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Answer to Analytics Geometry Problems #1-5

Answer to Analytics Geometry Problems #1-5

Compilation of Problems in Analytic Geometry COMPILATION OF MATH PROBLEMS ANALYTIC GEOMETRY PROBLEMS & SOLTUIONS: 1. Find the eq...

LET Reviewer in Math - Probability

<div class="separator"><div style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijzsYT9OmWmfvnqldC0XX0bT_hcAsz5L_m8peronNby56HuuvMCMdPX2HUK4F-Jesd-V6hcW_S_57m5jwu_Bj9lKeOIdqvaneknGida-ddxfWfUm55PX8L-iyrx3kuKurlFboR7Dvs8-q1IhE9FWO7hWeWwkFyBk8Tv8RZ0jgrAxeP1jw_iWzNvFJvxg/s320/math3.jpg" /></div></div><br /><div><br /></div><div><div><b>1. A coin is tossed 3 times. What is the probability of getting 2 heads and 1 tail?</b></div><div>a) 1/4</div><div>b) 3/8</div><div>c) 1/2</div><div>d) 5/8</div><div>e) none of the above</div><div><br /></div><div>Solution: The total number of possible outcomes is 2^3 = 8 (since there are 2 options for each toss). There are 3 ways to get 2 heads and 1 tail: HHT, HTH, THH. Each of these outcomes has a probability of (1/2)^3 = 1/8. Therefore, the probability of getting 2 heads and 1 tail is:</div><div><br /></div><div>3 x 1/8 = 3/8</div><div><br /></div><div>Answer: (b) 3/8.</div><div><br /></div><div><br /></div><div><b>2. A jar contains 5 red balls, 3 blue balls, and 2 green balls. If one ball is drawn at random, what is the probability of getting a blue ball?</b></div><div>a) 1/2</div><div>b) 1/4</div><div>c) 3/10</div><div>d) 3/4</div><div>e) none of the above</div><div><br /></div><div>Solution: There are 10 balls in total, so the probability of getting a blue ball is 3/10.</div><div><br /></div><div>Answer: (c) 3/10.</div><div><br /></div><div><br /></div><div><b>3. A fair 6-sided die is rolled twice. What is the probability of getting a sum of 7?</b></div><div>a) 1/6</div><div>b) 1/5</div><div>c) 1/4</div><div>d) 1/3</div><div>e) none of the above</div><div><br /></div><div>Solution: There are 36 possible outcomes when rolling two dice (since there are 6 options for each roll). There are 6 ways to get a sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Therefore, the probability of getting a sum of 7 is:</div><div><br /></div><div>6/36 = 1/6</div><div><br /></div><div>Answer: (a) 1/6.</div></div><div><br /></div><div><br /></div><div><div><b>4. In a certain town, it is known that 30% of the people have a college degree. If 4 people from this town are randomly selected, what is the probability that all 4 have a college degree?</b></div><div>a) 0.001</div><div>b) 0.008</div><div>c) 0.027</div><div>d) 0.064</div><div>e) none of the above</div><div><br /></div><div>Solution: The probability that any one person has a college degree is 0.3. Assuming independence between the people, the probability that all 4 have a college degree is:</div><div><br /></div><div>0.3 x 0.3 x 0.3 x 0.3 = 0.0081</div><div><br /></div><div>Answer: (b) 0.008.</div><div><br /></div><div><br /></div><div><b>5. A bag contains 8 red balls and 4 blue balls. Two balls are drawn at random without replacement. What is the probability that both balls are red?</b></div><div>a) 1/3</div><div>b) 2/7</div><div>c) 2/3</div><div>d) 4/7</div><div>e) none of the above</div><div><br /></div><div>Solution: The probability of drawing a red ball on the first draw is 8/12. If a red ball is drawn on the first draw, the probability of drawing another red ball on the second draw (without replacement) is 7/11. Therefore, the probability of drawing two red balls is:</div><div><br /></div><div>(8/12) x (7/11) = 14/33</div><div><br /></div><div>Answer: e) none of the given options.</div></div><div><br /></div><div><br /></div><div><div><b>6. A company produces light bulbs, and the probability of a bulb being defective is 0.05. A sample of 10 bulbs is chosen at random from the production line. What is the probability that at least 1 bulb is defective?</b></div><div>a) 0.401</div><div>b) 0.598</div><div>c) 0.950</div><div>d) 0.999</div><div>e) none of the above</div><div><br /></div><div>Solution: The probability that none of the bulbs is defective is:</div><div><br /></div><div>(0.95)^10 = 0.5987369392</div><div><br /></div><div>Therefore, the probability that at least 1 bulb is defective is:</div><div><br /></div><div>1 - 0.5987369392 = 0.4012630608</div><div><br /></div><div>Answer: (a) 0.401.</div></div><div><br /></div><div><br /></div><div><div><b>7. A class has 25 students, of which 10 are boys and 15 are girls. If 4 students are chosen at random, what is the probability that 2 are boys and 2 are girls?</b></div><div>a) 0.222</div><div>b) 0.329</div><div>c) 0.441</div><div>d) 0.556</div><div>e) none of the above</div><div><br /></div><div>Solution: The number of ways to choose 2 boys out of 10 is C(10,2) = 45, and the number of ways to choose 2 girls out of 15 is C(15,2) = 105. The total number of ways to choose 4 students out of 25 is C(25,4) = 12,650. Therefore, the probability of choosing 2 boys and 2 girls is:</div><div><br /></div><div>(45 x 105) / 12650 = 0.375</div><div><br /></div><div>Answer: e) none of the given options.</div></div><div><br /></div><div><br /></div><div><div><b>8. In a certain town, it is known that 40% of the people have brown eyes and 60% have blue eyes. If 3 people from this town are randomly selected, what is the probability that all 3 have brown eyes?</b></div><div>a) 0.064</div><div>b) 0.096</div><div>c) 0.216</div><div>d) 0.343</div><div>e) none of the above</div><div><br /></div><div>Solution: The probability that any one person has brown eyes is 0.4. Assuming independence between the people, the probability that all 3 have brown eyes is:</div><div><br /></div><div>0.4 x 0.4 x 0.4 = 0.064</div><div><br /></div><div>Answer: (a) 0.064.</div><div><br /></div><div><br /></div><div><b>9. A bowl contains 5 red marbles and 3 green marbles. If 2 marbles are drawn at random without replacement, what is the probability that both marbles are red?</b></div><div>a) 1/14</div><div>b) 5/14</div><div>c) 5/21</div><div>d) 10/21</div><div>e) none of the above</div><div><br /></div><div>Solution: The probability of drawing a red marble on the first draw is 5/8. If a red marble is drawn on the first draw, the probability of drawing another red marble on the second draw (without replacement) is 4/7. Therefore, the probability of drawing 2 red marbles is:</div><div><br /></div><div>(5/8) x (4/7) = 5/14</div><div><br /></div><div>Answer: (b) 5/14</div></div><div><br /></div><div><br /></div><div><div><b>10. A card is drawn at random from a standard deck of 52 playing cards. What is the probability that the card is a face card (jack, queen, or king) or a heart?</b></div><div>a) 13/52</div><div>b) 16/52</div><div>c) 20/52</div><div>d) 24/52</div><div>e) none of the above</div><div><br /></div><div>Solution: There are 12 face cards in a deck of 52 playing cards, and there are 13 hearts. However, the jack, queen, and king of hearts are counted twice since they are both face cards and hearts. Therefore, the total number of cards that are either face cards or hearts is:</div><div><br /></div><div>12 + 13 - 3 = 22</div><div><br /></div><div>The probability of drawing a face card or a heart is therefore:</div><div><br /></div><div>22/52 = 11/26</div><div><br /></div><div>Answer: e) none of the given options.</div></div>

LET Reviewer in Math - Probability

1. A coin is tossed 3 times. What is the probability of getting 2 heads and 1 tail? a) 1/4 b) 3/8 c) 1/2 d) 5/8 e) none of the above Solutio...

Number Theory Practice Test with Solution

<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjD142Y1yzILkOo-083w2DrSsyhr1fu1wZDuMcX8XhWoikJ9Ub2RfKKot7Q7pAO_JXzaMyPZcq06UhSD6ei5tynIe0pV5tGZ-Sm3u62aX4M_ksETVnqt5BayM3KQVp64KrQQBVhmwkT2z3Dxvy6YFFLtyRQTDwnqhrvmfZiYAcnBzOgjnml875lWb2NyA/s494/math4.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="328" data-original-width="494" height="185" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjD142Y1yzILkOo-083w2DrSsyhr1fu1wZDuMcX8XhWoikJ9Ub2RfKKot7Q7pAO_JXzaMyPZcq06UhSD6ei5tynIe0pV5tGZ-Sm3u62aX4M_ksETVnqt5BayM3KQVp64KrQQBVhmwkT2z3Dxvy6YFFLtyRQTDwnqhrvmfZiYAcnBzOgjnml875lWb2NyA/s280/math4.jpg" width="280" /></a></div><br /><div><br /></div><div><br /></div><div><b>1. Find the absolute value of the complex number 2 + 3i.</b></div><div><br /></div><div>Solution:</div><div>The absolute value of a complex number is the distance between the origin and the point representing the complex number on the complex plane. We can use the Pythagorean theorem to find the distance:</div><div>|2 + 3i| = sqrt(2^2 + 3^2) = sqrt(13)</div><div>Therefore, the absolute value of the complex number 2 + 3i is sqrt(13).</div><div><br /></div><div><br /></div><div><b>2. Find the square of the complex number 1 + i.</b></div><div><br /></div><div>Solution:</div><div>To square a complex number, we can use the formula:</div><div>(a + bi)^2 = a^2 + 2abi - b^2</div><div>where a and b are the real and imaginary parts of the complex number.</div><div>In this case, a = 1 and b = 1, so we have:</div><div>(1 + i)^2 = 1^2 + 2(1)(i) - 1^2 = 2i</div><div><br /></div><div>Therefore, the square of the complex number 1 + i is 2i.</div><div><br /></div><div><br /></div><div><b>3. Find all complex solutions to the equation z^3 = 1.</b></div><div><br /></div><div>Solution:</div><div>To solve this equation, we can write 1 as a complex number using polar coordinates:</div><div>1 = 1(cos(0) + i sin(0))</div><div>where 0 is any multiple of 2π.</div><div>Then, the cube roots of 1 are given by:</div><div>z = 1^(1/3) = cos(0/3) + i sin(0/3)</div><div>z = 1^(1/3) = cos(2π/3) + i sin(2π/3)</div><div>z = 1^(1/3) = cos(4π/3) + i sin(4π/3)</div><div><br /></div><div>Therefore, the three complex solutions to the equation z^3 = 1 are:</div><div>z = 1, z = -1/2 + i sqrt(3)/2, and z = -1/2 - i sqrt(3)/2.</div><div><br /></div><div><br /></div><div><b>4. Find all complex solutions to the equation z^4 = -16.</b></div><div><br /></div><div>Solution:</div><div>To solve this equation, we can write -16 as a complex number using polar coordinates:</div><div>-16 = 16(cos(π) + i sin(π))</div><div>Then, the fourth roots of -16 are given by:</div><div>z = (-16)^(1/4) = 2(cos(π/4) + i sin(π/4))</div><div>z = (-16)^(1/4) = 2(cos(3π/4) + i sin(3π/4))</div><div>z = (-16)^(1/4) = 2(cos(5π/4) + i sin(5π/4))</div><div>z = (-16)^(1/4) = 2(cos(7π/4) + i sin(7π/4))</div><div><br /></div><div>Therefore, the four complex solutions to the equation z^4 = -16 are:</div><div>z = 2(cos(π/4) + i sin(π/4)), </div><div>z = 2(cos(3π/4) + i sin(3π/4)), </div><div>z = 2(cos(5π/4) + i sin(5π/4)), and </div><div>z = 2(cos(7π/4) + i sin(7π/4)).</div><div><br /></div><div>Note that we can also write these solutions in rectangular form if desired.</div><div><br /></div><div><br /></div><div><div><b>5. Find all complex numbers z such that |z - 3 - 4i| = 5.</b></div><div><br /></div><div>Solution:</div><div>The equation |z - 3 - 4i| = 5 represents the set of complex numbers that are a distance of 5 units from the point (3,4) in the complex plane. Geometrically, this represents a circle with radius 5 centered at (3,4).</div><div>To find all complex numbers z that satisfy this equation, we can write z = x + yi and substitute into the equation, then use the distance formula for two points to obtain:</div><div><br /></div><div>sqrt((x-3)^2 + (y-4)^2) = 5</div><div><br /></div><div>Squaring both sides gives:</div><div><br /></div><div>(x-3)^2 + (y-4)^2 = 25</div><div><br /></div><div>Expanding and simplifying:</div><div><br /></div><div>x^2 - 6x + y^2 - 8y + 24 = 0</div><div><br /></div><div>Completing the square for both x and y:</div><div><br /></div><div>(x - 3)^2 + (y - 4)^2 = 1</div><div><br /></div><div>This is the equation of a circle with radius 1 centered at (3,4). Therefore, the set of complex numbers that satisfy |z - 3 - 4i| = 5 is the set of complex numbers that lie on the circle with radius 5 centered at (3,4) in the complex plane.</div><div><br /></div><div>In rectangular form, the equation of this circle is:</div><div><br /></div><div>(x - 3)^2 + (y - 4)^2 = 25</div><div><br /></div><div>or</div><div><br /></div><div>x^2 + y^2 - 6x - 8y + 24 = 0</div><div><br /></div><div>which is the same equation we obtained earlier by completing the square.</div></div>

Number Theory Practice Test with Solution

1. Find the absolute value of the complex number 2 + 3i. Solution: The absolute value of a complex number is the distance between the origin...

Math Reviewer: Differential Calculus - Problems and Solutions

<h3 style="text-align: left;"> Multiple Choice Questions in Differential Calculus </h3><p><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIl2yrjkH7Gj6Gy9ElHwyLV1JOOEaZW3f8q5vGeY_2X03cB9UpRScr-WdkxjPd-YdjZMpMbTFCVQJ_LjTqRZPWSDufuC9wVjjHZZxp84WbpltYbA50NpcQnWl7AjIEmKHjzvnlkRWQJoMFgmMyWlBHtLv_ynJ6OgGbBeNZrc3PPV0Q2AlzemrNj3tcNg/s600/mcqs-math.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="250" data-original-width="600" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIl2yrjkH7Gj6Gy9ElHwyLV1JOOEaZW3f8q5vGeY_2X03cB9UpRScr-WdkxjPd-YdjZMpMbTFCVQJ_LjTqRZPWSDufuC9wVjjHZZxp84WbpltYbA50NpcQnWl7AjIEmKHjzvnlkRWQJoMFgmMyWlBHtLv_ynJ6OgGbBeNZrc3PPV0Q2AlzemrNj3tcNg/s16000/mcqs-math.jpg" /></a></div><br /><p><br /></p><p>1. What is the derivative of f(x) = x^3 at x = 2?</p><p>a) 6</p><p>b) 12</p><p>c) 24</p><p>d) 8</p><p><br /></p><p>2. What is the second derivative of f(x) = sin(x)?</p><p>a) cos(x)</p><p>b) -sin(x)</p><p>c) -cos(x)</p><p>d) -cos(x)</p><p><br /></p><p>3. What is the critical point of f(x) = x^2 - 4x + 5?</p><p>a) x = 2</p><p>b) x = 4</p><p>c) x = -2</p><p>d) There are no critical points</p><p><br /></p><p>4. If f(x) = e^x + x^2, what is f'(x)?</p><p>a) e^x + 2x</p><p>b) e^x + x</p><p>c) 2e^x + x</p><p>d) e^x + 2</p><p><br /></p><p>5. What is the limit of (x^3 - 3x^2 + 2x + 1) / (x^2 - x - 2) as x approaches 2?</p><p>a) 0</p><p>b) 1</p><p>c) 2</p><p>d) the limit does not exist</p><p><br /></p><p>6. What is the derivative of f(x) = ln(x) + 2x?</p><p>a) 1/x + 2</p><p>b) 1/x + 2x</p><p>c) ln(x) + 2</p><p>d) ln(x) + 2x</p><p><br /></p><p>7. What is the critical point of f(x) = x^3 - 12x + 5?</p><p>a) x = 2</p><p>b) x = -2</p><p>c) Both a and b</p><p>d) There are no critical points</p><p><br /></p><p>8. If f(x) = 3x^4 + 2x^3 - 6x^2, what is f''(x)?</p><p>a) 24x^2 + 12x - 12</p><p>b) 12x^2 + 12x - 12</p><p>c) 12x^2 + 6x - 12</p><p>d) 24x^2 + 6x - 12</p><p><br /></p><p>9. What is the derivative of f(x) = sqrt(x) / (x^2 + 1)?</p><p>a) (1/2)x^(-1/2)(x^2+1)^(-1) - 2x(x^2+1)^(-2)</p><p>b) (1/2)x^(-1/2)(x^2+1)^(-1) - x(x^2+1)^(-2)</p><p>c) (1/2)x^(-1/2)(x^2+1)^(-2) - x(x^2+1)^(-1)</p><p>d) (1 - 3 x^2)/(2 sqrt(x) (1 + x^2)^2)</p><p><br /></p><p>10. What is the limit of (2x^2 - 3x + 1) / (x^2 - 1) as x approaches 1?</p><p>a) -2</p><p>b) -1</p><p>c) 1</p><p>d) 1/2</p><div><br /></div><h3 style="text-align: left;">Here are the solutions:</h3><div><br /></div><div><div><b>1.<span style="white-space: pre;"> </span>b) 12 </b></div><div>Using the power rule, f'(x) = 3x^2, so f'(2) = 12.)</div><div><br /></div><div><b>2.<span style="white-space: pre;"> </span>b) -sin(x)</b></div><div>Taking the derivative of f(x) = sin(x) gives us f'(x) = cos(x). Taking the derivative again gives us f''(x) = -sin(x)</div><div><br /></div><div><b>3.<span style="white-space: pre;"> </span>a) x = 2 </b></div><div>Taking the derivative of f(x) = x^2 - 4x + 5 gives us f'(x) = 2x - 4. Setting this equal to zero and solving for x, we get x = 2.</div><div><br /></div><div><b>4.<span style="white-space: pre;"> </span>a) e^x + 2x </b></div><div>Using the sum rule and the derivative of e^x, we get f'(x) = e^x + 2x.</div><div><br /></div><div><b>5.<span style="white-space: pre;"> </span>d) the limit does not exist</b></div><div>The limit of (x^3 - 3x^2 + 2x + 1)/(x^2 - x - 2) as x approaches 2 can be found by direct substitution: ((2)^3 - 3(2)^2 + 2(2) + 1)/((2)^2 - (2) - 2) = (8 - 12 + 4 + 1)/(4 - 2 - 2) = (1)/(0). Since division by zero is undefined, the limit does not exist</div><div><br /></div><div><b>6.<span style="white-space: pre;"> </span>a) 1/x + 2 </b></div><div>Using the sum rule and the derivative of ln(x), we get f'(x) = 1/x + 2.</div><div><br /></div><div><b>7.<span style="white-space: pre;"> c) </span>Both a and b</b></div><div>The derivative of f(x) = x^3 - 12x +5 is f’(x)=3x^2-12. To find the critical points we set f’(x)=0 and solve for x: 3x^2-12=0 => x^2-4=0 => (x+2)(x-2)=0 So we have two critical points: x=-2 and x=+2.</div><div><br /></div><div><b>8.<span style="white-space: pre;"> </span>(a)36x^2+12x-12</b></div><div>If f(x)=3x4+2x3-6x^2 then the first derivative of f(x),f’(x)=12x3+6x2-12x and the second derivative of f(x),f’'(x)=36x^2+12x-12. The correct answer is (a)36x^2+12x-12.</div><div><br /></div><div><b>9.<span style="white-space: pre;"> </span>d) (1 - 3 x^2)/(2 sqrt(x) (1 + x^2)^2)</b></div><div><div>Let's find the derivative of f(x) = sqrt(x) / (x^2 + 1) using the quotient rule and the chain rule:</div><div>f(x) = (x^(1/2)) / ((x^2 + 1))</div><div>f'(x) = [(1/2)(x^(-1/2))(x^2 + 1) - (x^(1/2))(2x)] / (x^2 + 1)^2</div><div>f'(x) = [x - 2x^(3/2)] / (2x^(3/2)(x^2 + 1)^2)</div><div>f'(x) = (x - 2x^2) / (2x^(5/2)(x^2 + 1)^2)</div><div>Simplifying this expression, we get:</div><div>f'(x) = (1 - 3x^2) / (2sqrt(x)(1 + x^2)^2)</div></div><div><br /></div><div><b>10.<span style="white-space: pre;"> </span>d) 1/2 </b></div><div>If you try to plug in x=1 directly, you'll end up with an indeterminate form of 0/0. So, you need to use another method to evaluate this limit. One possible method is to factorize both the numerator and the denominator and simplify the expression. (2x^2 - 3x + 1) / (x^2 - 1) = [(2x - 1)(x - 1)] / [(x - 1)(x + 1)]. Therefore, the limit of (2x^2 - 3x + 1) / (x^2 - 1) as x approaches 1 is 1/2.</div></div><div><br /></div>

$Math Reviewer: Differential Calculus - Problems and Solutions$

Math Reviewer: Differential Calculus - Problems and Solutions

Multiple Choice Questions in Differential Calculus 1. What is the derivative of f(x) = x^3 at x = 2? a) 6 b) 12 c) 24 d) 8 2. What is the ...

Reviewer in Math: Advanced Geometry - Problems and Solution Part 1

<h3 style="text-align: left;">Sample multiple-choice questions in advanced geometry:</h3><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIl2yrjkH7Gj6Gy9ElHwyLV1JOOEaZW3f8q5vGeY_2X03cB9UpRScr-WdkxjPd-YdjZMpMbTFCVQJ_LjTqRZPWSDufuC9wVjjHZZxp84WbpltYbA50NpcQnWl7AjIEmKHjzvnlkRWQJoMFgmMyWlBHtLv_ynJ6OgGbBeNZrc3PPV0Q2AlzemrNj3tcNg/s600/mcqs-math.jpg" imageanchor="1" style=""><img border="0" data-original-height="250" data-original-width="600" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIl2yrjkH7Gj6Gy9ElHwyLV1JOOEaZW3f8q5vGeY_2X03cB9UpRScr-WdkxjPd-YdjZMpMbTFCVQJ_LjTqRZPWSDufuC9wVjjHZZxp84WbpltYbA50NpcQnWl7AjIEmKHjzvnlkRWQJoMFgmMyWlBHtLv_ynJ6OgGbBeNZrc3PPV0Q2AlzemrNj3tcNg/s16000/mcqs-math.jpg" /></a></div><h4 style="text-align: left;">Multiple Choice Questions in Geometry </h4><p>1. Which of the following is a postulate of Euclidean geometry?</p><p>a) Two parallel lines intersect.</p><p>b) Two points determine a unique line.</p><p>c) The shortest distance between two points is a curved line.</p><p>d) The sum of the interior angles of a triangle is greater than 180 degrees.</p><p>Answer: b) Two points determine a unique line.</p><p><br /></p><p>2. A sphere has a diameter of 10 cm. What is its surface area?</p><p>a) 50π cm^2</p><p>b) 100π cm^2</p><p>c) 200π cm^2</p><p>d) 400π cm^2</p><p>Answer: b) 100π cm^2</p><p><br /></p><p>3. In triangle ABC, angle A = 60 degrees and angle B = 80 degrees. What is the measure of angle C?</p><p>a) 40 degrees</p><p>b) 60 degrees</p><p>c) 80 degrees</p><p>d) 100 degrees</p><p>Answer: d) 100 degrees. (The sum of the angles in a triangle is 180 degrees, so angle C = 180 - 60 - 80 = 40 degrees.)</p><p><br /></p><p>4. Which of the following is true about a line and a plane in three-dimensional space?</p><p>a) A line can intersect a plane at more than one point.</p><p>b) A line can be parallel to a plane.</p><p>c) A line can be perpendicular to a plane.</p><p>d) All of the above.</p><p>Answer: d) All of the above.</p><p><br /></p><p>5. A triangle has side lengths 3 cm, 4 cm, and 5 cm. Which of the following statements is true?</p><p>a) The triangle is acute.</p><p>b) The triangle is right.</p><p>c) The triangle is obtuse.</p><p>d) There is not enough information to determine the type of triangle.</p><p>Answer: b) The triangle is right. (This is a Pythagorean triple, which means the triangle is a right triangle.)</p><p><br /></p><p>6. In triangle ABC, the bisectors of angles A and B intersect at point O. If the measure of angle C is 120 degrees, what is the measure of angle O?</p><p>a) 60 degrees</p><p>b) 90 degrees</p><p>c) 120 degrees</p><p>d) 150 degrees</p><p>Answer: a) 60 degrees. (By the angle bisector theorem, the lengths of the segments formed by the bisectors of angles A and B are proportional to the lengths of the sides opposite those angles. Therefore, the measure of angle O is 1/2 the measure of angle C, or 60 degrees.)</p><p><br /></p><p>7. A cone has a height of 10 cm and a radius of 6 cm. What is the slant height of the cone?</p><p>a) 6 cm</p><p>b) 8 cm</p><p>c) 10 cm</p><p>d) 12 cm</p><p>Answer: d) 12 cm. (By the Pythagorean theorem, the slant height of a cone is given by √(r^2 + h^2), so in this case it is √(6^2 + 10^2) = √136 = 12 cm.)</p><p><br /></p><p>8. In a rectangular prism with dimensions 6 cm × 8 cm × 10 cm, what is the length of the diagonal?</p><p>a) 10 cm</p><p>b) 12 cm</p><p>c) 14 cm</p><p>d) 16 cm</p><p>Answer: d) 16 cm. (By the Pythagorean theorem, the length of the diagonal of a rectangular prism is given by √(l^2 + w^2 + h^2), so in this case it is √(6^2 + 8^2 + 10^2) = √260 = 16 cm.)</p><p><br /></p><p>9. A regular hexagon has a side length of 5 cm. What is its area?</p><p>a) 43.3 cm^2</p><p>b) 64.95 cm^2</p><p>c) 75.0 cm^2</p><p>d) 129.9 cm^2</p><p>Answer: b) 64.95 cm^2. (The formula for the area of a regular polygon is A = (1/2)ns^2 sin(360/n), where n is the number of sides and s is the side length. In this case, n = 6 and s = 5, so A = (1/2)6(5)^2 sin(360/6) = 64.95 cm^2.)</p><p><br /></p><p>10. In a circle with center O and radius 6 cm, chord AB has length 8 cm. What is the distance between O and AB?</p><p>a) 2 cm</p><p>b) 3 cm</p><p>c) 4 cm</p><p>d) 5 cm</p><p>Answer: b) 3 cm. (The distance between the center of a circle and a chord is given by (2rh)/(2r), where r is the radius of the circle and h is the distance from the center to the chord. In this case, r = 6 and AB is not a diameter, so h = √(r^2 - (AB/2)^2) = √(6^2 - 4^2) = √20. Therefore, the distance between O and AB is (2rh)/(2r) = h = √20 = 2√5 = approximately 3 cm.)</p><div><br /></div>

$Reviewer in Math: Advanced Geometry - Problems and Solution Part 1$

Reviewer in Math: Advanced Geometry - Problems and Solution Part 1

Sample multiple-choice questions in advanced geometry: Multiple Choice Questions in Geometry 1. Which of the following is a postulate of Eu...

LET Reviewer in Math: Sample Topics, Problems and Solution

<h3 style="text-align: left;">Here are some sample topics and questions that may be included in a licensure exam for Math teachers:</h3><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTzRhqBMeeuNaaF1r7uukqm23bSS8mifUiRegBHt9EnBwlOLO7Lfa2fGIcg2-MJ3ldjUw4j2-4qkDqBX4kD6FJsIJnKfift26XShF7hS_Ap8_KquUcF-Bceoo0RbmfkDUslxX1DPbQOClweHDd4XBXcoiJhiY2Ncjmn3Zo2bGsWwYsZEoucLUepWuvKw/s600/mcqs-math.jpg" imageanchor="1" style=""><img border="0" data-original-height="250" data-original-width="600" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTzRhqBMeeuNaaF1r7uukqm23bSS8mifUiRegBHt9EnBwlOLO7Lfa2fGIcg2-MJ3ldjUw4j2-4qkDqBX4kD6FJsIJnKfift26XShF7hS_Ap8_KquUcF-Bceoo0RbmfkDUslxX1DPbQOClweHDd4XBXcoiJhiY2Ncjmn3Zo2bGsWwYsZEoucLUepWuvKw/s16000/mcqs-math.jpg" /></a></div><div style="text-align: left;"><br /></div><h3 style="text-align: left;">1. Algebra and Functions</h3><p></p><ul style="text-align: left;"><li>Solving linear and quadratic equations</li><li>Graphing and analyzing linear and quadratic functions</li><li>Simplifying and solving rational expressions</li><li>Finding the domain and range of functions</li></ul><p></p><h3 style="text-align: left;">2. Geometry</h3><p></p><ul style="text-align: left;"><li>Properties of triangles, circles, and other polygons</li><li>Perimeter, area, and volume of geometric shapes</li><li>Similarity and congruence of triangles</li><li>Trigonometry and its applications</li></ul><p></p><h4 style="text-align: left;">3. Statistics and Probability</h4><p></p><ul style="text-align: left;"><li>Measures of central tendency and variation</li><li>Probability rules and concepts</li><li>Permutations and combinations</li><li>Sampling techniques and surveys</li></ul><p></p><h4 style="text-align: left;">4. Calculus</h4><p></p><ul style="text-align: left;"><li>Limits, derivatives, and integrals</li><li>Applications of calculus to real-world problems</li><li>Optimization problems and related rates</li></ul><p></p><p><br /></p><h3 style="text-align: left;">Sample Questions:</h3><p></p><ol style="text-align: left;"><li>Solve for x: 2x + 3 = 11</li><li>Graph the function f(x) = 2x - 5 and find its intercepts.</li><li>Find the perimeter of a regular hexagon with side length 6 cm.</li><li>A survey of 500 students found that 200 of them preferred math over science. What is the probability of randomly selecting a student who prefers math?</li><li>Find the derivative of f(x) = 3x^2 - 4x + 1.</li><li>An open box is to be made from a rectangular piece of cardboard measuring 24 cm by 36 cm by cutting out squares of equal area from each corner and folding up the sides. What size of square should be cut out to make the box have maximum volume?</li><li>If sin x = 0.6, what is the value of cos x?</li></ol><div><br /></div><div><h3 style="text-align: left;">Here are the solutions for the sample questions provided:</h3><div><br /></div><div>1. Solve for x: 2x + 3 = 11</div><div><ul style="text-align: left;"><li>Solution: Subtract 3 from both sides of the equation: 2x = 8. Divide both sides by 2: x = 4.</li></ul></div><div><br /></div><div>2. Graph the function f(x) = 2x - 5 and find its intercepts.</div><div><ul style="text-align: left;"><li>Solution: To find the y-intercept, set x = 0: f(0) = -5. So the y-intercept is (0, -5). To find the x-intercept, set f(x) = 0: 2x - 5 = 0. Solving for x gives x = 2.5. So the x-intercept is (2.5, 0). The graph of the function is a line with a slope of 2 and a y-intercept of -5.</li></ul></div><div><br /></div><div>3. Find the perimeter of a regular hexagon with side length 6 cm.</div><div><ul style="text-align: left;"><li>Solution: A regular hexagon has six sides of equal length. To find the perimeter, we multiply the length of one side by 6: 6 x 6 = 36 cm. So the perimeter of the hexagon is 36 cm.</li></ul></div><div><br /></div><div>4. A survey of 500 students found that 200 of them preferred math over science. What is the probability of randomly selecting a student who prefers math?</div><div><ul style="text-align: left;"><li>Solution: The probability of selecting a student who prefers math is the number of students who prefer math divided by the total number of students: 200/500 = 0.4 or 40%.</li></ul></div><div><br /></div><div>5. Find the derivative of f(x) = 3x^2 - 4x + 1.</div><div><ul style="text-align: left;"><li>Solution: The derivative of f(x) is f'(x), which is given by the power rule: f'(x) = 6x - 4.</li></ul></div><div><br /></div><div>6. An open box is to be made from a rectangular piece of cardboard measuring 24 cm by 36 cm by cutting out squares of equal area from each corner and folding up the sides. What size of square should be cut out to make the box have maximum volume?</div><div><ul style="text-align: left;"><li>Solution: Let x be the length of each side of the square that is cut out. Then the dimensions of the box will be (36-2x) x (24-2x) x x. The volume of the box is given by V(x) = x(36-2x)(24-2x). To find the maximum volume, we take the derivative of V(x) with respect to x, set it equal to zero, and solve for x. After some algebraic manipulations, we get x = 3 cm. So the size of the square that should be cut out to maximize the volume of the box is 3 cm.</li></ul></div><div><br /></div><div>7. If sin x = 0.6, what is the value of cos x?</div><div><ul style="text-align: left;"><li>Solution: To find cos x, we use the Pythagorean identity: cos^2 x + sin^2 x = 1. Substituting sin x = 0.6 gives: cos^2 x + (0.6)^2 = 1. Solving for cos x gives cos x = ±0.8. Since x is in the first or second quadrant (where cosine is positive), we take the positive value: cos x = 0.8.</li></ul></div></div><p></p>

$LET Reviewer in Math: Sample Topics, Problems and Solution$

LET Reviewer in Math: Sample Topics, Problems and Solution

Here are some sample topics and questions that may be included in a licensure exam for Math teachers: 1. Algebra and Functions Solving linea...

MCQs / Practice Test on Engineering Mathematics

<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDPwA8LDJJMlTn3yFwY1mfKK9AVGnYO-5AHMrbSCfTwYfKNXumO_L9Z0pF9xANvEzwgM98Bl5dttfIoxpP9DVyDXCxHiypSwb7_MN_Gh0zbbE0Fryq7F1CLaQiqPrsuVOOrPYnPdl0QObT/s1600/mcqs-eng-math.jpg" imageanchor="1" style=""><img alt="MCQs on Engineering Mathematics" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDPwA8LDJJMlTn3yFwY1mfKK9AVGnYO-5AHMrbSCfTwYfKNXumO_L9Z0pF9xANvEzwgM98Bl5dttfIoxpP9DVyDXCxHiypSwb7_MN_Gh0zbbE0Fryq7F1CLaQiqPrsuVOOrPYnPdl0QObT/s1600/mcqs-eng-math.jpg" title="MCQs on Engineering Mathematics" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">MCQs/Practice Test on Engineering Mathematics</td></tr>
</tbody></table>
<b>Multiple Choice Questions / Practice Test</b><br />
<b>Engineering Mathematics - Geometry</b><br />
<br />
1. Three circles C1, C2 and C3 are externally tangent to each other. Center to center distances are 10 cm between C1 and C2, 8 cm between C2 and C3 and 6 cm between C3 and C1. Determine the total areas of the circles.<br />
(a) 184.12 cm^2<br />
(b) 157.08 cm^2<br />
(c) 162.31 cm^2<br />
(d) 175.93 cm^2<br />
<br />
2. A rectangle is inscribed in a circle whose radius is 5 inches. The base of the rectangle is 8 inches. Find the area of the rectangle.<br />
(a) 44<br />
(b) 50<br />
(c) 48<br />
(d) 40<br />
<br />
3. The wall at one end of an attic takes the shape of trapezoid because of a slanted ceiling. The wall is 8 ft high at one end, 10 ft wide and only 3 ft high on the other end. Determine the area of the wall in sq. ft.<br />
(a) 50<br />
(b) 55<br />
(c) 60<br />
(d) 48<br />
<br />
4. The corner of a 2-meter square is cutoff to form a regular octagon. Determine the length of the resulting side of the octagon.<br />
(a) 0.834<br />
(b) 0.724<br />
(c) 0.732<br />
(d) 0.828<br />
<br />
5. A polygon has 170 diagonals. How many sides does it have?<span class="Apple-tab-span" style="white-space: pre;"> </span><br />
(a) 20<br />
(b) 30<br />
(c) 25<br />
(d) 28<br />
<br />
6. Find the length of the side of a pentagon if the line perpendicular to its side is 12 units from the center.<br />
(a) 14.74 units<br />
(b) 17.44 units<br />
(c) 71.44 units<br />
(d) 14.47 units<br />
<br />
7. A regular dodecagon is inscribed in a circle of radius 24. Find the perimeter of the dodecagon.<br />
(a) 149.08<br />
(b) 142.74<br />
(c) 145.53<br />
(d) 152.70<br />
<br />
8. The length of the side of a square is increased by 100%. The perimeter is increased by .<br />
(a) 25%<br />
(b) 100%<br />
(c) 200%<br />
(d) 150%<br />
<br />
9. The area of a rhombus is 168 m^2. If one side of its diagonal is 12 m, find the length of the sides of a rhombus.<br />
(a) 15.23 cm<br />
(b) 10.42 cm<br />
(c) 14.22 cm<br />
(d) 12.43 cm<br />
<br />
10. The perimeter of a sector is 9 and its radius is 3. What is the area of the sector?<br />
(a) 4.50<br />
(b) 6.52<br />
(c) 3.75<br />
(d) 5.25<br />
<br />
11. If the altitude and base of a parallelogram are each increased by 5 inches. If the altitude is increased by 3 inches and the base is decreased by 2 inches, the area will increase by 5 sq. inches. Determine the original values in inches of the altitude and base of the parallelogram.<br />
(a) alt = 3.2 in, base = 0.8 in<br />
(b) alt = 1.0 in, base = 3.0 in<br />
(c) alt = 0.8 in, base = 4.2 in<br />
(d) alt = 2.0 in, base = 2.0 in<br />
<br />
12. Three spheres of radii 10, 20 and 30 cm, respectively, are melted and formed into a single sphere. Find the volume of the single sphere.<br />
(a) 48,000π<br />
(b) 45,000π<br />
(c) 50,000π<br />
(d) 42,000π<br />
<br />
13. The bases of a right prism are regular hexagons with each side equal to 6 cm. The bases are 10 cm apart.<br />
What is the volume of the prism?<br />
(a) 935.31<br />
(b) 943.57<br />
(c) 783.22<br />
(d) 893.73<br />
<br />
14. What is the measure of the interior angle of a regular 2000-gon?<br />
(a) 180°<br />
(b) 220°<br />
(c) 200°<br />
(d) 178°<br />
<br />
15. Seven regular hexagons, each with 6-cm sides, are arranged so that they share some side and the centers of the six hexagons are equidistant from the seventh central hexagon. Determine the ratio of the area of one hexagon to the total outer perimeter enclosing the hexagons.<br />
(a) 0.8660<br />
(b) 1.0392<br />
(c) 0.6014<br />
(d) 0.7217<br />
<br />
16. The volume of a right prism with an altitude of 15 m and having an equilateral triangle as its base is equal to 234 m^3. Determine the length of the side of the triangular base.<br />
(a) 6<br />
(b) 7<br />
(c) 8<br />
(d) 5<br />
<br />
17. The interior angles of a polygon are in arithmetic progression. The least angle is 120° and the common difference is 5°. Find the number of sides.<br />
(a) 8<br />
(b) 9<br />
(c) 11<br />
(d) 10<br />
<br />
18. If the sides of a parallelogram and an included angle are 6, 10 and 100°, respectively, find the length of<br />
the shorter diagonal.<br />
(a) 10.73<br />
(b) 15.32<br />
(c) 12.53<br />
(d) 11.71<br />
<br />
19. A sphere of radius 10 m and a right circular cone of base radius 10 m and height 15 m stands on a table. At what height from the table should the two solids be cut in order to have equal circular sections?<br />
(a) 3.14 m<br />
(b) 3.24 m<br />
(c) 3.44 m<br />
(d) 3.54 m<br />
<br />
20. The sum of the interior angles of a polygon of n sides is 1080°. Find the value of n .<br />
(a) 10<br />
(b) 9<br />
(c) 8<br />
(d) 11<br />
<br />
21. A prism has an equilateral triangle with 20 cm on a side for its base, and an altitude of 30 cm. Determine the lateral area.<br />
(a) 1800<br />
(b) 1750<br />
(c) 1505<br />
(d) 2100<br />
<br />
22. Find the volume of a spherical wedge whose angle is 54° on a sphere of radius 27 cm.<br />
(a) 12,367.19 cm^3<br />
(b) 11,243.42 cm^3<br />
(c) 12,422.42 cm^3<br />
(d) 10,626.71 cm^3<br />
<br />
23. A metal sphere is melted and recast into a hollow spherical shell whose outer radius is 277 cm. The radius of the hollow interior of the shell is equal to the radius of the original sphere. Find the radius of the original sphere.<br />
(a) 200 cm<br />
(b) 225 cm<br />
(c) 220 cm<br />
(d) 240 cm<br />
<br />
24. Find the total surface area of a regular triangular pyramid if each edge of the base measures 6 inches and each lateral edge of the pyramid measures 5 inches.<br />
(a) 51.59 in^2<br />
(b) 50.23 in^2<br />
(c) 54.42 in^2<br />
(d) 50.21 in^2<br />
<br />
25. The altitude of an oblique circular cone is 8 inches and its longest element is 17 inches. Find the length of the shortest element if its volume is 54π cubic inches.<br />
(a) 8<br />
(b) 7<br />
(c) 9<br />
(d) 10<br />
<br />
26. Find the area of the spherical lune whose angle is 75° on a sphere of radius 30 cm.<br />
(a) 2356.19 cm^2<br />
(b) 2052.52 cm^2<br />
(c) 2054.34 cm^2<br />
(d) 2257.26 cm^2<br />
<br />
27. What is the name of a polygon that has 27 diagonals?<br />
(a) hexagon<br />
(b) decagon<br />
(c) nonagon<br />
(d) undecagon<br />
<br />
28. Water is poured to a depth of 12 cm into a hemispherical bowl of radius 20 cm. Find the volume of the water.<br />
(a) 2210π<br />
(b) 2304π<br />
(c) 2104π<br />
(d) 2540π<br />
<br />
29. The volume of a right circular cone is 36π. If its altitude is 3, find its radius.<br />
(a) 3<br />
(b) 5<br />
(c) 4<br />
(d) 6<br />
<br />
30. The volume of a regular pyramid whose base area is a regular hexagon is 156 cm^3. If the altitude of the pyramid is 5 cm, determine the length of the sides of the base.<br />
(a) 4 cm<br />
(b) 5 cm<br />
(c) 3 cm<br />
(d) 6 cm<br />
<br />
31. For a regular polygon of heptagon sides, find the number of degrees contained in each central angle.<br />
(a) 51.43°<br />
(b) 40°<br />
(c) 60°<br />
(d) 32.73°<br />
<br />
32. The great pyramid of Egypt has a square base 232 m on a side and 147 m high. Find its volume.<br />
(a) 2,768,334 m^3<br />
(b) 2,662,351 m^3<br />
(c) 2,632,242 m^3<br />
(d) 2,637,376 m^3<br />
<br />
33. A tetrahedron is a regular solid with a equilateral triangles for each of three surfaces. If each side is 10 cm, what is the volume of the tetrahedron?<br />
(a) 116.21 cu. cm<br />
(b) 117.85 cu. m<br />
(c) 91.67 cu. m<br />
(d) 83.33 cu. m<br />
<br />
34. A canonical vessel has a height of 24 cm and a base diameter of 12 cm. It holds water to depth of 18 cm above its vertex. Find the volume of its content in cm^3 .<br />
(a) 381.7<br />
(b) 392.2<br />
(c) 353.3<br />
(d) 363.2<br />
<br />
35. The area of a hexagon inscribed in a circle is 374.11 cm^2. Determine the area of the circle.<br />
(a) 452.39<br />
(b) 465.32<br />
(c) 423.34<br />
(d) 431.46<br />
<br />
36. A polygon has 170 diagonals. How many sides does it have?<br />
(a) 16<br />
(b) 22<br />
(c) 18<br />
(d) 20<br />
<br />
37. If the edge of the cube is increased by 20%, find the percentage increase in volume?<br />
(a) 72.8%<br />
(b) 83.2%<br />
(c) 76.3%<br />
(d) 63.3%<br />
<br />
38. The sum of the interior angles of a polygon is 540°. Find the number of sides.<br />
(a) 4<br />
(b) 6<br />
(c) 5<br />
(d) 7<br />
<br />
39. The height of a circular cone with circular base is h. If it contains water to a depth of 2/3 h, what is the ratio of the volume of the water to that of the cone?<br />
(a) 1:27<br />
(b) 8:27<br />
(c) 26:27<br />
(d) 24:27<br />
<br />
40. The diagonal of the face of a cube is 3sqrt(2) cm. Find the main diagonal of the cube.<br />
(a) 3sqrt(3)<br />
(b) 2sqrt(3)<br />
(c) 2sqrt(2)<br />
(d) 4sqrt(2)<br />
<br />
41. Find the length of the diagonal of a cube whose volume is 729 cubic cm.<br />
(a) 15.59 cm<br />
(b) 17.43 cm<br />
(c) 12.54 cm<br />
(d) 14.72 cm<br />
<br />
42. A room is 12 ft wide, 15 ft long and 8 ft high. If an air conditioner changes the air once every five minutes, how many cubic feet of air does it change per hour?<br />
(a) 17,280<br />
(b) 19,553<br />
(c) 14,522<br />
(d) 16,733<br />
<br />
43. The base areas of a frustum of a cone are 25 cm^2 volume. and 16 cm^2, respectively. If its altitude is 6 cm, find its<br />
(a) 120<br />
(b) 125<br />
(c) 122<br />
(d) 136<br />
<br />
44. Find the length of the side of a regular pentagon inscribed in a circle of radius 10 cm.<br />
(a) 10.34 cm<br />
(b) 11.76 cm<br />
(c) 12.42 cm<br />
(d) 35.22 cm<br />
<br />
45. The area of a zone of a spherical segment is 1/4 that of a sphere. What is the ratio of the radius to the altitude of the spherical zone?<br />
(a) 3:1<br />
(b) 2:1<br />
(c) 2:3<br />
(d) 3:4<br />
<br />
46. A cone and cylinder have the same height and the same volume. Find the ratio of the radius of the cone to the radius of the cylinder.<br />
(a) 2<br />
(b) sqrt(2)<br />
(c) 3<br />
(d) sqrt(3)<br />
<br />
47. If the edge of the cube is increased by 30%, by what percent is the surface area increased?<br />
(a) 45%<br />
(b) 90%<br />
(c) 70.7%<br />
(d) 69%<br />
<br />
48. Find the area of a regular pentagon whose side is 25 cm and apothem is 17.2 cm.<br />
(a) 1075<br />
(b) 1142<br />
(c) 1010<br />
(d) 1067<br />
<br />
49. The bases of a right prism are a hexagon with one side 6 cm long. If the volume of the prism is 450 cm^3, how apart are the bases?<br />
(a) 4.81 cm<br />
(b) 4.22 cm<br />
(c) 5.32 cm<br />
(d) 5.06 cm<br />
<br />
50. The circumference of the base of a right circular cylinder is 48 cm and its altitude is 15 cm. Determine its total surface area.<br />
(a) 1086.69 cm^2<br />
(b) 1042.52 cm^2<br />
(c) 1204.23 cm^2<br />
(d) 1102.62 cm^2<br />
<br />
51. A trapezoid has an area of 36 m^2 and altitude of 2 m. Its two bases have ratio of 4:5. What are the lengths of the longer base?<br />
(a) 20 m<br />
(b) 30 m<br />
(c) 16 m<br />
(d) 25 m<br />
<br />
52. The lateral area of a right circular cone is 4 times the area of the base. Find the angle at which an element of the cone is inclined to the base.<br />
(a) 76°45'<br />
(b) 73°36'<br />
(c) 75°31'<br />
(d) 77°42'<br />
<br />
53. Three spheres of radii 10, 20 and 30 cm, respectively, are melted and formed into a single sphere. Find the surface area of the single sphere.<br />
(a) 12,645.37 cm^2<br />
(b) 10,432.38 cm^2<br />
(c) 13,700.77 cm^2<br />
(d) 13,077.42 cm^2<br />
<br />
54. A sphere having a diameter of 30 cm is cut into 2 segments. The altitude of the first segment is 6 cm. What is the ratio of the area of the second segment to that of the first?<br />
(a) 2<br />
(b) 4<br />
(c) 3<br />
(d) 2.5<br />
<br />
55. What is the spherical excess of a spherical polygon of four sides whose angles are 95°, 112°, 134° and 78° ?<br />
(a) 60°<br />
(b) 54°<br />
(c) 67°<br />
(d) 59°<br />
<br />
56. Two equilateral triangles, each with 12 cm sides overlap each other to form a 6-point star. Determine the overlapping area.<br />
(a) 60°<br />
(b) 54°<br />
(c) 67°<br />
(d) 59°<br />
<br />
57. A reverse curve on a railroad track consists of two circular arcs. The central angle of one is 20° with radius 2500 ft and the central angle of the other is 25° with radius of 3000 ft. Find the total length of the two arcs.<br />
(a) 2182 ft<br />
(b) 2282 ft<br />
(c) 2382 ft<br />
(d) 2482 ft<br />
<br />
58. The area of a spherical lune is 90 cm^2. If the area of the sphere is 810 cm^2, what is the angle subtended by the lune?<br />
(a) 45°<br />
(b) 40°<br />
(c) 48°<br />
(d) 42°<br />
<br />
59. The diagonals of a rhombus are 10 cm and 8 cm, respectively. Find its area.<br />
(a) 20<br />
(b) 60<br />
(c) 40<br />
(d) 80<br />
<br />
60. Find the surface area of a right circular cone in which radius measures 14 inches while the slant height measures 20 inches.<br />
(a) 1520 in^2<br />
(b) 1453 in^2<br />
(c) 1496 in^2<br />
(d) 1562 in^2<br />
<br />
61. Find the volume of a spherical cone in a sphere of radius 25 cm, if the radius of the zone is 10 cm.<br />
(a) 2735.8<br />
(b) 2232.2<br />
(c) 2563.43<br />
(d) 2442.73<br />
<br />
62. The angle of a sector is 30° and the radius is 15 cm. What is the area of the sector?<br />
(a) 58.9 cm^2<br />
(b) 62.3 cm^2<br />
(c) 45.6 cm^2<br />
(d) 52.2 cm^2<br />
<br />
63. A regular octagon is inscribed in a circle of radius 10 inches. Find the area of the octagon.<br />
(a) 282.8<br />
(b) 265.5<br />
(c) 302.2<br />
(d) 294.8<br />
<br />
64. Find the volume of a hexagonal spherical pyramid whose base angles are 135°, 105°, 122°, 131° and 142° on a sphere of radius 20 in.<br />
(a) 1303.18 in^3<br />
(b) 1243.22 in^3<br />
(c) 1205.92 in^3<br />
(d) 1043.32 in^3<br />
<br />
65. A hemisphere whose radius is 12 inches is surmounted by a right circular cone with the same radius and altitude of 15 inches. Find the total volume.<br />
(a) 5422.84 in^3<br />
(b) 5032.62 in^3<br />
(c) 4252.32 in^3<br />
(d) 5881.06 in^3<br />
<div>
<br /></div>

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MCQs / Practice Test on Engineering Mathematics

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Compilation of Mathematics Problems - SOLID GEOMETRY Set 2

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<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXoNPE6zftTAMV5TSjpfs_3PIcA8MLfS6q_bMc9MN6zNsWHpbDI5BSz7156r2TIsxkzs8G1-tIrmXWd3-vxcM4w-8mAbHfIO_a3nZT1yMaey8-sPt2l0vUvrebTKyk-qH7V6WCXZFlfh15/s1600/Solids.jpg" imageanchor="1" style=""><img alt="Compilation of Mathematics Problems - SOLID GEOMETRY" border="0" height="280" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXoNPE6zftTAMV5TSjpfs_3PIcA8MLfS6q_bMc9MN6zNsWHpbDI5BSz7156r2TIsxkzs8G1-tIrmXWd3-vxcM4w-8mAbHfIO_a3nZT1yMaey8-sPt2l0vUvrebTKyk-qH7V6WCXZFlfh15/s280/Solids.jpg" title="Compilation of Mathematics Problems - SOLID GEOMETRY" width="280" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12.8000001907349px;">Compilation of Problems in Geometry</td></tr>
</tbody></table>
<div style="text-align: center;">
<b>COMPILATION OF MATH PROBLEMS</b></div>
<div style="text-align: center;">
<b>SOLID MENSURATION</b></div>
<br />
1.Determine the value of each interior angle of a regular dodecagon.<br />
<br />
2.How many sides has a polygon if the sum of the interior angles is 2520° ?<br />
<br />
3.How many sides has polygon if the sum of its interior angles equals the sum of its exterior angles?<br />
<br />
4.How many diagonals are there in a hexagon?<br />
<br />
5.The area of a circle circumscribing a regular octagon is 100π. Find the area of the octagon.<br />
<br />
6.Find the area of a pentagon whose apothem is 10 cm.<br />
<br />
7.Determine the area of a regular 6-point star polygon if the inner regular hexagon has 10 cm side.<br />
<br />
8.One side of a parallelogram is 10 cm and its diagonals are 16 cm and 24 cm, respectively. Find its area.<br />
<br />
9.A piece of wire is shaped to enclose a square whose area is 169 in^2. It is then reshaped to form a rectangle whose length is 15 in. Determine the area of the rectangle.<br />
<br />
10.A rhombus has an area of 320 cm^2. If one diagonal measures 20 cm, determine the length of each side.<br />
<br />
11.Find the area of a circle inscribed in a rhombus whose perimeter is 100 cm and whose longer diagonal is 40 cm.<br />
<br />
12.Find the area of an isosceles trapezoid if the measure of one angle is 135° and the lengths of the bases are 10 and 18.<br />
<br />
13.Assuming that the earth is a sphere whose radius is 6400 km, find the distance along a 3-degree arc at the equator of the earth’s surface.<br />
<br />
14.In a circle whose radius is 12, find the area of a minor segment whose arc has a central angle of 60° .<br />
<br />
15.A circle with radius 6 cm has half its area removed by cutting off a border of uniform width. Find the width of the border.<br />
<br />
16.What is the surface area of a cube whose space diagonal is 6.<br />
<br />
17.A parallelepiped has an altitude of 20 inches. The base is a parallelogram with adjacent sides 8 inches and 12 inches and included angle 60°. Find its volume.<br />
<br />
18.The base of a pyramid is a regular hexagon with each side equal to 6 m and an altitude of 10 m. What is the volume of water it can hold when full and inverted?<br />
<br />
19.The lateral area of the right circular water tank is 92 sq. cm. Its volume is 342 cu. m., determine its radius.<br />
<br />
20.Determine the volume of a right circular cone of radius 8 cm and slant height of 17 cm.<br />
<br />
21.The frustum of a pyramid has a lower base 50 m by 10 m and an upper base of 40 m by 8 m. If the altitude of the frustum is 6 m, determine its volume.<br />
<br />
22.Determine the total surface of an icosahedron if the length of each edge is 5 inches.<br />
<div>
<br /></div>

Compilation of Mathematics Problems - SOLID GEOMETRY Set 2

Compilation of Problems in Geometry COMPILATION OF MATH PROBLEMS SOLID MENSURATION 1.Determine the value of each interior angle of ...

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