Mathematics Olympiad Model Test Papers - 2 | Olympiad Preparation for Class 10 PDF Download

Ans: (b)
The line segment rotates through 90o and moves to the adjacent portion of the rhombus in an anti-clockwise direction in first, third, fifth, ... steps. The other symbol moves to the adjacent portion of the rhombus in an anti-clockwise direction and also gets replaced by a new symbol in second, fourth, sixth, ... steps.

Ans: (c)
In all other figures, the arrow and the V sign lie towards the black end of the main figure.

Ans: (d)
is the correct mirror image of the figure (X).

Ans: (a)
In each column, the second figure (middle figure) is obtained by removing the upper part of the first figure (uppermost figure) and the third figure (lowermost figure) is obtained by vertically inverting the upper part of the first figure.

Ans: (a)

• The input consists of words and numbers that need to be rearranged based on specific rules.
• In the steps provided, the numbers are moved to the front in ascending order while the words follow a specific sequence.
• For the new input, the fourth step involves moving the smallest number (22) to the front, followed by the next smallest (34), and then rearranging the remaining words and numbers accordingly.
• Thus, the correct fourth step for the new input is 22 the 34 pull cat 62 81 bell.

Ans: (c)
In each row, the third figure is a collection of the common elements (line segments) of the first and the second figures.

Ans: (d)

• To solve this, we need to replace the symbols with their meanings:
• For option (d): 19 * 5 % 8 @ 3 becomes 19 + 5 - 8 × 3.
• This simplifies to 19 + 5 - 24, which equals 0, not 10.
• Thus, option (d) is incorrect, while the others are correct when calculated.

Ans: (d)

Ans: (c)

• From the information, we know that Q is on the 1st floor.
• Since there are 4 families between P and Q, P must be on the 6th floor.
• N is just below P, so N is on the 5th floor.
• Family R is above M, and since O is on an even floor, O must be on the 2nd or 4th floor.
• Given the arrangement, O is on the 4th floor, making M on the 3rd floor and R on the 6th floor.
• Thus, the family between N (5th floor) and R (6th floor) is O (4th floor).

Ans: (a)

• To determine that U is the mother-in-law of P, we need to analyze the relationships given in the options.
• In option (a), P is the wife of Q, Q is the brother of T, T is the daughter of U, and U is the wife of W. This means U is the mother of T, making her the mother-in-law of P.
• The other options do not establish this relationship clearly, as they either change the roles or do not connect U as the mother-in-law of P.
• Thus, option (a) correctly indicates that U is the mother-in-law of P.

Ans: (d)

• The letters from the word GENEROUS are: E, R, O, and S (2nd, 5th, 6th, and 8th letters).
• These letters can be rearranged to form multiple words, such as ROSE and SORE.
• Since more than two valid words can be formed, the answer is More than 2.
• Thus, the correct option is (d).

Ans: (b)
In all other figures, there are two small line segments towards the pin and three small line segments towards the arrow.

Ans: (a)
The figure may be labelled as shown.

The simplest triangles are AHB, GHI, BJC, GFE, GIE, IJE, CEJ and CDE i.e. 8 in number.
The triangles composed of two components each are HEG, BEC, HBE, JGE and ICE i.e. 5 in number.
The triangles composed of three components each are FHE, GCE and BED i.e. 3 in number.
There is only one triangle i.e. AGC composed of four components.
There is only one triangle i.e. AFD composed of nine components.
Thus, there are 8 + 5 + 3 + 1 + 1 = 18 triangles in the given figure.

Ans: (a)
Hence required direction is South-West.

Ans: (c)

• In the code for 5KR7CM, we analyze each character based on the given conditions.
• The number 5 is followed by the letter K, so K is surrounded by numbers, which means it is coded as →.
• Next, R is surrounded by letters K and 7, so it is coded as ↑.
• Finally, C is surrounded by letters M and 7, so it is coded as ↑.
• Thus, the complete code for 5KR7CM becomes $©ζ+*@(.

Ans: (c)

• The given equation is a quadratic equation of the form ax² + bx + c = 0.
• To determine the nature of the roots, we can use the discriminant (D), which is calculated as D = b² - 4ac.
• For this equation, a = 13, b = 5, and c = -2. Plugging in these values gives D = 5² - 4(13)(-2) = 25 + 104 = 129.
• Since the discriminant is positive (D > 0), it indicates that the roots are real and distinct.

Ans: (b)

• The total number of cards in a standard deck is 52.
• There are 2 red queens in the deck: the queen of hearts and the queen of diamonds.
• The probability of drawing a red queen is calculated as the number of favorable outcomes (2) divided by the total outcomes (52).
• This gives us a probability of 2/52, which simplifies to 1/26.

Ans: (d)
CSA of the cone = Area of the rectangle = 56 × 22 = 1232 cm²

Ans: (c)

Ans: (a)

• When rolling two dice, the possible outcomes range from 2 (1+1) to 12 (6+6).
• The total number of outcomes when rolling two dice is 36 (6 sides on the first die multiplied by 6 sides on the second die).
• The sums that are less than or equal to 10 include all combinations except for the sums of 11 and 12.
• There are 33 outcomes that result in a sum of 10 or less, leading to a probability of 33/36, which simplifies to 11/12.

Ans: (d)

• To find the midpoint P of line segment AB, use the formula: P = ((x₁ + x₂)/2, (y₁ + y₂)/2). Here, A(-10, 4) and B(-2, 0) give P = ((-10 - 2)/2, (4 + 0)/2) = (-6, 2).
• Next, we need to determine how P divides the line segment CD. The coordinates of C and D are C(-9, -4) and D(-4, 6).
• Using the section formula, if P divides CD in the ratio m:n, then P = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)). Setting this equal to P(-6, 2) allows us to solve for the ratio.
• After calculations, we find that P divides CD in the ratio 3:2.

Ans: (c)
Let the point be P (x, y)
Its distance from (p, 0)
Its distance from the axis of y will be x

Ans: (d)
80% of 300 = 240

Ans: (d)
The time interval for them to meet at the starting point for the next time = LCM (10, 15, 12,18) = 180 minutes

Ans: (b)

• In an Arithmetic Progression (A.P.), the nth term can be expressed as: a + (n-1)d, where a is the first term and d is the common difference.
• Given the 5th term (a + 4d) is 46 and the 17th term (a + 16d) is 130.
• By setting up the equations: a + 4d = 46 and a + 16d = 130, we can solve for a and d.
• Subtracting the first equation from the second gives us 12d = 84, leading to d = 7. Substituting back, we find a = 46 - 28 = 18.
• Now, to find the 14th term: a + 13d = 18 + 13*7 = 18 + 91 = 109.

Ans: (b)

• The median is the middle value of a data set when arranged in order. In this case, we know the median is 35.
• To find the value of x, we need to analyze the frequencies associated with the class intervals.
• Given the options, we can determine that the correct frequency that allows the median to be 35 is 10.
• Thus, the value of x that satisfies this condition is 10.

Ans: (a)

• To simplify the expression, we can find a common denominator, which is (sin A - cos A)(sin A + cos A).
• After combining the fractions, we can simplify the numerator and denominator.
• Upon simplification, we find that the result is equal to 2 / (2sin² A - cos² A).
• This shows that the correct answer is option (a).

Ans: (b)

• The statement that 0.423442344423... is a rational number is incorrect because it is a non-terminating, repeating decimal. Rational numbers can be expressed as a fraction of two integers, and this number cannot be simplified into such a form.
• In contrast, the other statements are true: 17.6 can be expressed as 53/3, 16.29 as 19/1, and the sum of √25 (5) and √64 (8) is 13, which is also a rational number.

Ans: (a)

• The equation of a line can be represented in the form of Ax + By + C = 0.
• To determine which equation matches the line in the graph, we can check the intercepts and slope of each option.
• After analyzing, we find that option (a) x + 2y - 6 = 0 correctly represents the line shown in the graph.
• Thus, the correct answer is option (a).

Ans: (a)

• For a system of equations to have no solution, the lines represented by the equations must be parallel. This occurs when the ratios of the coefficients of x and y are equal, but the constant terms are not.
• In this case, we can set up the ratios from the equations: p/3 = 12/1. Solving this gives p = 36.
• Thus, when p equals 36, the two lines will be parallel, leading to no intersection point, which means no solution.
• Therefore, the correct value of p that results in no solution is 36.

Ans: (a)

• The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Here, the area ratio is 16:49, which means the ratio of the sides is √16:√49 = 4:7.
• The perimeter of ΔDSKY is 21 cm, so the corresponding perimeter of ΔDRED can be calculated using the side ratio. If we let the perimeter of ΔDRED be x, then 4/7 = x/21, leading to x = 12 cm.
• Now, the perimeter of ΔDRED is the sum of its sides: 3 cm + 5 cm + third side = 12 cm.
• Thus, the third side = 12 cm - (3 cm + 5 cm) = 12 cm - 8 cm = 4 cm.

Ans: (d)

• To determine the value of p, 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 the equation: p(2)² + 5(2) + r = 0.
• Substituting x = 1/2 gives us another equation: p(1/2)² + 5(1/2) + r = 0.
• Solving these equations simultaneously will show that p = r.

Ans: (c)
(p+1)y²−(p+1)by=(p−1)ay−c(p−1) 
(p+1)y²−y[(p+1)b+(p−1)a]+c(p−1)=0
The coefficient of y must equate to 0.
 [(p+1)b+(p−1)a]=0 
pb+b+pa−a=0 
p(a+b)=a−b

Ans: (d)
a + 2d = 21 and a + 6d = 37
So, a = 13 and d = 4

Ans: (d)
If sin sin A =sin sin B, where A = 120° −α and B = 120°−β 
A = B or A = π − B ,i.e.,A + B = π 
120° −𝛼 = 120°−𝛽 𝑜𝑟,120° −𝛼 + 120° − 𝛽 = 180°
α = β or α+β= 60°

Ans: (c)

• To find the total items needed for the year, multiply the desired average (4800) by 12 months, which equals 57600 items.
• For the first 4 months, the factory produces 4500 items each month, totaling 18000 items.
• Subtract the total produced in the first 4 months from the yearly total: 57600 - 18000 = 39600 items needed for the next 8 months.
• To find the average per month for the next 8 months, divide 39600 by 8, which equals 4950 items.

Ans: (b)

• Let the fare from A to B be represented as x and from A to C as y.
• From the first scenario: 2x + 3y = 795.
• From the second scenario: 3x + 5y = 1300.
• By solving these two equations, we find that x = 75 and y = 215.
• Thus, the fares from A to B and A to C are ₹75 and ₹215, respectively.

Ans: (b)
Let y be the no. of accidents, x is overtime hours, then y = ax + b where a and b are constants.
Substituting the values: 
1000a + b = 8 and 400a + b = 5
We get a  = 1/200 and b=3
For x = 0, y = 3.

Ans: (b)

• Let the original speed be x km/h and the distance be d km.
• From the problem, we know that increasing the speed by 12 km/h reduces the time by 1 hour, and increasing it by another 12 km/h (total increase of 24 km/h) reduces the time by an additional 45 minutes (or 0.75 hours).
• Using the formula: time = distance/speed, we can set up equations based on the information given.
• After solving these equations, we find that the distance d equals 504 km.

Ans: (c)

• The total investment is ₹2600, and the interest earned from each rate is the same.
• Let the amount invested at 6% be x. Then, the amounts invested at 4% and 8% would be ₹2600 - x.
• The interest from each investment can be calculated using the formula: Interest = Principal × Rate × Time.
• Setting up the equations for the interest earned at each rate and solving them leads to the conclusion that the amount invested at 6% is ₹800.

Ans: (c)

• First, calculate the total work done by women and children in terms of "work units".
• 10 women complete the work in 7 days, so their work rate is 10/7 work units per day.
• 10 children finish the work in 14 days, giving them a work rate of 10/14 work units per day.
• Now, find the combined work rate of 5 women and 10 children, which is (5 * 10/7) + (10 * 10/14).
• After calculating, you will find that they can complete the work in 7 days.

Ans: (a)

• The weight of a cube is related to the volume of the cube, which is calculated as side length cubed (side³).
• For the first cube, the volume is 3 cm x 3 cm x 3 cm = 27 cm³, and it weighs 12 kg.
• The second cube has a side length of 12 cm, so its volume is 12 cm x 12 cm x 12 cm = 1728 cm³.
• Since the cubes are similar and made of the same material, the weight ratio is the same as the volume ratio. The weight of the second cube is calculated as follows: (1728 cm³ / 27 cm³) x 12 kg = 768 kg.

Ans: (b)

• Let the original weight of the solution be x kg. Since it contains 42% salt, the weight of salt in the solution is 0.42x kg.
• After 25 kg of water evaporates, the new weight of the solution becomes (x - 25) kg.
• The weight of salt remains the same, but now it constitutes 56% of the new solution, so we can set up the equation: 0.42x = 0.56(x - 25).
• Solving this equation gives us x = 100 kg, which is the original quantity of the solution.

Ans: (a)

• To find the side of the square base, we first calculate the volume of the first tank: Volume = length × width × height = 15 cm × 12 cm × 8 cm = 1440 cm³.
• The second tank has a volume given by the formula for a square base: Volume = side² × height. Here, the height is 10 cm.
• Setting the volumes equal: side² × 10 cm = 1440 cm³. Solving for side² gives side² = 1440 cm³ / 10 cm = 144 cm².
• Taking the square root, we find side = √144 cm² = 12 cm.

Ans: (d)

• The total debt is ₹3600, and one-third of it is unpaid after 30 installments, which means ₹1200 is still owed.
• This indicates that ₹2400 has been paid through the 30 installments.
• Since there are 40 installments, the total amount paid can be expressed as the sum of an arithmetic series.
• Using the formula for the sum of an A.P., we can find the value of the 10th installment, which turns out to be ₹69.

Ans: (d)

• Statement I: The equation x² - x + 1 = 0 has roots that are not real because the discriminant (b² - 4ac) is negative (1 - 4 < 0).
• Statement II: The equation x² - 5x + 5 = 0 has real and distinct roots since its discriminant (25 - 20 = 5) is positive.
• Thus, Statement I is false and Statement II is true.
• Therefore, the correct answer is (d).

Ans: (c)
Total sum of the numbers written on the blackboard
When two numbers ‘a’ and ‘b’ are erased and replaced by a new number a + b – 1, the total sum of the numbers written on the blackboard is reduced by 1.
Since, this operation is repeated 39 times, therefore, the total sum of the numbers will be reduced by 1 × 39 = 39.
Therefore, after 39 operations there will be only 1 number that will be left on the blackboard and that will be 820 – 39 = 781.

Ans: (d)

• To find the volumes, we calculate: V₁ = x³, V₂ = (2x)³ = 8x³, V₃ = (3x)³ = 27x³, and V₄ = (4x)³ = 64x³.
• Now, we evaluate the statements: (1) V₁ + V₂ + 2V₃ = x³ + 8x³ + 54x³ = 63x³ < 64x³ = V₄, which is true.
• (2) V₁ + 4V₂ + V₃ = x³ + 32x³ + 27x³ = 60x³ < 64x³ = V₄, which is also true.
• (3) 2(V₁ + V₃) + V₂ = 2(x³ + 27x³) + 8x³ = 62x³ = V₄, which is true as well.
• Since all statements are correct, the answer is (1), (2), and (3).

Ans: (c)
Total number of terms in the sequence 17, 21, 25 …
417 is equal to 
Total number of terms in the sequence 16, 21, 26 …

nth term of the first sequence = 4n + 13. mth term of the second sequence = 5m + 11.
As per the information given in the question 4n + 13= 5m + 11 ⇒ 5m – 4n = 2.
Possible integral values of n that satisfy 5m = 2 + 4nare (2, 7, 12 … 97)

Ans: (a)

• The incorrect steps in the construction are identified as Step I and Step IV.
• In Step I, the radius should be 6 cm, not 3 cm as stated.
• In Step IV, the perpendicular from A to OP should be correctly drawn to ensure the tangent is accurate.
• Thus, both these steps do not align with the requirements for constructing the tangent correctly.