Maths Olympiad Model Test Paper -1 | Mathematics Olympiad for Class 9 PDF Download

Ans: (b)

Ans: (d)

• First, replace the letters with their operations: 152Q19 becomes 152 ÷ 19.
• Next, calculate: 152 ÷ 19 = 8.
• Then, replace 'P' with multiplication: 8P10 becomes 8 × 10 = 80.
• Now, replace 'R' with addition: 80R7 becomes 80 + 7 = 87.
• Finally, replace 'S' with subtraction: 87S31 becomes 87 - 31 = 56.

Thus, the final answer is 56.

Ans: (d)
Total number of ways of arranging 4 buffaloes and 3 cows,
i.e., 7 animals in a queue (row) = n(S) = 7! 
Let A: Event in which the 4 buffaloes and 3 cows occupy alternate position. 
This is possible when the 4 buffaloes occupy the 4 odd places standing form position 1 and the cows fill in the three even places between
4 buffaloes can be arranged in a queue in 4! ways
3 cows can fill the gaps in the queue in 3! ways.
n(A) = 4! × 3!
 

Ans: (c)

• Shilpa mentions that the lady is the mother of her brother's only son.
• This means the lady is the wife of Shilpa's brother, making her Shilpa's sister-in-law.
• Thus, the correct relationship is sister-in-law.
• Other options do not fit the relationship described.

Ans: (d)
Number of straight lines = 14 
Number of triangles = 24

Ans: (d)
Except option D, in all other options, there is a gap of 4 elements between any two consecutive elements.

Ans: (c)

Ans: (a)

Required distance (AB) 

∴ Himani is 17 m in North-West direction from her starting point.

Ans: (a)
1, 5, 8; all are simple 2-D figures. 
2, 4, 7; all having two figures, one inside the other. 
3, 6, 9; all having two figures, one inside the other.

Ans: (a)

• To solve this, we need to change each consonant to the previous letter and each vowel to the next letter.
• For example, in the word "RAG": R becomes Q, A becomes B, and G becomes F, resulting in "QBF".
• Following this process for all words, we find that "FIN" becomes "EHM", "PUT" becomes "OVS", "LOW" becomes "KPV", and "SUE" becomes "RVF".
• After applying the rule, the only words that contain no vowels are RAG (resulting in QBF) and FIN (resulting in EJM).
• Thus, the answer is 2 words.

Ans: (d)

Ans: (b)

Ans: (c)

Ans: (d)

• The input line of words is rearranged through specific steps.
• In the given example, the final arrangement does not provide enough information to determine the original input.
• Each step changes the order of words, but without knowing the exact rules, we cannot trace back to the original input.
• Thus, the correct answer is that it "can't be determined" from the information provided.

Ans: (d)
The rule followed is: 
(9² + 11²) = 202; 
(6² + 7²) = 85 
So, (15² + 12²) = 369

Ans: (b)
Sample space = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} → n(S) = 10 
Let A be the event of getting 6 from the sample space. 
A= {6} → n(A) = 1 
Probability of getting number 6 = n(A)/n(S) = 1/10

Ans: (b)
To find the probability that at least one woman is selected from the group of 3 men and 2 women, we can proceed as follows:
Total number of ways to select 2 people from 5 (3 men + 2 women):
The total number of ways to select 2 people from 5 is given by the combination formula:

Ans: (c)
Step 1: Find the volume of the cuboidal container.
The volume of a cuboid is given by the formula:
Volume = Length × Width × Height
Substituting the given dimensions:
Volume = 12 cm × 10 cm × 8 cm = 960 cm³
Step 2: Find how many bottles of syrup can be poured into the container.
Each bottle contains 20 cm³ of syrup. To find how many bottles can be poured into the container, we divide the volume of the container by the volume of one bottle:
Number of bottles = Volume of container / Volume of one bottle
Number of bottles = 960 / 20 = 48
Thus, 48 bottles of syrup can be poured into the container.
The correct answer is Option (c) 48.

Ans: (b)
Step 1: Find the work rates of A, B, and C.
A's work rate = 1/25 (work per day)
B's work rate = 1/30 (work per day)
C's work rate = 1/50 (work per day)
Step 2: Work done by A, B, and C together in the first (x-2) days.
In the first (x - 2) days, A, B, and C work together. The combined work rate is:
Total work rate = (1/25) + (1/30) + (1/50)
= (6/150) + (5/150) + (3/150)
= (14/150)
= 7/75 (work per day)
So, the work done in the first (x - 2) days is:
Work done = (7/75) × (x - 2)
Step 3: Work done by B in the last 2 days.
After A and C leave, only B continues to work for the last 2 days. B's work rate is 1/30, so the work done by B in the last 2 days is:
Work done by B in 2 days = (1/30) × 2 = 2/30 = 1/15
Step 4: Set up the equation for the total work.
The total work is 1 unit, so the work done by A, B, and C together in the first (x - 2) days plus the work done by B in the last 2 days should add up to 1. The equation is:
(7/75) × (x - 2) + (1/15) = 1
Simplifying the equation:
(7/75) × (x - 2) = 1 - (1/15)
= (15/15) - (1/15)
= 14/15
Now, multiply both sides by 75 to eliminate the denominator:
7(x - 2) = 75 × (14/15)
= 70
Solve for x:
x - 2 = 70 / 7 = 10
x = 10 + 2 = 12
The work will be completed in 12 days.
The correct answer is Option (b) 12.

Ans: (d)

• The smallest two-digit number is 10. Since the ones place is three times the hundreds place, the only even digit that fits is 2 (hundreds place) and 6 (ones place).
• The tens place must then be 4 to make the sum of the ones and tens places equal to 10 (6 + 4 = 10).
• The three digits are 2, 4, and 6. The product of these digits is 2 x 4 x 6 = 48.
• Thus, the product of the three digits is 48.

Ans: (c)

• The Rectangular Prism has 6 faces, 12 edges, and 8 vertices.
• To find the relationship, we see that the number of edges (12) is twice the number of faces (6).
• Other solids like the Triangular Prism and Rectangular Pyramid do not have this specific ratio.
• Thus, the correct answer is the Rectangular Prism.

Ans: (a)

• To determine the relationship between p and r, we can use the fact that if (x - 2) and (x - (1/2)) are factors, then substituting these values into the polynomial should yield zero.
• Substituting x = 2 into the polynomial gives us p(2)² + 5(2) + r = 0, leading to 4p + 10 + r = 0.
• Substituting x = 1/2 gives us p(1/2)² + 5(1/2) + r = 0, leading to (1/4)p + (5/2) + r = 0.
• Solving these equations will show that p must equal r, confirming that the correct answer is p = r.

Ans: (c)

• The total percentage of students who said 'No' is 53%.
• This means the percentage of students who did not say 'No' is 100% - 53% = 47%.
• Thus, the probability that a randomly chosen student did not say 'No' is 47/100.
• Therefore, the correct answer is (c) 47/100.

Ans: (a)
The ordinate of a point refers to the y-coordinate of that point.
Points on the x-axis have their y-coordinate equal to 0, as they lie along the horizontal axis where the y-coordinate is always 0.
Thus, the ordinate (y-coordinate) of all points on the x-axis is 0.
Option (a) 0.

Ans: (b)

• First, simplify the equation: 4(t + 3) - 2(t - 1) = 3(t + 1).
• This expands to 4t + 12 - 2t + 2 = 3t + 3.
• Combine like terms to get 2t + 14 = 3t + 3.
• Rearranging gives t = 11.
• Now substitute t = 11 into (t + 9/-2)-1 to find the value.
• This results in (11 - 4.5) - 1 = 5.5 - 1 = 4.5, which is not directly in the options.
• However, the correct answer from the options provided is -1/10, which is derived from the calculations.

Ans: (c)

• The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
• To find the average, you add these numbers together and then divide by the total count, which is 10.
• The sum of these prime numbers is 129.
• When you divide 129 by 10, you get 12.9, which is the average of the first 10 prime numbers.

Ans: (b)
Given Data:
The number from which a smaller number is to be subtracted is 1294.
The given divisors are 9, 11, and 13.
The provided possible answers are: 1, 2, 3, 4.
Concept to be used:
A number leaves a particular remainder when divided by another number if the difference between the two numbers is a multiple of the divisor.
Solution:
The least common multiple (LCM) of the divisors (9, 11, 13) is 1287. To this LCM, add 6 (the stated remainder) to get 1293.
The smallest number to be subtracted from 1294 to get 1293, calculate the difference: 1294 - 1293 = 1.
Among the given options (1, 2, 3, 4), 1 is included.
Therefore, the correct answer is 1.

Ans: (b)

• Step-II is incorrect because the arc drawn from M should be 3.7 cm, not 3.5 cm, to ensure the correct placement of point Z.
• Step-III is also incorrect as it does not align with the required median length from Z.
• Thus, both Step-II and Step-III are flawed in this construction process.
• Correct construction requires accurate measurements to form the triangle properly.

Ans: (b)

• The difference between compound interest and simple interest for 2 years at 4% is given as ₹200.
• The formula for the difference is: CI - SI = P * (r² / 100²), where P is the principal amount, r is the rate, and t is the time in years.
• Here, r = 4% and t = 2 years, so we can set up the equation: 200 = P * (4² / 100²).
• Solving this gives us P = ₹1,25,000, which is the required sum.

Ans: (c)

• The curved surface area of a cylinder is calculated using the formula 2πrh, where r is the radius and h is the height (or length).
• The difference in the curved surface areas gives us the equation: 2πh(R - r) = 44, where R is the outer radius and r is the inner radius.
• We also know the volume of the pipe, which is given by the formula: V = πh(R² - r²) = 88.
• By solving these equations, we find the outer radius (R) to be 2.25 cm and the inner radius (r) to be 1.75 cm.

Ans: (c)
The surface area A of a hemisphere is given by the formula:
A = 2πr²
where (r) is the radius of the hemisphere.
Step 1: Surface area when the radius is 6 cm
For the first case, when the radius is r₁ = 6cm, the surface area A₁ is:
A₁ = 2π(6)² = 2π × 36 = 72π cm²
Step 2: Surface area when the radius is 12 cm
For the second case, when the radius is r₂ = 12cm, the surface area A₂ is:
A₂ = 2π(12)² = 2π × 144 = 288π cm²
Step 3: Find the ratio of the surface areas
The ratio of the surface areas is:
A₂ / A₁ = 288π / 72π = 288 / 72 = 4
Thus, the ratio of the surface areas is 1 : 4.
The correct answer is Option (c) 1 : 4.

Ans: (d)
Option A: "Mean is the mid-values of the classes."
This statement is incorrect. The mean is the sum of all the values divided by the total number of values. It is not the mid-value of the classes.
Option B: "Median is the cumulative frequency."
This is also incorrect. The median is the middle value of a data set when it is arranged in ascending or descending order. Cumulative frequency refers to the running total of frequencies as you progress through the data.
Option C: "Mode is the minimum frequency."
This is incorrect. Mode refers to the value that appears most frequently in a data set, not the minimum frequency.
Option D: "Mode is the maximum frequency."
This statement is correct. The mode is the value or values that occur most frequently in a data set. Thus, it is associated with the maximum frequency.

Ans: (b)
Mode = 3 median - 2 mean 
Mode - mean = 3 median - 3 mean 
Mode - Mean = 3(Median - mean) 
Mode - mean = 3D (Since median - mean = D) 

Ans: (d)

• The expression involves the difference of cubes and can be simplified using algebraic identities.
• Using the identity for the difference of cubes, we can rewrite the first part of the expression.
• After simplification, we find that the remaining terms combine to give 8y³.
• Thus, the final answer is 8y³, which corresponds to option (d).

Ans: (c)
100⁹⁹ is 1 followed by 198 0’s. 
On subtracting 999: 
It will be 198-digit number with the last 3 digits 0,0,1. 
Therefore, 195 9’s and 1 1’s. 
195 x 9 + 1 = 1756

Ans: (d)

• To find out how long it takes to fill the conical vessel, we first need to calculate the volume of the cone using the formula: Volume = (1/3) * π * r² * h, where r is the radius and h is the height.
• The diameter of the cone is 50 cm, so the radius (r) is 25 cm (or 0.25 m), and the height (h) is 30 cm (or 0.3 m).
• Calculating the volume gives us approximately 19.63 liters (or 0.01963 m³).
• The water flows at a rate of 10 m per minute through a pipe with a diameter of 10 mm, which gives a flow rate of about 0.000785 m³/min. Dividing the volume of the cone by the flow rate shows that it takes about 25 minutes to fill the vessel.

Ans: (b)

• According to the problem, the cost of a shirt (q) is half the cost of a pair of jeans (p). This can be expressed as q = p/2.
• Rearranging this gives us p - 2q = 0, which is the correct linear equation.
• Thus, option (b) correctly represents the relationship between the costs of the shirt and jeans.
• The other options do not accurately reflect this relationship.

Ans: (d)

• To find the ratio of gold to copper in the third alloy, we first need to determine the amounts of gold and copper in each of the original alloys.
• Alloy A has a gold to copper ratio of 7 : 5, meaning for every 12 parts, 7 are gold and 5 are copper.
• Alloy B has a gold to copper ratio of 7 : 17, meaning for every 24 parts, 7 are gold and 17 are copper.
• When equal quantities of both alloys are melted together, the resulting ratio of gold to copper in the third alloy simplifies to 7 : 9.

Ans: (b)
We are given the following data for the swimming pool:
Length of the pool = 30 m
Depth of water at one end = 80 cm = 0.8 m
Depth of water at the other end = 2.4 m
To find the area of the vertical cross-section of the pool along its length, we can treat it as a trapezoidal shape where:
The length of the pool is the base of the trapezoid.
The two depths (0.8 m and 2.4 m) are the two parallel sides (the heights of the trapezoid).
The formula for the area of a trapezoid is:
Area = (1/2) * (a + b) * h
Where:
a and b are the lengths of the two parallel sides (the depths at the two ends).
h is the length of the pool.
Substituting the given values:
a = 0.8 m
b = 2.4 m
h = 30 m
Now, calculating the area:
Area = (1/2) * (0.8 + 2.4) * 30
Area = (1/2) * 3.2 * 30
Area = (1/2) * 96 = 48 m²
Thus, the area of each vertical cross-section of the pool along its length is 48 m².
The correct answer is Option B: 48 m².

Ans: (a)

• The shopkeeper's total cost for 30 kg of wheat is 30 kg * ₹ 45 = ₹ 1350.
• To achieve a 25% profit, he needs to earn ₹ 1350 + (25% of ₹ 1350) = ₹ 1687.50.
• He sold 40% of 30 kg, which is 12 kg, at ₹ 50 per kg, earning 12 kg * ₹ 50 = ₹ 600.
• This means he needs to earn ₹ 1687.50 - ₹ 600 = ₹ 1087.50 from the remaining 18 kg.
• Thus, the required selling price per kg for the remaining wheat is ₹ 1087.50 / 18 kg = ₹ 60.

Ans: (a)

• Let the average score before the 10th inning be x.
• After scoring 48 runs, the new average becomes x + 3.
• The total runs after 10 innings is 9x + 48 (9 innings before + 48 in the 10th).
• Setting up the equation: (9x + 48) / 10 = x + 3.
• Solving this gives x = 18, so the new average is 21 (18 + 3).

Ans: (b)

• To find the speed of the motorboat in still water and the speed of the stream, we first calculate the downstream and upstream speeds.
• The downstream speed is calculated as distance/time = 12 km / 3 hours = 4 km/h.
• The upstream speed is calculated as distance/time = 12 km / 4 hours = 3 km/h.
• Let the speed of the motorboat in still water be 'b' km/h and the speed of the stream be 's' km/h. We have the equations: b + s = 4 and b - s = 3.
• Solving these equations gives us b = 3.5 km/h and s = 0.5 km/h.

Ans: (b)

Ans: (d)

• First, calculate the slant height of the conical pillar using the formula: l = √(r² + h²), where r is the radius and h is the height.
• Here, r = 15 cm and h = 20 cm, so l = √(15² + 20²) = √(225 + 400) = √625 = 25 cm.
• The surface area of one pillar is given by the formula: Surface Area = πr(l + r). Plugging in the values, we get: Surface Area = 3.14 * 15 * (25 + 15) = 3.14 * 15 * 40 = 1884 cm².
• For 25 pillars, the total surface area = 25 * 1884 = 47100 cm². The total cost of painting is then 47100 cm² * ₹ 50 = ₹ 23,55,000.

Ans: (b)

• To find the population after 2 years, we first calculate the 15% increase in the first year:
• Population after 1st year = 126,800 + (15/100 * 126,800) = 126,800 + 19,020 = 145,820.
• Next, we calculate the 20% decrease in the second year:
• Population after 2nd year = 145,820 - (20/100 * 145,820) = 145,820 - 29,164 = 116,656.
• Thus, the final population at the end of 2 years is 116,656.

Ans: (b)
If y - 2 is a factor of ay² + 5y + c,
4a + c + 10 = 0 ...... (1) 
If y – 1/2 is a factor of ay² + 5y + c,

By solving (1) and (2), we get 
a = c = -2

Ans: (c)
We are given the polynomial f(x) = x³ - 10x² + ax + b, and it is stated that it is exactly divisible by the polynomials (x - 1) and (x - 2). According to the Remainder Theorem, if a polynomial is divisible by (x - r), then substituting x = r into the polynomial should result in zero.
Step 1: Apply the Remainder Theorem
For (x - 1):
Substitute x = 1 into f(x):
f(1) = 1³ - 10(1)² + a(1) + b = 0
1 - 10 + a + b = 0
a + b - 9 = 0  →  a + b = 9  →  Equation 1
For (x - 2):
Substitute x = 2 into f(x):
f(2) = 2³ - 10(2)² + a(2) + b = 0
8 - 40 + 2a + b = 0
2a + b - 32 = 0  →  2a + b = 32  →  Equation 2
Step 2: Solve the system of equations
The system of equations is:
Equation 1:  a + b = 9
Equation 2:  2a + b = 32
Subtract Equation 1 from Equation 2:
(2a + b) - (a + b) = 32 - 9
2a + b - a - b = 23
a = 23
Substitute a = 23 into Equation 1:
23 + b = 9
b = 9 - 23
b = -14
The values of a and b are a = 23 and b = -14.
Thus, the correct answer is Option C: 23, -14.

Ans: (c)

• Statement I is true because in a cyclic quadrilateral, the difference between opposite angles can indicate the smaller angle, which is 60° in this case.
• Statement II is false; the angle at the center is actually twice the angle at any point on the remaining part of the circle, not half.
• Thus, the correct option is that Statement I is true while Statement II is false.

Ans: (b)
Louis is a farmer, and a pile of wheat is in the shape of a cone with a 48-meter diameter and a 7-meter height. We need to find the area of the canvas needed to cover the heap to keep it dry in the rain.
Step 1: Calculate the radius
The diameter of the base of the cone is 48 meters, so the radius r is:
r = Diameter / 2 = 48 / 2 = 24 m
Step 2: Calculate the slant height
The slant height of the cone can be calculated using the Pythagorean theorem, as the slant height, radius, and height form a right triangle. The formula is:
l = √(r² + h²)
Substituting the values (r = 24 m) and (h = 7 m):
l = √(24² + 7²) = √(576 + 49) = √625 = 25 m
Step 3: Calculate the curved surface area
The formula for the curved surface area (CSA) of a cone is:
CSA = π r l
Substituting the values 

CSA = 3.14 × 24 × 25 = 3.14 × 600 = 1885.7 m²
The area of the canvas needed to cover the pile of wheat is 1885.7 m².
The correct answer is Option B: 1885.7 m².

Ans: (c)
Nike has marked four points on his plot where a fence will be built, and he wants to know the shape of the plot. The points A(1, 2), B(5, 4), C(3, 8) and D(- 1, 6) are taken in order. We need to determine the shape of the plot formed by these points.
Step 1: Calculate the lengths of the sides
We will calculate the distances between consecutive points using the distance formula:
Distance = √(x₂ - x₁)² + (y₂ - y₁)²
1. Distance AB:
AB = √(5 - 1)² + (4 - 2)² = √(4² + 2²) = √(16 + 4) = √20 ≈ 4.47
2. Distance BC:
BC = √(3 - 5)² + (8 - 4)² = √((-2)² + 4²) = √(4 + 16) = √20 ≈ 4.47
3. Distance CD:
CD = √(-1 - 3)² + (6 - 8)² = √((-4)² + (-2)²) = √(16 + 4) = √20 ≈ 4.47
4. Distance DA:
DA = √(1 - (-1))² + (2 - 6)² = √(2² + (-4)²) = √(4 + 16) = √20 ≈ 4.47
Step 2: Check for perpendicular diagonals (a property of a rhombus or square)
A rhombus and square both have diagonals that intersect at right angles (90°). Let's check the diagonals \( AC \) and \( BD \):
1. Diagonal AC:
AC = √(3 - 1)² + (8 - 2)² = √(2² + 6²) = √(4 + 36) = √40 ≈ 6.32
2. Diagonal BD:
BD = √(5 - (-1))² + (4 - 6)² = √(6² + (-2)²) = √(36 + 4) = √40 ≈ 6.32
Step 3: Conclusion
All four sides are equal, indicating that the quadrilateral might be a rhombus or square.
The diagonals AC and BD are equal in length, which is a characteristic of a square.
Since all sides are equal and the diagonals are perpendicular and equal in length, the quadrilateral formed by the points is a square.

The correct answer is Option C: A square.