Mathematics Olympiad Model Test Paper - 2 | Mathematics Olympiad Class 7 PDF Download

Ans: (b)

• Ishu is the brother of Bipin.
• Bipin is married to Aarti.
• This means that Ishu is Aarti's brother-in-law because he is the brother of her husband.
• Thus, the correct relationship is that Ishu is Aarti's brother-in-law.

Ans: (1.5)³ =3.375

Ans: (d)

• First, we need to replace the operations according to the given rules:
• 125 – 6 × 3 + 5 ÷ 18 becomes 125 + 6 ÷ 3 × 5 – 18.
• Now, we perform the operations in the correct order: division and multiplication first, then addition and subtraction.
• Calculating 6 ÷ 3 gives 2, then 2 × 5 gives 10. Finally, we compute 125 + 10 – 18, which equals 117.

Ans: (d)
Perpendicular distance of point (3, 4) from y-axis is 3.

Ans: (d)

• The code language involves a specific pattern of shifting letters. Each letter in “BIRTHDAY” is replaced by another letter a certain number of places down the alphabet.
• For example, 'B' becomes 'X', which is 22 letters back in the alphabet. This pattern continues for each letter.
• Applying the same shifting pattern to “PLEASURE” results in “LHASAQNA.”
• Thus, the correct answer is option (d) LHASAQNA.

Ans: (d)

Ans: (c) y-coordinates of all the points below the x-axis are negative.

Ans: (a)
864 × n is a perfect cube. 864 =
2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
⇒ n=2

Ans: (b) Abscissa of a point is positive in I and IV quadrants.

Ans: (b)

Ans: (d) On Friday, the temperature was high, so it was the hottest day.

Ans: (c)
Expressing 7200 as its prime factors
7200 = 2×2×2×2×2×3×3×5×5
7200 = (2×2×2)×(2×2)×(3×3)×(5×5)
We find that prime factors 2, 3 & 5 appear in groups of two, so to make the given number a perfect cube, we must multiply it with
2×3×5=30
∴ 7200 × 30
=(2×2×2)×(3×3×3)×(2×2×2)×(5×5×5) is a perfect cube.

Ans: (d) Number of dolls sold on Saturday = 40
Cost of 1 doll = ₹35
Total cost of 35 dolls = 40 × 35 = ₹1400

Ans: (b)
Since the unit’s place digit of xyz8 is 8
∴ unit’s place digit of cube of xyz8 is 2.

Ans: (d) Month with highest expenditure = February
Month with lowest expenditure = April
Difference in expenditure:
600 – 200 = 400

Ans: (c)

• First, calculate (-9) × (7) which equals -63.
• Next, compute (-15) ÷ (-3) which gives 5, and then multiply by 6 to get 30.
• Now, calculate 9 × 3 which equals 27.
• Finally, combine all parts: -63 + 30 + 27 = -6.

Ans: (a)
Point (–10, 0) lies on the negative direction of x-axis.

Ans: (b) Figure I is formed by the given coordinates.

Ans: (d)

• To find what to subtract, we set up the equation: (x³ + 4x² + 3x - 2) - ? = (x^3 - 2x² + 4).
• Rearranging gives us: ? = (x³ + 4x² + 3x - 2) - (x³ - 2x² + 4).
• When we simplify this, we combine like terms: 4x² + 2x² + 3x - 2 - 4 = 6x² + 3x - 6.
• Thus, the expression we need to subtract is (6x² + 3x - 6), which matches option (d).

Ans: (c)

• The construction of triangle PQR involves specific steps to ensure the correct angles and lengths are achieved.
• In this case, we need to create an angle of 120° at point Q and ensure both sides PQ and QR are 6.5 cm long.
• Step III is identified as incorrect, possibly due to a mistake in measuring or drawing the angle or sides.
• It's crucial to follow the construction steps accurately to form the triangle as specified.

Ans: (b)

• The total possible outcomes when rolling a die are 6: {1, 2, 3, 4, 5, 6}.
• The numbers greater than 2 are {3, 4, 5, 6}, which gives us 4 favorable outcomes.
• To find the probability, we use the formula: Probability = (Number of favorable outcomes) / (Total outcomes).
• So, the probability = 4/6 = 2/3.

Ans: (b) Abscissa and ordinate of a point are negative in III quadrants.

Ans: (c)

• Let the number be represented as x.
• According to the statement, we can write the equation: (1/4)x + 15 = x.
• To solve for x, first subtract (1/4)x from both sides: 15 = x - (1/4)x.
• This simplifies to 15 = (3/4)x, and then multiplying both sides by (4/3) gives x = 20.
• Thus, the number is 20.

Ans: (c)

• The statement "20 x 30 + 40 is 1" is incorrect because when you calculate it, 20 x 30 equals 600, and adding 40 gives you 640, not 1.
• Option (a) is correct as it follows the rules of exponents.
• Option (b) is also correct as it represents the proper conversion to standard form.
• Option (d) is valid since P³ × P⁵ ÷ P⁴ simplifies to P(^3+5-4) = P⁴.

Ans: (a)
The factors of 42875 = 5 × 5 × 5 × 7 × 7 × 7
Make the factors of the number by taking three identical numbers. Now multiply each number of the factors.

Ans: (c)

• Let the scores of team A and B be 5x and 6x respectively, based on the ratio 5:6.
• The total score is given as 154, so we can write the equation: 5x + 6x = 154.
• This simplifies to 11x = 154, leading to x = 14.
• Now, substituting x back, team A's score is 5x = 5 x 14 = 70, and team B's score is 6x = 6 x 14 = 84.

Ans: (a)

• To find the weight of the sixth student, we first calculate the total weight of all six students. Since the mean weight is 40.8 kg, the total weight is 40.8 kg x 6 = 244.8 kg.
• Next, we add the weights of the first five students: 42.5 kg + 35.7 kg + 38.9 kg + 40.2 kg + 44.5 kg = 201.8 kg.
• Now, we can find the weight of the sixth student by subtracting the total weight of the first five students from the total weight of all six: 244.8 kg - 201.8 kg = 43 kg.
• Thus, the weight of the sixth student is 43 kg.

Ans: (c) Point (0, –7) lies on the y-axis.

Ans: (c)

• Given the ratio (a : b = 2 : 3), we can express a and b in terms of a common variable. Let a = 2x and b = 3x.
• Now, substituting these values into the expression (3a + 2b) and (5a + 3b) gives us (3(2x) + 2(3x)) and (5(2x) + 3(3x)).
• This simplifies to (6x + 6x) : (10x + 9x), which is (12x : 19x).
• Thus, the ratio simplifies to 12/19, confirming that the answer is (c).

Ans: (d)

• For a number to be divisible by 6, it must be divisible by both 2 and 3.
• To check for divisibility by 2, the last digit (which is *) must be even. The options that are even are 2, 4, 6, and 8.
• To check for divisibility by 3, the sum of the digits must be divisible by 3. The sum of the digits in 478265* is 32 + *.
• Testing the options: If * = 4, the sum is 36, which is divisible by 3. Thus, the correct answer is 4.

Ans: (c)
Let the natural number be ‘x’.
∴ x³ −x² =48
⇒ x² (x−1)=48
⇒ 4² ( 4 − 1 ) = 48
∴ 𝑥 = 4 x=4

Ans: (d)
Since ‘a’ is the greatest single digit perfect cube, therefore a=8 according to the question,
2a−b=7 ⇒b=9

Ans: (b)

• The product of two negative integers results in a positive integer. This is because when you multiply two negative numbers, the negatives cancel each other out, leading to a positive result.
• For example, if you take -3 and -4, their product is 12, which is positive.
• In contrast, the other options are incorrect: the sum of a negative and a positive integer can be negative, -27 is not greater than -25, and there are actually three integers between -82 and -86.

Ans: (c)

• The cost price of 11 oranges is ₹10, so the cost price per orange is ₹10/11 = ₹0.91 (approximately).
• The selling price of 10 oranges is ₹11, so the selling price per orange is ₹11/10 = ₹1.10.
• The gain per orange is ₹1.10 - ₹0.91 = ₹0.19.
• The gain percentage is calculated as (Gain/Cost Price) x 100 = (0.19/0.91) x 100 ≈ 20.88%, which rounds to 21% gain.

Ans: (d) Abscissa is zero, so it will lie on y-axis, but value of ordinate is negative. So it will lie on negative y-axis.

Ans: (b) Least number of points required to draw graph of a line is two.

Ans: (c)

• Let Varun's current age be represented as V. Therefore, Karan's current age is 4V.
• Five years ago, Varun's age was V - 5 and Karan's age was 4V - 5.
• According to the problem, five years ago, Karan was 7 times Varun's age: 4V - 5 = 7(V - 5).
• Simplifying this gives: 4V - 5 = 7V - 35, leading to 3V = 30, so V = 10.
• Thus, Karan's current age is 4V = 4 x 10 = 40 years.

Ans: (a)

• To find the rate of interest, we can use the formula: Interest = Principal x Rate x Time.
• Here, the Principal is ₹24,200, the Interest earned is ₹2,904, and the Time is 3 years.
• Rearranging the formula to find the Rate gives us: Rate = (Interest / (Principal x Time)) x 100.
• Substituting the values: Rate = (2904 / (24200 x 3)) x 100 = 4%.

Ans: (d)

• Let the price of the ball be x. Then, the price of the bat can be expressed as 3x + 14.
• According to the problem, the cost of the bat is ₹176, so we set up the equation: 3x + 14 = 176.
• Solving for x, we find that 3x = 162, which gives x = 54. Thus, the price of the ball is ₹54.
• The total cost for 3 bats and 4 balls is calculated as follows: 3 x 176 + 4 x 54 = 528 + 216 = 744.

Ans: (b)

• First, calculate the number of boys in the school: 4620 total students - 2075 girls = 2545 boys.
• Next, find the number of girls going on the trip: 24% of 2075 = 0.24 x 2075 = 498.
• Then, calculate the number of boys going on the trip: 20% of 2545 = 0.20 x 2545 = 509.
• Thus, the number of boys and girls going on the trip is 509 and 498, respectively.

Ans: (d)

• To find the average height, you need to add all the heights together and then divide by the number of students.
• The total height is 5.4 + 4.8 + 4.6 + 3.8 + 5.7 + 3.6 + 4.2 + 4.7 + 3.8 + 3.2 = 43.8 feet.
• Now, divide this total by the number of students, which is 10: 43.8 / 10 = 4.38 feet.
• Thus, the average height of the students is 4.38 feet.

Ans: (a)

• To find the number of marbles left, we need to subtract the number of marbles Ranjana takes from the total number of marbles.
• The total marbles are 6xy - 4x² - y² + 5 and the marbles taken are x² - 3xy + 7y² - 2.
• Performing the subtraction: (6xy - 4x² - y² + 5) - (x² - 3xy + 7y² - 2) gives us -5x² - 8y² + 9xy + 7.
• Thus, the number of marbles left is (-5x² - 8y² + 9xy + 7).

Ans: (d)

• To find the length of the ladder, we can use the Pythagorean theorem. The distance from the boy to the wall (3 meters) and the height of the window (4 meters) form a right triangle with the ladder as the hypotenuse.
• According to the theorem, the length of the ladder (hypotenuse) can be calculated as: length = √(height² + distance²).
• Substituting the values: length = √(4² + 3²) = √(16 + 9) = √25 = 5 meters.
• Thus, the length of the ladder needed to reach the window is 5 meters.

Ans: (a)

• Let Ram's age be represented by x.
• According to the problem, Ram's father's age is 3 years more than twice Ram's age, which can be expressed as 2x + 3.
• Since Ram's father is 45 years old, we can set up the equation: 2x + 3 = 45.
• This equation allows us to solve for Ram's age.

Ans: (b)

• To find the number of medium-sized boxes, first, we need to understand the total parts in the ratio 2:8:5. This adds up to 2 + 8 + 5 = 15 parts.
• Next, we calculate the value of one part by dividing the total number of boxes (255) by the total parts (15): 255 ÷ 15 = 17.
• Now, to find the number of medium-sized boxes, we multiply the number of parts for medium boxes (8) by the value of one part: 8 x 17 = 136.
• Thus, the number of medium-sized boxes is 136.

Ans: (b)

=−(2×2×2×3) =−24

Ans: (c)
According to the definition of Armstrong number,
1³+5³+3³=1+125+27
=126+27=153

Ans: (C)

• The median is the middle value when the numbers are arranged in order. In this case, the median is greater than the mode, which is the most frequently occurring number.
• The mode here is 132, as it appears most often in the list.
• The mean is the average, but the question focuses on the relationship between median and mode.
• Thus, the correct statement is that the median of the observations is indeed more than the mode.

Ans: (b)

• Statement (i) is true because when you multiply a whole number (like 2) with a rational number (like 1/2), the result (1) is still a rational number.
• Statement (ii) is false because not all rational numbers are fractions; for example, the number 2 is rational but not a fraction.
• Statement (iii) is false because multiplying a rational number (like 1/2) by an integer (like 2) gives a rational number (1), not an integer.

Ans: (d)

• Statement I: To find the cost price, we can use the formula: Selling Price = Cost Price - Loss. Here, if the selling price is ₹24,700 and the loss is 5%, the cost price is actually ₹26,000, not ₹28,500. So, this statement is false.
• Statement II: 3% of 1 hour (which is 3600 seconds) is calculated as 0.03 x 3600 = 108 seconds. This statement is true.
• Since Statement I is false and Statement II is true, the correct option is (d).