Mathematics Olympiad Model Test Paper - 1

Ans: (a) (2, 7)

Ans: (c)

• To solve this problem, we need to look at the words represented by the numbers and determine their alphabetical order.
• Identify the words corresponding to each number and then arrange them from A to Z.
• After arranging, we find that the correct sequence is represented by option (c) which is 2, 5, 1, 3, 4.
• This means that option (c) correctly reflects the dictionary order of the words.

Ans: (a)

• First, we need to replace the symbols according to the given rules:
• 115 × 17 becomes 115 + 17 = 132.
• 132 ÷ 81 becomes 132 - 81 = 51.
• 51 - 3 becomes 51 ÷ 3 = 17.
• Finally, 17 + 4 becomes 17 × 4 = 68.
• However, we need to check the calculations again as we are looking for the final answer.
• After recalculating, we find that the correct final value is 24.

Ans: (c)
Resolving 3888 into its prime factors, we find that
3888 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3
3888 = (2 × 2) × (2 × 2) × (3 × 3) × (3 × 3)
Here we find that prime factor 3 is appearing alone.
So, if we divide 3888 by 3, we will get a perfect square number.
∴ 1296 = (2 × 2) × (2 × 2) × (3 × 3) × (3 × 3)

Ans: (b)
Let one number = a 
∴ Second number = 4
⇒4a×a=1936
⇒a²=484
a2=(2×2)×(11×11)
⇒a=2×11=22
and 4 𝑎 = 4 × 22 = 88
4a=4×22=88
∴ Numbers are 22 and 88.

Ans: (d)
Least number which is divisible by 4, 6, 10, 15 is LCM (4, 6, 10, 15)
LCM (4, 6, 10, 15) = 2 × 3 × 5 × 2
LCM (4, 6, 10, 15) = 60
60 = 2 × 2 × 3 × 5

Here we find that 3 and 5 occurs alone. So, if we multiply 60 by
3 × 5 = 15, we get a perfect square number.
∴ 60 × 3 × 15 = 900
900 is the least square number which is divisible by 4, 6, 10, 15.

Ans: (c)
Let the number of students sitting in a row = 'x'
∴ Number of rows in auditorium = 'x'
∴ Number of students sitting in auditorium
= x x x = x²
Now x² = 7396 Number of rows in auditorium = 86.

Ans: (b)

24136 > 155²
∴ Least number must be added to 24136 to make it a perfect square
(156)² -24136=24336-24136=200

Ans: (b)

Ans: (b) We can clearly see that the location of both points is different.

Ans: (a)
(-5, 4)

Ans: (c)

As 

Ans: (d)

• The statement indicates that the woman is the mother of Shashank's son's only daughter.
• This means she is the wife of Shashank's son, making her Shashank's daughter-in-law.
• Thus, the correct relationship is that she is Shashank's daughter-in-law.
• Other options do not fit the relationship described in the statement.

Ans: (c)
Resolving 3456 as a product of prime factors, we get

Clearly, to make it a perfect cube it should be multiplied by 2×2=4

Ans: (a)
x-axis shows time (in hours) and y-axis shows temperature (in °F).

Ans: (b)
So 

Ans: (a)

• The first 8 prime numbers are 2, 3, 5, 7, 11, 13, 17, and 19.
• To find the mean, add these numbers together: 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 = 77.
• Then, divide the total by the number of prime numbers, which is 8: 77 ÷ 8 = 9.625.
• Thus, the mean of the first 8 prime numbers is 9.625.

Ans: (c) 4 from the x-axis and 3 from the y-axis.

Ans: (d) Total watches sold
= (20 + 50 + 20 + 60 + 30) = 180

Ans: (b)

• To find the predecessor of the smallest 5-digit number, we first identify that the smallest 5-digit number is 10000.
• The predecessor of 10000 is 9999.
• Next, we need to determine what to subtract from 66486.67 to reach 9999. This can be calculated as 66486.67 - 9999 = 56487.67.
• Thus, the correct number to subtract is 56487.67, which corresponds to option (b).

Ans: (b)

• The equation requires finding a value of z that makes it true.
• Among the options, 3 is the correct value that satisfies the equation.
• To verify, you can substitute each option into the equation and see which one works.
• In this case, 3 is the only value that meets the condition of the equation.

Ans: (a)

• First, calculate the first part: (-17) x (-4) x (-1) x (-5). Multiplying these negative numbers results in a positive value.
• Next, the second part: (-15) x (-3) x 0 x (-13) x (-11). Since one of the factors is 0, the entire product equals 0.
• Now, we have: (positive value) - 0 = positive value.
• Calculating the first part gives us 340, so the final answer is 340.

Ans: (b)
(b) Let no. of students in the class = x
∴ Contribution of each student = ₹ x
x × x=1024

Ans: (a)
Price of chickens in towns M and N are the same on Tuesday.

Ans: (b)
= 2 x 2 x 2 x 3√2 = 24√2
∴ 2 is the smallest number by which 1152 is  divided.

Ans: (b) Coordinates of R are (1, -2).

Ans: (b)
Let us extract the square root from 24136. 
∴ 24136, is 111 more than (155)2. So if we
Subtract 111 from 24136, we will get a  perfect sq. number. 

Ans: (c)
The coordinates of the point A are (2, 0), and the coordinates of the point C are (4, 0). We can see that line AB lies at an equal distance from the origin (0, 0) and the point C.
Thus, the image of the point C under a reflection in the line AB is the origin, i.e., (0, 0).

Ans: (a)
(a) Let the side of the square field = 'a' m
∴ Area of square field = a² sq. m
= a² =22500 m²
⇒ 𝑎 = 150 m
 Speed of cycling = 15 km/hr
∴ 600 m is covered in
= 2 min 24 sec.

Ans: (b)
Number of Members

Ans: (b)
X-coordinate of point B is -10.

Ans: (a)

Ans: (a)

• A cone has one curved surface and two flat surfaces, making a total of two faces, not three.
• A sphere has one continuous curved surface, which is correct.
• A cube has six flat surfaces, which is also correct.
• A cylinder has two flat circular faces and one curved surface, totaling three faces.

Ans: (d)

Ans: (d)
Total expenditure over the first three months
= (500 + 600 + 300) = 1400

Ans: (b)

• First, calculate the total pages read by Aarav: 285 + 342 = 627 pages.
• Next, find out how many pages are left to read: 1083 - 627 = 456 pages.
• Now, to find the fraction of pages still to read, divide the remaining pages by the total pages: 456/1083.
• Simplifying this fraction gives us 8/19, which is the correct answer.

Ans: (d)

• To find the median, we first need to arrange the data in ascending order: 58, 58, 58, 64, 64, 67, 67, 72, 77.
• Since there are 9 values (an odd number), the median is the middle value, which is the 5th number in this ordered list.
• The 5th number is 64, so the median of the weekly pocket money is 64.
• Thus, the correct answer is (d) 64.

Ans: (c)
Resolve 136 into prime factors and make  group of two of each prime factor
136 = 2 x 2 x 2 x 17 
136 = (2 × 2) × 2 × 17
We find that 2 and 17 doesn't appear in group of two. So, 136 has to be multiplied with 34 to make it a perfect square.
∴ 136 × 2 × 17 = (2 × 2) × (2 × 2) × (17 × 17)
⇒ 4624 = (2 × 2) × (2 × 2) × (17 × 17)
⇒ Required square root = 2 × 2 × 17 = 68

Ans: (c)

• First, add the parts of the ratio: 5 + 4 + 3 = 12 parts.
• Next, determine the number of English questions: (3/12) x 180 = 45 questions.
• To find the percentage of English questions, use the formula: (45/180) x 100 = 25%.
• Thus, the percentage of English questions in the entrance test is 25%.

Ans: (b)

• Let the total height of the pole be H meters.
• According to the problem, one-fourth of the pole is red, which is H/4.
• Two-fifths of the pole is blue, which is 2H/5.
• The remaining height, which is painted green, is given as 21 m.
• So, we can set up the equation: H - (H/4 + 2H/5) = 21.
• To solve this, first find a common denominator for the fractions, which is 20.
• Rewriting the equation gives: H - (5H/20 + 8H/20) = 21.
• This simplifies to: H - 13H/20 = 21, or 7H/20 = 21.
• Multiplying both sides by 20/7 gives H = 60 m.
• Thus, the height of the pole is 60 m.

Ans: (c)

• The minimum distance they should walk is determined by finding the least common multiple (LCM) of their step lengths: 15 cm, 25 cm, and 30 cm.
• The LCM of these numbers is 150 cm, which means each person will take whole steps to cover this distance.
• For example, the person with a 15 cm step will take 10 steps, the one with a 25 cm step will take 6 steps, and the one with a 30 cm step will take 5 steps.
• Thus, 150 cm is the distance that allows all three to walk in complete steps.

Ans: (d)

• Calculate Team A's total score: 45 + (-50) + 75 = 70 points.
• Calculate Team B's total score: (-45) + 50 + 25 = 30 points.
• Compare the scores: Team A scored 70 points, while Team B scored 30 points.
• Determine the difference: 70 - 30 = 40 points.
• Conclusion: Team A scored more than Team B by 40 points.

Ans: (c)

• To find the distance traveled by car, first add the distances traveled by motorbike and train: 84.375 km + 514.260 km = 598.635 km.
• Next, subtract this total from the overall distance to the hilly area: 747.585 km - 598.635 km = 148.950 km.
• Thus, the distance Ashok traveled by car is 148.950 km, which corresponds to option (c).

Ans: (d)
Opposite sides of the parallelogram are equal.
Length of PQ = (abscissa of Q) - (abscissa of P)
= 5 - 1 = 4
Thus, the length of SR = 4
⇒ Abscissa of R is 6
Ordinate of R from X-axis = 3
Thus, the coordinates of R are (6, 3).

Ans: (c)

• Let the number of males be 7x and the number of females be 9x, based on the ratio.
• According to the problem, the number of males is 16 less than the number of females: 7x = 9x - 16.
• Simplifying this gives: 2x = 16, so x = 8.
• Now, the total number of males and females is 7x + 9x = 16x = 16 x 8 = 128.

Ans: (b)

• Statement (i): This is false because while all whole numbers are integers, not all integers are whole numbers (e.g., negative integers).
• Statement (ii): This is true. The product of -35 and 4 is indeed 140.
• Statement (iii): This is also true. -1 is greater than any other negative integer, such as -2 or -3.

Thus, the correct answer is (b) F T T.

Ans: (c)

Ans: (b)

• Statement-I: To check if Mukul's selling price for a 36% profit is correct, we calculate the cost price first. Selling price at 28% profit is ₹448. The cost price is ₹448 / (1 + 0.28) = ₹350. For a 36% profit, the selling price should be ₹350 x 1.36 = ₹476, not ₹480. So, Statement-I is false.
• Statement-II: The increase in marks is from 35 to 42, which is an increase of 7 marks. The percent increase is (7 / 35) x 100 = 20%, not 25%. Therefore, Statement-II is also false.
• Since both statements are incorrect, the correct option is (b).

Ans: (b)

• To solve this question, we need to match the items correctly based on the given pairs.
• Option (b) states that P matches with (i), Q with (iv), R with (ii), and S with (iii), which is the correct combination.
• It's important to carefully analyze each pairing to ensure they align with the criteria provided.
• Thus, option (b) is the right choice as it accurately reflects the correct matches.

Ans: (d)

• To solve this problem, we need to analyze the given options.
• Each option presents a pair of values, where the first value is in square meters (m²) and the second in square centimeters (cm²).
• We need to find the correct pair that fits the criteria of the question.
• Option (d) provides 80 m² and 308 cm², which is the correct answer based on the context of the problem.