Mathematics Olympiad Previous Year Papers - 1 | Mathematical Olympiad Class 8 PDF Download

Ans: (b)
We have,
Similarly,

Ans: (d)

Ans: (a)

Ans: (a)

Ans: (c)
Minimum number of straight lines required to draw the given figure is 12.

Ans: (a)

• First, arrange the digits of 745293 in ascending order: 2, 3, 4, 5, 7, 9.
• The second digit from the right in this arrangement is 7.
• The third digit from the left is 4.
• Now, calculate the difference: 7 - 4 = 3.
• Thus, the answer is 3.

Ans: (b)
The number of dots on opposite faces are: (1, 4), (2, 5) and (3, 6).

Ans: (b)

• First, we break down the expression ‘M + J * L – N’ step by step.
• ‘M + J’ means M is the sister of J.
• ‘J * L’ means J is the father of L.
• Now, we know M is the sister of J, and J is the father of L, which makes M the aunt of L.
• Finally, since N is the sibling of L, M is also the aunt of N.

Ans: (b)

Ans: (d)

So, Vikram is in South-East direction with respect to his home.

Ans: (d)

Ans: (b)

• To determine which operations to switch, we need to analyze the equation: 18 + 12 × 2 ÷ 5 – 20 = 28.
• By changing ÷ and ×, we can recalculate the left side: 18 + 12 ÷ 2 × 5 – 20.
• This results in 18 + 6 × 5 – 20, which simplifies to 18 + 30 – 20 = 28.
• Thus, the correct interchange of operations is ÷ and ×, making the equation true.

Ans: (b)

Ans: (a)

• To find the missing number, we first look at the relationship between 16 and 51.
• The relationship can be seen as: 16 multiplied by 3 plus 3 equals 51 (16 * 3 + 3 = 51).
• Now, applying the same logic to 18: 18 multiplied by 3 plus 3 gives us 57 (18 * 3 + 3 = 57).
• Thus, the missing number is 57, which corresponds to option (a).

Ans: (b)

• First, calculate 40% of 1640: 0.4 * 1640 = 656.
• Next, calculate 35% of 980: 0.35 * 980 = 343.
• Then, calculate 150% of 850: 1.5 * 850 = 1275.
• Now, set up the equation: 656 + X = 343 + 1275.
• Simplifying the right side gives: 656 + X = 1618.
• To find X, subtract 656 from 1618: X = 1618 - 656 = 962.

Ans: (a)

• The expression is a difference of squares, which can be factored using the formula a² - b² = (a - b)(a + b).
• Here, let a = 5(x + y) and b = 6(x - 2y). So, we rewrite the expression as (5(x + y) - 6(x - 2y))(5(x + y) + 6(x - 2y)).
• After simplifying, we find the factors to be (11x - 7y) and (17y - x).
• Thus, the correct factorization is (11x – 7y)(17y – x).

Ans: (b)
Convex polyhedron, as it is bounded by plane polygonal faces.

Ans: (d)

• First, simplify the left side: x - [3x - (2x - 0.5)] = x - [3x - 2x + 0.5] = x - [x + 0.5] = -0.5.
• Now, simplify the right side: 1/6 (2x - 57) - 5/3 = (2x - 57)/6 - 5/3 = (2x - 57)/6 - 10/6 = (2x - 67)/6.
• Set the two sides equal: -0.5 = (2x - 67)/6.
• Multiply both sides by 6: -3 = 2x - 67, leading to 2x = 64, so x = 32.

Ans: (b)

• Let the number be represented as x.
• The expression for the operation is (2/3)x * x - 61 = 9200.
• This simplifies to (2/3)x^2 - 61 = 9200.
• Adding 61 to both sides gives (2/3)x^2 = 9261.
• Multiplying both sides by (3/2) results in x^2 = 13891.5.
• Taking the square root gives x ≈ 63, which is the correct answer.

Ans: (a)

• To find the interest on `600 for 4 years at 7/2%, we use the formula: Interest = Principal × Rate × Time.
• Calculating this gives: Interest = 600 × (7/2)/100 × 4 = 84.
• Now, for `1200 over 2 years, we set up the equation: 1200 × Rate × 2 = 84.
• Solving for Rate gives us Rate = 84 / (1200 × 2) = 0.035 or 3.5%, which is equal to 7/2%.

Ans: (c)
Resolve 136 into prime factors and make  group of two of each prime factor
136 = 2 x 2 x 2 x 17 
136 = (2 × 2) × 2 × 17We find that 2 and 17 doesn't appear in group of two. So, 136 has to be multiplied with 34 to make it a perfect square.∴ 136 × 2 × 17 = (2 × 2) × (2 × 2) × (17 × 17)
⇒ 4624 = (2 × 2) × (2 × 2) × (17 × 17)
⇒ Required square root = 2 × 2 × 17 = 68

Ans: (d)

• First, calculate 3/8 - (-2/7). This is the same as 3/8 + 2/7.
• To add these fractions, find a common denominator, which is 56. Convert both fractions: 3/8 becomes 21/56 and 2/7 becomes 16/56.
• Now, add them: 21/56 + 16/56 = 37/56.
• Next, set up the equation: 37/56 + x = 5/14. Convert 5/14 to a fraction with a denominator of 56, which is 20/56.
• Now, solve for x: x = 20/56 - 37/56 = -17/56.

Ans: (c)
Distance on map for an actual distance of 4 m = 1 mm.
Distance on map for actual distance of 52 m =

Thus, distance on map for actual distance of 52 m is 13 mm.

Ans: (d)

• To find the ratio, first, add the marks Ashima scored in Mathematics for Test I and Test III.
• Next, add the marks she scored in Science for Test II and Test IV.
• Finally, divide the total marks in Mathematics by the total marks in Science to get the ratio.
• The resulting ratio simplifies to 17:14, which is the correct answer.

Ans: (a)

• When adding integers, the order does not affect the sum. This is known as the commutative property of addition.
• The statement in option (b) is incorrect because (–71 + 52) equals –19, which is indeed to the left of (71 – 52) which equals 19.
• In option (c), if p is a prime number (like 2) and q is a composite number (like 4), then pq can be even (2 * 4 = 8).
• Thus, the only correct statement is option (a).

Ans: (b)

• First, calculate x: x = (3/2)² x (2/3)⁻⁴.
• Convert (2/3)⁻⁴ to its reciprocal: (2/3)⁻⁴ = (3/2)⁴.
• Now, x becomes: x = (3/2)² x (3/2)⁴ = (3/2)^(2+4) = (3/2)⁶.
• Next, find x – 2: x – 2 = (3/2)⁶ – 2. To express 2 in terms of (3/2), we can rewrite it as (2/3)¹².
• Thus, the value of x – 2 simplifies to (2/3)¹².

Ans: (d)

• To find how much x³ + 3x²y – 3xy² – y³ is less than 4x³ – 2x²y + 5xy² – y³, we subtract the first expression from the second.
• This gives us: (4x³ – 2x²y + 5xy² – y³) - (x³ + 3x²y – 3xy² – y³).
• Simplifying this results in 3x³ - 5x²y + 8xy², which is option (d).
• Thus, the answer is 3x³ - 5x²y + 8xy², indicating how much less the first expression is compared to the second.

Ans: (d)

Ans:  (c)
A square pyramid has 5 faces, 8 edges, and 5 vertices.
So, 5+5−8=2.

Ans: (c)

• First, calculate m: m = 3√(140 x 2450). This simplifies to m = 3√(343000) which equals 70.
• Next, calculate n: n = 3√(-216). Since the cube root of a negative number is negative, n = -6.
• Now, find m + n: 70 + (-6) = 64.
• Finally, calculate 3√(64), which equals 4.

Ans: (d)

• To find the total amount after 4 years, we first determine the rate of interest using the formula for compound interest.
• From 1600 to 2000 in 2 years, we can calculate the interest rate. The formula for compound interest is A = P(1 + r)^n, where A is the amount, P is the principal, r is the rate, and n is the number of years.
• Using the values, we find that the rate of interest leads to an amount of 2500 after 4 years, as the interest compounds.
• Thus, the total amount after 4 years is 2500.

Ans: (a)

• To find the number of sides in a regular polygon, we can use the formula for the interior angle: Interior Angle = (n - 2) * 180° / n, where n is the number of sides.
• Setting the interior angle to 120°, we have: 120 = (n - 2) * 180 / n.
• By solving this equation, we find that n = 6, meaning the polygon has 6 sides.
• Thus, the correct answer is (a) 6, which corresponds to a regular hexagon.

Ans: (c)

• When a die is rolled, it has 6 faces numbered from 1 to 6.
• The numbers that are less than or equal to 4 are 1, 2, 3, and 4. This gives us 4 favorable outcomes.
• The total number of possible outcomes when rolling a die is 6.
• Thus, the probability is calculated as the number of favorable outcomes divided by the total outcomes: 4/6, which simplifies to 2/3.

Ans: (c)

• To determine if x and y vary directly, we check if the ratio of y to x is constant.
• In option (c), the ratios are: 28/7 = 4, 84/21 = 4, and 168/42 = 4, which are all equal.
• This shows that as x increases, y also increases proportionally, confirming a direct variation.
• Options (a) and (b) do not maintain a constant ratio, hence they do not show direct variation.

Ans: (b)

• The total cost of the land is 300000.
• He sells one-third (100000) at a 20% loss, which means he receives 80000 for it.
• He sells two-fifths (120000) at a 25% gain, which means he receives 150000 for it.
• Now, he has sold 220000 in total, and to achieve a 10% profit, he needs to sell the remaining land for 330000 in total.
• This means he must sell the remaining land for 110000. Since he has already sold for 220000, he needs to sell the remaining land for 100000 to reach his goal.

Ans: (b)

• To find the ratio of the size of a red blood cell to that of a plant cell, we divide the size of the red blood cell (0.000007 m) by the size of the plant cell (0.00001275 m).
• This gives us a ratio of 0.000007 m / 0.00001275 m, which simplifies to approximately 0.549.
• When we convert this decimal into a ratio, we find that it is equivalent to 28:51.
• Thus, the correct answer is (b) 28:51, which represents the size comparison between the two types of cells.

Ans: (a)

• Let the amount lent at 3% be x. Then, the amount lent at 2% is 3000 - x.
• The interest from the amount at 3% for 7/2 years is calculated as: Interest = Principal × Rate × Time = x × 3/100 × 7/2.
• The interest from the amount at 2% for 4 years is: Interest = (3000 - x) × 2/100 × 4.
• Setting up the equation: (x × 3/100 × 7/2) + ((3000 - x) × 2/100 × 4) = 280.
• Solving this equation gives x = 1600, which is the amount lent at 3% per annum.

Ans: (c)

• First, convert the speed of the train from km/h to m/s: 54 km/h = 54 * (1000/3600) = 15 m/s.
• The length of the train can be calculated using the formula: Distance = Speed × Time. Here, Distance = 15 m/s × 20 s = 300 m.
• To cross the bridge, the train needs to cover its own length plus the length of the bridge: 300 m + 900 m = 1200 m.
• Now, calculate the time to cross the bridge: Time = Distance / Speed = 1200 m / 15 m/s = 80 seconds.

Ans: (a)

• First, calculate the work done by A in 4 days. A's rate is 1/12, so in 4 days, A completes 4/12 = 1/3 of the work.
• Next, B works for 2 days. B's rate is 1/18, so in 2 days, B completes 2/18 = 1/9 of the work.
• Now, total work done by A and B is 1/3 + 1/9. To add these, find a common denominator (which is 9): 1/3 = 3/9, so 3/9 + 1/9 = 4/9.
• The remaining work is 1 - 4/9 = 5/9. C's rate is 1/9, so to find out how long C takes to finish 5/9 of the work, use the formula: time = work/rate = (5/9) / (1/9) = 5 days.

Ans: (c)

• First, calculate the volume of the reservoir: Volume = length × breadth × height = 8 m × 6 m × 3 m = 144 m³.
• Convert the volume from cubic meters to liters: 1 m³ = 1000 liters, so 144 m³ = 144,000 liters.
• Next, determine the time taken to fill the reservoir: Time = Total volume / Rate of flow = 144,000 liters / 60 liters/min = 2400 minutes.
• Convert minutes to hours: 2400 minutes ÷ 60 = 40 hours.

Ans: (c)

• The average age of the teachers is calculated based on the total age of all teachers divided by the number of teachers.
• When the 60-year-old teacher retires, the total age decreases significantly, while the 30-year-old teacher's age is much lower.
• This change results in a net decrease in the total age of the teachers, leading to a decrease in average age.
• Specifically, the average age will decrease by 1 year due to this replacement.

Ans: (c)

• Let Shyam's current age be S and Ram's current age be R.
• Four years ago, Shyam's age was S - 4 and Ram's age was R - 4. The equation is: S - 4 = (3/4)(R - 4).
• In four years, Shyam's age will be S + 4 and Ram's age will be R + 4. The equation is: S + 4 = (5/6)(R + 4).
• Solving these two equations gives S = 16, which means Shyam is currently 16 years old.

Ans: (c)

• First, calculate the height of pole B using the height of pole A: height of B = (4/3) * (97/3) = 388/9 m.
• Next, find the height of pole C: height of C = (2/3) * (388/9) = 776/27 m.
• Convert 776/27 m to a mixed number: 776 ÷ 27 = 28 with a remainder of 20, so height of C = 28 20/27 m.
• Thus, the height of pole C is 28 20/27 m.

Ans: (b)

• The playground is a parallelogram, which means opposite sides are equal and parallel.
• In a rhombus, the diagonals bisect each other at right angles and are not necessarily equal, which fits the given diagonal lengths.
• A rectangle has equal diagonals, which is not the case here.
• A kite has two pairs of adjacent sides that are equal, but does not fit the diagonal criteria.

Ans: (a)

• Even numbers: The even numbers on the cube are 2, 2, and 4. There are 3 even numbers out of 6 faces, so the probability is 3/6 = 1/2.
• Even prime numbers: The only even prime number is 2. There are 2 faces with the number 2, so the probability is 2/6 = 1/3.
• Multiples of 3: The only multiple of 3 on the cube is 3. There is 1 face with the number 3, so the probability is 1/6.

Ans: (d)

Thus, scale is 1 : 100.

Ans: (b)
∴ F = 4 and V = 3
We have Euler's formula as:
F + V = E + 2
⇒ 4 + 3 = E + 2
⇒ E = 5

Ans: (d)

• All the statements provided are true regarding the properties of shapes.
• The lengths of the diagonals in a rectangle are indeed equal.
• It is correct that a square, rectangle, and rhombus are all different types of parallelograms.
• Additionally, the diagonals of a square do intersect at right angles and are perpendicular bisectors of each other.
• Since all statements are accurate, the correct answer is All of the above.

Ans: (C)

• Statement (i): The product of (4x + y) and (6x² + 5xy - y²) results in 5 terms, which is true.
• Statement (ii): When substituting p = 1 and q = -2 into 4q² x (5pq + p² - 6q²), the value is indeed -528, making this statement true.
• Statement (iii): The multiplication of (5b + 2bc - c - 3) by 5abc does not yield the stated expression, hence this statement is false.