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Question 1Mathematics

The least number which, when divided by 12, 15 and 18, leaves a remainder of 5 in each case is:

  1. 185 Correct answer.
  2. 175
  3. 180
  4. 365

Explanation

Such a number must be 5 more than a common multiple of 12, 15 and 18, and the least one is built on the LCM. LCM(12, 15, 18) = 180, so the required number is 180 + 5 = 185. Checking: 185 ÷ 12, 185 ÷ 15 and 185 ÷ 18 each leave remainder 5.

Question 2Mathematics

A rectangular parade ground in a Sainik School measures 45 m by 30 m. A cadet runs one complete round along its boundary. What distance does the cadet cover?

  1. 1350 m
  2. 150 m Correct answer.
  3. 300 m
  4. 75 m

Explanation

One complete round of a rectangle equals its perimeter, which is 2 × (length + breadth). Here 2 × (45 + 30) = 2 × 75 = 150 m.

Question 3Mathematics

The marks scored by a cadet in five subjects are 72, 85, 68, 90 and 80. The average mark per subject is:

  1. 81
  2. 79 Correct answer.
  3. 80
  4. 78

Explanation

The average equals the sum of all observations divided by their number. The sum is 72 + 85 + 68 + 90 + 80 = 395, and 395 ÷ 5 = 79.

Question 4Mathematics

A shopkeeper buys 40 notebooks at ₹25 each and sells the whole lot for ₹1,200. His profit per cent is:

  1. 16⅔%
  2. 20% Correct answer.
  3. 25%
  4. 12%

Explanation

Cost price = 40 × ₹25 = ₹1,000, and profit = ₹1,200 − ₹1,000 = ₹200. Profit per cent is profit divided by cost price multiplied by 100, so (200 ÷ 1000) × 100 = 20%.

Question 5Mathematics

A sum of ₹8,000 is deposited in a bank at 7.5% per annum simple interest for 2 years. What total amount becomes payable at the end of 2 years?

  1. ₹9,200 Correct answer.
  2. ₹8,600
  3. ₹8,120
  4. ₹9,600

Explanation

Simple interest is given by I = (P × R × T) ÷ 100 = (8000 × 7.5 × 2) ÷ 100 = ₹1,200. The amount payable is principal plus interest, that is ₹8,000 + ₹1,200 = ₹9,200.

Question 6Mathematics

Two numbers are in the ratio 5 : 7. If their difference is 24, then the smaller number is:

  1. 48
  2. 60 Correct answer.
  3. 84
  4. 72

Explanation

The difference of the ratio terms is 7 − 5 = 2 parts, and these 2 parts equal 24. So one part is 12, and the smaller number is 5 × 12 = 60, while the larger is 7 × 12 = 84.

Question 7Mathematics

The value of 2/3 + 3/4 − 5/6 is:

  1. 3/4
  2. 5/12
  3. 7/12 Correct answer.
  4. 1/2

Explanation

Take the LCM of the denominators 3, 4 and 6, which is 12, and convert each fraction: 8/12 + 9/12 − 10/12. Combining the numerators gives 8 + 9 − 10 = 7, so the result is 7/12.

Question 8Mathematics

If 3(x − 4) = 2x + 5, then the value of x is:

  1. 17 Correct answer.
  2. -7
  3. 7
  4. 9

Explanation

Expand the bracket to get 3x − 12 = 2x + 5. Collecting the x-terms on one side and the numbers on the other gives 3x − 2x = 5 + 12, so x = 17. Checking: 3(17 − 4) = 39 and 2(17) + 5 = 39.

Question 9Mathematics

A can finish a piece of work in 12 days and B in 18 days. They work together for 4 days, after which A leaves. In how many more days will B finish the remaining work?

  1. 10 days
  2. 9 days
  3. 8 days Correct answer.
  4. 6 days

Explanation

In one day A does 1/12 and B does 1/18 of the work, so together they do 5/36. In 4 days they complete 4 × 5/36 = 5/9, leaving 4/9 of the work. B alone does 1/18 per day, so B needs (4/9) × 18 = 8 days.

Question 10Mathematics

The area of a triangle whose sides measure 13 cm, 14 cm and 15 cm is:

  1. 78 cm²
  2. 84 cm² Correct answer.
  3. 91 cm²
  4. 90 cm²

Explanation

With three sides given, apply Heron's formula. The semi-perimeter is s = (13 + 14 + 15) ÷ 2 = 21, so the area is √(21 × 8 × 7 × 6) = √7056 = 84 cm².

Question 11Mathematics

Which of the following numbers is exactly divisible by 8?

  1. 4,518
  2. 2,316
  3. 3,512 Correct answer.
  4. 1,234

Explanation

A number is divisible by 8 exactly when the number formed by its last three digits is divisible by 8. For 3,512 the last three digits are 512, and 512 ÷ 8 = 64, so 3,512 is divisible by 8. The other choices all leave a remainder.

Question 12Mathematics

A cadet scored 45 marks out of 60 in a mathematics test. His score expressed as a percentage is:

  1. 75% Correct answer.
  2. 80%
  3. 65%
  4. 70%

Explanation

To express a score as a percentage, divide the marks obtained by the total marks and multiply by 100. Here 45 ÷ 60 = 0.75, and 0.75 × 100 = 75. So the cadet scored 75 per cent.

Question 13Mathematics

In a triangle, two of the angles measure 48° and 67°. The measure of the third angle is:

  1. 75°
  2. 55°
  3. 85°
  4. 65° Correct answer.

Explanation

The three interior angles of any triangle add up to 180°. The two given angles total 48° + 67° = 115°, so the third angle is 180° − 115° = 65°.

Question 14Mathematics

A cadet cycles at 12 km/h for 45 minutes and then at 8 km/h for 30 minutes. The total distance covered by him is:

  1. 15 km
  2. 11 km
  3. 13 km Correct answer.
  4. 12 km

Explanation

Convert each time into hours: 45 minutes = 3/4 hour and 30 minutes = 1/2 hour. Distance = speed × time, so the first part is 12 × 3/4 = 9 km and the second part is 8 × 1/2 = 4 km. The total distance is 9 + 4 = 13 km.

Question 15Mathematics

The circumference of a circular ground is 44 m. Taking π = 22/7, the area of the ground is:

  1. 168 m²
  2. 144 m²
  3. 154 m² Correct answer.
  4. 196 m²

Explanation

Circumference = 2πr, so 44 = 2 × 22/7 × r, which gives r = 7 m. Area = πr² = 22/7 × 7 × 7 = 154 m².

Question 16Mathematics

The value of (2⁵ × 2³) ÷ 2⁶ is:

  1. 4 Correct answer.
  2. 8
  3. 16
  4. 2

Explanation

When powers with the same base are multiplied, the indices are added: 2⁵ × 2³ = 2⁸. Dividing by 2⁶ means subtracting the indices: 2⁸ ÷ 2⁶ = 2² = 4.

Question 17Mathematics

A shopkeeper marks an article at ₹500 and allows a discount of 10% on it. If he still gains 12.5% on the deal, his cost price for the article is:

  1. ₹450
  2. ₹375
  3. ₹420
  4. ₹400 Correct answer.

Explanation

The discount is calculated on the marked price, so the selling price is 500 − 10% of 500 = ₹450. A gain of 12.5% means selling price = cost price × 1.125. Hence cost price = 450 ÷ 1.125 = ₹400.

Question 18Mathematics

The weights, in kg, of seven cadets are 42, 45, 40, 48, 45, 43 and 47. The median weight of the group is:

  1. 45 kg Correct answer.
  2. 43 kg
  3. 47 kg
  4. 44 kg

Explanation

Arrange the seven weights in ascending order: 40, 42, 43, 45, 45, 47, 48. With an odd number of observations, the median is the middle value, which is the fourth value in the ordered list, namely 45 kg.

Question 19Mathematics

The greatest number which divides 245 and 1,029 leaving a remainder of 5 in each case is:

  1. 24
  2. 16 Correct answer.
  3. 32
  4. 8

Explanation

Subtract the remainder from each number: 245 − 5 = 240 and 1,029 − 5 = 1,024. The required number is the HCF of 240 and 1,024. Since 240 = 2⁴ × 3 × 5 and 1,024 = 2¹⁰, the HCF is 2⁴ = 16.

Question 20Mathematics

A cube of side 6 cm is painted on all its faces and then cut into small cubes of side 1 cm. The number of small cubes having exactly two painted faces is:

  1. 48 Correct answer.
  2. 72
  3. 60
  4. 36

Explanation

In a cube of side n cm cut into 1 cm cubes, the cubes with exactly two painted faces lie along the edges but not at the corners. Each of the 12 edges contributes n − 2 such cubes, so the count is 12 × (6 − 2) = 12 × 4 = 48.

Question 21Mathematics

How many prime numbers lie between 60 and 80?

  1. 4
  2. 5 Correct answer.
  3. 6
  4. 7

Explanation

The numbers between 60 and 80 that have exactly two factors are 61, 67, 71, 73 and 79. Counting these prime numbers gives 5, while the other numbers in the interval are composite.

Question 22Mathematics

The perimeter of a square field is 48 m. The area of the field is:

  1. 144 m² Correct answer.
  2. 576 m²
  3. 120 m²
  4. 96 m²

Explanation

For a square, perimeter = 4 × side, so each side is 48 ÷ 4 = 12 m. The area is side² = 12 × 12 = 144 m².

Question 23Mathematics

If 6 identical notebooks cost ₹90, what is the cost of 10 such notebooks?

  1. ₹150 Correct answer.
  2. ₹180
  3. ₹120
  4. ₹135

Explanation

The cost of one notebook is ₹90 ÷ 6 = ₹15. Therefore, 10 notebooks cost 10 × ₹15 = ₹150 by the unitary method.

Question 24Mathematics

A cadet deposits ₹10,000 in a bank at 10% per annum compounded annually. What amount will he receive after 2 years?

  1. ₹11,000
  2. ₹12,000
  3. ₹12,100 Correct answer.
  4. ₹12,210

Explanation

For compound interest, amount = P(1 + r/100)ⁿ. Here P = ₹10,000, r = 10 and n = 2, so amount = 10,000 × (1.1)² = ₹12,100.

Question 25Mathematics

If x + 1/x = 5, then x² + 1/x² equals:

  1. 21
  2. 25
  3. 27
  4. 23 Correct answer.

Explanation

Squaring both sides gives x² + 2 + 1/x² = 25. Subtracting 2 gives x² + 1/x² = 23.

Question 26Mathematics

A trapezium has parallel sides 12 cm and 8 cm and the perpendicular distance between them is 5 cm. Its area is:

  1. 60 cm²
  2. 40 cm²
  3. 100 cm²
  4. 50 cm² Correct answer.

Explanation

The area of a trapezium is ½ × (sum of parallel sides) × height. Here it is ½ × (12 + 8) × 5 = ½ × 20 × 5 = 50 cm².

Question 27Mathematics

The mean of five numbers is 24. When one number, 30, is removed, the mean of the remaining four numbers is:

  1. 23.0
  2. 24.0
  3. 21.0
  4. 22.5 Correct answer.

Explanation

The total of the five numbers is 5 × 24 = 120. After removing 30, the remaining total is 90, so the new mean is 90 ÷ 4 = 22.5.

Question 28Mathematics

A number is increased by 20% and then the result is decreased by 20%. The net change in the number is:

  1. no change
  2. a 4% increase
  3. a 2% decrease
  4. a 4% decrease Correct answer.

Explanation

Taking the number as 100, a 20% increase gives 120. A 20% decrease on 120 gives 96, which is 4 less than 100, so the net change is a 4% decrease.

Question 29Mathematics

The largest 4-digit number exactly divisible by 12, 15 and 18 is:

  1. 9990
  2. 9720
  3. 9900 Correct answer.
  4. 9960

Explanation

The LCM of 12, 15 and 18 is 180. Dividing 9999 by 180 gives quotient 55 and remainder 99, so the largest required multiple is 55 × 180 = 9900.

Question 30Mathematics

A can complete a piece of work in 20 days. B is 25% more efficient than A. In how many days can B alone complete the work?

  1. 16 days Correct answer.
  2. 25 days
  3. 15 days
  4. 18 days

Explanation

A's rate is 1/20 of the work per day. B is 25% more efficient, so B's rate is 1.25 × 1/20 = 1/16 of the work per day, meaning B alone takes 16 days.

Question 31Mathematics

A cadet reads 2/5 of a book on Monday and 1/3 of the same book on Tuesday. What fraction of the book is still unread?

  1. 4/15 Correct answer.
  2. 7/15
  3. 11/15
  4. 2/15

Explanation

The part read is 2/5 + 1/3 = 6/15 + 5/15 = 11/15 of the book. The whole book is 1, so the unread part is 1 − 11/15 = 4/15. Always convert the fractions to a common denominator before adding.

Question 32Mathematics

A cadet deposits ₹4,500 in a post office scheme that pays simple interest at 8% per annum. What total interest will he earn in 3 years?

  1. ₹1,140
  2. ₹1,080 Correct answer.
  3. ₹540
  4. ₹360

Explanation

Simple interest is calculated as SI = (P × R × T)/100, where P is the principal, R the yearly rate and T the time in years. Here SI = (4500 × 8 × 3)/100 = ₹1,080. Simple interest grows equally each year, so the three yearly amounts of ₹360 add to ₹1,080.

Question 33Mathematics

The value of √0.0144 is:

  1. 0.012
  2. 1.2
  3. 0.12 Correct answer.
  4. 0.24

Explanation

Write 0.0144 as 144/10000. Then √144 = 12 and √10000 = 100, so the square root is 12/100 = 0.12. Checking, 0.12 × 0.12 = 0.0144, which confirms the answer.

Question 34Mathematics

Two numbers are in the ratio 5 : 7. When 6 is subtracted from each of them, the ratio becomes 3 : 5. The smaller of the two numbers is:

  1. 21
  2. 25
  3. 15 Correct answer.
  4. 20

Explanation

Let the numbers be 5x and 7x. The condition gives (5x − 6)/(7x − 6) = 3/5, so 25x − 30 = 21x − 18, giving 4x = 12 and x = 3. The numbers are 15 and 21, and the smaller is 15; indeed 9 : 15 = 3 : 5.

Question 35Mathematics

A father is three times as old as his son. After 12 years he will be twice as old as his son. The son's present age is:

  1. 16 years
  2. 10 years
  3. 14 years
  4. 12 years Correct answer.

Explanation

Let the son's age be x years, so the father's age is 3x years. In 12 years, 3x + 12 = 2(x + 12), which gives 3x + 12 = 2x + 24, so x = 12. Checking: father is 36 now, and after 12 years they are 48 and 24, exactly twice.

Question 36Mathematics

The sum of the interior angles of a regular polygon is 1080°. The number of sides of this polygon is:

  1. 6
  2. 7
  3. 10
  4. 8 Correct answer.

Explanation

For a polygon with n sides, the interior angles add up to (n − 2) × 180°. Setting (n − 2) × 180 = 1080 gives n − 2 = 6, hence n = 8. The polygon is a regular octagon.

Question 37Mathematics

A right circular cylinder has radius 7 cm and height 10 cm. Taking π = 22/7, its volume is:

  1. 2200 cm³
  2. 1540 cm³ Correct answer.
  3. 1440 cm³
  4. 770 cm³

Explanation

The volume of a cylinder is πr²h. Substituting, V = (22/7) × 7 × 7 × 10 = 22 × 7 × 10 = 1540 cm³. The radius must be squared before multiplying by the height.

Question 38Mathematics

A bag contains 5 red balls and 7 blue balls. One ball is drawn at random from the bag. The probability that the drawn ball is red is:

  1. 5/7
  2. 7/12
  3. 5/12 Correct answer.
  4. 1/2

Explanation

Probability equals favourable outcomes divided by total outcomes. There are 5 red balls out of 5 + 7 = 12 balls, so P(red) = 5/12. The result must lie between 0 and 1, and 5/12 satisfies this.

Question 39Mathematics

A solid metallic sphere of radius 6 cm is melted and recast into small spheres, each of radius 2 cm. The number of small spheres formed is:

  1. 18
  2. 3
  3. 9
  4. 27 Correct answer.

Explanation

The volume of a sphere is (4/3)πr³, so volumes are in the ratio of the cubes of the radii. The ratio (6/2)³ = 27 gives the number of small spheres, since the metal volume is conserved during melting. Thus 27 small spheres of radius 2 cm are formed.

Question 40Mathematics

The angle between the hour hand and the minute hand of a clock at 3:30 is:

  1. 85°
  2. 60°
  3. 75° Correct answer.
  4. 90°

Explanation

At 3:30 the minute hand points at 6, that is 180° from 12. The hour hand has moved 3 × 30° + 30 × 0.5° = 105° from 12. The difference is 180° − 105° = 75°, so the hands are 75° apart.

Question 41Mathematics

Simplify: 12 + 6 ÷ 3 × 2 − 4.

  1. 10
  2. 16
  3. 8
  4. 12 Correct answer.

Explanation

Using BODMAS, division and multiplication are performed before addition and subtraction, from left to right: 6 ÷ 3 = 2, then 2 × 2 = 4, so 12 + 4 − 4 = 12.

Question 42Mathematics

Find the median of the data: 7, 9, 4, 6, 8, 11, 5.

  1. 7 Correct answer.
  2. 8
  3. 9
  4. 6

Explanation

Arrange the observations in ascending order: 4, 5, 6, 7, 8, 9, 11. There are seven values, so the median is the fourth value, which is 7.

Question 43Mathematics

The angles of a triangle are in the ratio 2 : 3 : 4. What is the measure of the largest angle?

  1. 40°
  2. 100°
  3. 80° Correct answer.
  4. 60°

Explanation

Let the angles be 2x, 3x and 4x. Since the angle sum of a triangle is 180°, 9x = 180°, so x = 20°; the largest angle is 4x = 80°.

Question 44Mathematics

Three bells ring at intervals of 6, 8 and 10 minutes. If they ring together at 8:00 a.m., at what time will they next ring together?

  1. 10:30 a.m.
  2. 10:00 a.m. Correct answer.
  3. 9:00 a.m.
  4. 9:30 a.m.

Explanation

They next ring together after the LCM of 6, 8 and 10 minutes. The LCM is 120 minutes, which is 2 hours, so the next time is 10:00 a.m.

Question 45Mathematics

A shopkeeper marks an article at ₹750 and allows a discount of 20%. He still makes a profit of 25%. What is the cost price of the article?

  1. ₹480 Correct answer.
  2. ₹500
  3. ₹560
  4. ₹450

Explanation

The selling price after the discount is 0.80 × 750 = ₹600. Since ₹600 is 25% above cost price, cost price = 600 ÷ 1.25 = ₹480.

Question 46Mathematics

A cadet travels the first 60 km of a journey at 30 km/h and the next 60 km at 20 km/h. What is his average speed for the whole journey?

  1. 25 km/h
  2. 26 km/h
  3. 22 km/h
  4. 24 km/h Correct answer.

Explanation

Average speed is total distance divided by total time. The times are 60/30 = 2 h and 60/20 = 3 h, so average = 120 ÷ 5 = 24 km/h.

Question 47Mathematics

The circumference of a circle is 44 cm. Taking π = 22/7, what is its area?

  1. 154 cm² Correct answer.
  2. 176 cm²
  3. 132 cm²
  4. 144 cm²

Explanation

Circumference = 2πr = 44 cm, so 2 × 22/7 × r = 44, giving r = 7 cm. Area = πr² = 22/7 × 49 = 154 cm².

Question 48Mathematics

How many positive factors does 72 have?

  1. 16
  2. 12 Correct answer.
  3. 10
  4. 8

Explanation

Write 72 in prime factor form: 72 = 2³ × 3². The number of positive factors is (3 + 1)(2 + 1) = 4 × 3 = 12.

Question 49Mathematics

Three pipes A, B and C can fill a tank in 12 hours, 15 hours and 20 hours respectively. Pipe C is an outlet pipe that empties the tank. If all three pipes are opened together, in how many hours will the tank be filled?

  1. 12 hours
  2. 8 hours
  3. 10 hours Correct answer.
  4. 15 hours

Explanation

Work done per hour is 1/12 + 1/15 − 1/20 = 5/60 + 4/60 − 3/60 = 6/60 = 1/10. Since one-tenth of the tank is filled each hour, it takes 10 hours.

Question 50Mathematics

A train 150 m long is running at a speed of 54 km/h. It crosses a platform completely in 30 seconds. What is the length of the platform?

  1. 250 m
  2. 300 m Correct answer.
  3. 400 m
  4. 350 m

Explanation

Convert 54 km/h to 15 m/s by multiplying by 5/18. In 30 seconds the train covers 15 × 30 = 450 m, which equals train length plus platform length; therefore platform length = 450 − 150 = 300 m.