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TG POLYCET practice questions

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Question 1Mathematics

In an arithmetic progression the 5th term is 14 and the 9th term is 26. What is the common difference?

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

Explanation

For an AP, a9 − a5 = (9 − 5)d, so 26 − 14 = 4d, giving d = 3. The first term is then a = 14 − 4(3) = 2.

Question 2Mathematics

The sum of the first 20 terms of the AP 3, 7, 11, 15, … is

  1. 800
  2. 760
  3. 820 Correct answer.
  4. 840

Explanation

Here a = 3 and d = 4, so a20 = 3 + 19(4) = 79. The sum is S20 = (20/2)(3 + 79) = 10 × 82 = 820.

Question 3Mathematics

The 3rd term of a geometric progression is 12 and the 6th term is 96. What is the common ratio?

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

Explanation

Dividing a6 by a3 gives r³ = 96/12 = 8, so r = 2. The first term is then 12/2² = 3.

Question 4Mathematics

The value of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + … is

  1. 3/2 Correct answer.
  2. 2/3
  3. 3
  4. 9/4

Explanation

This is a GP with first term a = 1 and ratio r = 1/3, and |r| < 1. The infinite sum is a/(1 − r) = 1/(1 − 1/3) = 3/2.

Question 5Mathematics

If the roots of x² − 7x + k = 0 are equal, then k equals

  1. 7/2
  2. 49/4 Correct answer.
  3. 14
  4. 49

Explanation

Equal roots require the discriminant b² − 4ac to vanish: 49 − 4k = 0, so k = 49/4.

Question 6Mathematics

The quadratic equation whose roots are 2 + √3 and 2 − √3 is

  1. x² + 4x + 1 = 0
  2. x² − 4x − 1 = 0
  3. x² − 4x + 1 = 0 Correct answer.
  4. x² − 2x + 1 = 0

Explanation

Sum of roots = 4 and product = 4 − 3 = 1. The equation is x² − (sum)x + product = 0, that is x² − 4x + 1 = 0.

Question 7Mathematics

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

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

Explanation

Using the distance formula, √[(3+1)² + (−2−1)²] = √(16 + 9) = √25 = 5.

Question 8Mathematics

The slope of the line passing through (2, 5) and (6, 13) is

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

Explanation

Slope = (13 − 5)/(6 − 2) = 8/4 = 2. Rise over run is computed from differences in y over differences in x.

Question 9Mathematics

The equation of the line perpendicular to 2x + 3y = 6 and passing through the origin is

  1. 3x + 2y = 0
  2. 2x + 3y = 0
  3. 3x − 2y = 0 Correct answer.
  4. 2x − 3y = 0

Explanation

The given line has slope −2/3, so a perpendicular line has slope 3/2. Through the origin this is y = (3/2)x, i.e. 3x − 2y = 0.

Question 10Mathematics

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

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

Explanation

The expression matches sin(A + B) with A = 30° and B = 60°, so it equals sin 90° = 1.

Question 11Mathematics

If tan θ = 3/4 and θ lies in the first quadrant, then sin θ equals

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

Explanation

With opposite 3 and adjacent 4, the hypotenuse is 5. In the first quadrant sin θ = opposite/hypotenuse = 3/5.

Question 12Mathematics

The value of sin² 25° + sin² 65° is

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

Explanation

Since 65° = 90° − 25°, we have sin 65° = cos 25°. Then sin²25° + cos²25° = 1 by the Pythagorean identity.

Question 13Mathematics

A ladder 10 m long leans against a wall making an angle of 60° with the ground. The height reached on the wall is

  1. 10√3 m
  2. 5√3 m Correct answer.
  3. 10/√3 m
  4. 5 m

Explanation

The height is opposite the 60° angle, so height = 10 sin 60° = 10 × (√3/2) = 5√3 m.

Question 14Mathematics

The mean of the observations 4, 6, 8, 10, 12 is

  1. 9
  2. 7
  3. 10
  4. 8 Correct answer.

Explanation

The sum is 40 and there are 5 observations, so the mean is 40/5 = 8.

Question 15Mathematics

The median of the data 7, 3, 9, 5, 11, 13 is

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

Explanation

Arranged in order: 3, 5, 7, 9, 11, 13. With an even count, the median is the average of the two middle values (7 + 9)/2 = 8.

Question 16Mathematics

A bag contains 5 red and 7 blue balls. Two balls are drawn one after another without replacement. The probability that both are red is

  1. 25/144
  2. 10/33
  3. 5/12
  4. 5/33 Correct answer.

Explanation

P(first red) = 5/12 and P(second red | first red) = 4/11. Their product is 20/132 = 5/33.

Question 17Mathematics

The probability of getting a sum of 9 when two fair dice are thrown is

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

Explanation

Favourable outcomes are (3,6), (4,5), (5,4), (6,3), giving 4 outcomes. So the probability is 4/36 = 1/9.

Question 18Mathematics

The total surface area of a solid sphere of radius 7 cm, in cm², is (take π = 22/7)

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

Explanation

The total surface area of a sphere is 4πr² = 4 × (22/7) × 49 = 616 cm².

Question 19Mathematics

A cone has base radius 3 cm and height 4 cm. Its slant height and curved surface area are respectively

  1. 4 cm and 12π cm²
  2. 5 cm and 12π cm²
  3. 7 cm and 21π cm²
  4. 5 cm and 15π cm² Correct answer.

Explanation

Slant height l = √(r² + h²) = √(9 + 16) = 5 cm. Curved surface area = πrl = π × 3 × 5 = 15π cm².

Question 20Mathematics

If the volume of a cube is 512 cm³, its total surface area in cm² is

  1. 256
  2. 512
  3. 384 Correct answer.
  4. 64

Explanation

Side = ∛512 = 8 cm. Total surface area = 6a² = 6 × 64 = 384 cm².

Question 21Mathematics

The HCF of 84 and 126 is

  1. 63
  2. 21
  3. 14
  4. 42 Correct answer.

Explanation

Using Euclid's algorithm, 126 = 84 × 1 + 42 and 84 = 42 × 2 + 0, so HCF = 42. Prime factorisation gives 2² × 3 × 7 for 84 and 2 × 3² × 7 for 126, whose common part is 42.

Question 22Mathematics

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

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

Explanation

The LCM of 12, 15 and 18 is 180. The required number is LCM + 5 = 185, since the remainder 5 must be added to a common multiple.

Question 23Mathematics

The value of log₂ 32 + log₃ 27 is

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

Explanation

Since 32 = 2⁵, log₂ 32 = 5; since 27 = 3³, log₃ 27 = 3. Their sum is 8.

Question 24Mathematics

If log₁₀ 2 = 0.3010, then the number of digits in 2³⁰ is

  1. 9
  2. 30
  3. 11
  4. 10 Correct answer.

Explanation

log₁₀(2³⁰) = 30 × 0.3010 = 9.03. A number with characteristic 9 has 9 + 1 = 10 digits.

Question 25Mathematics

If α and β are the roots of 2x² − 5x + 3 = 0, then the value of α² + β² is

  1. 13/4 Correct answer.
  2. 5/2
  3. 25/4
  4. 1/4

Explanation

Here α + β = 5/2 and αβ = 3/2. Then α² + β² = (α + β)² − 2αβ = 25/4 − 3 = 13/4.

Question 26Mathematics

If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then the number of elements in A ∩ B is

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

Explanation

The intersection A ∩ B collects elements present in both sets, namely 3 and 4. Counting them gives exactly 2 elements, while the union A ∪ B would have 6 elements.

Question 27Mathematics

The 10th term of the arithmetic progression 3, 7, 11, 15, … is

  1. 37
  2. 39 Correct answer.
  3. 41
  4. 43

Explanation

Here the first term a = 3 and the common difference d = 4. The nth term is a + (n − 1)d, so the 10th term is 3 + 9 × 4 = 39.

Question 28Mathematics

The sum of the first 20 terms of the arithmetic progression 5, 9, 13, … is

  1. 800
  2. 820
  3. 840 Correct answer.
  4. 860

Explanation

With a = 5, d = 4 and n = 20, the sum is (n/2)[2a + (n − 1)d] = 10 × (10 + 76) = 10 × 84 = 840. The 20th term is 5 + 19 × 4 = 81, and the average of first and last terms times 20 also gives 840.

Question 29Mathematics

The 4th term of the geometric progression 2, 6, 18, … is

  1. 72
  2. 36
  3. 54 Correct answer.
  4. 48

Explanation

The first term is 2 and each term is multiplied by the common ratio 3. Continuing the pattern gives 2, 6, 18, 54, so the 4th term is 54, which also equals 2 × 3³.

Question 30Mathematics

The point that divides the line segment joining (1, 2) and (7, 8) internally in the ratio 1 : 2 is

  1. (5, 6)
  2. (3, 4) Correct answer.
  3. (2, 3)
  4. (4, 5)

Explanation

Using the section formula with ratio m : n = 1 : 2, the x-coordinate is (1×7 + 2×1)/3 = 3 and the y-coordinate is (1×8 + 2×2)/3 = 4. So the point is (3, 4).

Question 31Mathematics

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

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

Explanation

The two legs along the axes have lengths 4 and 3, and they meet at a right angle at the origin. The area of a right triangle is half the product of the legs, so it is ½ × 4 × 3 = 6 square units.

Question 32Mathematics

If the line 3x + 4y = 12 meets the x-axis at P and the y-axis at Q, then the length PQ is

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

Explanation

Setting y = 0 gives P = (4, 0), and setting x = 0 gives Q = (0, 3). The distance PQ is √(16 + 9) = √25 = 5.

Question 33Mathematics

If the roots of x² − 6x + k = 0 are equal, then the value of k is

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

Explanation

Equal roots require the discriminant b² − 4ac to vanish. Here 36 − 4k = 0, giving k = 9, and the equation then becomes (x − 3)² = 0.

Question 34Mathematics

The quadratic polynomial whose sum of zeroes is −3 and product of zeroes is −10 is

  1. x² − 3x + 10
  2. x² − 3x − 10
  3. x² + 3x + 10
  4. x² + 3x − 10 Correct answer.

Explanation

A quadratic with sum S and product P of zeroes is x² − Sx + P. Substituting S = −3 and P = −10 gives x² + 3x − 10, whose zeroes are 2 and −5.

Question 35Mathematics

If the polynomial p(x) = x³ − 4x² + x + 6 is divided by (x − 2), the remainder is

  1. 2
  2. −4
  3. 6
  4. 0 Correct answer.

Explanation

By the remainder theorem, the remainder on dividing by x − 2 is p(2). Computing 8 − 16 + 2 + 6 gives 0, so x − 2 is a factor of the polynomial.

Question 36Mathematics

The value of (1 + tan² θ) cos² θ is

  1. sec² θ
  2. 1 Correct answer.
  3. sin² θ
  4. cos² θ

Explanation

Since 1 + tan² θ = sec² θ, the expression becomes sec² θ × cos² θ. Because sec θ = 1/cos θ, this product simplifies to exactly 1 for every θ where the functions are defined.

Question 37Mathematics

If cosec θ − cot θ = 1/2, then cosec θ + cot θ equals

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

Explanation

The identity cosec² θ − cot² θ = 1 factors as (cosec θ − cot θ)(cosec θ + cot θ) = 1. Dividing 1 by the given value 1/2 gives cosec θ + cot θ = 2.

Question 38Mathematics

The angle of elevation of the top of a tower from a point 30 m away from its foot on level ground is 45°. The height of the tower is

  1. 15 m
  2. 60 m
  3. 30√3 m
  4. 30 m Correct answer.

Explanation

In the right triangle formed, tan 45° equals the height divided by the horizontal distance. Since tan 45° = 1, the height equals the distance, namely 30 m.

Question 39Mathematics

The value of cos 0° + sin 90° + tan 45° is

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

Explanation

The standard values are cos 0° = 1, sin 90° = 1 and tan 45° = 1. Adding these three values gives 3.

Question 40Mathematics

A sector of a circle of radius 7 cm has a central angle of 90°. Its area, in cm², is (take π = 22/7)

  1. 154
  2. 24.5
  3. 77
  4. 38.5 Correct answer.

Explanation

The sector area is (θ/360) × πr², so it is (90/360) × (22/7) × 49 = ¼ × 154 = 38.5 cm². This is one quarter of the full circle area.

Question 41Mathematics

The curved surface area of a cylinder of radius 7 cm and height 10 cm, in cm², is (take π = 22/7)

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

Explanation

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

Question 42Mathematics

The volume of a hemisphere of radius 3 cm, in cm³, is (take π = 22/7)

  1. 396/7 Correct answer.
  2. 1188/7
  3. 132/7
  4. 198/7

Explanation

The volume of a sphere is (4/3)πr³, so a hemisphere has volume (2/3)πr³. With r = 3 this is (2/3) × (22/7) × 27 = 396/7 cm³.

Question 43Mathematics

If the mean of 5 observations is 12 and four of them are 8, 10, 14 and 16, then the fifth observation is

  1. 12 Correct answer.
  2. 10
  3. 18
  4. 14

Explanation

The total of the five observations is 5 × 12 = 60. The four known observations add to 48, so the fifth is 60 − 48 = 12.

Question 44Mathematics

The mode of the data 3, 5, 3, 7, 9, 3, 5 is

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

Explanation

The mode is the observation occurring most frequently. The value 3 appears three times, while 5 appears twice and 7 and 9 appear once each, so the mode is 3.

Question 45Mathematics

A card is drawn at random from a well-shuffled pack of 52 playing cards. The probability that it is a face card is

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

Explanation

There are 3 face cards (jack, queen, king) in each of the 4 suits, giving 12 face cards in total. The probability is therefore 12/52, which simplifies to 3/13.

Question 46Mathematics

Two coins are tossed together. The probability of getting at least one head is

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

Explanation

The sample space is HH, HT, TH, TT, and three of these four outcomes contain at least one head. Hence the probability is 3/4, which is also 1 minus the probability 1/4 of getting no head.

Question 47Mathematics

The value of ³√(0.027) is

  1. 0.3 Correct answer.
  2. 0.03
  3. 3
  4. 0.9

Explanation

Write 0.027 as 27/1000, whose cube root is 3/10 because 3³ = 27 and 10³ = 1000. Hence the cube root is 0.3.

Question 48Mathematics

If 2^(x+1) = 8^(x−1), then the value of x is

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

Explanation

Since 8 = 2³, the right side equals 2^(3x−3). Equating the exponents gives x + 1 = 3x − 3, so 2x = 4 and x = 2.

Question 49Mathematics

The sum of the first 10 odd natural numbers is

  1. 55
  2. 90
  3. 110
  4. 100 Correct answer.

Explanation

The sum of the first n odd natural numbers is n². For n = 10 this gives 100, which can be checked by adding 1 + 3 + 5 + … + 19.

Question 50Mathematics

If the sum of the roots of x² − (k + 2)x + 8 = 0 is equal to their product, then k equals

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

Explanation

The sum of the roots is k + 2 and the product is 8. Setting them equal gives k + 2 = 8, so k = 6.