Synopsis IAS

Answer

Solution

UPSC Prelims

Mathematics

Theme:

Permutation and Combination / क्रमचय और संचय

Q63 2021, Permutation and Combination

There are three points P, Q and R on a straight line such that PQ QR = 3: 5. If n is the number of possible values of PQ: PR, then what is n equal to ?

तीन बिंदु P, Q तथा R एक सरल रेखा पर इस प्रकार स्थित हैं कि PQ QR 3: 5 है। यदि PQ: PR के संभाव्य मानों की संख्या n है, तो n किसके बराबर है ?

Explanation

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Option B: b
To solve this problem, we need to determine the number of possible values for the ratio

P Q : P R

$PQ:PR$ given the ratio

P Q : Q R = 3 : 5

$PQ:QR = 3:5$ .
     Step-by-step Solution:
     1. Understanding the Ratios:
            ○ We have three points

P

$P$ ,

Q

$Q$ , and

R

$R$ on a straight line.
○ The ratio

P Q : Q R = 3 : 5

$PQ:QR = 3:5$ implies that if

P Q = 3 x

$PQ = 3x$ , then

Q R = 5 x

$QR = 5x$ .
2. Finding $PR$ :
○ Since

P

$P$ ,

Q

$Q$ , and

R

$R$ are collinear, the total distance

P R = P Q + Q R

$PR = PQ + QR$ .
○ Therefore,

P R = 3 x + 5 x = 8 x

$PR = 3x + 5x = 8x$ .
3. Calculating $PQ:PR$ :
○ The ratio

P Q : P R = 3 x : 8 x

$PQ:PR = 3x:8x$ .
○ Simplifying this ratio gives

P Q : P R = 3 : 8

$PQ:PR = 3:8$ .
4. Considering Possible Arrangements:
○ The problem does not specify the order of points, so we must consider different arrangements of

P

$P$ ,

Q

$Q$ , and

R

$R$ on the line.
○ The possible arrangements are:
○

P

$P$ is between

Q

$Q$ and

R

$R$ .
○

Q

$Q$ is between

P

$P$ and

R

$R$ .
○

R

$R$ is between

P

$P$ and

Q

$Q$ .
5. Analyzing Each Arrangement:
● Arrangement 1:

P

$P$ is between

Q

$Q$ and

R

$R$ .
○ This is not possible because

P Q < Q R

$PQ < QR$ and

P

$P$ cannot be between

Q

$Q$ and

R

$R$ .
● Arrangement 2:

Q

$Q$ is between

P

$P$ and

R

$R$ .
○ This is the scenario we calculated:

P Q : P R = 3 : 8

$PQ:PR = 3:8$ .
● Arrangement 3:

R

$R$ is between

P

$P$ and

Q

$Q$ .
○ In this case,

P R = P Q - Q R = 3 x - 5 x = - 2 x

$PR = PQ - QR = 3x - 5x = -2x$ , which is not possible as distances cannot be negative.
6. Conclusion:
○ The only valid arrangement is when

Q

$Q$ is between

P

$P$ and

R

$R$ , giving us the ratio

P Q : P R = 3 : 8

$PQ:PR = 3:8$ .
○ Therefore, there is only one possible value for

P Q : P R

$PQ:PR$ .
Hence, the number of possible values of

P Q : P R

$PQ:PR$ is 1, which corresponds to Option B: b.

Option B: b To solve this problem, we need to determine the number of possible values for the ratio <mi>P</mi><mi>Q</mi><mo>:</mo><mi>P</mi><mi>R</mi></math>" role="presentation" style="font-size: 121%; position: relative;">PQ:PR<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>Q</mi><mo>:</mo><mi>P</mi><mi>R</mi></math><script type="math/tex" id="MathJax-Element-1"> PQ:PR </script> given the ratio <mi>P</mi><mi>Q</mi><mo>:</mo><mi>Q</mi><mi>R</mi><mo>=</mo><mn>3</mn><mo>:</mo><mn>5</mn></math>" role="presentation" style="font-size: 121%; position: relative;">PQ:QR=3:5<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>Q</mi><mo>:</mo><mi>Q</mi><mi>R</mi><mo>=</mo><mn>3</mn><mo>:</mo><mn>5</mn></math><script type="math/tex" id="MathJax-Element-2"> PQ:QR = 3:5 </script>.      Step-by-step Solution:      1. Understanding the Ratios:             ○ We have three points <mi>P</mi></math>" role="presentation" style="font-size: 121%; position: relative;">P<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math><script type="math/tex" id="MathJax-Element-3"> P </script>, <mi>Q</mi></math>" role="presentation" style="font-size: 121%; position: relative;">Q<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math><script type="math/tex" id="MathJax-Element-4"> Q </script>, and <mi>R</mi></math>" role="presentation" style="font-size: 121%; position: relative;">R<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math><script type="math/tex" id="MathJax-Element-5"> R </script> on a straight line.             ○ The ratio <mi>P</mi><mi>Q</mi><mo>:</mo><mi>Q</mi><mi>R</mi><mo>=</mo><mn>3</mn><mo>:</mo><mn>5</mn></math>" role="presentation" style="font-size: 121%; position: relative;">PQ:QR=3:5<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>Q</mi><mo>:</mo><mi>Q</mi><mi>R</mi><mo>=</mo><mn>3</mn><mo>:</mo><mn>5</mn></math><script type="math/tex" id="MathJax-Element-6"> PQ:QR = 3:5 </script> implies that if <mi>P</mi><mi>Q</mi><mo>=</mo><mn>3</mn><mi>x</mi></math>" role="presentation" style="font-size: 121%; position: relative;">PQ=3x<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>Q</mi><mo>=</mo><mn>3</mn><mi>x</mi></math><script type="math/tex" id="MathJax-Element-7"> PQ = 3x </script>, then <mi>Q</mi><mi>R</mi><mo>=</mo><mn>5</mn><mi>x</mi></math>" role="presentation" style="font-size: 121%; position: relative;">QR=5x<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mi>R</mi><mo>=</mo><mn>5</mn><mi>x</mi></math><script type="math/tex" id="MathJax-Element-8"> QR = 5x </script>.      2. Finding <mi>P</mi><mi>R</mi></math>" role="presentation" style="font-size: 121%; position: relative;">PR<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>R</mi></math><script type="math/tex" id="MathJax-Element-9"> PR </script>:             ○ Since <mi>P</mi></math>" role="presentation" style="font-size: 121%; position: relative;">P<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math><script type="math/tex" id="MathJax-Element-10"> P </script>, <mi>Q</mi></math>" role="presentation" style="font-size: 121%; position: relative;">Q<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math><script type="math/tex" id="MathJax-Element-11"> Q </script>, and <mi>R</mi></math>" role="presentation" style="font-size: 121%; position: relative;">R<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math><script type="math/tex" id="MathJax-Element-12"> R </script> are collinear, the total distance <mi>P</mi><mi>R</mi><mo>=</mo><mi>P</mi><mi>Q</mi><mo>+</mo><mi>Q</mi><mi>R</mi></math>" role="presentation" style="font-size: 121%; position: relative;">PR=PQ+QR<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>R</mi><mo>=</mo><mi>P</mi><mi>Q</mi><mo>+</mo><mi>Q</mi><mi>R</mi></math><script type="math/tex" id="MathJax-Element-13"> PR = PQ + QR </script>.             ○ Therefore, <mi>P</mi><mi>R</mi><mo>=</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>x</mi><mo>=</mo><mn>8</mn><mi>x</mi></math>" role="presentation" style="font-size: 121%; position: relative;">PR=3x+5x=8x<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>R</mi><mo>=</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>x</mi><mo>=</mo><mn>8</mn><mi>x</mi></math><script type="math/tex" id="MathJax-Element-14"> PR = 3x + 5x = 8x </script>.      3. Calculating <mi>P</mi><mi>Q</mi><mo>:</mo><mi>P</mi><mi>R</mi></math>" role="presentation" style="font-size: 121%; position: relative;">PQ:PR<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>Q</mi><mo>:</mo><mi>P</mi><mi>R</mi></math><script type="math/tex" id="MathJax-Element-15"> PQ:PR </script>:             ○ The ratio <mi>P</mi><mi>Q</mi><mo>:</mo><mi>P</mi><mi>R</mi><mo>=</mo><mn>3</mn><mi>x</mi><mo>:</mo><mn>8</mn><mi>x</mi></math>" role="presentation" style="font-size: 121%; position: relative;">PQ:PR=3x:8x<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>Q</mi><mo>:</mo><mi>P</mi><mi>R</mi><mo>=</mo><mn>3</mn><mi>x</mi><mo>:</mo><mn>8</mn><mi>x</mi></math><script type="math/tex" id="MathJax-Element-16"> PQ:PR = 3x:8x </script>.             ○ Simplifying this ratio gives <mi>P</mi><mi>Q</mi><mo>:</mo><mi>P</mi><mi>R</mi><mo>=</mo><mn>3</mn><mo>:</mo><mn>8</mn></math>" role="presentation" style="font-size: 121%; position: relative;">PQ:PR=3:8<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>Q</mi><mo>:</mo><mi>P</mi><mi>R</mi><mo>=</mo><mn>3</mn><mo>:</mo><mn>8</mn></math><script type="math/tex" id="MathJax-Element-17"> PQ:PR = 3:8 </script>.      4. Considering Possible Arrangements:             ○ The problem does not specify the order of points, so we must consider different arrangements of <mi>P</mi></math>" role="presentation" style="font-size: 121%; position: relative;">P<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math><script type="math/tex" id="MathJax-Element-18"> P </script>, <mi>Q</mi></math>" role="presentation" style="font-size: 121%; position: relative;">Q<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math><script type="math/tex" id="MathJax-Element-19"> Q </script>, and <mi>R</mi></math>" role="presentation" style="font-size: 121%; position: relative;">R<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math><script type="math/tex" id="MathJax-Element-20"> R </script> on the line.             ○ The possible arrangements are:               ○ <mi>P</mi></math>" role="presentation" style="font-size: 121%; position: relative;">P<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math><script type="math/tex" id="MathJax-Element-21"> P </script> is between <mi>Q</mi></math>" role="presentation" style="font-size: 121%; position: relative;">Q<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math><script type="math/tex" id="MathJax-Element-22"> Q </script> and <mi>R</mi></math>" role="presentation" style="font-size: 121%; position: relative;">R<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math><script type="math/tex" id="MathJax-Element-23"> R </script>.               ○ <mi>Q</mi></math>" role="presentation" style="font-size: 121%; position: relative;">Q<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math><script type="math/tex" id="MathJax-Element-24"> Q </script> is between <mi>P</mi></math>" role="presentation" style="font-size: 121%; position: relative;">P<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math><script type="math/tex" id="MathJax-Element-25"> P </script> and <mi>R</mi></math>" role="presentation" style="font-size: 121%; position: relative;">R<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math><script type="math/tex" id="MathJax-Element-26"> R </script>.               ○ <mi>R</mi></math>" role="presentation" style="font-size: 121%; position: relative;">R<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math><script type="math/tex" id="MathJax-Element-27"> R </script> is between <mi>P</mi></math>" role="presentation" style="font-size: 121%; position: relative;">P<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math><script type="math/tex" id="MathJax-Element-28"> P </script> and <mi>Q</mi></math>" role="presentation" style="font-size: 121%; position: relative;">Q<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math><script type="math/tex" id="MathJax-Element-29"> Q </script>.      5. Analyzing Each Arrangement:         ● Arrangement 1: <mi>P</mi></math>" role="presentation" style="font-size: 121%; position: relative;">P<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math><script type="math/tex" id="MathJax-Element-30"> P </script> is between <mi>Q</mi></math>" role="presentation" style="font-size: 121%; position: relative;">Q<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math><script type="math/tex" id="MathJax-Element-31"> Q </script> and <mi>R</mi></math>" role="presentation" style="font-size: 121%; position: relative;">R<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math><script type="math/tex" id="MathJax-Element-32"> R </script>.
                 ○ This is not possible because <mi>P</mi><mi>Q</mi><mo>&lt;</mo><mi>Q</mi><mi>R</mi></math>" role="presentation" style="font-size: 121%; position: relative;">PQ<QR<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>Q</mi><mo><</mo><mi>Q</mi><mi>R</mi></math><script type="math/tex" id="MathJax-Element-33"> PQ < QR </script> and <mi>P</mi></math>" role="presentation" style="font-size: 121%; position: relative;">P<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math><script type="math/tex" id="MathJax-Element-34"> P </script> cannot be between <mi>Q</mi></math>" role="presentation" style="font-size: 121%; position: relative;">Q<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math><script type="math/tex" id="MathJax-Element-35"> Q </script> and <mi>R</mi></math>" role="presentation" style="font-size: 121%; position: relative;">R<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math><script type="math/tex" id="MathJax-Element-36"> R </script>.         ● Arrangement 2: <mi>Q</mi></math>" role="presentation" style="font-size: 121%; position: relative;">Q<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math><script type="math/tex" id="MathJax-Element-37"> Q </script> is between <mi>P</mi></math>" role="presentation" style="font-size: 121%; position: relative;">P<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math><script type="math/tex" id="MathJax-Element-38"> P </script> and <mi>R</mi></math>" role="presentation" style="font-size: 121%; position: relative;">R<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math><script type="math/tex" id="MathJax-Element-39"> R </script>.
                 ○ This is the scenario we calculated: <mi>P</mi><mi>Q</mi><mo>:</mo><mi>P</mi><mi>R</mi><mo>=</mo><mn>3</mn><mo>:</mo><mn>8</mn></math>" role="presentation" style="font-size: 121%; position: relative;">PQ:PR=3:8<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>Q</mi><mo>:</mo><mi>P</mi><mi>R</mi><mo>=</mo><mn>3</mn><mo>:</mo><mn>8</mn></math><script type="math/tex" id="MathJax-Element-40"> PQ:PR = 3:8 </script>.         ● Arrangement 3: <mi>R</mi></math>" role="presentation" style="font-size: 121%; position: relative;">R<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math><script type="math/tex" id="MathJax-Element-41"> R </script> is between <mi>P</mi></math>" role="presentation" style="font-size: 121%; position: relative;">P<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math><script type="math/tex" id="MathJax-Element-42"> P </script> and <mi>Q</mi></math>" role="presentation" style="font-size: 121%; position: relative;">Q<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math><script type="math/tex" id="MathJax-Element-43"> Q </script>.
                 ○ In this case, <mi>P</mi><mi>R</mi><mo>=</mo><mi>P</mi><mi>Q</mi><mo>&#x2212;</mo><mi>Q</mi><mi>R</mi><mo>=</mo><mn>3</mn><mi>x</mi><mo>&#x2212;</mo><mn>5</mn><mi>x</mi><mo>=</mo><mo>&#x2212;</mo><mn>2</mn><mi>x</mi></math>" role="presentation" style="font-size: 121%; position: relative;">PR=PQ−QR=3x−5x=−2x<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>R</mi><mo>=</mo><mi>P</mi><mi>Q</mi><mo>−</mo><mi>Q</mi><mi>R</mi><mo>=</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>x</mi><mo>=</mo><mo>−</mo><mn>2</mn><mi>x</mi></math><script type="math/tex" id="MathJax-Element-44"> PR = PQ - QR = 3x - 5x = -2x </script>, which is not possible as distances cannot be negative.      6. Conclusion:             ○ The only valid arrangement is when <mi>Q</mi></math>" role="presentation" style="font-size: 121%; position: relative;">Q<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math><script type="math/tex" id="MathJax-Element-45"> Q </script> is between <mi>P</mi></math>" role="presentation" style="font-size: 121%; position: relative;">P<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math><script type="math/tex" id="MathJax-Element-46"> P </script> and <mi>R</mi></math>" role="presentation" style="font-size: 121%; position: relative;">R<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math><script type="math/tex" id="MathJax-Element-47"> R </script>, giving us the ratio <mi>P</mi><mi>Q</mi><mo>:</mo><mi>P</mi><mi>R</mi><mo>=</mo><mn>3</mn><mo>:</mo><mn>8</mn></math>" role="presentation" style="font-size: 121%; position: relative;">PQ:PR=3:8<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>Q</mi><mo>:</mo><mi>P</mi><mi>R</mi><mo>=</mo><mn>3</mn><mo>:</mo><mn>8</mn></math><script type="math/tex" id="MathJax-Element-48"> PQ:PR = 3:8 </script>.             ○ Therefore, there is only one possible value for <mi>P</mi><mi>Q</mi><mo>:</mo><mi>P</mi><mi>R</mi></math>" role="presentation" style="font-size: 121%; position: relative;">PQ:PR<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>Q</mi><mo>:</mo><mi>P</mi><mi>R</mi></math><script type="math/tex" id="MathJax-Element-49"> PQ:PR </script>.      Hence, the number of possible values of <mi>P</mi><mi>Q</mi><mo>:</mo><mi>P</mi><mi>R</mi></math>" role="presentation" style="font-size: 121%; position: relative;">PQ:PR<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>Q</mi><mo>:</mo><mi>P</mi><mi>R</mi></math><script type="math/tex" id="MathJax-Element-50"> PQ:PR </script> is 1, which corresponds to Option B: b.

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Q 63. 2021, Permutation and Combination

Mathematics

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Mathematics

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