Unicus Non-Routine Mathematics Olympiad (UNRMO) Class 10 Sample Paper

Sample Paper
(2020-21)

Class 9 & 10
Unicus Non-Routine Mathematics Olympiad

Section – Class* Total Marks per
Total Marks
*Syllabus covered Questions Question
Classic Section – Class 9 & 10 10 3 30

Scholar Section – Class 9 & 10 10 6 60

Grand Total 20 90
Note: There will be negative marking of 1/3 rd of the marks allotted for that question if the answer is incorrect.

1. The pth term of an A.P is 20 and qth term is 10. Find the sum of first (p+q)th terms.
a) 5(p+q/p-q){(3p-(q-1))}
b) 5(p+q/p-q){(3p-(q+1))}
c) 5(p+q/p-q){(3p-(-q-1))}
d) 5(p+q/p-q){(3p-(1-q))}

Correct Answer: b 3 Marks

2. It α and β are the roots of the equation x 2 - px + q = 0 and α > 0, β > 0. then the value of
α1/4 + β1/4 =
a) [P + √q + 4q1/4 √(P + √q)]4
b) [P + 6√q + 4q1/4 √(P + 2√q)]4
c) [P + √q + 4q1/4 √(P + 4√q)]4
d) [P + 6√q + 4q1/4 √(P + 4√q)]4

Correct Answer: b 3 Marks

3. Let α, β, γ are the roots of x3 + qx + r = 0, then the equation whose roots are β2 + βγ +
γ2; γ2 + γ α + α2 and α2 + αβ + β2 is.
a) (y - q)3 = 0
b) (y + q)3 = 0
c) (y + 2q)3 = 0
d) (y - 2q)3 = 0

Correct Answer: b 3 Marks

4. The area of a square inscribed in a semicircle to the area inscribed in a quadrant of the
same circle.
a) 2 : 1
b) 3 : 2
c) 5 : 3
d) 8 : 5

Correct Answer: d 3 Marks

2

5. BC is the diameter of a semi circle. The sides AB and AC of a triangle ABC meet the
semi circle in p and q respectively. PQ subtends 140 o at the centre of the semi circle
then ∠A = ?
a) 100
b) 200
c) 300
d) 400

Correct Answer: b 3 Marks

6. Let the circum radius of ΔABC be 4 and the in radius of XYZ be 2 of the area of ABC =
32, then area of XYZ = ?

a) 8
b) 16
c) 4
d) 20

Correct Answer: b 3 Marks

7. It cos x + cos2x = 1, then sin12x + 3sin10x + 3sin8x + sin6x =
a) 0
b) √2
c) 1
d) 2

Correct Answer: c 3 Marks

3

8. The angle of elevation of the top of a tower from a point A due south of the tower is x
and from B due east of the tower is y. If AB = h, then calculate the height of the tower.
a) h/√(cot2x + cot2y)
b) h/√(cot2x - cot2y)
c) 2h/√(cot2x - cot2y)
d) 2 tan x/√(cot2x + cot2y)

Correct Answer: a 3 Marks

9. It the point {x1 + t (x2 - x1), y1 + t (y2 - y1)} divides the join of (x1, y1) and (x2, y2) internally
then the condition of t will be.
a) t < 0
b) t = 1
c) 0 ∠ t ∠1
d) None of these

Correct Answer: c 3 Marks

10. If the mean of a frequency distribution is 8.1 and Σ fixi : = 132 + 5x, Σfi = 20, then x = ?
a) 3
b) 4
c) 5
d) 6

Correct Answer: d 3 Marks

11. Solve the equation (x - 1)4 + (x - 5)4 = 82.
a) x = ± 1, 4, 2
b) x = 4, 2, -3 -5i, 2 + i
c) x = 3 ± 5i, 4, 2
d) x = 3 ± 5i, ± 1

Correct Answer: a 6 Marks

4

12. Simplify [∛(6√a9)]4 [6√(∛a9)]4 is
a) a16
b) a12
c) a8
d) a4

Correct Answer: d 6 Marks

13. Given that x6 + 4x5 + 6x4 + 6x3 + 4x2 + 2x + 1 can be factorized as (x2 + ax + 1) (x4 + bx3
+ cx2 + dx + 1) then (a + b) = ?
a) 1
b) 2
c) 3
d) 4

Correct Answer: d 6 Marks

14. Four circles of r = 1, are each tangent of two sides of a square and externally tangent to
a circle of r = 2. It the area of the square is A, then A - 12√2 ?
a) 14
b) 21
c) 22
d) 24

Correct Answer: c 6 Marks

15. Two circle with centres A and B intersect at points P and Q so that ∠PAQ = 60o and
∠PBQ = 90o. What is the ratio of the area of the circle with centre A to the area of the
circle with centre B?
a) 3 : 1
b) 3 : 2
c) 4 : 3
d) 2 : 1

Correct Answer: d 6 Marks

5

16. Square ABCD has an area 4, E is the midpoint of AB. Similarity F, G, H and I are mid
points of DE, CF, DG and CH, then area ΔIDC = ?

a) 1/4
b) 1/8
c) 1/16
d) 1/32

Correct Answer: b 6 Marks

17. It tan θ = 1 - e2, then sec θ + tan3θ cosecθ = ?
a) (1 - e2)3/2
b) (2 - e2)1/2
c) (2 - e2)3/2
d) None of these

Correct Answer: c 6 Marks

18. The value of (1 + cos π/8) (1 + cos 3π/8). (1 + cos 5π/8) (1 + cos 7π/8) is equal to
a) 1/8
b) -1/8
c) 1/4
d) -1/4

Correct Answer: a 6 Marks

6

19. If Sn = ∑ tr = 1/6 n (2n2 + 9n + 13), then ∑ √tr = ?
a) 1/2 n (n + 1)
b) 1/2 n (n + 2)
c) 1/2 n (n + 3)
d) 1/2 n (n + 5)

Correct Answer: c 6 Marks

20. If ui = (xi – 25)/10, Σ fi ui = 20, Σ fi = 100, then 𝑥̅ = ?
a) 23
b) 24
c) 27
d) 25

Correct Answer: c 6 Marks

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