Maths Olympiad Model Test Paper -3 | Mathematics Olympiad for Class 9 PDF Download

Ans: (c)

• In the expression B + A ÷ C – D:
• B + A indicates that B is the father of A.
• A ÷ C shows that A is the mother of C.
• C – D means that C is the sister of D.
• Thus, A is the mother of C, and since C is D's sister, A is also the mother of D.

Ans: (a)

Ans: (b)

Ans: (c)

• In the first row, the numbers are 48, 12, and 253. The operations are performed from left to right.
• First, divide 48 by 12, which gives 4.
• Next, multiply 4 by 253, resulting in 1012. This is the resultant number for the first row.
• For the second row, the operations are similar. You need to apply the same rules to find the resultant number.
• After performing the calculations, the resultant number for the second row is found to be 69.

Ans: (c)

Ans: (b)

Ans: (b)

Ans: (b)
Number of married poets who are not graduates is 6.

Ans: (d)
The correct sitting arrangement is
So, P and K are the immediate neighbours of D.

Ans: (c)

Ans: (a)

• The input consists of words and numbers: jockey, firm, 36, 43, growth, chart, 22, 45.
• Following the pattern from the previous example, the machine prioritizes numbers first, then words in alphabetical order.
• In the first step, the highest number is moved to the front, followed by the next highest, and so on.
• After several steps, the final arrangement will be similar to Step V from the previous example, which is the last step for this input.

Ans: (a)
Input : jockey firm 36  43  growth chart 22 45 
Step I : 45 jockey firm 36 43 growth chart 22 
Step II : 45 jockey 43 firm 36 growth chart 22 
Step III : 45 jockey 43 growth firm 36 chart 22 
Step IV : 45 jockey 43 growth 36 firm chart 22 
Step V : 45 jockey 43 growth 36 firm 22 chart 

Ans: (c)

Ans: (b)

• To find the count of 9’s that meet the criteria, we need to look for 9’s that are not preceded by a 5 and are immediately followed by either a 2 or a 3.
• In the series, the valid 9’s are: the second 9 (followed by 3), the third 9 (followed by 3), and the fifth 9 (followed by 2).
• Thus, there are a total of three 9’s that satisfy the conditions.
• Therefore, the answer is three.

Ans: (d)
Triangles formed are: B, D, H, I, J, K, L, AB, BC, DE, EI, IJ, GH, GL, KL, BFJ, BFK, CGL, AEI, FGHJ, DEFK, DEFGH, BFJCG, AEFKB, ABEFIJ, CBFGKL, DEFGLK, .....
So, number of triangles formed is more than 26.

Ans: (c)
We have,

Ans: (c)

Ans: (d)

• The statement "Every real number is a rational number" is incorrect because real numbers include both rational numbers (like fractions) and irrational numbers (like the square root of 2 or pi).
• On the other hand, integers are indeed rational numbers, as they can be expressed as a fraction (e.g., 5 can be written as 5/1).
• Natural numbers are a subset of integers, and they are also real numbers.
• Thus, the only false statement among the options is (d).

Ans: (b)

• To find the cost price (CP) of the bike, we can use the loss percentage. If selling price (SP) is ₹ 38,000 and the loss is 20%, then:
• CP = SP / (1 - Loss%) = 38,000 / (1 - 0.20) = 38,000 / 0.80 = ₹ 47,500.
• Now, to find the selling price for a 15% gain, we calculate:
• SP = CP × (1 + Gain%) = 47,500 × (1 + 0.15) = 47,500 × 1.15 = ₹ 54,625.

Ans: (c)

• To determine the values of a and b, we need to ensure that the polynomial is divisible by x² - 1. This means that both x = 1 and x = -1 should be roots of the polynomial.
• Substituting x = 1 into the polynomial gives us the equation: 1 + a + 2 - 3 + b = 0, simplifying to a + b = 0.
• Substituting x = -1 gives us: 1 - a + 2 + 3 + b = 0, simplifying to -a + b + 6 = 0.
• Solving these two equations, we find that a = 3 and b = -3 satisfy both conditions, confirming that option (c) is correct.

Ans: (b)

Ans: (d)

Ans: (b)

• To check if a number is divisible by 3, we need to look at the sum of its digits.
• The digits of 1578n are 1, 5, 7, 8, and n. The sum of the known digits is 1 + 5 + 7 + 8 = 21.
• Now, we add n to this sum: 21 + n. For this to be divisible by 3, 21 + n must also be divisible by 3.
• Since 21 is already divisible by 3, n must also be a multiple of 3. The possible values for n from the options are 3, 6, 9.
• The greatest value among these options is 9, making it the correct answer.

Ans: (d)

• To find the total of the 7 numbers, multiply the mean by the number of items: 81 * 7 = 567.
• After removing 'X', the total of the remaining 6 numbers is 78 * 6 = 468.
• Now, to find 'X', subtract the total of the remaining numbers from the original total: 567 - 468 = 99.
• Thus, the value of 'X' is 99.

Ans: (b)

• To find the point, we start with the equation 2x + 5y = 18.
• We know that the abscissa (x) is 1/2 of the ordinate (y), so we can express this as x = (1/2)y.
• Substituting x in the equation gives us 2(1/2)y + 5y = 18, simplifying to y + 5y = 18, or 6y = 18.
• Solving for y gives y = 3, and substituting back gives x = (1/2)(3) = 3/2.
• Thus, the point is (3/2, 3), which matches option (b).

Ans: (b)

• Complementary angles are two angles that add up to 90°.
• Given that ∠A is x, we can express ∠B as 90° - x.
• This means that to find ∠B, we subtract the measure of ∠A from 90°.
• Thus, the correct equation to find ∠B is y = (90° - x).

Ans: (a)

Ans: (d)

• The volume of a sphere is given by the formula (4/3)πr³.
• The volume of a cone is calculated using the formula (1/3)πr²h.
• When the sphere is melted, its volume remains the same, so we set the volume of the sphere equal to the volume of the cone.
• By substituting the height of the cone (r) into the cone's volume formula and solving for the radius of the cone's base, we find that the radius is 2r.

Ans: (c)

• To uniquely construct a quadrilateral, you need 5 measurements. This can include the lengths of the sides and the angles.
• With 4 measurements, you might not have enough information to determine the shape and size of the quadrilateral.
• 3 measurements are insufficient as they do not provide enough data to define all four sides and angles.
• Thus, the correct answer is 5 measurements to ensure a unique construction of a quadrilateral.

Ans: (c)

• Let the length of the equal sides be 3x and the base be 2x. The perimeter is given by the equation: 3x + 3x + 2x = 32.
• Simplifying this gives 8x = 32, leading to x = 4. Thus, the equal sides are 12 cm and the base is 8 cm.
• To find the area, we can use the formula: Area = 1/2 * base * height. First, we need to find the height using the Pythagorean theorem.
• The height can be calculated as √(12² - 4²) = √(144 - 16) = √128 = 8√2. Therefore, the area is 1/2 * 8 * 8√2 = 32√2 cm².

Ans: (a)

• To find the probability of getting at least one head, we first need to determine the total outcomes from the 500 tosses.
• From the results, if we denote the number of times we got no heads as X, then the probability of getting at least one head is calculated as 1 - P(no heads).
• Assuming the number of times no heads appeared is 100, then the probability of no heads is 100/500 = 1/5.
• Thus, the probability of getting at least one head is 1 - 1/5 = 4/5, which simplifies to 98/125.

Ans: (a)

• To find the area of a rectangle, we multiply its length and breadth.
• The length is given as x² + 2xy + y², and the breadth is x² - 2xy + y².
• When we multiply these two expressions, we can use the difference of squares formula, which helps simplify the calculation.
• After performing the multiplication, we arrive at the result: x⁴ + y⁴ - 2x²y².

Ans: (b)

• The angles of a quadrilateral always add up to 360 degrees.
• Given the ratio 11:19:21:9, we can find the actual angles by calculating their total parts: 11 + 19 + 21 + 9 = 60 parts.
• Each part equals 360/60 = 6 degrees, so the angles are 66, 114, 126, and 54 degrees respectively.
• Since one pair of angles (A and D) are not equal, but the other pair (B and C) are not equal either, this indicates that ABCD is a trapezium with AD parallel to BC.

Ans: (b)

• The problem states that Karan squared a number and then subtracted 25, but he should have subtracted 25 first and then squared the result.
• Let’s denote the original number as x. According to Karan's method, he calculated x² - 25.
• The correct method would be (x - 25)². For Karan to get the same result, we set the two expressions equal: x² - 25 = (x - 25)².
• Solving this equation leads us to find that x = 13, which is the original number he worked with.

Ans: (a)

• To calculate the interest earned, we first find the amount after the first year with half-yearly compounding.
• For the first 6 months, the interest is calculated on ₹ 20,000 at 10% (half of 20%), which gives ₹ 2,000.
• After 6 months, the new principal becomes ₹ 22,000. For the next 6 months, the interest is again 10% on ₹ 22,000, resulting in ₹ 2,200.
• At the end of the first year, the total amount is ₹ 24,200. In the second year, the interest is compounded yearly on this amount at 20%, giving ₹ 4,840.
• Thus, the total interest earned over 2 years is ₹ 9,040.

Ans: (c)

• The average weight of the 4 men increases by 3 kg. This means the total weight of the 4 men increases by 4 x 3 = 12 kg.
• Initially, the total weight of the 4 men was 4 times the average. After replacing the 120 kg man, the new total weight becomes original total + 12 kg.
• If we denote the weight of the new man as x, we can set up the equation: (total weight - 120 + x) = total weight + 12.
• Solving this gives us x = 132 kg, which is the weight of the new man.

Ans: (a)

• First, calculate the total usable onions: 20% of 240 kg is 48 kg, so usable onions = 240 kg - 48 kg = 192 kg.
• Next, determine the cost price of the usable onions: The total cost is ₹ 380, so the cost price per kg = ₹ 380 / 240 kg = ₹ 1.58.
• To achieve a 25% profit, the selling price per kg must be: Cost price per kg + 25% of cost price = ₹ 1.58 + (25/100 * ₹ 1.58) = ₹ 1.58 + ₹ 0.395 = ₹ 1.975.
• Finally, to find the average selling price for the 192 kg to make a profit of 25%, the total selling price needed is: Total cost + Profit = ₹ 380 + (25% of ₹ 380) = ₹ 380 + ₹ 95 = ₹ 475. Therefore, average price per kg = ₹ 475 / 192 kg = ₹ 2.47.

Ans: (c)

• To find the height of the cone, we first calculate the volume of the original cylinder using the formula: Volume = πr²h, where r is the radius and h is the height.
• The volume of the cylinder is: Volume = π(2 m)²(8 m) = 32π m³.
• Next, we use the volume of the cone formula: Volume = (1/3)πr²h, where r is the radius of the cone and h is its height.
• Setting the volumes equal (since the lead is recast), we have: 32π = (1/3)π(1.5 m)²h. Simplifying gives h = 42.67 m.

Ans: (b)

• The area of the rectangle is given as (14x² - 11x - 15) m².
• To find the length and breadth, we need to factor this expression.
• Factoring gives us (7x + 5) and (2x - 3), which matches option (b).
• Thus, the possible expressions for the dimensions of the field are (7x + 5) m and (2x - 3) m.

Ans: (c)

• First, calculate the work done by 1 man in a day: 8 men complete the work in 20 days, so 1 man does 1/160 of the work per day.
• Next, find the work done by 1 woman: 8 women finish the work in 32 days, meaning 1 woman does 1/256 of the work per day.
• Now, calculate the combined work of 5 men and 8 women: 5 men do 5/160 = 1/32 of the work, and 8 women do 8/256 = 1/32 of the work.
• Adding these, 5 men and 8 women together complete 1/32 + 1/32 = 1/16 of the work in one day.
• Thus, they will finish the work in 16 days (since 1 divided by 1/16 equals 16).

Ans: (c)

• Let the current age of the younger person be Y years. Then, the age of the older person is Y + 10 years.
• Fifteen years ago, the younger person's age was Y - 15, and the older person's age was (Y + 10) - 15 = Y - 5.
• According to the problem, fifteen years ago, the older person was twice the age of the younger person: Y - 5 = 2(Y - 15).
• Simplifying this gives Y - 5 = 2Y - 30, leading to Y = 25. Therefore, the older person's age is 25 + 10 = 35 years.

Ans: (d)

• The student needs to achieve 55% to pass.
• Since the student failed by 5%, they actually needed 60% to pass.
• Let the maximum aggregate marks be x. Then, 60% of x = 520.
• From this, we can calculate x = 520 / 0.60 = 1040.
• Thus, the maximum aggregate marks are 1040.

Ans: (a)

• To find the number of boys, first calculate the number of girls in the class. The probability of selecting a girl is 3/8, which means that out of every 8 students, 3 are girls.
• Using the total number of students (360), the number of girls can be calculated as: (3/8) * 360 = 135.
• Now, to find the number of boys, subtract the number of girls from the total number of students: 360 - 135 = 225.
• Thus, the number of boys in the class is 225.

Ans: (b)

Ans: (c)

Ans: (c)

Ans: (b)

Ans: (b)

Ans: (a)

Ans: (c)