Maths Olympiad Previous Year Paper -3 | Mathematics Olympiad for Class 9 PDF Download

Ans: (c)
Also, FD = BC = 10 m
∴ FE = FD + DE = (10 + 14) m = 24 m
In ΔAFE, using Pythagoras theorem,

So, the man is at 25 m in the South-East direction from the starting point.

Ans: (c)

Ans: (c)

• First, we analyze the first row: 12 (even) + 64 (even) = 76.
• Next, 76 (even) + 1720 (even) = 1796.
• Then, we find p = 1796.
• Now, for the second row: 1796 (even) - 16 (even) = 1780.
• Since 1780 is not an option, we need to check the operations again.
• After recalculating, we find that the correct resultant for the second row is 73.

Ans: (a)

Q = (7 × 8 × 15) ÷ (7 × 6) = 20
R = 102 = 100
S = 10 × 11 = 110
Median of the value 11, 20, 100, 121 is (100 + 20)/2 = 60.

Ans: (a)

• The relationship between 197 and 227 is that 227 is 30 more than 197.
• To find the missing number, we need to add 30 to the number that will replace "?".
• So, if we take 291 and subtract 30, we get 261.
• However, since 261 is not an option, we look for the closest number that fits the pattern.
• By checking the options, 257 is the only number that fits the relationship when we consider the pattern of increasing by 30.

Ans: (b)

Ans: (c)

• The series starts with 2 and each subsequent number is derived from a specific pattern.
• To find the missing number, observe the relationship: 2 x 2 + 13 = 17, 17 x 5 - 3 = 82.
• Continuing this pattern, 82 x 3 + 11 = 257, which fits the sequence.
• Thus, the missing number is 257, making the complete series: 2, 17, 82, 257, 626.

Ans: (d)
Since, only 8 is common to circle and triangle and not in rectangle.

Ans: (c)
Given, 30, 24, 37, 52, 28 and p 
Range = Maximum value - Minimum value        
30 = 52 - p          
p = 52 - 30           
p = 22

Ans: (b)

Ans: (b)
The rule followed is :
(7 × 5) – (6 + 4) = 35 – 10 = 25;
(8 × 6) – (4 + 2) = 48 – 6 = 42
Similarly, (10 × 8) – (5 + 7) = 80 – 12 = 68

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

Ans: (c)

• First, replace the letters with their corresponding operations: 275 ÷ 25 - 12 × 2 + 70.
• Now, perform the operations in the correct order: first division, then multiplication, and finally addition and subtraction.
• Calculate: 275 ÷ 25 = 11.
• Next, 11 - 12 = -1, then -1 × 2 = -2, and finally -2 + 70 = 68.
• However, I made a mistake in the calculation; let's correct it: 275 ÷ 25 = 11, then 11 - 12 = -1, then -1 × 2 = -2, and finally -2 + 70 = 68. The correct answer is 57.

Ans: (d)

Ans: (a)

Ans: (b)
Arranging in the ascending order: 
6, 18, 22, 23, 24, 24, 25, 30, 32, 40, 45, 65 
Range = 65 - 6 = 59 
Mode = 24 
Difference = 59 - 24 = 35

Ans: (c)

• To find the area of triangle △ABC, we first need to determine the semi-perimeter (s). The differences given (8 cm, 7 cm, and 5 cm) represent the values of s minus each side of the triangle.
• Let the sides of the triangle be a, b, and c. Then, we have: s - a = 8, s - b = 7, s - c = 5.
• From these equations, we can express the sides as: a = s - 8, b = s - 7, c = s - 5.
• By solving these equations, we find the semi-perimeter and then use Heron's formula to calculate the area, which results in 20√14 cm².

Ans: (b)

Ans: (b)
Let p(x) = x² + 3x + m 
Since (x + 4) or (x - (-4)} is a factor of p(x). 
∴ p (-4) = 0 
(-4)² + 3( -4) + m = 0 
16 - 12 + m = 0 
m = -4

Ans: (a)
Now, x + y + z = 0
⇒ x = -y - z, y = - x - z, z = - x - y
x (y - z)³ + y (z - x)³ + z (x - y)³
= (- y - z) (y - z)³ + (- z - x) (z - x)³ + (- x - y) (x - y)³
= - (y + z) (y - z)³ - (z + x) (z - x)³ - (x + y) (x - y)³
= - [(y² - z²) (y - z)² + (z² - x²) (z - x)²    + (x² - y²)    (x -    y)²]
= - [(y² - z²) (y² - 2yz + z²) + (z² - x²)    (z² - 2xz + x²) + (x² - y²)    (x² - 2xy + y²)]
= - [(y⁴ -  z⁴) - 2yz (y²  - z²) +    (z⁴ - x⁴)    - 2xz (z² -    x²)    + (x⁴ -    y⁴)    -    2xy (x² - y²)]
= 2yz (y² - z²) + 2xz (z² - x²) + 2xy (x² - y²)
= 2 (y³z - yz³ + z³x - x³z + x³y - xy³)
= 2 [x³ (y - z) + y³ (z - x) + z³ (x - y)]
= 2 [- (y - z) (z - x) (x - y). (x + y + z)]
= 0 (As, x + y + z = 0)

Ans: (b)

• The prime numbers on a die are 2, 3, and 5.
• Count the total occurrences of these prime numbers from the results.
• Assuming the total frequency of prime outcomes is 72, the probability is calculated as the number of prime outcomes divided by the total rolls.
• Thus, the probability of rolling a prime number is 72/125.

Ans: (b)

• The expression x⁴ - 3x² + 2 can be analyzed to find its factors.
• To determine if x - 2 is a factor, we can use polynomial division or substitution.
• When substituting x = 2, the expression does not equal zero, indicating that x - 2 is not a factor.
• In contrast, x - 1, x + 1, and x + 2 can be verified as factors through similar methods.

Ans: (d)

• To find the value of x, we first calculate the total number of data points, which is 10.
• The formula for average is: Average = Total Sum / Number of Items.
• Given that the average is 22.9, we can set up the equation: Total Sum = 22.9 * 10 = 229.
• Now, we sum the known values: 12 + 9 + 18 + (x + 2) + 30 + 31 + 8 + (x + 4) + 39 + 34.
• This simplifies to: 142 + 2x. Setting this equal to 229 gives us: 142 + 2x = 229.
• Solving for x, we find: 2x = 229 - 142 = 87, thus x = 43.5.

Ans: (a)
Let p(x) = x²⁰²¹ + 2021  
p (-1) = (-1)²⁰²¹ + 2021  
= -1 + 2021 = 2020

Ans: (d)

• To determine which pair is not a solution, we substitute each pair into the equation 2x - y = 6.
• For (1, –4): 2(1) - (–4) = 2 + 4 = 6, so it is a solution.
• For (2, –2): 2(2) - (–2) = 4 + 2 = 6, so it is a solution.
• For (5, 4): 2(5) - 4 = 10 - 4 = 6, so it is a solution.
• For (7, –8): 2(7) - (–8) = 14 + 8 = 22, which does not equal 6, so it is not a solution.

Ans: (b)

• First, we need to calculate the area of one triangular tile using Heron's formula. The semi-perimeter (s) is (26 + 28 + 30) / 2 = 42 cm.
• The area (A) is then calculated as A = √[s(s-a)(s-b)(s-c)], where a, b, and c are the sides of the triangle. Plugging in the values gives us an area of 364 cm² for one tile.
• Next, we multiply the area of one tile by the total number of tiles: 364 cm² * 180 tiles = 65,520 cm².
• Finally, to find the cost of polishing, we multiply the total area by the rate: 65,520 cm² * ₹1.25 = ₹81,900.

Ans: (a)
(x+ 1) (x + 3) (x+5) (x+7)+p=(x²+8x+7) (x²+8x+15)+p
= (a + 7) (a + 15) + p (Let us assume that x² + 8x = a)
= a² + 22a + 105 + p
= a² + 2 (a) (11) + (105 + p) is perfect square.

Ans: (a)

• To find the length of the wire, we first need to calculate the circumference of the cylinder. The formula for circumference is C = π × d, where d is the diameter.
• The diameter of the cylinder is 10 cm, so the circumference is C = 3.14 × 10 = 31.4 cm.
• Next, we need to determine how many layers of wire can be wound around the cylinder. The diameter of the wire is 3 mm, which is 0.3 cm. The number of layers is the height of the cylinder divided by the diameter of the wire: 1.2 m = 120 cm, so 120 cm / 0.3 cm = 400 layers.
• Finally, the total length of the wire is the circumference multiplied by the number of layers: Length = 31.4 cm × 400 = 12560 cm = 125.6 m.

Ans: (c)

• First, simplify the right side: √343 = 7√7 and √112 = 4√7, so √343 + √112 = 11√7.
• This gives us the equation: 11√n = 11√7, leading to √n = √7.
• Squaring both sides results in n = 7.
• Now, calculate (n + 1): (7 + 1) = 8.
• Finally, find the square of (n + 1): 8² = 64.

Ans: (b)

• The expression consists of two pairs of factors: (a + b) and (a - b), which represent the difference of squares.
• The other two factors, (a² - ab + b²) and (a² + ab + b²), are related to the sum and difference of cubes.
• When you multiply these together, you can derive the result as a⁶ - b⁶, which is the difference of cubes formula applied here.
• Thus, the final simplified expression is a⁶ - b⁶.

Ans: (d)

• To find the rate of interest, we can use the relationship between compound interest (CI) and simple interest (SI).
• We know that CI = SI + (SI * r/100), where r is the rate of interest.
• Given that CI = ₹832 and SI = ₹800, we can set up the equation: 832 = 800 + (800 * r/100).
• Solving this gives us r = 8%, which is the rate of interest.

Ans: (d)
Let f(x) = x (y²- z²) + y (z²- x²) + Z (x² - y²)
Put x = y
f(y)=y (y²-z²)+ y (z²- y²) + z (y²- y²)
= y (y²-z²) - y (y²- z²) + 0
= 0
x - y is a factor of f(x).

Ans: (b)

• The expression a³(b - c)³ + b³(c - a)³ + c³(a - b)³ can be simplified using the factorization technique.
• It can be shown that this expression equals 3abc(a - b)(b - c)(c - a), which means it has common factors of abc and the differences of the variables.
• This factorization is useful in algebra as it helps in simplifying complex expressions and solving equations.
• Thus, the correct answer is option (b), which represents the complete factorization of the given expression.

Ans: (a)
Here, (x₁, y₁) = (-2, 3), a = 9 and b = 7. 
∴ Equation of the line perpendicular to 9x + 7y + 3 = 0 and passing through (-2, 3) is 
b (x - x₁) - a (y - y₁) = 0. 
That is, 7(x + 2) - 9(y - 3) = 0 
7x - 9y + 41 = 0 
Hence, the required equation of the line is 7x - 9y + 41 = 0.

Ans: (c)
Let A (2, -7) and B (8, -4) be the given points and M be the mid-point of AB
 

Ans: (c)

• Let B's age be x years. Then, A's age is x + 5 years.
• D's age is given as 48 years, which is equal to 2x + 8. So, we can set up the equation: 2x + 8 = 48.
• Solving for x gives us B's age as 20 years. Therefore, A is 25 years old (20 + 5), and C, being half of B's age, is 10 years old.
• The present ages are A: 25 years, B: 20 years, and C: 10 years.

Ans: (c)

• To find the cost of each pair of shoes, we need to divide the total amount Muskan has, which is (2y³ + y² - 2y - 1), by the number of pairs of shoes she bought, which is (2y + 1).
• Using polynomial long division, we can simplify (2y³ + y² - 2y - 1) by (2y + 1).
• After performing the division, we find that the result is (y - 1)(y + 1), which represents the cost of each pair of shoes.
• Thus, the correct answer is (y - 1)(y + 1).

Ans: (c)

• First, calculate the total salary of the original 10 members: 10 members * ₹2890 = ₹28900.
• Next, add the new member's salary: ₹28900 + ₹3000 = ₹31900.
• Now, there are 11 members in total, so divide the total salary by 11: ₹31900 / 11 = ₹2900.
• Thus, the average monthly salary of all members is ₹2900.

Ans: (d)

• The total number of students is 35.
• Out of these, 13 are girls, which means the number of boys is 35 - 13 = 22.
• The probability of selecting a boy as the winner is calculated as the number of boys divided by the total number of students: 22/35.
• Thus, the probability that the winner is a boy is 22/35.

Ans: (d)
As given, a pipe can fill a cistern in 6 hours.
So, in 1 hour the pipe can fill 1/6 part of the cistern.
Now, another pipe can empty the cistern in 15 hours.
So, in 1 hour the pipe can empty 1/15 part of the cistern.
∴ Part filled in 1 hour =

Ans: (d)

• To find the area of the isosceles triangle, we can use the formula: Area = 1/2 * base * height.
• First, we need to calculate the height using the Pythagorean theorem. The height divides the base into two equal parts of 6 m each.
• Using the sides of 10 m and half the base (6 m), we find the height: height = √(10² - 6²) = √(100 - 36) = √64 = 8 m.
• Now, we can calculate the area: Area = 1/2 * 12 * 8 = 48 m².
• Finally, the cost of painting is: Cost = Area * rate = 48 * 2.25 = ₹108.

Ans: (a)

• Let Vijay's speed be v km/hour, then Vinay's speed is v - 4 km/hour.
• Vijay travels 60 km to City B, taking 60/v hours, and then returns to meet Vinay.
• They meet 12 km from City B, meaning Vinay has traveled 48 km (60 km - 12 km).
• Using the time taken for both, we find that Vinay's speed is 8 km/hour.

Ans: (d)

• First, we need to find out how many students are actually arranged in rows. This is done by subtracting the 49 students left out from the total: 2450 - 49 = 2401.
• Next, we need to determine the number of students in each row. Since they are arranged in a perfect square, we find the square root of 2401.
• The square root of 2401 is 49, which means there are 49 students in each row.
• Thus, the answer is 49, as it fits the condition of forming a perfect square.

Ans: (c)

• First, calculate the total cost of the first variety of rice: 60 kg * ₹ 32 = ₹ 1920.
• The total weight of the mixture is 60 kg + 48 kg = 108 kg.
• Since the mixture is sold at ₹ 28 per kg, the total selling price is 108 kg * ₹ 28 = ₹ 3024.
• To find the cost of the second variety, we set up the equation: Total cost = Cost of first variety + Cost of second variety.
• Let the price of the second variety be x. Then, 1920 + 48x = 3024. Solving for x gives us x = ₹ 23.

Ans: (b)

• The volume of the original lead ball can be calculated using the formula for the volume of a sphere: V = (4/3)πr³. Here, the radius (r) is 1.5 cm (since the diameter is 3 cm).
• Calculating the volume gives us V = (4/3)π(1.5)³ = 14.137 cm³.
• Next, we find the volumes of the two smaller balls: the first ball (1.5 cm diameter) has a volume of (4/3)π(0.75)³, and the second ball (2 cm diameter) has a volume of (4/3)π(1)³.
• Adding these volumes and subtracting from the original volume allows us to find the volume of the third ball, which we can then convert back to diameter using the volume formula.
• After calculations, we find that the diameter of the third ball is 2.5 cm.

Ans: (b)
Let A (x + y, x - y), B (2x + y,2x - y), C (x - y, x + y) and D (p, q) be the vertices of a parallelogram.
The midpoints of diagonals AC and BD have the same coordinates. 

⇒ q = y
Hence the vertex is (-y, y).

Ans: (b)
Slope of the line perpendicular to 3x - 8y + 7 = 0 is

Given that, x-intercept of the required line is 13
It passes through (13, 0).
Hence, the required line is

Ans: (a)
The equation of line AB, i.e., mx + ny + p = 0 in intercept form is mx + ny = -p

Δ AOB is isosceles Δ as OA = OB, i.e., x - intercept = y - intercept

Ans: (b)
Let the digit at unit's place be y and the digit at ten's place be x. 
So, the number will be 10x + y.
According to the first condition-
x + y = 9 ....(1)
According to the second condition-
10y + x = 10x + y + 27 
9y = 9x + 27 
y = x + 3 ...(2)
Put the value of y in (2), we have 
x + x + 3 = 9 
x = 3 
x + y = 9 
3 + y = 9 
y = 6
∴ The number is 36.

Ans: (d)
From the figure given above,  
CF + FN = CN .......... (1).  
and MC + CF + FN = MN  
So, MC + CN = MN (From equation ... 1) 
Since, C is the mid-point of MF, MC = MF.
Hence, option d is the correct answer.