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AP POLYCET study materials
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AP POLYCET practice questions

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Question 1Mathematics

If A = {1, 2, 3}, then the number of subsets of A is

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

Explanation

A set with n elements has 2^n subsets, counting the empty set and the set itself. Here n = 3, so the number of subsets is 2^3 = 8, namely ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3} and {1,2,3}.

Question 2Mathematics

The LCM of 12 and 18 is

  1. 36 Correct answer.
  2. 216
  3. 72
  4. 6

Explanation

Write 12 = 2² × 3 and 18 = 2 × 3²; the LCM takes the highest power of each prime, giving 2² × 3² = 36. Equivalently, LCM = (12 × 18)/HCF = 216/6 = 36.

Question 3Mathematics

The 10th term of the arithmetic progression 3, 7, 11, ... is

  1. 39 Correct answer.
  2. 43
  3. 7
  4. 35

Explanation

In an AP the nth term is a + (n − 1)d. Here a = 3 and d = 7 − 3 = 4, so the 10th term is 3 + (10 − 1)(4) = 3 + 36 = 39.

Question 4Mathematics

If 2 is a root of the equation x² − 7x + k = 0, then the value of k is

  1. 14
  2. 10 Correct answer.
  3. 5
  4. −10

Explanation

A root makes the polynomial equal to zero, so substituting x = 2 gives 4 − 14 + k = 0. Solving, k = 10. Alternatively, the roots are 2 and 5, whose product 10 equals the constant term k.

Question 5Mathematics

The points (1, 2), (3, k) and (5, 6) are collinear when k equals

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

Explanation

Three points are collinear when the slope between every pair of them is the same. The slope from (1, 2) to (5, 6) is (6 − 2)/(5 − 1) = 1, so the line is y = x + 1, and putting x = 3 gives k = 4.

Question 6Mathematics

If θ is an acute angle and sin θ = 3/5, then tan θ equals

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

Explanation

Take a right triangle with opposite side 3 and hypotenuse 5. The adjacent side is √(5² − 3²) = 4, so tan θ = opposite/adjacent = 3/4.

Question 7Mathematics

The total surface area of a sphere of radius 7 cm (taking π = 22/7) is

  1. 308 cm²
  2. 616 cm² Correct answer.
  3. 1232 cm²
  4. 154 cm²

Explanation

The total surface area of a sphere is 4πr². With r = 7 cm and π = 22/7, the area is 4 × (22/7) × 49 = 4 × 154 = 616 cm².

Question 8Mathematics

A bag contains 3 red balls and 5 black balls. One ball is drawn at random. The probability that it is red is

  1. 3/5
  2. 5/8
  3. 3/8 Correct answer.
  4. 1/3

Explanation

Probability of an event = favourable outcomes ÷ total outcomes. There are 3 red balls among 3 + 5 = 8 balls in all, so P(red) = 3/8.

Question 9Mathematics

The variance of the observations 1, 2, 3, 4, 5 is

  1. 2 Correct answer.
  2. 2.5
  3. 10
  4. 3

Explanation

Variance = Σ(xᵢ − x̄)² / n. The mean is 3, the squared deviations are 4, 1, 0, 1, 4, and their sum is 10, so variance = 10/5 = 2.

Question 10Mathematics

If α and β are the roots of x² − 6x + 8 = 0, then α² + β² equals

  1. 28
  2. 36
  3. 16
  4. 20 Correct answer.

Explanation

For ax² + bx + c = 0, α + β = −b/a and αβ = c/a, so here α + β = 6 and αβ = 8. Then α² + β² = (α + β)² − 2αβ = 36 − 16 = 20.

Question 11Mathematics

If n(A) = 17, n(B) = 13 and n(A ∩ B) = 5, then n(A ∪ B) equals

  1. 20
  2. 35
  3. 30
  4. 25 Correct answer.

Explanation

For any two finite sets, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Substituting the given values gives 17 + 13 − 5 = 25. The intersection is counted in both n(A) and n(B), so it must be subtracted once.

Question 12Mathematics

The HCF of 36 and 84 is

  1. 12 Correct answer.
  2. 6
  3. 24
  4. 18

Explanation

Writing the prime factorisations, 36 = 2² × 3² and 84 = 2² × 3 × 7. The highest common factor uses the least power of each common prime, so HCF = 2² × 3 = 12.

Question 13Mathematics

The 10th term of the geometric progression 2, 6, 18, ... is

  1. 39366 Correct answer.
  2. 118098
  3. 13122
  4. 4374

Explanation

Here the first term a = 2 and the common ratio r = 3. The nth term of a geometric progression is a·r^(n−1), so the 10th term is 2 × 3⁹ = 2 × 19683 = 39366.

Question 14Mathematics

If the quadratic equation x² + kx + 9 = 0 has equal roots, then the positive value of k is

  1. 6 Correct answer.
  2. 12
  3. 9
  4. 3

Explanation

A quadratic has equal roots exactly when its discriminant b² − 4ac is zero. Here b = k, a = 1 and c = 9, so k² − 36 = 0, giving k = ±6. The positive value is therefore 6.

Question 15Mathematics

The distance between the points (3, −2) and (−1, 1) is

  1. 25
  2. √7
  3. 7
  4. 5 Correct answer.

Explanation

Using the distance formula, the differences are x₂ − x₁ = −4 and y₂ − y₁ = 3. The distance is √((−4)² + 3²) = √(16 + 9) = √25 = 5.

Question 16Mathematics

If θ is an acute angle and tan θ = 3/4, then sin θ + cos θ equals

  1. 7/5 Correct answer.
  2. 12/25
  3. 1
  4. 5/7

Explanation

For an acute angle with tan θ = 3/4, a right triangle with opposite side 3 and adjacent side 4 has hypotenuse 5. Hence sin θ = 3/5 and cos θ = 4/5, whose sum is 7/5.

Question 17Mathematics

The volume of a cone of radius 6 cm and height 7 cm, taking π = 22/7, is

  1. 792 cm³
  2. 132 cm³
  3. 264 cm³ Correct answer.
  4. 528 cm³

Explanation

The volume of a cone is (1/3)πr²h. Substituting r = 6, h = 7 and π = 22/7 gives (1/3) × (22/7) × 36 × 7 = 264 cm³.

Question 18Mathematics

Two fair dice are thrown together. The probability that the sum of the numbers shown is a prime number is

  1. 1/2
  2. 7/18
  3. 1/3
  4. 5/12 Correct answer.

Explanation

The possible prime sums are 2, 3, 5, 7 and 11, occurring in 1, 2, 4, 6 and 2 ways respectively. This gives 15 favourable outcomes out of 36 equally likely outcomes, so the probability is 15/36 = 5/12.

Question 19Mathematics

For a moderately skewed distribution, the mean is 30 and the median is 28. Using the empirical relation, the mode is

  1. 24 Correct answer.
  2. 29
  3. 26
  4. 32

Explanation

The empirical relation for a moderately skewed distribution is mode = 3 median − 2 mean. Substituting the values gives 3 × 28 − 2 × 30 = 84 − 60 = 24.

Question 20Mathematics

The sum of the first 20 terms of an arithmetic progression whose nth term is 3n + 1 is

  1. 700
  2. 670
  3. 620
  4. 650 Correct answer.

Explanation

The first term is 3(1) + 1 = 4 and the second term is 7, so the common difference is 3. Using Sₙ = (n/2)[2a + (n − 1)d], we get S₂₀ = 10[8 + 19 × 3] = 10 × 65 = 650.

Question 21Mathematics

If A = {2, 4, 6, 8} and B = {4, 8, 10}, then the number of elements in the symmetric difference A △ B is

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

Explanation

The symmetric difference A △ B contains elements that belong to exactly one of A and B. Removing the common elements 4 and 8 leaves 2, 6 and 10, so there are three elements.

Question 22Mathematics

The least number which when divided by 12, 15 and 20 leaves a remainder of 5 in each case is

  1. 75
  2. 55
  3. 60
  4. 65 Correct answer.

Explanation

The required number is 5 more than a common multiple of 12, 15 and 20. Their LCM is 60, so the least such number is 60 + 5 = 65.

Question 23Mathematics

Three numbers are in arithmetic progression. Their sum is 24 and the sum of their squares is 200. The positive common difference of the progression is

  1. 2 Correct answer.
  2. 1
  3. 3
  4. 4

Explanation

Let the numbers be a − d, a and a + d. Their sum is 3a = 24, so a = 8. Then (8 − d)² + 8² + (8 + d)² = 200 gives 192 + 2d² = 200, so d² = 4 and the positive value is d = 2.

Question 24Mathematics

If α and β are the roots of 3x² − 10x + 3 = 0, then 1/α + 1/β equals

  1. 3/10
  2. 1
  3. 13/3
  4. 10/3 Correct answer.

Explanation

For 3x² − 10x + 3 = 0, the sum of the roots is α + β = 10/3 and the product is αβ = 1. Since 1/α + 1/β = (α + β)/(αβ), the value is (10/3)/1 = 10/3.

Question 25Mathematics

The midpoint of the line segment joining the points (−3, 5) and (7, −1) is

  1. (2, 3)
  2. (4, 3)
  3. (5, 2)
  4. (2, 2) Correct answer.

Explanation

The midpoint is ((−3 + 7)/2, (5 + (−1))/2). This simplifies to (4/2, 4/2) = (2, 2).

Question 26Mathematics

If θ is an acute angle and sin θ + cos θ = √2, then sin θ cos θ equals

  1. 1/2 Correct answer.
  2. √2/2
  3. 1
  4. 1/4

Explanation

Squaring sin θ + cos θ = √2 gives sin²θ + cos²θ + 2 sin θ cos θ = 2. Using sin²θ + cos²θ = 1, we get 2 sin θ cos θ = 1, so sin θ cos θ = 1/2.

Question 27Mathematics

The curved surface area of a cylinder of radius 7 cm and height 10 cm, taking π = 22/7, is

  1. 440 cm² Correct answer.
  2. 1540 cm²
  3. 880 cm²
  4. 220 cm²

Explanation

The curved surface area of a cylinder is 2πrh. Substituting r = 7, h = 10 and π = 22/7 gives 2 × (22/7) × 7 × 10 = 440 cm².

Question 28Mathematics

From a well-shuffled deck of 52 playing cards, one card is drawn at random. The probability that it is either a face card or an ace is

  1. 5/13
  2. 3/13
  3. 4/13 Correct answer.
  4. 1/13

Explanation

A deck has 12 face cards and 4 aces, giving 16 favourable cards in all. Hence the probability is 16/52 = 4/13.

Question 29Mathematics

The largest number that divides 245 and 1029, leaving a remainder of 5 in each case, is

  1. 8
  2. 16 Correct answer.
  3. 24
  4. 12

Explanation

The required divisor must divide 245 − 5 = 240 and 1029 − 5 = 1024 exactly. The HCF of 240 and 1024 is 16, and 16 leaves remainder 5 in both divisions.

Question 30Mathematics

Two circles of radii 5 cm and 3 cm touch each other externally. The length of their common external tangent is

  1. 2√15 cm Correct answer.
  2. 4 cm
  3. √55 cm
  4. 8 cm

Explanation

For two circles touching externally, the distance between their centres is 5 + 3 = 8 cm. The external common tangent length is √(d² − (r₁ − r₂)²) = √(64 − 4) = √60 = 2√15 cm.

Question 31Mathematics

The number of proper subsets of a set containing 4 elements is

  1. 15 Correct answer.
  2. 8
  3. 16
  4. 14

Explanation

A set with n elements has 2^n subsets in total, so four elements give 16 subsets. Proper subsets exclude the original set itself, leaving 16 − 1 = 15.

Question 32Mathematics

The decimal expansion of 13/125 terminates after how many decimal places?

  1. 4
  2. 1
  3. 2
  4. 3 Correct answer.

Explanation

Since 125 = 5^3, multiplying numerator and denominator by 8 gives 13/125 = 104/1000 = 0.104. Thus the decimal terminates after three places.

Question 33Mathematics

The median of the observations 7, 3, 9, 5, 11 is

  1. 7 Correct answer.
  2. 5
  3. 9
  4. 8

Explanation

Arranging the five observations in ascending order gives 3, 5, 7, 9, 11. The median is the middle value, which is 7.

Question 34Mathematics

If one root of the quadratic equation x² − 7x + k = 0 is 3, then k equals

  1. 10
  2. 21
  3. 12 Correct answer.
  4. −12

Explanation

Substitute x = 3 into x² − 7x + k = 0: 9 − 21 + k = 0, so k = 12. The other root is 4, whose product with 3 is also 12.

Question 35Mathematics

If θ is an acute angle and sin θ = 5/13, then tan θ equals

  1. 12/5
  2. 13/5
  3. 13/12
  4. 5/12 Correct answer.

Explanation

For acute θ, cos θ = √(1 − sin²θ) = √(1 − 25/169) = 12/13. Therefore tan θ = sin θ / cos θ = (5/13)/(12/13) = 5/12.

Question 36Mathematics

The area of the triangle whose vertices are (0, 0), (4, 0) and (0, 3) is

  1. 12 square units
  2. 5 square units
  3. 6 square units Correct answer.
  4. 7 square units

Explanation

The sides along the axes have lengths 4 and 3 and are perpendicular. Hence area = 1/2 × 4 × 3 = 6 square units.

Question 37Mathematics

The total surface area of a sphere of radius 7 cm, taking π = 22/7, is

  1. 154 cm²
  2. 308 cm²
  3. 1232 cm²
  4. 616 cm² Correct answer.

Explanation

Total surface area of a sphere is 4πr². Substituting r = 7 and π = 22/7 gives 4 × 22/7 × 49 = 616 cm².

Question 38Mathematics

A bag contains 5 red and 3 blue balls. Two balls are drawn at random without replacement. The probability that both are red is

  1. 5/28
  2. 5/14 Correct answer.
  3. 15/28
  4. 3/28

Explanation

The number of ways to choose two red balls is C(5,2) = 10, and the total number of ways to choose two balls is C(8,2) = 28. Hence the probability is 10/28 = 5/14.

Question 39Mathematics

If the sum of the first n terms of an arithmetic progression is S_n = n² + 3n, then its 10th term is

  1. 24
  2. 20
  3. 22 Correct answer.
  4. 21

Explanation

The 10th term equals S10 − S9. Now S10 = 100 + 30 = 130 and S9 = 81 + 27 = 108, so t10 = 130 − 108 = 22.

Question 40Mathematics

If two similar triangles have areas in the ratio 16 : 25, then the ratio of their corresponding medians is

  1. 16 : 25
  2. 2 : 3
  3. 5 : 4
  4. 4 : 5 Correct answer.

Explanation

In similar triangles, the ratio of areas equals the square of the ratio of corresponding linear measures. Taking the square root of 16 : 25 gives 4 : 5.

Question 41Mathematics

The HCF of two positive integers is 6 and their LCM is 72. If one of the integers is 24, then the other integer is

  1. 16
  2. 36
  3. 12
  4. 18 Correct answer.

Explanation

For any two positive integers, HCF × LCM equals the product of the two numbers. Here the product is 6 × 72 = 432, so the second number is 432 ÷ 24 = 18. Checking, HCF(24, 18) = 6 and LCM(24, 18) = 72, so the value is consistent.

Question 42Mathematics

The sum of the zeroes of the polynomial 2x² − 8x + 5 is

  1. 5/2
  2. 8
  3. 4 Correct answer.
  4. −4

Explanation

For a quadratic ax² + bx + c, the sum of the zeroes equals −b/a. Here a = 2 and b = −8, so the sum is −(−8)/2 = 4. The constant term 5 affects only the product of the zeroes, not their sum.

Question 43Mathematics

The value of sin 30° cos 60° + cos 30° sin 60° is

  1. 1 Correct answer.
  2. √3/2
  3. 1/2
  4. 0

Explanation

Using standard values, sin 30° = cos 60° = 1/2 and cos 30° = sin 60° = √3/2. The expression therefore equals 1/4 + 3/4 = 1. This also agrees with the identity sin(A + B) = sin A cos B + cos A sin B with A = 30° and B = 60°.

Question 44Mathematics

The point that divides the line segment joining (1, 2) and (4, 5) internally in the ratio 2 : 1 is

  1. (5/2, 7/2)
  2. (3, 4) Correct answer.
  3. (3, 3)
  4. (2, 3)

Explanation

The section formula for internal division in ratio m : n gives ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). With m = 2, n = 1, x₁ = 1, y₁ = 2, x₂ = 4, y₂ = 5, the point is ((8 + 1)/3, (10 + 2)/3) = (3, 4). Since the ratio is 2 : 1, the point lies closer to (4, 5), which (3, 4) satisfies.

Question 45Mathematics

The volume of a right circular cone of radius 3 cm and height 7 cm, taking π = 22/7, is

  1. 22 cm³
  2. 198 cm³
  3. 66 cm³ Correct answer.
  4. 132 cm³

Explanation

The volume of a cone is (1/3)πr²h. Substituting r = 3 cm, h = 7 cm and π = 22/7 gives (1/3) × (22/7) × 9 × 7 = 66 cm³. The factor 1/3 is what distinguishes this from the cylinder of the same radius and height, whose volume is 198 cm³.

Question 46Mathematics

Two fair dice are thrown together. The probability that the sum of the numbers on the top faces is 9 is

  1. 5/36
  2. 1/6
  3. 1/12
  4. 1/9 Correct answer.

Explanation

There are 6 × 6 = 36 equally likely outcomes. A sum of 9 occurs for (3, 6), (4, 5), (5, 4) and (6, 3), giving 4 favourable outcomes. Hence the probability is 4/36, which simplifies to 1/9.

Question 47Mathematics

The mean of five observations is 12. If the observation 20 is removed, then the mean of the remaining four observations is

  1. 12
  2. 8
  3. 9
  4. 10 Correct answer.

Explanation

The total of the five observations is 5 × 12 = 60. Removing 20 leaves a total of 40 shared among four observations, so the new mean is 40 ÷ 4 = 10. Since the removed value is above the original mean, the new mean must fall below 12.

Question 48Mathematics

Which term of the arithmetic progression 3, 7, 11, … is equal to 83?

  1. 20th
  2. 21st Correct answer.
  3. 19th
  4. 22nd

Explanation

Here the first term a = 3 and the common difference d = 4. Using aₙ = a + (n − 1)d gives 3 + (n − 1) × 4 = 83, so (n − 1) × 4 = 80 and n − 1 = 20, hence n = 21. Checking, the 21st term is 3 + 20 × 4 = 83.

Question 49Mathematics

A solid metallic sphere of radius 6 cm is melted and recast into small cones, each of radius 2 cm and height 3 cm. The number of cones formed is

  1. 108
  2. 72 Correct answer.
  3. 36
  4. 24

Explanation

The volume of the sphere is (4/3)π(6)³ = 288π cm³, and the volume of one cone is (1/3)π(2)²(3) = 4π cm³. Dividing the sphere's volume by the cone's volume gives 288π ÷ 4π = 72. The factor π cancels, so the numerical value of π is not needed.

Question 50Mathematics

If θ is an acute angle and sec θ + tan θ = 5, then sec θ − tan θ equals

  1. 1/25
  2. 1/5 Correct answer.
  3. 5
  4. 1/4

Explanation

The identity sec²θ − tan²θ = 1 can be written as (sec θ + tan θ)(sec θ − tan θ) = 1. Substituting sec θ + tan θ = 5 gives sec θ − tan θ = 1/5. This avoids solving for θ itself and holds for every acute angle satisfying the given condition.