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CTET Maths MCQ with Answers, Paper 2 (Classes VI-VIII)
234 practice questions on Mathematics, Classes VI-VIII (Paper II) for CTET Paper II (Mathematics and Science), with answers; most also have an explanation or a worked solution. Try each one before you open its answer.
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Q.1
At simple interest, a sum of ₹6,500 amounts to ₹8,190 in 4 years. What is the rate of interest per annum? (A) 5.2% (B) 6.5% (C) 13% (D) 26%
Show answer
Answer: (B) 6.5%. SI = 8190 − 6500 = ₹1,690. Rate = (1690 × 100) ÷ (6500 × 4) = 6.5% per annum.
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Q.2
At noon the temperature at a hill station is 6 °C. It falls steadily by 2 °C every hour. At what time will the temperature be −8 °C?
- A 3 p.m.
- B 4 p.m.
- C 1 p.m.
- D 7 p.m.
Show answer
Correct answer: (D) 7 p.m.
The fall needed is 6 − (−8) = 14 °C, which takes 14 ÷ 2 = 7 hours, so the temperature is −8 °C at 7 p.m. It reaches 0 °C at 3 p.m.; working with 8 − 6 = 2 °C or 8 ÷ 2 = 4 hours ignores the change of sign.
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Q.3
Ramesh bought an old cycle for ₹2,200 and spent ₹300 on its repair. He then sold it for ₹2,825. What is his gain per cent?
- A 11.5%
- B 13%
- C 14.8%
- D 28.4%
Show answer
Correct answer: (B) 13%
The repair cost is part of the cost price: CP = 2200 + 300 = ₹2,500. Gain = 2825 − 2500 = ₹325, and gain % = (325 ÷ 2500) × 100 = 13%.
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Q.4
The sum of the interior angles of a regular polygon is 1440°. What is the measure of each interior angle?
- A 135°
- B 140°
- C 144°
- D 160°
Show answer
Correct answer: (C) 144°
(n − 2) × 180° = 1440° gives n − 2 = 8, so n = 10. Each interior angle = 1440° ÷ 10 = 144° (check: each exterior angle is 36° and 10 × 36° = 360°).
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Q.5
Which is the expanded form of 7,05,402 using powers of 10?
- A 7 × 10⁶ + 5 × 10⁴ + 4 × 10³ + 2 × 10¹
- B 7 × 10⁵ + 5 × 10⁴ + 4 × 10² + 2 × 10⁰
- C 7 × 10⁵ + 5 × 10³ + 4 × 10² + 2 × 10⁰
- D 7 × 10⁵ + 5 × 10³ + 4 × 10¹ + 2 × 10⁰
Show answer
Correct answer: (C) 7 × 10⁵ + 5 × 10³ + 4 × 10² + 2 × 10⁰
7,05,402 = 7,00,000 + 5,000 + 400 + 2 = 7 × 10⁵ + 5 × 10³ + 4 × 10² + 2 × 10⁰, since 10⁰ = 1. The zeros in the ten-thousands and tens places add nothing.
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Q.6
A shopkeeper sold shoes of these sizes in a day: 6, 7, 8, 7, 9, 6, 7, 10, 8, 7, 6. Which size should he keep in the largest number?
- A Size 6
- B Size 7
- C Size 8
- D Size 10
Show answer
Correct answer: (B) Size 7
Size 6 was sold 3 times, size 7 four times, size 8 twice, and sizes 9 and 10 once each. The mode, size 7, is in the greatest demand.
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Q.7
Which of the following fractions is NOT equivalent to 15/35?
- A 45/105
- B 6/14
- C 21/49
- D 9/28
Show answer
Correct answer: (D) 9/28
15/35 in its lowest terms is 3/7. 6/14, 21/49 and 45/105 all reduce to 3/7, but 9/28 does not (3/7 = 12/28).
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Q.8
A right-angled triangle has sides 9 cm, 12 cm and 15 cm. What is the length of the perpendicular drawn from the right-angle vertex to the hypotenuse?
- A 6 cm
- B 7.2 cm
- C 7.5 cm
- D 10.8 cm
Show answer
Correct answer: (B) 7.2 cm
Area = 1/2 × 9 × 12 = 54 cm². Taking the hypotenuse as base, 1/2 × 15 × h = 54, so h = 108 ÷ 15 = 7.2 cm.
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Q.9
In triangle PQR, angle Q = 90° and angle P is 26° more than angle R. What is angle P?
- A 32°
- B 58°
- C 64°
- D 77°
Show answer
Correct answer: (B) 58°
Angles P and R add up to 180° − 90° = 90°. With P − R = 26°, we get 2P = 116°, so P = 58° (and R = 32°).
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Q.10
6 identical pipes can fill a tank in 1 hour 20 minutes. How long will 5 such pipes take to fill the same tank?
- A 1 hour 7 minutes
- B 1 hour 26 minutes
- C 1 hour 36 minutes
- D 1 hour 44 minutes
Show answer
Correct answer: (C) 1 hour 36 minutes
1 hour 20 minutes = 80 minutes. Pipes × time stays the same: 6 × 80 = 480, so 5 pipes take 480 ÷ 5 = 96 minutes = 1 hour 36 minutes.
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Q.11
Which of the following is a reflex angle?
- A 90°
- B 179°
- C 180°
- D 181°
Show answer
Correct answer: (D) 181°
A reflex angle is greater than 180° and less than 360°. 90° is a right angle, 179° is obtuse and 180° is a straight angle, so only 181° is reflex.
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Q.12
What is the value of (−2/3)⁻³?
- A −2/9
- B −8/27
- C −9/2
- D −27/8
Show answer
Correct answer: (D) −27/8
A negative exponent means taking the reciprocal: (−2/3)⁻³ = (−3/2)³ = −27/8, which is negative because the power is odd. Forgetting to take the reciprocal gives −8/27.
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Q.13
Neha has only ₹50 and ₹20 notes, 30 notes in all, with a total value of ₹1,170. How many ₹50 notes does she have?
- A 11
- B 15
- C 19
- D 23
Show answer
Correct answer: (C) 19
If there are x notes of ₹50, then 50x + 20(30 − x) = 1170, so 30x + 600 = 1170 and x = 19. Check: 19 × 50 + 11 × 20 = 950 + 220 = 1170.
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Q.14
Using each of the digits 3, 0, 8 and 5 exactly once, what is the difference between the greatest and the smallest 4-digit numbers that can be formed?
- A 5,445
- B 8,172
- C 5,472
- D 5,572
Show answer
Correct answer: (C) 5,472
Greatest = 8,530 and smallest = 3,058, because 0 cannot be the first digit of a 4-digit number. 8,530 − 3,058 = 5,472; using 0358 as the smallest wrongly gives 8,172.
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Q.15
Find the simple interest on ₹8,000 at 7.5% per annum for 3 years.
- A ₹180
- B ₹600
- C ₹9,800
- D ₹1,800
Show answer
Correct answer: (D) ₹1,800
SI = P × R × T ÷ 100 = 8000 × 7.5 × 3 ÷ 100 = ₹1,800. (₹600 is the interest for one year only, and ₹9,800 is the amount.)
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Q.16
Consider the following statements. 1. A triangle can have two obtuse angles. 2. Each angle of an equilateral triangle is 60°. 3. An exterior angle of a triangle is greater than each of its interior opposite angles. Which of the statements are correct?
- A 2 and 3 only
- B 1 and 2 only
- C 1 and 3 only
- D 1, 2 and 3
Show answer
Correct answer: (A) 2 and 3 only
Two obtuse angles alone would add up to more than 180°, so 1 is false. Statement 2 is true, and 3 is true because the exterior angle equals the sum of the two interior opposite angles.
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Q.17
Which of the following Roman numerals is NOT written according to the rules?
- A CDXC
- B XCIX
- C XLIV
- D VX
Show answer
Correct answer: (D) VX
V, L and D are never written to the left of a greater symbol to subtract, so VX is wrong (5 is simply V). XCIX = 99, XLIV = 44 and CDXC = 490 are all correct.
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Q.18
18 workers can build a wall in 25 days. Working at the same rate, how many days will 30 workers take to build the same wall?
- A 13 days
- B 15 days
- C 8⅓ days
- D 41⅔ days
Show answer
Correct answer: (B) 15 days
More workers take fewer days (inverse proportion), so workers × days stays the same: 18 × 25 = 450, and 450 ÷ 30 = 15 days. Cutting 25 days by two-thirds, because the workers increased by two-thirds, gives the wrong 8⅓.
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Q.19
The cube of a natural number n can be written as the sum of n consecutive odd numbers. Which of the following correctly shows 5³ in this way?
- A 21 + 23 + 25 + 27 + 29
- B 19 + 21 + 23 + 25 + 27
- C 23 + 25 + 27 + 29 + 31
- D 1 + 3 + 5 + 7 + 9
Show answer
Correct answer: (A) 21 + 23 + 25 + 27 + 29
21 + 23 + 25 + 27 + 29 = 125 = 5³. The second sum is 115 and the third is 135, while 1 + 3 + 5 + 7 + 9 = 25 = 5², not 5³.
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Q.20
Using prime factorisation, find the square root of 6,084.
- A 88
- B 72
- C 78
- D 68
Show answer
Correct answer: (C) 78
6,084 = 2 × 2 × 3 × 3 × 13 × 13 = (2 × 3 × 13)², so √6,084 = 2 × 3 × 13 = 78. Check: 78² = 6,084.
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Q.21
In a school of 720 students, 200 come by bus, 220 walk and 300 come by bicycle. In a pie chart of this data, what is the central angle of the sector for the students who walk?
- A 30.6°
- B 55°
- C 110°
- D 220°
Show answer
Correct answer: (C) 110°
Central angle = (220/720) × 360° = 110°. Check: the bus and bicycle sectors are 100° and 150°, and 100° + 110° + 150° = 360°.
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Q.22
Shabnam bought 3 kg of rice and 2 kg of dal for ₹438. One kg of dal costs ₹34 more than one kg of rice. What is the price of 1 kg of rice?
- A ₹74
- B ₹80.80
- C ₹87.60
- D ₹108
Show answer
Correct answer: (A) ₹74
If rice costs ₹x per kg, then 3x + 2(x + 34) = 438, so 5x + 68 = 438 and x = 74. Check: 3 × 74 + 2 × 108 = 222 + 216 = 438.
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Q.23
A family spends ₹ 36,000 a month. In a pie chart of its expenses, the sector for rent has a central angle of 70°. How much does the family spend on rent?
- A ₹ 3,500
- B ₹ 7,000
- C ₹ 7,200
- D ₹ 25,200
Show answer
Correct answer: (B) ₹ 7,000
Rent is 70/360 of the total, so rent = 70/360 × ₹ 36,000 = ₹ 7,000. (₹ 25,200 wrongly treats 70° as 70%.)
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Q.24
Consider the following statements:
1. The ratio 3 : 4 is the same as the ratio 4 : 3.
2. A ratio compares two quantities of the same kind and has no unit.
3. 15 : 20 and 9 : 12 are equivalent ratios.
Which of these are correct?
- A 1 and 2 only
- B 2 and 3 only
- C 1 and 3 only
- D 1, 2 and 3
Show answer
Correct answer: (B) 2 and 3 only
Order matters in a ratio, so 3 : 4 is not the same as 4 : 3 and statement 1 is wrong. Statement 2 is correct, and statement 3 is correct because both ratios reduce to 3 : 4.
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Q.25
Consider the following statements about the equation 2x + 5 = 11:
1. x = 3 is its solution.
2. Transposing 5 to the other side gives 2x = 16.
Which of these is/are correct?
- A Only 1
- B Only 2
- C Both 1 and 2
- D Neither 1 nor 2
Show answer
Correct answer: (A) Only 1
2 × 3 + 5 = 11, so x = 3 is the solution. When 5 is transposed to the other side its sign changes, so 2x = 11 − 5 = 6, not 16; statement 2 is wrong.
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Q.26
Which of the following arranges 5/8, 7/12, 2/3 and 11/18 in ascending order?
- A 7/12, 11/18, 5/8, 2/3
- B 7/12, 5/8, 11/18, 2/3
- C 2/3, 5/8, 11/18, 7/12
- D 11/18, 7/12, 5/8, 2/3
Show answer
Correct answer: (A) 7/12, 11/18, 5/8, 2/3
With the common denominator 72: 5/8 = 45/72, 7/12 = 42/72, 2/3 = 48/72 and 11/18 = 44/72. The numerators in order are 42, 44, 45, 48, so 7/12 < 11/18 < 5/8 < 2/3.
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Q.27
The average (mean) weight of 5 children is 28 kg. When one child leaves the group, the mean weight of the remaining 4 becomes 25 kg. What is the weight of the child who left?
- A 3 kg
- B 25 kg
- C 28 kg
- D 40 kg
Show answer
Correct answer: (D) 40 kg
Total weight of the 5 children = 5 × 28 = 140 kg, and of the remaining 4 = 4 × 25 = 100 kg. The child who left weighs 140 − 100 = 40 kg.
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Q.28
In the expression 6x²y − 23xy + 4, the numerical coefficient of the xy term is:
- A −23xy
- B −23
- C 6
- D −23y
Show answer
Correct answer: (B) −23
The xy term is −23xy; its numerical coefficient is the number multiplying xy, together with its sign, which is −23.
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Q.29
Which of the following numbers is divisible by 8?
- A 3,12,612
- B 5,94,236
- C 7,31,416
- D 8,60,524
Show answer
Correct answer: (C) 7,31,416
A number is divisible by 8 if the number formed by its last three digits is divisible by 8. 416 = 8 × 52, but 236, 612 and 524 leave remainder 4 when divided by 8 (they are divisible by 4, not by 8).
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Q.30
Seven negative integers and four positive integers, none of them zero, are multiplied together. The product is:
- A Always positive
- B Always negative
- C Always zero
- D Positive or negative, depending on the sizes of the numbers
Show answer
Correct answer: (B) Always negative
The sign depends only on how many factors are negative. Seven is odd, so the product of the seven negative integers is negative, and multiplying by positive integers does not change the sign.
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Q.31
A fair die is thrown once. What is the probability of getting a number that is a factor of 6?
- A 1/6
- B 1/3
- C 1/2
- D 2/3
Show answer
Correct answer: (D) 2/3
The factors of 6 on a die are 1, 2, 3 and 6, which is 4 of the 6 equally likely outcomes. Probability = 4/6 = 2/3.
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Q.32
The product of 3x²y and −4xy³ is:
- A −12x³y⁴
- B −12x²y³
- C −x³y⁴
- D −7x³y⁴
Show answer
Correct answer: (A) −12x³y⁴
Multiply the coefficients and add the powers of the same variable: 3 × (−4) = −12, x² × x = x³ and y × y³ = y⁴, giving −12x³y⁴.
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Q.33
The number 1,23,45x, where x is its units digit, is divisible by 6. Which values can x take?
- A 3 or 9
- B 0 or 6
- C 6 only
- D 0, 3, 6 or 9
Show answer
Correct answer: (B) 0 or 6
To be divisible by 6, the number must be divisible by both 2 and 3. The digit sum 15 + x is a multiple of 3 for x = 0, 3, 6 or 9, and x must also be even, which leaves x = 0 or 6.
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Q.34
Triangles are made in a row with matchsticks so that each new triangle shares one side with the one before it. 1 triangle needs 3 matchsticks, 2 triangles need 5 and 3 triangles need 7. How many matchsticks are needed for 25 triangles?
- A 75
- B 50
- C 53
- D 51
Show answer
Correct answer: (D) 51
The first triangle needs 3 sticks and each new triangle adds 2, so n triangles need 2n + 1 sticks; for 25 triangles, 2 × 25 + 1 = 51.
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Q.35
At point O on a line AB, a student constructs a 90° angle and a 120° angle, both measured from ray OB and on the same side of AB. She then bisects the angle between the two new arms. What angle does this bisector make with ray OB?
- A 15°
- B 75°
- C 105°
- D 210°
Show answer
Correct answer: (C) 105°
The two arms are at 90° and 120° from OB, so the angle between them is 30°. Its bisector lies 15° beyond the 90° arm, that is at 90° + 15° = 105° from OB (and 75° from OA).
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Q.36
Which of the following rational numbers lies between −1/3 and −1/4?
- A −7/24
- B −5/12
- C −1/5
- D −3/8
Show answer
Correct answer: (A) −7/24
With denominator 24: −1/3 = −8/24 and −1/4 = −6/24, so −7/24 lies between them. −5/12 = −10/24 and −3/8 = −9/24 are less than −1/3, while −1/5 = −0.2 is greater than −1/4.
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Q.37
Which of the following sets of faces (F), vertices (V) and edges (E) CANNOT belong to any polyhedron, because it does not satisfy Euler's formula?
- A F = 6, V = 6, E = 9
- B F = 6, V = 8, E = 12
- C F = 5, V = 5, E = 8
- D F = 7, V = 10, E = 15
Show answer
Correct answer: (A) F = 6, V = 6, E = 9
Euler's formula needs F + V − E = 2. The other sets give 6 + 8 − 12 = 2 (cube), 5 + 5 − 8 = 2 (square pyramid) and 7 + 10 − 15 = 2 (pentagonal prism), but 6 + 6 − 9 = 3.
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Q.38
Which list contains exactly the prime numbers between 60 and 90, with none missing and none extra?
- A 61, 67, 71, 77, 79, 83, 89
- B 61, 67, 71, 73, 79, 83, 87, 89
- C 61, 67, 71, 73, 79, 83, 89
- D 61, 67, 69, 71, 73, 79, 83, 89
Show answer
Correct answer: (C) 61, 67, 71, 73, 79, 83, 89
The primes between 60 and 90 are 61, 67, 71, 73, 79, 83 and 89. 87 = 3 × 29, 77 = 7 × 11 and 69 = 3 × 23 are composite, and the third list also leaves out 73.
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Q.39
Which of the following statements about whole numbers is FALSE?
- A Whole numbers are closed under subtraction
- B 0 is the additive identity for whole numbers
- C 1 is the multiplicative identity for whole numbers
- D Division of a whole number by 0 is not defined
Show answer
Correct answer: (A) Whole numbers are closed under subtraction
3 − 5 is not a whole number, so whole numbers are not closed under subtraction. The other three statements are true: a + 0 = a, a × 1 = a, and a ÷ 0 has no meaning.
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Q.40
If each edge of a cube is doubled, its volume becomes how many times the original volume?
- A 2 times
- B 4 times
- C 6 times
- D 8 times
Show answer
Correct answer: (D) 8 times
New volume = (2a)³ = 8a³, so the volume becomes 8 times. (The surface area becomes 4 times, since (2a)² = 4a².)
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Q.41
If a + b = 11 and ab = 24, then a² + b² equals:
- A 73
- B 97
- C 145
- D 169
Show answer
Correct answer: (A) 73
a² + b² = (a + b)² − 2ab = 121 − 48 = 73. Check with a = 3 and b = 8: 9 + 64 = 73.
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Q.42
A father is now three times as old as his son. Seven years ago, he was five times as old as his son. What is the father's present age?
- A 42 years
- B 14 years
- C 35 years
- D 49 years
Show answer
Correct answer: (A) 42 years
Let the son be x years old. Then 3x − 7 = 5(x − 7), so 3x − 7 = 5x − 35 and x = 14; the father is 42. Check: seven years ago they were 7 and 35, and 35 = 5 × 7.
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Q.43
A number is divisible by both 6 and 15. Which of the following is NOT necessarily true about it?
- A It is divisible by 10
- B It is divisible by 30
- C It is divisible by 90
- D It is divisible by 5
Show answer
Correct answer: (C) It is divisible by 90
Being divisible by 6 and by 15 means being divisible by their LCM, 30, and hence also by 10 and 5. It need not be divisible by 6 × 15 = 90: the number 30 itself is a counter-example.
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Q.44
Which of the following statements about a circle is NOT correct?
- A A diameter is the longest chord of the circle
- B All radii of the same circle are equal
- C A diameter divides the circle into two semicircles
- D Every chord of a circle passes through its centre
Show answer
Correct answer: (D) Every chord of a circle passes through its centre
A chord is any segment joining two points on the circle; only the chords through the centre are diameters. The other three statements are true.
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Q.45
A hostel has enough food for 90 students for 28 days. If 30 more students join, how long will the same food last?
- A 18⅔ days
- B 21 days
- C 25 days
- D 37⅓ days
Show answer
Correct answer: (B) 21 days
Students × days stays the same: 90 × 28 = 2520. With 120 students, the food lasts 2520 ÷ 120 = 21 days. Cutting 28 days by one-third, because one-third more students joined, gives the wrong 18⅔.
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Q.46
The diagonals of a rhombus are 16 cm and 30 cm long. What is the length of each side?
- A 17 cm
- B 23 cm
- C 34 cm
- D 46 cm
Show answer
Correct answer: (A) 17 cm
The diagonals bisect each other at right angles, giving right triangles with legs 8 cm and 15 cm. Side = √(8² + 15²) = √289 = 17 cm.
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Q.47
A cylindrical tin has a base radius of 7 cm and a height of 20 cm. What is its volume? (Take π = 22/7.)
- A 440 cm³
- B 880 cm³
- C 1540 cm³
- D 3080 cm³
Show answer
Correct answer: (D) 3080 cm³
Volume of a cylinder = πr²h = 22/7 × 7 × 7 × 20 = 3080 cm³. (880 is the number for the curved surface area 2πrh, in cm².)
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Q.48
Which of the following pairs of quantities are in inverse proportion?
- A Number of children and each child's share when ₹600 is shared equally
- B Weight of rice bought and its cost, at a fixed price per kg
- C Pages of a 200-page book already read and the pages still left to read
- D Hours worked and the wages earned, at a fixed rate per hour
Show answer
Correct answer: (A) Number of children and each child's share when ₹600 is shared equally
When ₹600 is shared equally, number of children × each share = 600 stays constant, so doubling the children halves each share: inverse proportion. Rice and its cost, and hours and wages, are in direct proportion. Pages read and pages left always add up to 200, so one falls as the other rises, but their product is not constant (for example, 50 × 150 is not equal to 100 × 100), so they are not in inverse proportion.
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Q.49
On a map, 2 cm represents 15 km. Two towns are 7.4 cm apart on the map. What is the actual distance between them?
- A 37 km
- B 51.8 km
- C 55.5 km
- D 111 km
Show answer
Correct answer: (C) 55.5 km
1 cm represents 15 ÷ 2 = 7.5 km, so 7.4 cm represents 7.4 × 7.5 = 55.5 km (map distance and actual distance are in direct proportion).
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Q.50
Using only a ruler and compasses (no protractor), which of the following angles CANNOT be constructed?
- A 22.5°
- B 40°
- C 52.5°
- D 67.5°
Show answer
Correct answer: (B) 40°
Starting from 60° and 90° and bisecting, we get 22.5° (half of 45°), 52.5° (half of 105° = 90° + 15°) and 67.5° (half of 135°). 40° cannot be obtained this way and needs a protractor.
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