b^{2}+c^{2} x^{2}} \, dx$

See Solution

Problem 1196

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 1197

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

See Solution

Problem 1198

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 1199

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 1200

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

See Solution

1 ...9 10 11 12 13 14 15 ...28

Natural Numbers

Problem 1101

Convert y=3x2+12x−1 y=3 x^{2}+12 x-1 y=3x2+12x−1 to vertex form and select the correct option from:1. y=3(x−4)2+1 y=3(x-4)^{2}+1 y=3(x−4)2+12. y=3(x+2)2−13 y=3(x+2)^{2}-13 y=3(x+2)2−133. y=3(x−2)2+13 y=3(x-2)^{2}+13 y=3(x−2)2+134. y=3(x+4)2−1 y=3(x+4)^{2}-1 y=3(x+4)2−1

Problem 1102

Differentiate xex x^{e^{x}} xex with respect to x x x.

Problem 1103

What is the value of 6÷2(1+2) 6 \div 2(1+2) 6÷2(1+2)?

Problem 1104

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

Problem 1105

Problem 1106

Prove that if BP B P BP and PQ P Q PQ are tangents, then (i) AB∥PQ A B \parallel P Q AB∥PQ and (ii) MP×AM=BM×MQ M P \times A M = B M \times M Q MP×AM=BM×MQ.

Problem 1107

Solve 60÷2(3+7) 60 \div 2(3+7) 60÷2(3+7).

Problem 1108

Prove that in a given diagram, (i) AB A B AB is parallel to PQ P Q PQ and (ii) MP×AM=BM×MQ M P \times A M = B M \times M Q MP×AM=BM×MQ.

Problem 1109

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

Problem 1110

Prove that if BP B P BP and PQ P Q PQ are tangents, then (i) AB∥PQ A B \parallel P Q AB∥PQ and (ii) MP×AM=BM×MQ M P \times A M = B M \times M Q MP×AM=BM×MQ.

Problem 1111

Show that the roots of mx2+2x+1=m m x^{2}+2 x+1=m mx2+2x+1=m are always real for any real constant m m m.

Problem 1112

Find k k k such that x2+k(x+2)+3(x+1)>0 x^{2}+k(x+2)+3(x+1)>0 x2+k(x+2)+3(x+1)>0 for all x x x. What is the range of k k k?

Problem 1113

Show that the roots of the equation mx2+2x+1=0 m x^{2}+2 x+1=0 mx2+2x+1=0 are real for a real constant m m m.

Problem 1114

Prove that if BP B P BP and PQ P Q PQ are tangents to circles, then (i) AB∥PQ A B \parallel P Q AB∥PQ and (ii) MP×AM=BM×MQ M P \times A M = B M \times M Q MP×AM=BM×MQ.

Problem 1115

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

Problem 1116

Find the largest prime k k k such that x2+6x+25−2kx>0 x^{2} + 6x + 25 - 2kx > 0 x2+6x+25−2kx>0 for all real x x x.

Problem 1117

Calculate (14×22+33)−25÷4 (1^{4} \times 2^{2} + 3^{3}) - 2^{5} \div 4 (14×22+33)−25÷4.

Problem 1118

Find the derivative of x2 x^{2} x2 with respect to x x x.

Problem 1119

Calculate 31−32÷4 31 - 32 \div 4 31−32÷4.

Problem 1120

Solve the system of equations: 5x−14y=−23 5x - 14y = -23 5x−14y=−23 and −6x+7y=8 -6x + 7y = 8 −6x+7y=8.

Problem 1121

In triangle DEF D E F DEF, find the longest side given m∠D=59∘,m∠E=76∘,m∠F=45∘ m \angle D=59^{\circ}, m \angle E=76^{\circ}, m \angle F=45^{\circ} m∠D=59∘,m∠E=76∘,m∠F=45∘.

Problem 1122

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

Problem 1123

Calculate (4+27)−32÷4 (4+27)-32 \div 4 (4+27)−32÷4.

Problem 1124

Multiply and simplify (2x−5)(3x−3)(2 x-5)(3 x-3)(2x−5)(3x−3).

Problem 1125

Find the ball's instantaneous velocity at t=10.0 t=10.0 t=10.0 s given x(t)=0.000015t5−0.004t3+0.4t x(t)=0.000015 t^{5}-0.004 t^{3}+0.4 t x(t)=0.000015t5−0.004t3+0.4t.

Problem 1126

Determine if the series converges or diverges: ∑n=1∞[(67)n−32n]\sum_{n=1}^{\infty}\left[\left(\frac{6}{7}\right)^{n}-\frac{3}{2^{n}}\right]n=1∑∞​[(76​)n−2n3​]. If it converges, find the sum.

Problem 1127

Find the sum of the series ∑n=1∞1n(n+2)\sum_{n=1}^{\infty} \frac{1}{n(n+2)}∑n=1∞​n(n+2)1​.

Problem 1128

Find the limit of the sequence an=n2+3n−na_{n}=\sqrt{n^{2}+3 n}-nan​=n2+3n​−n. Options: A) 3 B) 2 C) 1/2 1 / 2 1/2 D) 3/2 3 / 2 3/2 E) 0 F) 1 G) ∞ \infty ∞ H) does not exist.

Problem 1129

Problem 1130

Problem 1131

Problem 1132

Solve the equation −3h2/2−24h−45/2=0-3 h^{2} / 2 - 24 h - 45 / 2 = 0−3h2/2−24h−45/2=0.

Problem 1133

Problem 1134

Prove that n7−n n^{7}-n n7−n is divisible by 42 for all positive integers n n n and that primes other than 2 or 5 divide infinitely many of 1,11,111,1111, 1, 11, 111, 1111, 1,11,111,1111, etc.

Problem 1135

Find the values of k k k such that the curve y=x2−4x+kx+3 y=x^{2}-4x+kx+3 y=x2−4x+kx+3 is above the line y+3x=2 y+3x=2 y+3x=2.

Problem 1136

What time is nearest to midnight: A. 11:55 a.m., B. 12:06 a.m., C. 11:50 a.m., D. 12:03 a.m.?

Problem 1137

Prove that for any positive integer n n n, n7−n n^{7}-n n7−n is divisible by 42. Also, show p2≡1( mod 24) p^{2} \equiv 1(\bmod 24) p2≡1(mod24) for primes p>3 p>3 p>3.

Problem 1138

Prove that n7−n n^{7}-n n7−n is divisible by 42 for all positive integers n n n and p>3 p>3 p>3 prime, p2≡1( mod 24) p^{2} \equiv 1(\bmod 24) p2≡1(mod24).

Problem 1139

If FB‾∥EC‾ \overline{F B} \| \overline{E C} FB∥EC and m∠EFB=103∘ m \angle E F B=103^{\circ} m∠EFB=103∘, find m∠CEF m \angle C E F m∠CEF.

Problem 1140

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

Problem 1141

Find the range of values of k k k for which the curve y=x2−4x+kx+3 y=x^{2}-4 x+k x+3 y=x2−4x+kx+3 is above the line y+3x=2 y+3 x=2 y+3x=2.

Problem 1142

Find a a a and b b b if the line y=x+h y=x+h y=x+h is tangent to y=k1−x y=\frac{k}{1-x} y=1−xk​ and k=(h+a)2b k=\frac{(h+a)^{2}}{b} k=b(h+a)2​.

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. $y=3(x+4)^{2}-1$

Differentiate $x^{e^{x}}$ with respect to $x$ .

What is the value of $6 \div 2(1+2)$ ?

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. $(x-2)^{2}=y-12$

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

Solve $60 \div 2(3+7)$ .

Prove that in a given diagram, (i) $A B$ is parallel to $P Q$ and (ii) $M P \times A M = B M \times M Q$ .

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 $m$ .

Find $k$ such that $x^{2}+k(x+2)+3(x+1)>0$ for all $x$ . What is the range of $k$ ?

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

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

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 $x$ .

Calculate $(1^{4} \times 2^{2} + 3^{3}) - 2^{5} \div 4$ .

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

Calculate $31 - 32 \div 4$ .

Solve the system of equations: $5x - 14y = -23$ and $-6x + 7y = 8$ .

In triangle $D E F$ , find the longest side given $m \angle D=59^{\circ}, m \angle E=76^{\circ}, m \angle F=45^{\circ}$ .

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?

Calculate $(4+27)-32 \div 4$ .

Multiply and simplify $(2 x-5)(3 x-3)$ .

Find the ball's instantaneous velocity at $t=10.0$ s given $x(t)=0.000015 t^{5}-0.004 t^{3}+0.4 t$ .

Determine if the series converges or diverges: $\sum_{n=1}^{\infty}\left[\left(\frac{6}{7}\right)^{n}-\frac{3}{2^{n}}\right]$ . If it converges, find the sum.

Find the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n(n+2)}$ .

Find the limit of the sequence $a_{n}=\sqrt{n^{2}+3 n}-n$ . Options: A) 3 B) 2 C) $1 / 2$ D) $3 / 2$ E) 0 F) 1 G) $\infty$ H) does not exist.

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 the range of values of $k$ for which the curve $y=x^{2}-4 x+k x+3$ is above the line $y+3 x=2$ .

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

甲、乙兩地相距 $7 \frac{1}{2}$ 公里，爸爸駕車走了 $1 \frac{1}{2}$ 公里，求他走了全程的幾分之幾？

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？

A ball's position is given by $x(t)=0.000015 t^5 - 0.004 t^3 + 0.4 t$ . Find its velocity at $t=10.0$ s.

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

Find values of $k$ so that $2 x^{2}+k^{2}+2-2(k+2) x > 0$ for all $x$ .

A ball's position is given by $x(t)=0.000015 t^5 - 0.004 t^3 + 0.4 t$ . Find its velocity at $t=10.0$ s.

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.

Prove $x^{2}-x+2 \geq \frac{7}{4}$ for all values of $x$ .

Find $x$ such that $1+x^{2} \geq-\frac{7 x-x^{2}}{3}$ and determine the minimum of $y=1+x^{2}+\frac{7 x-x^{2}}{3}$ .

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

Find all solutions for the function $f(x)=2 x^{3}-6 x^{2}+9 x-27$ .

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

Calculate the integral $\int_{1}^{5} \frac{x-1}{x} dx$ and choose the correct answer: (a) $5-\ln 5$ , (b) $4-\ln 5$ , (c) $2-\ln 5$ , (d) $1-\ln 5$ .

Find the limit: $\lim _{x \rightarrow 1} \frac{2 \cdot \ln (x)}{\mathrm{e}^{x}-1}$ . Options: (a) $\frac{2}{e}$ (b) 1 (c) 0 (d) nonexistent.

Find $f(1)$ given $f^{\prime}(x)=3 x^{2}$ and $f(2)=3$ . Choices: (a) -7 (b) -4 (c) 7 (d) 10

17) Find the area of the region between $y=2x$ and $y=x^2$ . 18) If a cube's side length $S=2$ in decreases at 5 in³/min, find the rate of change of $S$ in in/min.

Find the derivative $\frac{d}{d x}(\tan (\ln (x)))$ . What is $\frac{d y}{d x}$ for $\sqrt{x}+y^{2}=x y+2$ at (4,0)?

A particle's velocity is $v(t)=e^{t-1}-3 \sin (t-1)$ . What is its motion at $t=1$ ? (a) speeding up, (b) slowing down, (c) neither, (d) at rest.

At $t=2$ , what does $R^{\prime}(2)=4$ mean? (a) Depth is 4 inches. (b) Rate is 4 in/hr. (c) Rate of change is 4 in/hr². (d) Depth increased by 4 inches.

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 limit: $\lim _{x \rightarrow \infty} \frac{4 \cdot \ln (x)+4}{3 x}$ . Choose (a) 2, (b) -2, (c) 0, or (d) nonexistent.

Find $f'(x)$ if $f(x)=\frac{x^{2}+1}{3 x}$ and evaluate $\int(2 x+3)(x^{2}+3 x)^{4} dx$ .