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SAMPAD Grade 10 Entrance Exam study materials
SAMPAD Grade 10 Entrance Exam study resources

SAMPAD Grade 10 Entrance Exam practice questions

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Question 1Mathematics

Evaluate the expression 3 + 4 × 2 − 6 ÷ 3, applying the standard order of operations.

  1. 9 Correct answer.
  2. 11
  3. 5
  4. 7

Explanation

Multiplication and division are performed before addition and subtraction, so 4 × 2 = 8 and 6 ÷ 3 = 2. The expression becomes 3 + 8 − 2, which equals 9.

Question 2Mathematics

A number n is multiplied by 5 and then 12 is subtracted from the result. Which expression represents this sequence of operations?

  1. 5n + 12
  2. 12 − 5n
  3. 5n − 12 Correct answer.
  4. 5(n − 12)

Explanation

Multiplying n by 5 gives 5n, and subtracting 12 afterwards gives 5n − 12. The subtraction applies only to the product, not to n before multiplication.

Question 3Mathematics

What is the value of x in the equation 4x − 7 = 21?

  1. 28
  2. 7 Correct answer.
  3. 14
  4. 3.5

Explanation

Add 7 to both sides to get 4x = 28, then divide both sides by 4 to obtain x = 7. Substituting 7 back gives 28 − 7 = 21, confirming the solution.

Question 4Mathematics

A straight line passes through the points (0, 3) and (4, 11). What is the slope of this line?

  1. 8
  2. 4
  3. 0.5
  4. 2 Correct answer.

Explanation

Slope equals the change in y divided by the change in x, so (11 − 3) ÷ (4 − 0) = 8 ÷ 4 = 2. The line rises 2 units for every 1 unit moved to the right.

Question 5Mathematics

Which of the following numbers is irrational?

  1. −3
  2. √2 Correct answer.
  3. 0.75
  4. √16

Explanation

An irrational number cannot be written as a ratio of two integers, and √2 has a non-terminating, non-repeating decimal expansion. The other three values are all rational.

Question 6Mathematics

A triangle has interior angles measuring 48° and 67°. What is the measure of the third interior angle?

  1. 55°
  2. 115°
  3. 65° Correct answer.
  4. 245°

Explanation

The interior angles of any triangle sum to 180°. Subtracting the two known angles gives 180° − 48° − 67° = 65°.

Question 7Mathematics

What is the mean of the data set 4, 8, 6, 10, 12?

  1. 40
  2. 10
  3. 6
  4. 8 Correct answer.

Explanation

The mean is the sum of the values divided by the number of values. The sum is 40 and there are 5 values, so 40 ÷ 5 = 8.

Question 8Mathematics

Simplify the expression (2x³)(5x⁴).

  1. 7x⁷
  2. 10x⁷ Correct answer.
  3. 10x¹²
  4. 10x²

Explanation

Multiply the coefficients: 2 × 5 = 10. When multiplying powers with the same base, add the exponents: x³ · x⁴ = x⁷, giving 10x⁷.

Question 9Mathematics

A bag contains 5 red marbles and 15 blue marbles. One marble is drawn at random. What is the probability that it is red?

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

Explanation

There are 5 red marbles out of 20 total, so the probability is 5/20, which simplifies to 1/4. Probability is favourable outcomes divided by total outcomes.

Question 10Mathematics

Factor the quadratic expression x² − 5x + 6.

  1. (x + 2)(x + 3)
  2. (x − 1)(x − 6)
  3. (x − 2)(x − 3) Correct answer.
  4. (x − 6)(x + 1)

Explanation

We need two numbers whose product is 6 and whose sum is −5. The numbers −2 and −3 satisfy both conditions, so the factorisation is (x − 2)(x − 3).

Question 11Mathematics

Solve the inequality 3x + 5 < 20 for x.

  1. x < 25/3
  2. x < 5 Correct answer.
  3. x > 25/3
  4. x > 5

Explanation

Subtract 5 from both sides to get 3x < 15, then divide by the positive number 3, which preserves the inequality direction, giving x < 5.

Question 12Mathematics

In a right triangle, the two legs have lengths 6 cm and 8 cm. What is the length of the hypotenuse?

  1. √28 cm
  2. 14 cm
  3. 10 cm Correct answer.
  4. 48 cm

Explanation

By the Pythagorean theorem, the hypotenuse squared equals 6² + 8² = 36 + 64 = 100. Taking the positive square root gives 10 cm.

Question 13Mathematics

The nth term of an arithmetic sequence is given by aₙ = 4 + 3(n − 1). What is the 10th term?

  1. 34
  2. 30
  3. 31 Correct answer.
  4. 40

Explanation

Substitute n = 10 into the formula: a₁₀ = 4 + 3(10 − 1) = 4 + 27 = 31. The first term is 4 and each step adds 3.

Question 14Mathematics

A rectangle has a perimeter of 34 cm and a length of 11 cm. What is its area?

  1. 121 cm²
  2. 44 cm²
  3. 187 cm²
  4. 66 cm² Correct answer.

Explanation

The perimeter formula gives 2(11 + w) = 34, so 11 + w = 17 and w = 6 cm. The area is length × width = 11 × 6 = 66 cm².

Question 15Mathematics

Which transformation maps the graph of y = x² onto the graph of y = (x − 3)² + 2?

  1. A shift 3 units right and 2 units down
  2. A shift 3 units right and 2 units up Correct answer.
  3. A shift 3 units left and 2 units up
  4. A shift 3 units left and 2 units down

Explanation

Replacing x with (x − 3) shifts the graph 3 units to the right, because the input must be 3 larger to produce the same output. Adding 2 outside the square shifts the graph 2 units upward.

Question 16Mathematics

Two similar triangles have corresponding sides in the ratio 3 : 5. What is the ratio of their areas?

  1. 9 : 25 Correct answer.
  2. 6 : 10
  3. 3 : 5
  4. 27 : 125

Explanation

For similar figures, the ratio of areas equals the square of the ratio of corresponding lengths. Squaring 3 : 5 gives 9 : 25.

Question 17Mathematics

What is the value of sin 30°?

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

Explanation

In a 30-60-90 right triangle the side opposite the 30° angle is half the hypotenuse, so sin 30° equals 1/2. Sine is opposite over hypotenuse.

Question 18Mathematics

A circle has a radius of 7 cm. What is its circumference, using π ≈ 22/7?

  1. 44 cm Correct answer.
  2. 154 cm
  3. 22 cm
  4. 49 cm

Explanation

Circumference equals 2πr, so 2 × (22/7) × 7 = 44 cm. The factor of 7 cancels with the denominator of the approximation for π.

Question 19Mathematics

A system of two linear equations has no solution. What must be true of the graphs of the two equations?

  1. They intersect at exactly one point
  2. They are the same line
  3. They intersect at exactly two points
  4. They are parallel lines with different y-intercepts Correct answer.

Explanation

No solution means no point satisfies both equations. Two distinct parallel lines never meet, so they share no point; identical slopes with different intercepts guarantee this.

Question 20Mathematics

The quadratic function f(x) = x² − 6x + 5 has its vertex at which point?

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

Explanation

For f(x) = ax² + bx + c, the vertex x-coordinate is −b/(2a) = 6/2 = 3. Substituting x = 3 gives 9 − 18 + 5 = −4, so the vertex is (3, −4).

Question 21Mathematics

A cone and a cylinder have the same base radius and the same height. What is the ratio of the cone's volume to the cylinder's volume?

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

Explanation

The volume of a cone is (1/3)πr²h, while the volume of a cylinder is πr²h. With equal radius and height, the cone's volume is one third of the cylinder's, giving 1 : 3.

Question 22Mathematics

If log₂(x) = 5, what is the value of x?

  1. 25
  2. 10
  3. 7
  4. 32 Correct answer.

Explanation

The equation log₂(x) = 5 means 2 raised to the power 5 equals x. Since 2⁵ = 32, we have x = 32.

Question 23Mathematics

The sum of the first n terms of an arithmetic series is Sₙ = n² + 2n. What is the 5th term of the series?

  1. 11 Correct answer.
  2. 35
  3. 13
  4. 9

Explanation

The nth term equals Sₙ − Sₙ₋₁. Here S₅ = 25 + 10 = 35 and S₄ = 16 + 8 = 24, so the 5th term is 35 − 24 = 11.

Question 24Mathematics

A polynomial P(x) is divided by (x − 2) and the remainder is 0. Which statement must be true?

  1. The degree of P(x) is exactly 2
  2. P(0) = 2
  3. P(2) = 0, so (x − 2) is a factor of P(x) Correct answer.
  4. P(−2) = 0, so (x + 2) is a factor of P(x)

Explanation

By the factor theorem, dividing P(x) by (x − 2) leaves remainder P(2). A remainder of 0 therefore means P(2) = 0 and (x − 2) is a factor.

Question 25Mathematics

For what values of k does the equation x² + kx + 9 = 0 have exactly one real solution?

  1. k = 6 or k = −6 Correct answer.
  2. k = 3 or k = −3
  3. k = 0 only
  4. k = 9 only

Explanation

A quadratic has exactly one real solution when its discriminant b² − 4ac equals zero. Here k² − 36 = 0, so k² = 36 and k = 6 or k = −6.

Question 26Mathematics

A shopkeeper buys a watch for ₹1,200 and sells it for ₹1,500. What is the profit percentage?

  1. 25% Correct answer.
  2. 30%
  3. 20%
  4. 15%

Explanation

Profit equals selling price minus cost price, so ₹1,500 − ₹1,200 = ₹300. Profit percentage is profit divided by cost price, multiplied by 100: (300/1200) × 100 = 25%. The percentage is always taken on the cost price, not the selling price.

Question 27Mathematics

What is the value of the expression 3² + 4 × 2 − 6 ÷ 3, following the standard order of operations?

  1. 21
  2. 11
  3. 15 Correct answer.
  4. 17

Explanation

Powers are evaluated first (3² = 9), then multiplication and division from left to right (4 × 2 = 8 and 6 ÷ 3 = 2). The expression becomes 9 + 8 − 2 = 15.

Question 28Mathematics

A number is increased by 20% and the result is 96. What was the original number?

  1. 76
  2. 80 Correct answer.
  3. 90
  4. 82

Explanation

If the original number is x, then 1.20x = 96. Dividing both sides by 1.20 gives x = 80. Checking: 20% of 80 is 16, and 80 + 16 = 96.

Question 29Mathematics

Which of these numbers is irrational?

  1. 0.75
  2. −3/5
  3. √7 Correct answer.
  4. √16

Explanation

An irrational number cannot be written as a fraction of two integers. Since 7 is not a perfect square, √7 has a non-terminating, non-repeating decimal expansion and is irrational. The other three values are all rational.

Question 30Mathematics

The mean of five numbers is 12. If four of the numbers are 8, 10, 14, and 15, what is the fifth number?

  1. 12
  2. 16
  3. 11
  4. 13 Correct answer.

Explanation

The total of all five numbers is 5 × 12 = 60. The four known numbers sum to 8 + 10 + 14 + 15 = 47. The fifth number is 60 − 47 = 13.

Question 31Mathematics

A right triangle has one acute angle of 35°. What is the measure of the other acute angle?

  1. 65°
  2. 145°
  3. 55° Correct answer.
  4. 45°

Explanation

The two acute angles in a right triangle must add to 90° because the third angle is 90°. Therefore the other acute angle is 90° − 35° = 55°.

Question 32Mathematics

What is the value of (2³)² ÷ 2⁴?

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

Explanation

Apply the power rule: (2³)² = 2⁶. Then divide using the quotient rule: 2⁶ ÷ 2⁴ = 2² = 4.

Question 33Mathematics

A train travels 240 km in 3 hours. At the same average speed, how far will it travel in 5 hours?

  1. 480 km
  2. 400 km Correct answer.
  3. 360 km
  4. 420 km

Explanation

The average speed is 240 ÷ 3 = 80 km/h. In 5 hours the distance is 80 × 5 = 400 km. This is a direct proportion: distance is proportional to time at constant speed.

Question 34Mathematics

Solve the simultaneous equations 2x + y = 11 and x − y = 1. What is the value of x + y?

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

Explanation

Adding the two equations eliminates y: 3x = 12, so x = 4. Substituting into x − y = 1 gives y = 3. Then x + y = 4 + 3 = 7.

Question 35Mathematics

The product of two consecutive positive integers is 156. What is the larger integer?

  1. 13 Correct answer.
  2. 14
  3. 11
  4. 12

Explanation

Let the integers be n and n + 1, so n(n + 1) = 156. This gives n² + n − 156 = 0, which factors as (n − 12)(n + 13) = 0. The positive solution is n = 12, so the larger integer is 13. Checking: 12 × 13 = 156.

Question 36Mathematics

A sum of ₹5,000 is invested at 8% simple interest per year. What is the total amount after 3 years?

  1. ₹6,400
  2. ₹5,400
  3. ₹5,800
  4. ₹6,200 Correct answer.

Explanation

Simple interest is PRT/100 = (5000 × 8 × 3)/100 = ₹1,200. The total amount is the principal plus interest: 5,000 + 1,200 = ₹6,200.

Question 37Mathematics

In a class of 40 students, 60% play cricket and 45% play football, while 10 students play both games. How many students play neither game?

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

Explanation

Cricket players number 60% of 40 = 24, and football players number 45% of 40 = 18. By inclusion–exclusion, those playing at least one game are 24 + 18 − 10 = 32. So 40 − 32 = 6 students play neither game.

Question 38Mathematics

A sector of a circle has a radius of 9 cm and a central angle of 80°. Using π ≈ 3.14, what is the area of the sector, to the nearest square centimetre?

  1. 57 cm² Correct answer.
  2. 113 cm²
  3. 28 cm²
  4. 254 cm²

Explanation

Sector area = (θ/360) × πr². Here (80/360) × 3.14 × 81 = (2/9) × 254.34 ≈ 56.5, which rounds to 57 cm².

Question 39Mathematics

The surface area of a sphere is 144π cm². What is its volume?

  1. 216π cm³
  2. 72π cm³
  3. 288π cm³ Correct answer.
  4. 144π cm³

Explanation

For a sphere, surface area = 4πr², so 4πr² = 144π gives r² = 36 and r = 6. Volume = (4/3)πr³ = (4/3)π(216) = 288π cm³.

Question 40Mathematics

If tan θ = 3/4 and θ is an acute angle, what is the value of sin θ + cos θ?

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

Explanation

With tan θ = 3/4, take the opposite side as 3 and the adjacent side as 4, giving hypotenuse √(9 + 16) = 5. Then sin θ = 3/5 and cos θ = 4/5, so sin θ + cos θ = 7/5.

Question 41Mathematics

What is the 8th term of the geometric sequence 3, 6, 12, 24, …?

  1. 384 Correct answer.
  2. 768
  3. 192
  4. 96

Explanation

The first term is 3 and the common ratio is 2. The nth term is a·r^(n−1), so the 8th term is 3 × 2⁷ = 3 × 128 = 384.

Question 42Mathematics

The median of the data set 7, 3, 9, 5, 11, 13 is which value?

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

Explanation

Arrange the values in order: 3, 5, 7, 9, 11, 13. With an even count of six values, the median is the mean of the two middle values, 7 and 9, which is 8.

Question 43Mathematics

A card is drawn at random from a standard deck of 52 playing cards. What is the probability that the card is a face card (jack, queen, or king)?

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

Explanation

There are 4 suits, each with 3 face cards, giving 12 face cards in total. The probability is 12/52, which simplifies to 3/13.

Question 44Mathematics

The point (3, −2) is reflected across the x-axis. What are the coordinates of the image?

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

Explanation

Reflection across the x-axis keeps the x-coordinate unchanged and changes the sign of the y-coordinate. Applying this to (3, −2) gives (3, 2).

Question 45Mathematics

A cylindrical water tank has a radius of 7 m and a height of 10 m. Using π ≈ 22/7, what is its capacity in cubic metres?

  1. 440 m³
  2. 770 m³
  3. 1,540 m³ Correct answer.
  4. 2,200 m³

Explanation

The volume of a cylinder is πr²h. Substituting gives (22/7) × 49 × 10 = 22 × 7 × 10 = 1,540 m³.

Question 46Mathematics

If f(x) = 2x² − 3x + 1, what is the value of f(−2)?

  1. 3
  2. −1
  3. 11
  4. 15 Correct answer.

Explanation

Substitute x = −2: 2(−2)² − 3(−2) + 1 = 2(4) + 6 + 1 = 15. The square of −2 is positive, and multiplying −3 by −2 gives a positive 6.

Question 47Mathematics

For what value of p does the system px + 2y = 5 and 4x + 8y = 10 have infinitely many solutions?

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

Explanation

Infinitely many solutions occur when the two equations are proportional. Comparing coefficients, 4/p = 8/2 = 10/5 must all be equal. Since 8/2 = 4 and 10/5 = 2, the constants rule out proportionality, so instead we need the coefficient ratio p/4 = 2/8 = 5/10, giving p = 1.

Question 48Mathematics

The polynomial x³ − 4x² + x + 6 has a factor (x − 2). What are the other two linear factors?

  1. (x − 6)(x + 1)
  2. (x + 2)(x − 3)
  3. (x + 3)(x − 1)
  4. (x − 3)(x + 1) Correct answer.

Explanation

Dividing by (x − 2) gives the quadratic x² − 2x − 3. This factors as (x − 3)(x + 1), since −3 and 1 multiply to −3 and add to −2. Expanding the three factors reproduces the original cubic.

Question 49Mathematics

In a geometric series the first term is 5 and the sum to infinity is 20. What is the common ratio?

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

Explanation

For a convergent geometric series, S∞ = a/(1 − r). Setting 5/(1 − r) = 20 gives 1 − r = 1/4, so r = 3/4. Since |r| < 1, the series does converge.

Question 50Mathematics

A vertical pole stands on level ground. From a point 12 m away from its foot, the angle of elevation of the top is 60°. What is the height of the pole?

  1. 12√3 m Correct answer.
  2. 4√3 m
  3. 24 m
  4. 12/√3 m

Explanation

The pole, the ground distance, and the line of sight form a right triangle with the 60° angle at the observation point. Using tan 60° = height/12, the height is 12 × √3 = 12√3 m.