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Arithmetic (UPSC) Part- IV

📆 April 18, 2025

Total Questions: 50

41. For what value of n, the sum of digits in the number (10ⁿ + 1) is 2? [2020-II]

Correct Answer: (b) For any whole number n
Solution:
Value of nNumberSum of digits
n = 0(10)⁰ + 1 = 1 + 1 = 22
n = 1(10)¹ + 1 = 10 + 1 = 111 + 1 = 2
n = 2(10)² + 1 = 100 + 1 = 1011 + 0 + 1 = 2
n = 3(10)³ + 1 = 1000 + 1 = 10011 + 0 + 0 + 1 = 2

Hence, for n = whole number, sum of digits of number
((10)ⁿ + 1) is always 2.

42. In a class, there are three groups, A, B and C. If one student from group A and two students from group B are shifted to group C, then what happens to the average weight for the students of the class? [2020-II]

Correct Answer: (c) It remains the same.
Solution:Average weight of the students
= Sum of weight of the students / Number of students

Dividing the students into several groups does not change total weight of the students and also number of students in the class remains unchanged. So, average weight of the class remains the same.

43. A shop owner offers the following discount options on an article to a customer: [2020-II]

  1. Successive discounts of 10% and 20%, and then pay a service tax of 10%.
  2. Successive discounts of 20% and 10%, and then pay a service tax of 10%.
  3. Pay a service tax of 10% first, then successive discounts of 20% and 10%.

 

Which one of the following is correct?

Correct Answer: (c) 3 only is the best option for the customer.
Solution:Let the original price of the article = ₹100.
1. 100 × (100 – 10 / 100) × (100 – 20 / 100) × (100 + 10 / 100)
= 100 × 90 / 100 × 80 / 100 × 110 / 100 = 79.2.

2. 100 × (100 – 20 / 100)(100 – 10 / 100)(100 + 10 / 100)
= 100 × 80 / 100 × 90 / 100 × 110 / 100 = 79.2.

3. 100 × (100 + 10 / 100)(100 – 20 / 100)(100 – 10 / 100)
= 100 × 110 / 100 × 80 / 100 × 90 / 100 = 79.2.

Thus, all the options are equally good for the customer.

44. What is the largest number among the following? [2020-II]

Correct Answer: (c) (1/3)⁻⁴
Solution:

45. There are two Classes A and B having 25 and 30 students respectively. In Class-A the highest score is 21 and lowest score is 17. In Class-B the highest score is 30 and lowest score is 22. Four students are shifted from Class-A to Class-B. [2021-II]

Consider the following statements:

  1. The average score of Class-B will definitely decrease.
  2. The average score of Class-A will definitely increase.
    Which of the above statements is/are correct?
Correct Answer: (a) 1 only 
Solution:
ClassTotal No. of studentsHighest marksLowest marks
A252117
B303022

So, average of class A is more than 17 but less than 21.
and average of class B is more than 22 but less than 30.
When 4 students of class A shifted to class B.
Let marks of each students is equal to maximum marks of class A = 21
So, new average of class B
= (30×30 + 21×4) / (30 + 4)
= 984 / 34 < 30×30 / 30

Thus, average score of class B will definitely decreases.
Again, let 4 students of class A that shifted to class B, each have minimum marks 17.
So, new average
= (17×25 – 17×4) / (25 – 4)
= 17×21 / 21
= 17 < Initial average

Thus, average score of class A not increases.
Hence, only statement I is true.

46. A student appeared in 6 papers. The maximum marks are the same for each paper. His marks in these papers are in the proportion of 5:6:7:8:9:10. Overall he scored 60%. In how many number of papers did he score less than 60% of the maximum marks? [2021-II]

Correct Answer: (b) 3 
Solution:Let total marks in each subject be 100.
Therefore, total marks for all 6 subjects = 600
Overall marks scored = 60% of 600 = 360
According to the question,
5x + 6x + 7x + 8x + 9x + 10x = 360
or 45x = 360
or x = 8
So, marks in the given 6 subjects must be:
5 × 8 = 40 ⇒ 6 × 8 = 48 ⇒ 7 × 8 = 56
8 × 8 = 64 ⇒ 9 × 8 = 72 ⇒ 10 × 8 = 80
Hence, in 3 subjects the student has scored less than 60% marks.

47. Integers are listed from 700 to 1000. In how many integers is the sum of the digits 10? [2021-II]

Correct Answer: (d) 9
Solution:Number between 700 to 1000 in which sum of its
digits = 10 are, 703, 712, 721, 730, 802, 811, 820, 901, 910.
Hence, Number of numbers = 9.

48. f 3²⁰¹⁹ is divided by 10, then what is the remainder? [2021-II]

Correct Answer: (c) 7
Solution:Periodicity of 3 = 4
Now 2019 = 504 × 4 + 3
Now, (3)²⁰¹⁹ = (3)⁵⁰⁴×⁴⁺³ = 27
When this number is divided by 10.
then, remainder = 7

49. The number 3798125P369 is divisible by 7. What is the value of the digit P? [2021-II]

Correct Answer: (a) 1
Solution:For P = 6,
Number = 37981256369.
Divisor = 7
Q = 37981256369 ÷ 7
= 5425893767
Hence, P = 6.

50. A biology class at high school predictor that a local population of animals will double in size every 12 years. The population at the beginning of the year 2021 was estimated to be 50 animals. If P represents the population after n years, then which one of the following equations represents the model of the class for the population? [2021-II]

Correct Answer: (d) P = 50(2)ⁿ/¹²
Solution:
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