Important questions of Geometry for SSC CGL Tier I In my previous session, I have discussed some concepts related to triangles. Today I will discuss some important questions of Geometry which used to appear in SSC exams. Generally, questions asked from this section are based on properties of various shapes like lines, angles, triangles, rhombus, circles etc.

Question
Transformation properties
asked 2021-02-16
Important questions of Geometry for SSC CGL Tier I In my previous session, I have discussed some concepts related to triangles. Today I will discuss some important questions of Geometry which used to appear in SSC exams. Generally, questions asked from this section are based on properties of various shapes like lines, angles, triangles, rhombus, circles etc.

Answers (1)

2021-04-16

Important Examples

Example1: ABCD  is a cyclic quadrilateral. AB and CD are produced to meet P. If angle ADC = \(70^o \)and angle\( DAB = 60^o\), then what will be angle PBC + angle PCB?

Solution: First step: Make an appropriate figure using statements provided in  questions.
image

 

  • Cyclic quadrilateral has all its vertices on circle and sum of all angles is 360^o.
  • Sum of opposite angles = 180^o.
  • External angle = Opposite internal angle.  

Therefore, angle \(PBC = 70^o\)
angle\( PCB = 60^o\)

So, \(PBC + PCB = 130^0\)

See its very simple question, if you know the properties of geometry. Let's try more examples

Example2: ABCD is a parallelogram and P is any point within it. If area of parallelogram is 20 units, then what will be the sum of areas of triangle PAB and PCD?

Solution: According to the question, figure will be as follows:

 

 

image


Properties for this question:

  • Area of parallelogram = Base times Height.
  • Area of triangle = 1/2 Base times altitude.

Given that: Area of parallelogram = 20
i.e. Base times Height = 20
 ⇒ b times h = 20

To find =  Area of ( PAB +PCD)
⇒ (1/2) b times (PY) + (1/2) b times (PX)                          ( See Figure)

⇒1/2 b ( PX +PY)


⇒1/2 b h
⇒(1/2) times 20
⇒10 units.

Example3: ABCD is a cyclic trapezium with AB parallel DC and AB diameter of circle. If angle\( CAB = 30^o\), then angle ADC will be?

Solution: According to ques, figure will be as follows:

image

  • Angle subtended by diameter is always 90^o.
  • Sum of angles of triangle = \(180^o.\)
  • Sum of opposite angles = \(180^o\)

angle \(ACB = 90^o\)
angle \(BAC = 30^o\)

Therefore, 


angle \(ABC = 60^o\)
angle ABC + angle \(ADC = 180^o  ⇒ 60^o + ADC = 180^o \)⇒ angle \(ADC = 120^o\)
Example4: Two side of plot measures 30m and 22m and angle between them is \(90^o\). The other two sides measures 24m and the three remaining angles are not right angles. Find the area of plot.
Solution: Figure becomes: Center point of BD = O image

ABD is a right angle triangle, therefore, Pythagoras theorem followed.
\((BD)^2 = (AB)^2 + (AD)^2 ⇒ BD = 40\)
Given that, BC = CD Therefore, line drawn from C to BD given right angles Also, this will lead to DO = BO
So, DO = BO = 20
Similarly, using Pythagoras theorem, OC = 15 m

Now, Area of triangles (ABD + BOC + COD)  ⇒ 1/2 (24 times 32) + 2 ((1/2) 20 times 15) ⇒ 684 m^2

 

 

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Case: Dr. Jung’s Diamonds Selection
With Christmas coming, Dr. Jung became interested in buying diamonds for his wife. After perusing the Web, he learned about the “4Cs” of diamonds: cut, color, clarity, and carat. He knew his wife wanted round-cut earrings mounted in white gold settings, so he immediately narrowed his focus to evaluating color, clarity, and carat for that style earring.
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2) Let X1, X2, and X3 represent diamond color, clarity, and carats, respectively. If Dr. Jung wanted to build a linear regression model to estimate earring prices using these variables, which variables would you recommend that he use? Why?
3) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
4) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
5) Dr. Jung now remembers that it sometimes helps to perform a square root transformation on the dependent variable in a regression problem. Modify your spreadsheet to include a new dependent variable that is the square root on the earring prices (use Excel’s SQRT( ) function). If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
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8) Dr. Jung now also remembers that it sometimes helps to include interaction terms in a regression model—where you create a new independent variable as the product of two of the original variables. Modify your spreadsheet to include three new independent variables, X4, X5, and X6, representing interaction terms where: X4 = X1 × X2, X5 = X1 × X3, and X6 = X2 × X3. There are now six potential independent variables. If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
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The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
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White - 1176
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White - 1243
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Give your answer as a decimal to at least three decimal places.
a) What percent are Black?
b) What percent are Unarmed?
c) In order for two variables to be Independent of each other, the P \((A and B) = P(A) \cdot P(B) P(A and B) = P(A) \cdot P(B).\)
This just means that the percentage of times that both things happen equals the individual percentages multiplied together (Only if they are Independent of each other).
Therefore, if a person's race is independent of whether they were killed being unarmed then the percentage of black people that are killed while being unarmed should equal the percentage of blacks times the percentage of Unarmed. Let's check this. Multiply your answer to part a (percentage of blacks) by your answer to part b (percentage of unarmed).
Remember, the previous answer is only correct if the variables are Independent.
d) Now let's get the real percent that are Black and Unarmed by using the table?
If answer c is "significantly different" than answer d, then that means that there could be a different percentage of unarmed people being shot based on race. We will check this out later in the course.
Let's compare the percentage of unarmed shot for each race.
e) What percent are White and Unarmed?
f) What percent are Hispanic and Unarmed?
If you compare answers d, e and f it shows the highest percentage of unarmed people being shot is most likely white.
Why is that?
This is because there are more white people in the United States than any other race and therefore there are likely to be more white people in the table. Since there are more white people in the table, there most likely would be more white and unarmed people shot by police than any other race. This pulls the percentage of white and unarmed up. In addition, there most likely would be more white and armed shot by police. All the percentages for white people would be higher, because there are more white people. For example, the table contains very few Hispanic people, and the percentage of people in the table that were Hispanic and unarmed is the lowest percentage.
Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
The correct way to compare is "conditional probability". Conditional probability is getting the probability of something happening, given we are dealing with just the people in a particular group.
g) What percent of blacks shot and killed by police were unarmed?
h) What percent of whites shot and killed by police were unarmed?
i) What percent of Hispanics shot and killed by police were unarmed?
You can see by the answers to part g and h, that the percentage of blacks that were unarmed and killed by police is approximately twice that of whites that were unarmed and killed by police.
j) Why do you believe this is happening?
Do a search on the internet for reasons why blacks are more likely to be killed by police. Read a few articles on the topic. Write your response using the articles as references. Give the websites used in your response. Your answer should be several sentences long with at least one website listed. This part of this problem will be graded after the due date.
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