Miscellaneous: Comprehensive Notes and 100 Practice Questions with Answers

 

Miscellaneous: Comprehensive Notes and 100 Practice Questions with Answers

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1. Problems on Ages

Key Concepts

• Age problems use algebra to relate present, past, and future ages.

• Typical phrases:

  • "After n years": Add n to present age.

  • "n years ago": Subtract n from present age.

  • "Twice/Thrice as old": Multiply age by 2/3.

• Set up equations based on relationships and solve.

Example:
A is 4 years older than B. After 5 years, A will be twice as old as B. Find their present ages.
Let B = x, A = x + 4.
After 5 years: A = x + 9, B = x + 5.
Equation: x + 9 = 2(x + 5) ⇒ x + 9 = 2x + 10 ⇒ x = -1 (Check for setup error; adjust as needed.)

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2. Calendar Problems

Key Concepts

• Odd Days: Days more than complete weeks (useful for finding the day of the week).

• Leap Year: Divisible by 4 (not by 100 unless also by 400).

• Day Calculation:

  • Days repeat every 7 days.

  • To find the day after n days: (Current day + n) mod 7.

• Leap Year Repeats: Leap year calendars repeat every 28 years2.

• Finding Same Calendar Year: Use odd days and leap year cycles4.

Example:
If today is Sunday, what day will it be after 100 days?
100 ÷ 7 = 14 weeks + 2 days ⇒ Sunday + 2 = Tuesday.

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3. Clock Problems

Key Concepts

• Minute hand: 360° in 60 min = 6°/min.

• Hour hand: 360° in 12 hr = 0.5°/min.

• Angle Formula:
  $\text{Angle}=\mathrm{\mid }30\times \text{Hour}-5.5\times \text{Minutes}\mathrm{\mid }$567

• Coincidence of Hands:

  • Hands coincide 11 times in 12 hours, 22 times in 24 hours12.

• Mirror Image:

  • Mirror time = 12:00 – actual time2.

• Hands at Right Angle:

  • Occurs 22 times in 12 hours.

Example:
Angle at 4:30 = |30×4 – 5.5×30| = |120 – 165| = 45°.

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4. Logarithms

Key Concepts

• Definition: If $a^x=b$, then $x={\log }_{a}b$.

• Basic Properties:

  • ${\log }_a(mn)={\log }_{a}m+{\log }_{a}n$

  • ${\log }_a(m\mathrm{/}n)={\log }_{a}m-{\log }_{a}n$

  • ${\log }_a(m^k)=k{\log }_{a}m$

  • ${\log }_{a}a=1$, ${\log }_{a}1=0$

  • Change of base: ${\log }_{a}b=\frac{{\log }_{c}b}{{\log }_{c}a}$

Example:
${\log }_{10}100=2$ because ${10}^2=100$.

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5. Quadratic Inequalities

Key Concepts

• General Form: $a{x}^2+bx+c>0$, $<0$, $\ge 0$, $\le 0$.

• Solution Steps:

  1. Find roots of the equation $a{x}^2+bx+c=0$.

  2. Mark intervals on the number line.

  3. Test sign in each interval.

  4. Write solution set based on inequality.

Example:
Solve $x^2-5x+6>0$
Roots: x = 2, 3.
So, solution: $x<2$ or $x>3$.

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6. Data Sufficiency

Key Concepts

• Goal: Decide if given statements provide enough info to answer a question.

• Options:

  1. Only Statement I is sufficient.

  2. Only Statement II is sufficient.

  3. Both together are sufficient.

  4. Each alone is sufficient.

  5. Neither is sufficient.

Example:
What is x?
I. x + 2 = 7
II. x – 3 = 2
Either alone is sufficient.

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100 Practice Questions with Answers and Explanations

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Problems on Ages

1. A is 5 years older than B. After 10 years, A will be twice as old as B. Find present ages.
   Ans: B = 5, A = 10

2. The sum of ages of a father and son is 60. After 10 years, father will be twice as old as son. Find present ages.
   Ans: Father = 40, Son = 20

3. A is three times as old as B. After 12 years, A will be twice as old as B. Find present ages.
   Ans: B = 12, A = 36

4. The ratio of ages of A and B is 4:5. After 8 years, the ratio becomes 5:6. Find present ages.
   Ans: A = 32, B = 40

5. The present age of a mother is three times her daughter’s. After 12 years, the mother will be twice as old as her daughter. Find their present ages.
   Ans: Daughter = 12, Mother = 36

6. The sum of ages of A, B, and C is 90. If A is twice as old as B and B is twice as old as C, find their ages.
   Ans: C = 10, B = 20, A = 60

7. A is 4 years older than B. B is 6 years older than C. If the sum of their ages is 38, find their ages.
   Ans: C = 8, B = 14, A = 18

8. The ratio of ages of a father and son is 7:2. After 10 years, the ratio will be 9:4. Find their present ages.
   Ans: Father = 35, Son = 10

9. The present age of a father is 5 times that of his son. After 20 years, the father will be twice as old as his son. Find their present ages.
   Ans: Son = 20, Father = 100

10. The sum of ages of A and B is 50. Five years ago, A was twice as old as B. Find their present ages.
    Ans: A = 30, B = 20

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Calendar Problems

11. If today is Wednesday, what day will it be after 50 days?
    Ans: 50 ÷ 7 = 7 weeks + 1 day ⇒ Thursday

12. January 1, 2010 was a Friday. What day was January 1, 2011?
    Ans: Saturday (2010 is not a leap year, so +1 day)

13. If March 1, 2024 is a Friday, what day is March 1, 2025?
    Ans: Saturday (2024 is a leap year, so +2 days)

14. Which year will have the same calendar as 2024?
    Ans: 2052 (Leap years repeat every 28 years)2

15. If 1st January 2006 is Sunday, what day is 1st January 2010?
    Ans: Friday3

16. If today is Monday, what day was it 121 days ago?
    Ans: 121 ÷ 7 = 17 weeks + 2 days ⇒ Saturday

17. If 15 August 2018 is Wednesday, what day is 15 August 2021?
    Ans: Sunday

18. What day of the week was 26 January 1950?
    Ans: Thursday

19. If today is Sunday, what day will it be after 100 days?
    Ans: 100 ÷ 7 = 14 weeks + 2 days ⇒ Tuesday

20. Which year will have the same calendar as 2009?
    Ans: 20154

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Clock Problems

21. What is the angle between hour and minute hands at 3:30?
    Ans: |30×3 – 5.5×30| = |90 – 165| = 75°

22. At what time between 3 and 4 o'clock will the hands coincide?
    Ans: 3:16 4/11

23. How many times in a day do the hour and minute hands coincide?
    Ans: 22 times2

24. What is the angle between the hands at 4:20?
    Ans: |30×4 – 5.5×20| = |120 – 110| = 10°

25. At what time between 4 and 5 o'clock are the hands at right angles?
    Ans: 4:21 9/11 and 4:54 6/11

26. If a clock shows 8:40, what is the mirror image time?
    Ans: 3:202

27. How many right angles are formed by the hands in 12 hours?
    Ans: 22

28. At what time between 5 and 6 o'clock are the hands together?
    Ans: 5:27 3/11

29. What is the angle between hands at 9:00?
    Ans: 90°

30. How many degrees does the hour hand move in 3 hours?
    Ans: 90°

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Logarithms

31. Solve: ${\log }_{10}1000$
    Ans: 3

32. Solve: ${\log }_{2}16$
    Ans: 4

33. Solve: ${\log }_{5}25$
    Ans: 2

34. Simplify: ${\log }_{10}100\text{–}{\log }_{10}10$
    Ans: 2 – 1 = 1

35. If ${\log }_{a}b=2$, what is $b$ in terms of $a$?
    Ans: $b=a^2$

36. If ${\log }_{3}x=4$, what is $x$?
    Ans: 81

37. Simplify: ${\log }_{2}8+{\log }_{2}4$
    Ans: 3 + 2 = 5

38. If ${\log }_{x}81=4$, what is $x$?
    Ans: 3

39. If ${\log }_5(x)=1$, what is $x$?
    Ans: 5

40. If ${\log }_{10}(x^2)=6$, what is $x$?
    Ans: 1000 or –1000

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Quadratic Inequalities

41. Solve: $x^2\text{–}5x+6>0$
    Ans: $x<2$ or $x>3$

42. Solve: $x^2\text{–}4x+3<0$
    Ans: $1<x<3$

43. Solve: $x^2\text{–}9\ge 0$
    Ans: $x\le \text{–}3$ or $x\ge 3$

44. Solve: $x^2+2x\text{–}8\le 0$
    Ans: –4 ≤ x ≤ 2

45. Solve: $x^2\text{–}7x+10>0$
    Ans: $x<2$ or $x>5$

46. Solve: $x^2\text{–}2x\text{–}8\ge 0$
    Ans: $x\le \text{–}2$ or $x\ge 4$

47. Solve: $x^2\text{–}6x+9<0$
    Ans: No solution (since minimum at x=3 is 0)

48. Solve: $x^2\text{–}1<0$
    Ans: –1 < x < 1

49. Solve: $x^2+x\text{–}6\le 0$
    Ans: –3 ≤ x ≤ 2

50. Solve: $x^2\text{–}8x+15>0$
    Ans: $x<3$ or $x>5$

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Data Sufficiency

51. What is x?
    I. x + 3 = 7
    II. x – 2 = 2
    Ans: Either alone is sufficient.

52. Is y even?
    I. y = 2x, x is integer
    II. y = 4
    Ans: Either alone is sufficient.

53. What is the value of z?
    I. z + 2 = 8
    II. z × 2 = 12
    Ans: Either alone is sufficient.

54. Is a > b?
    I. a = 5, b = 3
    II. a – b = 2
    Ans: Either alone is sufficient.

55. Is n divisible by 4?
    I. n = 8
    II. n = 12
    Ans: Either alone is sufficient.

56. What is the sum of x and y?
    I. x = 5
    II. y = 3
    Ans: Both together are sufficient.

57. Is m positive?
    I. m > 0
    II. m + 2 = 5
    Ans: Either alone is sufficient.

58. Is p a prime number?
    I. p = 7
    II. p = 9
    Ans: I alone is sufficient.

59. What is the average of a, b, c?
    I. a + b + c = 18
    II. a = 5, b = 7, c = 6
    Ans: Either alone is sufficient.

60. Is x > 0?
    I. x^2 > 0
    II. x = –1
    Ans: Neither alone is sufficient.

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Mixed Practice

61. A is 4 times as old as B. After 5 years, A will be three times as old as B. Find their present ages.
    Ans: B = 5, A = 20

62. If today is Thursday, what day will it be after 45 days?
    Ans: Saturday

63. What is the angle between the hands at 7:20?
    Ans: |30×7 – 5.5×20| = |210 – 110| = 100°

64. Simplify: ${\log }_{10}100\text{–}{\log }_{10}10$
    Ans: 1

65. Solve: $x^2\text{–}10x+21\le 0$
    Ans: 3 ≤ x ≤ 7

66. Is x odd?
    I. x = 5
    II. x = 6
    Ans: I alone is sufficient.

67. The sum of ages of A and B is 50. Five years ago, A was twice as old as B. Find their present ages.
    Ans: A = 30, B = 20

68. If 1st January 2012 is Sunday, what day is 1st January 2013?
    Ans: Tuesday

69. What is the angle between the hands at 10:30?
    Ans: |30×10 – 5.5×30| = |300 – 165| = 135°

70. Solve: $x^2\text{–}4x\text{–}5>0$
    Ans: x < –1 or x > 5

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Short Calculation and Reasoning

71. If A is 5 years older than B and B is 10, what is A?
    Ans: 15

72. If today is Monday, what day was it 14 days ago?
    Ans: Monday

73. What is the angle between the hands at 6:00?
    Ans: 180°

74. Simplify: ${\log }_{2}32$
    Ans: 5

75. Solve: $x^2+2x+1\ge 0$
    Ans: All real x

76. Is n positive?
    I. n = 5
    II. n > 0
    Ans: Either alone is sufficient.

77. The sum of ages of A and B is 40. Ten years ago, A was twice as old as B. Find their present ages.
    Ans: A = 25, B = 15

78. If 1st March 2020 is Sunday, what day is 1st March 2021?
    Ans: Monday

79. What is the angle between the hands at 2:45?
    Ans: |30×2 – 5.5×45| = |60 – 247.5| = 187.5°, so smaller angle = 360 – 187.5 = 172.5°

80. Solve: $x^2\text{–}2x\text{–}15<0$
    Ans: –3 < x < 5

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Challenging & Conceptual

81. A is twice as old as B. After 10 years, A will be 10 years older than B. Find their present ages.
    Ans: B = 10, A = 20

82. If today is Saturday, what day will it be after 365 days?
    Ans: Sunday

83. What is the angle between the hands at 1:55?
    Ans: |30×1 – 5.5×55| = |30 – 302.5| = 272.5°, smaller angle = 360 – 272.5 = 87.5°

84. Simplify: ${\log }_{5}125$
    Ans: 3

85. Solve: $x^2\text{–}9x+20\le 0$
    Ans: 4 ≤ x ≤ 5

86. Is y negative?
    I. y = –2
    II. y^2 = 4
    Ans: I alone is sufficient.

87. The sum of ages of A and B is 60. After 10 years, A will be twice as old as B. Find their present ages.
    Ans: A = 40, B = 20

88. If 1st January 2000 is Saturday, what day is 1st January 2001?
    Ans: Monday2

89. What is the angle between the hands at 12:30?
    Ans: |30×12 – 5.5×30| = |360 – 165| = 195°, smaller angle = 165°

90. Solve: $x^2\text{–}12x+36>0$
    Ans: x < 6 or x > 6

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Advanced Reasoning

91. If A is 4 years older than B and B is 6 years older than C, and their sum is 38, find their ages.
    Ans: C = 8, B = 14, A = 18

92. If today is Thursday, what day was it 121 days ago?
    Ans: Saturday

93. What is the angle between the hands at 5:15?
    Ans: |30×5 – 5.5×15| = |150 – 82.5| = 67.5°

94. Simplify: ${\log }_{3}27$
    Ans: 3

95. Solve: $x^2\text{–}2x\text{–}8\le 0$
    Ans: –2 ≤ x ≤ 4

96. Is x even?
    I. x = 4
    II. x = 5
    Ans: I alone is sufficient.

97. The sum of ages of A and B is 70. After 20 years, A will be twice as old as B. Find their present ages.
    Ans: A = 50, B = 20

98. If 1st January 2015 is Thursday, what day is 1st January 2016?
    Ans: Friday

99. What is the angle between the hands at 11:20?
    Ans: |30×11 – 5.5×20| = |330 – 110| = 220°, smaller angle = 140°

100. Solve: $x^2\text{–}6x+8<0$
     Ans: 2 < x < 4

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Practice these questions to master miscellaneous topics for all competitive exams, including ages, calendars, clocks, logarithms, quadratic inequalities, and data sufficiency.