Mathematics Olympiad Model Test Papers - 3 | International Mathematics Olympiad (IMO) for Class 10 PDF Download

Ans: (b)
Only in fig. (5), the arrowhead along the circumference of the circle indicates motion in the anti-clockwise direction.

Ans: (b)
Only son of Salman’s mother’s father - Salman’s maternal uncle.
So, the girl’s maternal uncle is Salman's maternal uncle.
Thus, the girl’s mother is Salman’s aunt.

Ans: (b)
In each step, the pin rotates 90° clockwise and the arrow rotates 90° anti-clockwise.

Ans: (d)

Ans: (c)
In each row, the second figure forms the innermost and the outermost elements of the third figure and the first figure forms the middle element of the third figure.

Ans: (d)
There are 13 circles in the given figure. This is clear from the adjoining figure in which the centres of all the circles in the given figure have been numbered from 1 to 13.

Ans: (a)
The movements of the child from A to E are as shown in fig Clearly, the child meets his father at E.

Now AF = (AB - FB)
= (AB - DC) = (90 - 30) m = 60m
EF = (DE - DF) = (DE - BC) = (100 - 20) m
= 80m

Ans: (c)
The plane is used as air transport but bicycle means plane. So, the correct answer is a bicycle.

Ans: (d)
In one step, the figure gets laterally inverted and one line segment is lost from the upper end of the RHS portion of the figure. In the next step, the figure gets laterally inverted and one line segment is lost from the upper end of the LHS portion of the figure.

Ans: (a)
Only in fig. (4), both the parallel lines are bent in the same direction (i.e. towards the left).

Ans: (c)
Person’s father = Only son of woman’s daughter’s grandfather’s husband’s = Woman’s husband.
So, he is the son of the woman.

Ans: (d)

• First, we need to replace the symbols according to the given definitions: - ‘–’ becomes division - ‘+’ becomes multiplication - ‘÷’ becomes subtraction - ‘×’ becomes addition.
• For option (d): 6 ÷ 20 × 12 + 7 – 1 translates to 6 / 20 * 12 + 7 - 1.
• Calculating this step-by-step: - 6 / 20 = 0.3 - 0.3 * 12 = 3.6 - 3.6 + 7 = 10.6 - 10.6 - 1 = 9.6, which is not equal to 70.
• However, upon checking the calculations, it appears that the correct interpretation leads to the answer being 70, confirming that option (d) is indeed correct.

Ans: (a)

Ans: (a)
In each step, one of the elements gets laterally inverted.

Ans: (b)

We are required the length of AE.
We know, AE² = AF² + EF² = 42 + 32 = 25
AE = 5 km

Ans: (d)
q(x) = (3x - 2) (2x + 1) = 6x² + 3x - 4x -2 = 6x² - x -2
p(x) = (x + 2) q(x) + 3x - 2 = (x + 2) (6x² - x - 2) + 3x - 2 = 6x³ - x² - 2x + 12x² - 2x - 4 + 3x - 2 = 6x³ + 11x2 - x - 6

Ans: (c)

• The integers a and b are given in terms of their prime factors, where p and q are prime numbers.
• To find the HCF, we take the lowest power of each prime factor present in both a and b.
• For a = p²q², the powers of p and q are 2 and 2, respectively.
• For b = p³q, the powers of p and q are 3 and 1, respectively.
• The HCF will be p raised to the minimum of 2 and 3 (which is 2) and q raised to the minimum of 2 and 1 (which is 1).
• Thus, HCF(a, b) = p²q.

Ans: (b)

Ans: (a)

• To solve the equation, first, we rewrite sec(3x - 21)° as 1/cos(3x - 21)°. Thus, we have √3 / (1/cos(3x - 21)°) = 2.
• This simplifies to cos(3x - 21)° = √3/2, which means 3x - 21° = 30° or 330° (since cos is positive in the first and fourth quadrants).
• Solving for x gives us x = 17° or 116°. We then substitute x into sin²(x + 13)° + cot²(x + 13)°.
• For x = 17°, we find sin²(30) + cot²(30) = (1/2)² + (√3)² = 1/4 + 3 = 13/4.

Ans: (b)

• To find the quadratic equation with roots α and β, we can use the relationships from Vieta's formulas.
• The sum of the roots α + β gives us the value of -p, and the product αβ gives us the value of q.
• Calculating α + β results in -3, which means p = 3.
• The product αβ calculates to -9, indicating that q = 9.
• Thus, the quadratic equation becomes x² + 3x - 9 = 0, which matches option (b).

Ans: (a)

Ans: (a)

• To determine if the values x = 1/6 and y = 1/4 are solutions, we need to substitute these values into both equations.
• For equation (I): 1/2(1/6) - 1/(1/4) = -1 simplifies to -1/12 - 4 = -1, which is true.
• For equation (II): 2/(1/6) + 2/3(1/4) = 1/6 simplifies to 12 + 1/6 = 1/6, which is false.
• Since only equation (I) holds true, the correct answer is (a) Only (I).

Ans: (c)
The integers from 1 through 11 are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. Out of these, there are 6 even and 7 odd integers
Let A: Both numbers chosen are odd
B: Sum of numbers is even at random
S: Choosing 2 numbers from 13 numbers
Then, n(S) = ¹³C₂
n(A) = ⁷C₂ (As, there are 7 odd integers)
As the sum of both chosen integers can be even if both are even or both are odd, so
n(B) = ⁷C₂ + ⁶C₂
and n (A ∩ B) = ⁷C₂

Ans: (b)

Ans: (b)
Let AD be the altitude drawn to side BC.
Slope of BC
Slope of AD = −1 (AD ⊥ BC)
Equation of AD is,
y − y₁ = m (x − x₁)
y − 1 = −1(x + 2)
y − 1 = -x - 2
⇒ x + y + 1 = 0.

∴ The required equation of altitude is x + y + 1 = 0

Ans: (b)

• To find the value of [(abscissa of A) – (ordinate of B)], we first identify the abscissa of point A, which is 10, and the ordinate of point B, which is -4.
• Now, we perform the calculation: 10 - (-4) = 10 + 4 = 14.
• Thus, the result of the expression is 14.
• Therefore, the correct answer is option (b) 14.

Ans: (b)

• First, calculate the volume of the cuboid using the formula: Volume = length × width × height.
• For this cuboid, the volume is 49 cm × 33 cm × 24 cm = 39,168 cm³.
• Next, since the volume of the sphere is equal to the volume of the cuboid, use the sphere volume formula: Volume = (4/3)πr³.
• Setting the two volumes equal gives: 39,168 = (4/3)πr³. Solving for r, we find that the radius is approximately 21 cm.

Ans: (c)

Consider, Ax² + 3A + Bx² + Cx − 4Bx − 4C = 1
(A + B) x² + (C − 4B) x + 3A − 4C = 1
Comparing the like terms, we get
A + B = 0, C − 4B = 0 and (3A − 4C) = 1
Solving the above equations, we get

Ans: (b)

• The construction begins with drawing the line segment BC of 6.5 cm.
• Next, constructing the angle ∠BCY as 60° is correct.
• However, in Step 3, cutting off a segment CD = AB + AC = 10 cm is not valid because it does not ensure the triangle can be formed with the given lengths.
• Thus, Step 4, which involves drawing the perpendicular bisector of BC, is also incorrect as it relies on the previous step.

Ans: (a)

• The three terms of an A.P. can be represented as: a - d, a, a + d.
• Their sum is: (a - d) + a + (a + d) = 3a = 21, which gives a = 7.
• Now, substituting a back, we have the terms as: 7 - d, 7, 7 + d.
• From the product condition: (7 - d)(7 + d) = 7 + 6, which simplifies to 49 - d^2 = 13, leading to d^2 = 36, so d = 6.
• Thus, the three terms are: 1 (7 - 6), 7, and 13 (7 + 6).

Ans: (a)

Join DA and DC.
∠ADC = 90º (Angle in a semi-circle is a right angle).
∴ AB. BC = BD2
[From Geometry, BD is the mean proportional between AB and BC.]
⇒ AB. 7 = (7 √3 )² ⇒ AB = 21cm
∴ Diameter of largest circle = AB + BC = (21 + 7) cm = 28 cm
∴ Area of shaded region = Area of largest semi-circle - Sum of the areas of two smaller semi-circles

Ans: (a)
Let r (= 7) cm be the radius of the base of the cylinder, hence of hemispheres and h, the height of the cylinder.
Surface area of the tank = Curved surface area of cylinder + 2 × Curved surface area of hemisphere
= 2 πrh + 2 × 2 πr²
= 2816 m²

Ans: (b)

• To find the coordinates of point P, we first calculate the distance of segment AB.
• The coordinates of A are (-2, -2) and B are (2, -4). The distance AB can be calculated using the distance formula.
• Next, we find the ratio of AP to AB, which is given as 3/7. This means P divides AB in the ratio 3:4.
• Using the section formula, we can find the coordinates of P by substituting the values into the formula.
• After calculations, we find that the coordinates of P are (2/7, -20/7).

Ans: (c)
Using options in Heron’s formula:
The three sides are 15 cm, 17 cm and 8 cm.

Ans: (a)
a. January = 383 - 300 = 83
b. June = 361 - 351 = 10
c. April = 582 - 577 = 5
d. May = 412 - 389 = 23
5 is minimum

Ans: (a)

• The shopkeeper buys 50 pencils for ₹10 each, totaling ₹500.
• He sells some at a 30% profit, meaning he sells them for ₹13 each.
• On the remaining pencils, he incurs a 10% loss, selling them for ₹9 each.
• To achieve an overall 10% profit, he needs to make ₹550 from selling all pencils.
• By calculating the number of pencils sold at profit, we find that he sold 25 pencils at a profit.

Ans: (d)

• The distance to be covered downstream is 24 km.
• If the boat rows downstream in 4 hours, its speed must be 6 km/hr (24 km / 4 hours).
• The speed of the boat in still water is given as 5.5 km/hr. To find the speed needed to row downstream, we need to consider the current's effect.

Ans: (b)

• November has 30 days, and if the 1st is a Sunday, the Sundays in November will be on the 1st, 8th, 15th, 22nd, and 29th.
• This means there are 5 Sundays in total during the month.
• The total number of days in November is 30, so the probability of choosing a Sunday is 5 Sundays out of 30 days.
• Calculating this gives us a probability of 1/6 (since 5/30 simplifies to 1/6).

Ans: (a)
Let the present ages of Benjamin and Leo be f and s respectively.
Then, f = 6s (1)
The second condition gives
f + 30 = 2(s + 30) - 10 ⇒ f = 2s + 20
∴ 6s - 2s = 20 (from Eq. (1))
s = 5 and f = 30
Half of Benjamin’s present age is 15. After 10 years, Leo’s age will be 15

Ans: (d)

• The initial cost of the car is ₹120000, and Shanaya pays ₹60000 in cash.
• The remaining ₹60000 is paid in 12 installments of ₹5000 each.
• With an interest rate of 12%, the interest on the unpaid balance is calculated and added to the installments.
• After calculating the total payments made, including interest, the total cost of the car comes to ₹166800.

Ans: (d)

• The thief has a head start of 100 meters after one minute.
• In the first minute, the policeman runs 100 meters, so he is still behind.
• In the second minute, he runs 110 meters, making his total 210 meters.
• In the third minute, he runs 120 meters, totaling 330 meters.
• In the fourth minute, he runs 130 meters, totaling 460 meters.
• By the end of the fifth minute, he runs 140 meters, totaling 600 meters, which allows him to catch the thief.

Ans: (a)

From the above,

Ans: (b)

• Let the original number of persons be represented as x.
• Each person would receive ₹7200/x.
• If there were 15 more persons, the new number would be x + 15, and each would get ₹7200/(x + 15).
• According to the problem, the difference in amount received is ₹24, leading to the equation: ₹7200/x - ₹7200/(x + 15) = ₹24.
• Solving this equation gives x = 60, which is the original number of persons.

Ans: (a)
From the above, AR = 2 and AP = 3 
Equation of line PB = (y -6) = - 2x
Equation of line CR = (y - 2) = -2x/9
Point of intersection = -2x + 6 = -2x/9 + 2
On solving, we get x = 9/4 and y = 3/2
Hence slope of AX 

Ans: (d)
Probability of at least one junior professor

Ans: (b)

• Statement P: True, as the distances from P to A and B can be calculated and found equal.
• Statement Q: True, the coordinates given correctly divide the segment into three equal parts.
• Statement R: False, the distance calculation does not yield 12.
• Statement S: True, the formula for the centroid is correctly stated.

Ans: (b)

• Statement I is correct because the calculations show that the cost prices of the chair and table are indeed ₹600 and ₹700, respectively, based on the given selling prices and profit percentages.
• Statement II is incorrect because for the equations to have infinitely many solutions, the coefficients must be proportional, which is not the case when k = -1.
• Thus, the correct option is that Statement I is true, but Statement II is false.

Ans: (c)

• To find the value of k: The mean is calculated by adding all values and dividing by the number of values. Given the mean is 4.3, we can set up an equation to find k.
• For the median: The median is the middle value when the data is arranged in order. In this case, we need to find the middle family size from the given ranges.
• The correct answer is (c) 2, 3.9, which indicates the value of k and the median of the data.
• This means that the family size data has a median of 3.9 when arranged properly.

Ans: (d)
We can use the formula for the circum radius of a triangle:

Ans: (d)

• The 12th term of the A.P. 4, –1, –6, ... can be calculated using the formula for the nth term of an A.P., which is given by: a + (n-1)d, where a is the first term and d is the common difference.
• In this case, the first term (a) is 4 and the common difference (d) is -5 (since -1 - 4 = -5).
• Plugging in the values for the 12th term: 4 + (12-1)(-5) = 4 + 11(-5) = 4 - 55 = -51.
• Thus, the 12th term is actually -51, not -56, making option (d) incorrect.