Maths Olympiad Model Test Paper -2 | Mathematics Olympiad for Class 9 PDF Download

Ans: (d)
After arranging the numbers in descending order, we get
Descending order

Ans: (b)
The alphabets on the opposite faces are : (A, D), (B, E) and (C, F)

Ans: (b)

Ans: (d)

• To find the code for ‘parrots’, we analyze the given codes.
• From ‘348’ (she likes apples) and ‘748’ (she likes parrots), we see that ‘8’ represents ‘likes’ and ‘3’ represents ‘she’.
• In ‘8375’ (parrots likes apples lot), since ‘8’ is already known as ‘likes’, we can deduce that ‘7’ must represent ‘parrots’.
• Thus, the code for ‘parrots’ is ‘7’.

Ans: (b)

AB = (20 – 13) m = 7 m
BM = (18 + 6) m = 24 m
Now, in ΔABM, using Pythagoras theorem,
AM² = AB² + BM²
= 7² + 24²
= 49 + 576 
= 625
⇒ AM = 25 m
So, Savita is 25 m away and in South-West direction with respect to the starting point.

Ans: (c)
Except figure in option (C), all other figures can be obtained by rotating each other.

Ans: (d)

Ans: (b)

• To solve this, we need to determine the arrangement of the individuals on the floors based on the clues provided.
• P is on the fourth floor and R is on the third floor. Since Q cannot be above or below R, Q must be on the first or second floor.
• U cannot be on the second or seventh floor, so U must be on the fifth or sixth floor. This means T is on the fourth or fifth floor.
• Since V cannot be below T, and T is below U, V must be on the seventh floor.
• Thus, the only arrangement that fits all clues is that V lives on the topmost floor (seventh floor).

Ans: (d) 

Ans: (c) 
Every element moves half and full step alternatively. Also one new element is added at right and left position of the existing element(s) alternatively.

Ans: (c)

Ans: (c)

Ans: (b)

Ans: (a)

• In the expression L £ M ★ N, L is the wife of M and M is the father of N. This means L is the mother of N.
• The symbol £ indicates that L is the wife, and ★ indicates that M is the father.
• Thus, L being the wife of M and M being the father confirms that L is indeed the mother of N.
• The other options do not correctly represent L as the mother of N based on the given symbols.

Ans: (b)

Ans: (b)

• Substituting x = k² and y = k into the equation x - 5y + 6 = 0 gives k² - 5k + 6 = 0.
• This is a quadratic equation that can be factored as (k - 2)(k - 3) = 0.
• Setting each factor to zero gives the solutions k = 2 and k = 3.
• Thus, the values of k that satisfy the equation are 2 and 3.

Ans: (b)
Let f (x) = 3z³ - kz² + 4z + 16
Then, f (z) will be divisible 

⇒ (k+ 4)(k² - 4k+ 32)=0
⇒ k + 4 = 0 = k = - 4

Ans: (c)
Let p(x) = 3y³ + 8y² + 8y + 3 + 5k
- 2 + 5k = 0 (Since, it is given that y² + y + 1 is a factor of 3y³ + 8y² + 8y + 3 + 5k, the remainder is equal to 0)  
k = 2/5

Ans: (c)

Ans: (a)
Slope of the line parallel to 7x + 10y - 9 = 0 is -7/10
Given that the x-intercept of the required line is -7
It passes through (-7, 0). Hence, the required line is
= 7x + 10y + 49 = 0

Ans: (c)

• To construct triangle △PQR, we need to ensure that the sum of the lengths of sides QR and PR is greater than the length of side PQ, which is 7 cm.
• If we take QR as 8 cm, then regardless of the length of PR (as long as it is a positive value), the sum QR + PR will always be greater than 7 cm.
• For example, if QR = 8 cm and PR = 1 cm, then QR + PR = 9 cm, which satisfies the condition.
• In contrast, lengths like 6 cm or 5 cm for QR would not satisfy the condition when added to any positive length for PR.

Ans: (c)
7x + 9y - 63 = 0

Intercepts are 9 and 7 and their product is 9 x 7 = 63.

Ans: (a)
Let the digits in the hundred's place, Ten's place and in the unit's, place be x, y, and z, respectively.
x + y + Z = 12
Given that:
100x + 10y + z + 99 = 100z + 10y + X
99x - 99z =- 99
X - Z =- 1
Given that (x + Z) x (1/3) = y
x + Z = 3y and x + y + Z = 12
⇒ x + z = 12 - y
⇒ 12 -y =3y
⇒ 12 = 4y
⇒ y = 3 
⇒ = x + Z = 3 × 3 
⇒ x + z = 9 
⇒ x - z =- 1 
⇒ 2x = 8 ⇒ x = 4 ⇒ z = 5 Product of the digits = 4 x 3 x 5 = 60

Ans: (a)
3(4a + 5b + 9c) = (36)3
⇒12a + 15b + 27c = 108 ...........(1) 

(1) - (2) ⇒ 5c = 10 ⇒ c = 2

Ans: (c)

• To factorise the expression, we start with the difference of squares: (x + y + z)² – (x – y – z)².
• This simplifies to 4y(x + z) using the identity a² - b² = (a - b)(a + b).
• Next, we combine this with the remaining terms: 4y(x + z) + 4y² - 4z².
• Factoring out 4 gives us 4[(y + z)(x + y - z)], which leads us to the final factorisation: 4(y + z)(x + y - z).

Ans: (d)

Ans: (b)
We know that they are linear angle.
So, 2x°+ 5y° = 180° 
Given: y = 24° 
So, 2x° = 180 - 5 × 24  2x° = 180 - 120 
x° = 60/2 = 30° 

Ans: (a)

• To find the median, first, arrange the numbers in ascending order: 13, 16, 17, 18, 19, 20, 24, 24, 25, 26, 28. Since there are 11 numbers, the median is the 6th number, which is 20.
• To find the mode, identify the number that appears most frequently. Here, 24 appears twice, more than any other number, making it the mode.
• Thus, the median is 20 and the mode is 24.

Ans: (b)

• To find the probability of a family having at least one pet, we first need to determine the total number of families surveyed.
• Next, we count the families that have at least one pet. This includes all families except those with zero pets.
• The probability is then calculated as the number of families with at least one pet divided by the total number of families.
• In this case, the probability comes out to be 18 out of 25 families, which simplifies to 18/25.

Ans: (a)

Ans: (c)
By basic proportionality theorem: 

Ans: (a)

Ans: (c)

Ans: (b)
In ∆ABC, AB = AC.
∠ABC = ∠ACB and
∠ACB = ∠ABC = 70° (given)
In ∆BCD, ∠BDC = 180° - (∠DBC + ∠DCB)
∠BDC = 70°

Ans: (c)

• To find the height of the hall, we first calculate the area of the floor and the roof. The area of the floor is length × width = 20 m × 16 m = 320 m². Since the roof has the same area, the total area of the floor and roof is 320 m² + 320 m² = 640 m².
• Next, we calculate the area of the four walls. The area of the walls is given by the formula: 2 × (length + width) × height. This equals 2 × (20 m + 16 m) × height = 72 m × height.
• According to the problem, the area of the floor and roof (640 m²) equals the area of the walls (72 m × height). Setting these equal gives us the equation: 640 = 72 × height.
• Solving for height, we find height = 640 / 72 = 8.89 m.

Ans: (c)
AP = AS, BS = BR, CR = CQ, and DQ = DP 
(Since Tangents drawn from an external point to a circle are equal.) 

Ans: (c)

• 12 years ago, the ratio of ages of P and Q was 3:4. This means if P was 3 parts, Q was 4 parts.
• Let’s say 12 years ago, P's age was 3x and Q's age was 4x. So, currently, P is 3x + 12 and Q is 4x + 12.
• We know that P's current age is (3 3/5) times R's age. Since R is 10 years old, P's age is (3 3/5) * 10 = 36 years.
• Now, we can set up the equation: 3x + 12 = 36, which gives us 3x = 24, so x = 8.
• Now, Q's current age is 4x + 12 = 4(8) + 12 = 32 + 12 = 44 years.

Ans: (a)

• First, calculate the total marks based on the average: 24 students * 56 average = 1344 total marks.
• Next, find the incorrect total by adding the misread marks: 44 + 45 + 61 = 150.
• Now, calculate the correct total by replacing the misread marks with the actual marks: 48 + 59 + 67 = 174.
• So, the correct total marks = 1344 - 150 + 174 = 1368.
• Finally, to find the correct average, divide the correct total by the number of students: 1368 / 24 = 57.

Ans: (c)

• To find the population after 3 years, we use the formula for exponential decay: Future Population = Present Population × (1 - Rate of Decrease)^Number of Years.
• Here, the present population is 32,000, the rate of decrease is 15% (or 0.15), and the number of years is 3.
• Calculating it: Future Population = 32,000 × (1 - 0.15)^3 = 32,000 × (0.85)^3 ≈ 19,652.
• Thus, the population after 3 years will be approximately 19,652.

Ans: (d)

• The batsman hits a boundary 5 times out of 40 balls.
• This means he does not hit a boundary 35 times (40 - 5 = 35).
• The probability of not hitting a boundary is calculated as the number of non-boundary hits divided by the total number of balls: 35/40.
• Simplifying 35/40 gives 7/8, which is the correct answer.

Ans: (b)

• To find the least number of cans, we first need to determine the greatest common divisor (GCD) of the three amounts of paint: 162, 126, and 180.
• The GCD of these numbers is 18. This means each can should hold 18 liters of paint.
• Now, we divide each amount of paint by the can size: 162/18 = 9 cans for red, 126/18 = 7 cans for blue, and 180/18 = 10 cans for yellow.
• Adding these gives us a total of 9 + 7 + 10 = 26 cans required.

Ans: (c)

• The taxi fare starts with a fixed charge of ₹ 25 for the first kilometer.
• For every additional kilometer, the charge is ₹ 12.50.
• Thus, for a total distance of p kilometers, the total fare can be expressed as 25 + 12.50(p - 1), which simplifies to 25 + 12.50p - 12.50.
• This leads to the equation 25 + 12.50p = q, which correctly represents the total fare paid.

Ans: (a)

• First, find the total parts of the ratio: 25 + 17 + 12 = 54 parts.
• Next, calculate the length of each side: - Side 1 = (25/54) * 540 = 250 m, - Side 2 = (17/54) * 540 = 170 m, - Side 3 = (12/54) * 540 = 120 m.
• Now, find the area of the triangle using Heron's formula: - Semi-perimeter (s) = 540/2 = 270 m, - Area = √[s(s-a)(s-b)(s-c)] = √[270(270-250)(270-170)(270-120)] = 2700 m².
• Finally, calculate the cost of ploughing: - Cost = Area * Rate = 2700 m² * ₹ 5 = ₹ 13,500.

Ans: (b)

• Let the cost price (CP) of the shirt be 100.
• The marked price (MP) is 20% higher, so MP = 100 + 20% of 100 = 120.
• A 20% discount on the marked price means the selling price (SP) = MP - 20% of MP = 120 - 24 = 96.
• Now, to find the loss percent, we calculate: Loss = CP - SP = 100 - 96 = 4.
• The loss percent is (Loss/CP) * 100 = (4/100) * 100 = 4%.

Ans: (a)

• To determine A's share of the profit, we first calculate the capital contribution of both A and B over the time they invested.
• A invests ₹ 6000 for 12 months, giving a total contribution of ₹ 6000 * 12 = ₹ 72,000.
• B invests ₹ 8000 for 8 months (since he joins 4 months later), resulting in ₹ 8000 * 8 = ₹ 64,000.
• The total capital contribution is ₹ 72,000 + ₹ 64,000 = ₹ 136,000.
• A's share of the profit is then calculated as (A's contribution / Total contribution) * Total profit = (₹ 72,000 / ₹ 136,000) * ₹ 34,000 = ₹ 18,000.

Ans: (a)
In △ABC, PQ = (1/2) AC, and in △ADC, SR = (1/2) AC. 
So, PQ + SR = AC 
In an isosceles trapezium, the diagonals are equal. 
So, SP + RQ = AC 
So, PQ + QR + RS + SP = 2AC = 30 cm

Ans: (c)

VXYZ is a parallelogram and AYZ is an equilateral triangle. VZ = 2YZ = 2AZ
A is the midpoint of VZ. Join AB such that  AB || VX.
Now VABX and ABYZ are two congruent rhombuses. 
∠VAB = 60°⇒ ∠XAB = 30° 
∠BAZ = 120°⇒ ∠BAY = 60° 
So, ∠XAY = 30° + 60° = 90°

Ans: (c)
Since diagonals of a parallelogram bisect each other, JKLM must be a parallelogram. 

Ans: (b)
Number of diagonals of an n-sided polygon =
21-sided figure = n = 21 
So, diagonals = 189 
23-sided figure = n = 23 
So, diagonals = 230 
Sum = 189 + 230 = 419

Ans: (b)
We know that: Exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. 
∠ CDE =∠ABE 
∠ ABE is given 100° 
So, ∠CDE = 100°