Unicus Mathematics Olympiad (UMO) Class 10 Sample Paper

Sample Paper
(2020-21)

Class 10
Unicus Mathematics Olympiad

Section – Class* Total Marks per Total
*Syllabus covered Questions Questions Questions
Classic Section – Class 9 28 1 28
Classic Section – Class 8 12 1 12
Scholar Section – Class 9 7 2 14
Scholar Section – Class 8 3 2 6
Grand Total 50 60

1. The ratio of milk and water is 3: 2 in a milkman's container. He adds some water in the
solution and the volume of the content increases by 50%. Then he replaces 30 L of the
resultant solution with. The resultant ratio of milk to water in the final solution is 3:7.
Find the original volume of the solution:

a) 80 L b) 130 L
c) 65 L d) 160 L

Correct Answer: a 1 Mark

2. The given figure ABCD is a parallelogram. If ∠QPB = 84° and ∠AQP is one-fourth of
∠QPB, then find ∠AQD:

a) 49° b) 55°
c) 63° d) 72°

Correct Answer: c 1 Mark

3. The sum of the present ages of a person and his daughter is 42 years. Six years hence
the ratio of their ages will be 7: 2. Find the difference between their present ages:

a) 38 years b) 30 years
c) 24 years d) 20 years

Correct Answer: b 1 Mark

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4. Shivam went to the market and bought an iron sheet. Using this sheet he made a
cylindrical tube open at both ends. The outer diameter is 16 cm and its length is 100
cm. Find how many cubic cm of iron has been used in making the tube if the iron sheet
was 2 cm thick?

a) 1,000 cm3 b) 2,200 cm3
c) 4,400 cm3 d) 8,800 cm3

Correct Answer: d 1 Mark

5. Look at the following polynomial given below:
f(x) = x4 + 3x3 + 2x2 – ax + 3a + 7
If it is divided by x + 1, then it leaves the remainder 19. Find the value of a and also find
the remainder when p(x) is divided by x - 2:

a) 3, 58 b) 5, 31
c) 10, 62 d) 12, 72

Correct Answer: a 1 Mark

6. The members of a company went for a picnic. They collected some money for picnic by
contribution. The money contributed by each member was equal to the cube of the total
number of members. If the total collected amount was Rs. 74,088, then find the total
number of members:

a) 22 b) 28
c) 42 d) 58

Correct Answer: c 1 Mark

7. Jai’s present age is 2/7th of his father’s present age. Jai’s brother is 3 years older than
Jai. The respective ratio between the present ages of Jai’s father and Jai’s brother is
14: 5. Find Jai’s present age:

a) 6 year b) 12 year
c) 18 year d) 24 year
Correct Answer: b 1 Mark

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8. Look at the statements given below:
I. Point (3, 0) lies in the first quadrant.
II. The coordinates of point whose ordinate is -1/2 and abscissa is 1 are (-1/2, 1).
III. If we multiply or divide both sides of a linear equation with a non-zero number, then
the solution of the linear equation changes in case of multiplication only.
IV. If angles A, B, C and D of a quadrilateral ABCD, taken in order are in the ratio 3: 7:
6: 4, then ABCD is a trapezium.

Which of the following statements is/are correct?

a) Only I and III b) Only II
c) Only II and IV d) Only IV

Correct Answer: d 1 Mark

9. Let PQRS is a parallelogram. Let x and y be positive integers such that y < x < 2y. Let
PR = 2xy and QS = x2 - y2 and PQ = (x2 + y2)/2?

Statement I. PR > QS
Statements II. PQRS is a rhombus.

Which one of the following is correct in respect of the above statements?
a) Both the statement I and II are true and statement II is the correct explanation of
statement I.
b) Both the statement I and II are true but statement II is not the correct explanation of
statement I.
c) Statement I is true but statement II is false.
d) Statement II is true but statement I is false.
Correct Answer: b 2 Marks

10. Arnav deposits some money in his account. He earns Rs. 540 as simple interest in 3
Years. If he earns a compound interest of Rs. 376.20 at the same rate of interest in 2
years, then find the amount:

a) Rs. 1,600 b) Rs. 1,800
c) Rs. 2,000 d) Rs. 2,100

Correct Answer: c 2 Marks

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