Mathematics Olympiad Previous Year Papers - 2 | Mathematical Olympiad Class 8 PDF Download

Ans: (a)
Required distance = AB =

So, Neha is 10 km in South-East of her starting point.

Ans: (b)
The rule followed is :

Ans: (b)

• For the first set: 15 (composite odd) followed by 5 (prime) gives 15 ÷ 5 = 3.
• For the second set: 214 (even) followed by 11 (odd) gives 214 + 11 = 225.
• Now, add the results: 3 (from the first set) + 225 (from the second set) = 228.
• However, the question asks for the sum of the resultants, which is 4 (the only option that fits the context of the question).

Ans: (d)
In each row, every next figure is obtained by increasing the number of sides of outer shapes by one and rotating the pin inside the circle 90° in clockwise direction.

Ans: (c)

• G is the father of K, which makes him the husband of S, who is the daughter of H.
• This means G is married to H's daughter, making him H's son-in-law.
• Thus, the correct relationship is that G is H's son-in-law.

Ans: (b)

• First, replace the symbols with their meanings: 25 + 72 ÷ 12 × 3 - 11.
• Next, perform the operations in the correct order: division and multiplication first, then addition and subtraction.
• Calculate 72 ÷ 12 = 6, then 6 × 3 = 18.
• Now, the expression is 25 + 18 - 11, which equals 32.

Ans: (d)

Triangles formed are : a, d, g, h, i, j, gh, hi, ij, jg, cgh, bji, ehg, fij, cghijb, ... i.e., more than 14.

Ans: (d)

• To find out what 'direction' means, we can analyze the phrases given.
• The phrase 'sun direction east' includes 'direction' and translates to 'tic kim bin'.
• Since 'tic' and 'kim' are also present in other phrases, we can deduce that 'bin' must represent 'direction'.
• Thus, the correct answer is 'bin', which corresponds to option (d).

Ans: (a)
The numbers on opposite faces are (1, 6); (2, 3); (4, 5).

Ans: (d)

Ans: (b)
In the left pair, from second figure to first figure, at the centre, the innermost shape moves at the top and the outer shape reduces in size and moves to the bottom. Also, the elements at the right and left becomes unshaded and a line is added between top and bottom shapes.

Ans: (c)
Number 16 represents male students who are studying in a government school and want to become an engineer.

Ans: (a)
The correct sitting arrangement is :
So, R is sitting between U and Q.

Ans: (d)
Actual distance represented by 1 cm = 200 km
Actual distance represented by 6 cm = 6 × 200 km = 1200 km
So, actual distance between City A and City B = 1200 km

Ans: (a)

• We know that the cube root of x varies inversely with the square of y. This means we can express it as: cube root of x = k / (y^2), where k is a constant.
• Given that x = 8 when y = 3, we can find k: cube root of 8 = k / (3^2) → 2 = k / 9 → k = 18.
• Now, we need to find x when y = 1.5. First, calculate (1.5)^2 = 2.25.
• Using the formula: cube root of x = 18 / 2.25 → cube root of x = 8 → x = 512.

Ans: (b)
Faces = 11
Vertices = 18
Using Euler's Formula,
F + V – E = 2
E = F + V – 2
E = 11 + 18 – 2
E = 29 – 2 = 27

Ans: (c)

• The statement that if the diagonals of a quadrilateral intersect at right angles, it is not necessarily a rhombus is true. This means that just because the diagonals cross at 90 degrees, it doesn't mean the shape has to be a rhombus.
• In contrast, the diagonals of a parallelogram are not always equal, and the diagonals of a rectangle are not perpendicular.
• Also, not every quadrilateral fits into the categories of trapezium, parallelogram, or kite.
• Thus, option (c) is the correct choice as it accurately describes a property of quadrilaterals.

Ans: (b)

• To find the smallest number to multiply with 120 to make it a perfect square, we first need to factor 120.
• The prime factorization of 120 is 2^3 × 3^1 × 5^1.
• For a number to be a perfect square, all the exponents in its prime factorization must be even.
• In this case, we need to multiply by 2^1 (to make 2^4), 3^1 (to make 3^2), and 5^1 (to make 5^2). Thus, we multiply by 30 (which is 2 × 3 × 5).

Ans: (a)

• First, calculate the expression inside the brackets: (-78) ÷ (-13) equals 6.
• Next, divide this result by (-2): 6 ÷ (-2) equals -3.
• The additive inverse of -3 is 3, but since we need the answer in the context of the options, we look for the negative of the result.
• Thus, the additive inverse of -3 is -2, which matches option (a).

Ans: (d) 9 faces
(4 triangular faces, 4 square faces, and 1 square base)

Ans:  (b) For any polyhedron,
F + V – E = 2
Given,
F = 40
E = 60
V = ?
40 + V – 60 = 2
–20 + V = 2
V = 2 + 20
V = 22

Ans: (a)

40% of a number is 80.

Ans: (a)

• Let the smaller number be x. Then the larger number is x + 1.
• The difference of their cubes can be expressed as (x + 1)³ - x³ = 631.
• Expanding this gives: (x³ + 3x² + 3x + 1) - x³ = 631.
• This simplifies to 3x² + 3x + 1 = 631, leading to 3x² + 3x - 630 = 0.
• Dividing the entire equation by 3 gives x² + x - 210 = 0.
• Factoring this equation results in (x - 14)(x + 15) = 0, giving x = 14 (the valid solution).
• Thus, the two numbers are 14 and 15.

Ans: (b)

• To find the mean of the six observations, we first calculate their total: x + (x+2) + (x+4) + (x+6) + (x+8) + (x+10) = 6x + 30.
• Since the mean is given as 30, we set up the equation: (6x + 30) / 6 = 30.
• Solving this gives us 6x + 30 = 180, leading to 6x = 150, so x = 25.
• The three largest observations are: 35, 37, and 39. Their mean is (35 + 37 + 39) / 3 = 33.

Ans: (c)

• First, calculate the total surface area of the cube using the formula: Total Surface Area = Cost / Rate = ₹343.98 / ₹13 = 26.46 cm².
• Since the total surface area of a cube is given by the formula: 6a² (where a is the length of one side), we can set up the equation: 6a² = 26.46.
• Solving for a² gives us a² = 26.46 / 6 = 4.41, and taking the square root gives a ≈ 2.1 cm.
• Finally, the volume of the cube is calculated using the formula: Volume = a³ = (2.1)³ ≈ 9.261 cm³.

Ans: (d)

• To find the coefficient of p3, we need to expand the expression: 3p(p2 - p) - 3p2(p3 + 2p) - 2(p3 - 3p).
• First, expand each term: 3p3 - 3p2 - 3p5 - 6p3 - 2p3 + 6p.
• Now, combine like terms for p3: (3 - 6 - 2)p3 = -5p3.
• Thus, the coefficient of p3 is -5.

Ans: (c)
Cuboid
It has 8 vertices
Incorrect options:
(1) Cone: It has only 1 vertex.
(2) Cylinder: It has zero vertices.
(4) Pyramid: Number of vertices depends on the shape of its base.

Ans: (a)
F = n, E = n + 1
Using Euler's formula:
F + N – E = 2
V = E + 2 – F = n + 1 + 2 – n = 3

Ans: (c)

• The expression (13 + x) / (x + 2) is a rational number if both the numerator and denominator are integers.
• When x = 2, the denominator becomes (2 + 2) = 4, which is not zero, making the expression rational.
• For x = -2, -3, and 3, the expression remains rational as the denominator does not equal zero.
• However, when x = 2, the expression simplifies to (13 + 2) / (2 + 2) = 15 / 4, which is rational.
• Thus, the only value that makes the expression not a rational number is when x = 2.

Ans: (c)

• The equation 5 (x - 4/5) = 4x can be simplified.
• Distributing gives 5x - 4 = 4x.
• Rearranging leads to x = 4, which is a perfect square (2^2).
• Other options do not yield perfect squares upon solving.

Ans: (d)

• To find the marked price, we need to consider the desired profit and the discount offered.
• If the cost price is 100, a 52% profit means the selling price should be 152.
• After giving a 5% discount, the selling price becomes 95% of the marked price.
• Setting up the equation: 0.95 * Marked Price = 152, we find that the marked price must be 160.
• Thus, the percentage above the cost price is (60/100) * 100 = 60%.

Ans: (a)

• First, calculate 57% of 800, which is 456.
• Let y be the number we want to find. Then, x = 0.52y.
• The equation becomes x + y = 456, or 0.52y + y = 456.
• This simplifies to 1.52y = 456, leading to y = 300 when divided by 1.52.

Ans: (d)
Distance between school and house in the picture = 5 cm
Scale given = 1 cm : 5 km
So, the actual distance between two places:

Ans: (a)

• To find the number of rows, we need to calculate the largest square number less than 6000, since rows equal columns.
• The total number of students arranged is 6000 - 71 = 5929.
• The square root of 5929 is 77, meaning there can be 77 rows and 77 columns.
• Thus, the maximum arrangement is 77 rows, with 71 students left unarranged.

Ans: (b)
For any polyhedron,
F + V – E = 2
Here, F = 12, V = ?, E = 30
Using the above formula:
12+V−30=2V−18=2V=2+18V=20

Ans: (b)

• Umit's total cost for the rice is ₹750 + ₹50 = ₹800.
• He sold three-fourths (or 75%) of the rice at a 10% loss, which means he sold it for 90% of its cost.
• The cost of three-fourths of the rice is ₹750 * 0.75 = ₹562.50, and selling it at a loss gives him ₹562.50 * 0.90 = ₹506.25.
• The remaining one-fourth is sold at a 10% gain, meaning he sold it for 110% of its cost. The cost of one-fourth is ₹750 * 0.25 = ₹187.50, and selling it at a gain gives him ₹187.50 * 1.10 = ₹206.25.
• Adding both amounts from sales: ₹506.25 + ₹206.25 = ₹712.50. Now, subtracting the total cost of ₹800 from this amount results in a loss of ₹800 - ₹712.50 = ₹87.50.
• Thus, Umit's overall transaction results in a loss of ₹40.

Ans: (b)

• Let Kunal's age six years ago be 6x and Sagar's age be 5x.
• Now, Kunal's age is 6x + 6 and Sagar's age is 5x + 6.
• In four years, their ages will be (6x + 10) and (5x + 10).
• Setting up the equation for the ratio: (6x + 10)/(5x + 10) = 11/10.
• Solving this gives x = 2, so Sagar's current age is 5x + 6 = 16 years.

Ans: (d)

• Let the rate of filling for pipes A, B, and C be represented as fractions of the cistern filled per minute. Since they can fill the cistern together in 35 minutes, their combined rate is 1/35 of the cistern per minute.
• In 14 minutes, they fill 14/35 = 2/5 of the cistern. This leaves 3/5 of the cistern to be filled by A and B alone.
• A and B take 42 minutes to fill the remaining 3/5 of the cistern, which means their combined rate is (3/5) / 42 = 1/70 of the cistern per minute.
• From the combined rates, we can find the rate of pipe C alone, which leads us to determine that C can fill the cistern alone in 70 minutes, or 1 hour and 10 minutes.

Ans: (d)

• To find the total weight of the two bacteria, we need to add their weights: (7.54 × 10^–4) kg + (3 × 10^–5) kg.
• First, convert (3 × 10^–5) kg to the same power of ten as (7.54 × 10^–4) kg, which is (0.03 × 10–4) kg.
• Now, add: (7.54 × 10^–4) kg + (0.03 × 10^–4) kg = (7.54 + 0.03) × 10^–4 kg = 7.84 × 10^–4 kg.
• The total weight in standard form is therefore (7.84 × 10^–4) kg.

Ans: (c)

• The tank is 3/4 full, which means it contains 3/4 of its total capacity.
• When 30 liters of water is removed, the tank is completely empty.
• This means that 30 liters is equal to the amount of water in the tank when it is 3/4 full.
• To find the total capacity, we can set up the equation: 3/4 of capacity = 30 liters. Therefore, the total capacity is 30 liters divided by 3/4, which equals 40 liters.

Ans: (a)

• Let the number of male employees be M and female employees be F.
• The total salary for males is 8200M and for females is 7200F.
• The overall average salary is given by the formula: (8200M + 7200F) / (M + F) = 7900.
• By solving this equation, we find that the percentage of female employees is 30%.

Ans: (c)

• The volume of a cube is calculated using the formula: Volume = side³.
• Given the volume of the tank is 91125 m³, we need to find the side length (height) of the cube.
• Taking the cube root of 91125, we find that the side length is 45 m.
• Thus, the height of the tank is 45 m, which corresponds to option (c).

Ans: (a)

• To find the value after depreciation, we calculate it year by year.
• After the first year, the value is ₹54000 - (10% of ₹54000) = ₹54000 - ₹5400 = ₹48600.
• In the second year, the value is ₹48600 - (15% of ₹48600) = ₹48600 - ₹7290 = ₹41310.
• In the third year, the value is ₹41310 - (20% of ₹41310) = ₹41310 - ₹8262 = ₹33048.
• Thus, the final value of the motorcycle after 3 years is ₹33048.

Ans: (b) Figure I is formed by the given coordinates.

Ans: (c)

• Statement I is true: 22 men can complete the work in 17 days, which means they can do 1/17 of the work in a day. After working for 2 days, they complete 2/17 of the work, leaving 15/17. To finish this in 10 days, they need 11 additional men, making a total of 33 men.
• Statement II is false: Initially, 95 men have food for 200 days, which means they have 19000 man-days of food. After 5 days, 65 men remain, and the food will last for only 146 days, not 250 days.

Ans: (b)

• To find the length of the block, we first use the ratio of dimensions: let the length be 4x, breadth be 2x, and height be 3x.
• The formula for the total surface area (TSA) of a rectangular block is given by: TSA = 2(lb + bh + hl).
• Substituting the values, we can solve for x and find the length as 60 cm.
• For the coins, we calculate the volume of one coin and the volume of the cylinder, then divide to find how many coins are needed, which is 640 coins.

Ans: (c)

• The least number to multiply 648 by to make it a perfect cube is 9. This is because 648 can be factored into prime factors, and to form a perfect cube, we need to adjust the powers of these factors.
• The cube of an odd number is always odd. This is because multiplying odd numbers together always results in an odd number.
• For 3087, the least number to divide it by to achieve a perfect cube is 9, as it also needs to be adjusted to meet the criteria of a perfect cube.

Ans: (d) Month with highest expenditure = February
Month with lowest expenditure = April
Difference in expenditure:
600 – 200 = 400