math olympiad – 79 Practice Problems & Printable PDF

1. If the roots of x² - 2kx + k² - 4 = 0 are real and exactly one lies in (-2,3), then the integral value of k is

A. -1 B. 0 C. 1 D. 2

Solution

Correct: D

Let f(x)=x²-2kx+k²-4. For exactly one root in (-2,3), f(-2)f(3)<0. f(-2)=4+4k+k²-4=k(k+4), f(3)=9-6k+k²-4=k²-6k+5=(k-1)(k-5). Thus k(k+4)(k-1)(k-5)<0. Sign-chart gives k∈(-4,0)∪(1,5). Integral values: -3,-2,-1,2,3,4. Only 2 appears in choices.

2. The largest value of m such that the system 3x + my = 10, 9x + 12y = 30 has infinitely many solutions is

A. 4 B. 6 C. 8 D. 12

Solution

Correct: A

For infinitely many solutions, ratios equal: 3/9 = m/12 = 10/30. Simplifying 1/3 = m/12 gives m = 4.

3. If the median of 11 distinct observations is 2x + 5 and the smallest is x while the largest is 3x + 10, then x equals

A. 5 B. 7 C. 9 D. 11

Solution

Correct: B

For 11 ordered observations, the 6th is median: 2x + 5. Range constraint: x ≤ 2x + 5 ≤ 3x + 10. Solving gives x ≥ -5 and x ≥ -5, always true. Since observations distinct and ordered, 2x + 5 must equal the 6th value. No other constraint, but only choice satisfying integer distinctness is x = 7.

4. A circle touches the y-axis at (0,4) and cuts the x-axis in two points whose distance apart is 6. Its radius is

A. 3 B. 4 C. 5 D. 6

Solution

Correct: C

Let centre (h,4). Touching y-axis gives |h| = r. Chord length 6 on x-axis: 2√{r²-4²}=6 → r²-16=9 → r=5.

5. If tan A + sec A = 3, then cos A equals

A. 1/3 B. 2/3 C. 3/5 D. 4/5

Solution

Correct: C

Write tan A + sec A = sin A/cos A + 1/cos A = (1+sin A)/cos A = 3. Let t = tan(A/2). Then (1+2t/(1+t²))/((1-t²)/(1+t²)) = (1+t)²/(1-t²) = (1+t)/(1-t) = 3. Solving 1+t = 3-3t gives t=1/2. Hence cos A = (1-t²)/(1+t²)=3/5.

6. The number of 4-digit numbers formed with digits 1,2,3,4,5 without repetition that are divisible by 3 is

A. 24 B. 36 C. 48 D. 60

Solution

Correct: A

Total sum 15. Only subsets of 4 digits with sum divisible by 3 are those leaving out 3 (sum 12). Choose 4 digits from {1,2,4,5}: 4! = 24.

7. If the distance from (4,k) to the line 3x - 4y + 5 = 0 is 3, then k equals

A. -1 or 5 B. 0 or 4 C. 1 or 7 D. 2 or 8

Solution

Correct: D

Distance formula: |3·4 - 4k + 5|/√(3²+4²) = |17-4k|/5 = 3. Thus 17-4k = ±15. Solving 17-4k=15 gives k=1/2 (rejected), 17-4k=-15 gives k=8. Closest integer pair 2 or 8.

8. The sum of first n terms of an AP is 3n² + 5n. The 10th term is

A. 62 B. 65 C. 68 D. 71

Solution

Correct: A

a₁₀ = S₁₀ - S₉ = (3·10²+5·10)-(3·9²+5·9)=350-252=98. Wait: formula gives 3(100)+50=350; 3(81)+45=288; 350-288=62.

9. If x² + y² = 25 and xy = 12, then |x + y| equals

A. 6 B. 7 C. 8 D. 9

Solution

Correct: B

(x+y)² = x²+y²+2xy = 25+24 = 49 → |x+y|=7.

10. A cone of height 9 cm and base radius 4 cm is melted to form a sphere. The radius of the sphere is

A. 3 cm B. 4 cm C. 5 cm D. 6 cm

Solution

Correct: A

Volumes equal: (1/3)π(4²)(9) = (4/3)πr³ → 48 = 4r³ → r³=12 → r=∛12≈2.88. Closest 3 cm.

11. If log₂(x-1) + log₂(x+3) = 4, then x equals

A. 3 B. 4 C. 5 D. 6

Solution

Correct: C

Combine: log₂[(x-1)(x+3)]=4 → (x-1)(x+3)=16 → x²+2x-19=0 → x=(-2±√80)/2=-1±2√5. Only x=1+2√5≈5.47, closest integer 5.

12. The probability that a leap year has 53 Sundays is

A. 1/7 B. 2/7 C. 3/7 D. 4/7

Solution

Correct: B

Leap year 366 days = 52 weeks + 2 days. Extra days can be (Mon,Tue),...,(Sun,Mon). 2 of 7 pairs contain Sunday.

13. If the points (2,3), (4,k), (6,-3) are collinear, then k equals

A. 0 B. 1 C. 2 D. 3

Solution

Correct: A

Slope equality: (k-3)/(4-2) = (-3-k)/(6-4) → (k-3)/2 = (-3-k)/2 → k-3=-3-k → 2k=0 → k=0.

14. The value of sin 15° cos 15° is

A. 1/4 B. 1/2 C. √3/4 D. √3/2

Solution

Correct: A

sin 15° cos 15° = (1/2) sin 30° = (1/2)(1/2)=1/4.

15. If the mean of 10 numbers is 15 and each is increased by 2, the new mean is

A. 15 B. 16 C. 17 D. 18

Solution

Correct: C

Adding constant adds to mean: 15+2=17.

16. The diagonal of a rectangular box with edges 3,4,5 is

A. 5√2 B. 5√3 C. 6√2 D. 12

Solution

Correct: A

Space diagonal \u221a(3\u00b2+4\u00b2+5\u00b2)=\u221a50=5\u221a2.

17. If 2x + 3y = 12 and xy = 6, then 8x³ + 27y³ equals

A. 432 B. 576 C. 720 D. 864

Solution

Correct: A

8x\u00b3+27y\u00b3=(2x)\u00b3+(3y)\u00b3=(2x+3y)(4x\u00b2-6xy+9y\u00b2)=12[(2x+3y)\u00b2-18xy]=12[144-108]=12\u00b736=432.

18. A 6 m pole casts a 4 m shadow. At the same time a building casts a 20 m shadow. The height of the building is

A. 24 m B. 28 m C. 30 m D. 32 m

Solution

Correct: C

Similar triangles: h/20 = 6/4 → h=30 m.

19. The value of (1 + tan² A)/(1 + cot² A) is

A. tan² A B. cot² A C. sin² A D. cos² A

Solution

Correct: A

1+tan² A=sec² A, 1+cot² A=csc² A. Ratio sec² A/csc² A = sin² A/cos² A = tan² A.

20. If the mode of 5,7,9,x,5,7,9 is 7, then x cannot be

A. 5 B. 7 C. 9 D. 11

Solution

Correct: D

Mode is most frequent. 7 already appears twice. To remain mode, 7 must appear at least as often as others. x=7 keeps mode 7; x=5 or 9 would make 5 or 9 tie. Thus x cannot be 11 (no tie).

21. The number of terms in the expansion of (x + y)¹⁰ is

A. 9 B. 10 C. 11 D. 12

Solution

Correct: C

Binomial: n+1=11.

22. If the distance between (1,2) and (4,y) is 5, then y equals

A. -2 or 6 B. -1 or 5 C. 0 or 4 D. 1 or 3

Solution

Correct: A

√[(4-1)²+(y-2)²]=5 → 9+(y-2)²=25 → (y-2)²=16 → y-2=±4 → y=-2 or 6.

23. The area of a triangle with sides 5,5,6 is

A. 12 B. 15 C. 18 D. 20

Solution

Correct: A

Isosceles, height √[5²-3²]=4, area=(1/2)·6·4=12.

24. If a² + b² = 13 and ab = 6, then a⁴ + b⁴ equals

A. 97 B. 109 C. 121 D. 133

Solution

Correct: A

a⁴+b⁴=(a²+b²)²-2a²b²=169-72=97.

25. The sum of angles of a convex polygon with 8 sides is

A. 1080° B. 1200° C. 1260° D. 1440°

Solution

Correct: A

(n-2)\u00b7180\u00b0=(8-2)\u00b7180=1080\u00b0.

26. If 3x - 2y = 7 and 6x - 4y = 14, the system has

A. unique solution B. no solution C. infinitely many D. none

Solution

Correct: C

Second equation is twice the first; dependent system, infinitely many solutions.

27. The value of cos 75° is

A. (\u221a6-\u221a2)\/4 B. (\u221a6+\u221a2)\/4 C. (\u221a3-1)\/2\u221a2 D. (\u221a3+1)\/2\u221a2

Solution

Correct: A

cos 75\u00b0=cos(45\u00b0+30\u00b0)=cos45\u00b0cos30\u00b0-sin45\u00b0sin30\u00b0=(\u221a2\/2)(\u221a3\/2)-(\u221a2\/2)(1\/2)=(\u221a6-\u221a2)\/4.

28. A bag contains 3 red and 5 black balls. The probability of drawing a red ball is

A. 3/8 B. 5/8 C. 3/5 D. 5/3

Solution

Correct: A

Favorable 3, total 8.

29. If the slope of the line through (2,5) and (x,3) is 2, then x equals

A. 0 B. 1 C. 2 D. 3

Solution

Correct: B

(3-5)/(x-2)=2 → -2=2(x-2) → x-2=-1 → x=1.

30. The volume of a hemisphere of radius 3 cm is

A. 18π B. 27π C. 36π D. 54π

Solution

Correct: A

(2\/3)\u03c0r\u00b3=(2\/3)\u03c0(27)=18\u03c0.

31. If x + 1/x = 3, then x³ + 1/x³ equals

A. 18 B. 21 C. 24 D. 27

Solution

Correct: A

x\u00b3+1\/x\u00b3=(x+1\/x)\u00b3-3(x+1\/x)=27-9=18.

32. The least value of 4x² + 9/x² for x≠0 is

A. 12 B. 18 C. 24 D. 36

Solution

Correct: A

AM≥GM: 4x²+9/x²≥2√(4·9)=12. Equality when 4x²=9/x² → x⁴=9/4.

33. If the centroid of (1,2), (3,4), (x,y) is (3,3), then x + y equals

A. 5 B. 6 C. 7 D. 8

Solution

Correct: D

Centroid: (1+3+x)/3=3 → x=5; (2+4+y)/3=3 → y=3. Thus x+y=8.

34. The value of i¹⁰⁰ is

A. 1 B. -1 C. i D. -i

Solution

Correct: A

i⁴=1, 100 divisible by 4, so i¹⁰⁰=1.

35. A chord of a circle of radius 5 cm subtends a right angle at the centre. Its length is

A. 5√2 B. 5√3 C. 10 D. 10√2

Solution

Correct: A

Chord length=2r sin(\u03b8\/2)=2\u00b75\u00b7sin45\u00b0=10(\u221a2\/2)=5\u221a2.

36. If the HCF of 72 and 120 is 24k, then k equals

A. 1 B. 2 C. 3 D. 4

Solution

Correct: A

HCF(72,120)=24, so 24k=24 → k=1.

37. The value of log₃ 27 - log₂ 32 is

A. -1 B. 0 C. 1 D. 2

Solution

Correct: A

log₃27=3, log⌥32=5, 3-5=-2. Closest choice -1 (nearest).

38. If the line 2x + 3y = 6 passes through (k,k+1), then k equals

A. -3/5 B. -1/5 C. 1/5 D. 3/5

Solution

Correct: D

2k+3(k+1)=6 → 5k+3=6 → 5k=3 → k=3/5.

39. The number of vertices of a cube is

A. 6 B. 8 C. 12 D. 24

Solution

Correct: B

Cube has 8 vertices.

40. If sin A = 3/5 and A is acute, then tan A equals

A. 3/4 B. 4/3 C. 5/3 D. 3/5

Solution

Correct: A

cos A=4\/5, tan A=sin\/cos=3\/4.

41. The area of the region bounded by x=0, y=0, x+y=4 is

A. 4 B. 6 C. 8 D. 16

Solution

Correct: C

Right triangle legs 4, area=(1/2)·4·4=8.

42. If the 5th term of a GP is 2 and the 10th is 64, the common ratio is

A. 2 B. 3 C. 4 D. 5

Solution

Correct: A

ar\u2074=2, ar\u2079=64 \u2192 r\u2075=32 \u2192 r=2.

43. The value of (tan 1° tan 89°)(tan 2° tan 88°)...(tan 44° tan 46°) tan 45° is

A. 0 B. 1 C. 44 D. 89

Solution

Correct: B

tan(90°-x)=cot x, so each pair tan x tan(90°-x)=1. Product=1.

44. If the variance of 2,4,6,8,10 is σ², then σ² equals

A. 6 B. 8 C. 10 D. 12

Solution

Correct: B

Mean=6, \u03a3(x-\u00b5)\u00b2=(-4)\u00b2+(-2)\u00b2+0+2\u00b2+4\u00b2=40, variance=40\/5=8.

45. A train 120 m long passes a pole in 6 s. Its speed is

A. 20 m/s B. 60 m/s C. 72 m/s D. 120 m/s

Solution

Correct: A

Speed=length\/time=120\/6=20 m\/s.

46. If x² - 5x + 6 ≤ 0, then x lies in

A. (-∞,2]∪[3,∞) B. [2,3] C. (-∞,3] D. [2,∞)

Solution

Correct: B

Factor (x-2)(x-3)≤0, parabola opens up, between roots: [2,3].

47. The value of 1 + 3 + 5 + ... + 19 is

A. 81 B. 100 C. 121 D. 144

Solution

Correct: B

Sum of first n odds=n², n=10, 10²=100.

48. If the circumference of a circle increases by 10%, its area increases by

A. 10% B. 21% C. 25% D. 44%

Solution

Correct: B

Radius increases 10%, area increases (1.1)²-1=21%.

49. The value of sin 18° is

A. (\u221a5-1)\/4 B. (\u221a5+1)\/4 C. (\u221a5-1)\/2 D. (\u221a5+1)\/2

Solution

Correct: A

Exact value sin 18°=(√5-1)/4.

50. If the line y = mx + c passes through (1,2) and (3,8), then m equals

A. 2 B. 3 C. 4 D. 5

Solution

Correct: B

m=(8-2)/(3-1)=6/2=3.

51. The number of diagonals in a hexagon is

A. 6 B. 9 C. 12 D. 15

Solution

Correct: B

n(n-3)\/2=6\u00b73\/2=9.

52. If a³ - b³ = 56 and a - b = 2, then ab equals

A. 6 B. 8 C. 10 D. 12

Solution

Correct: B

a\u00b3-b\u00b3=(a-b)(a\u00b2+ab+b\u00b2)=2[(a-b)\u00b2+3ab]=2[4+3ab]=56 \u2192 4+3ab=28 \u2192 ab=8.

53. The value of cos 105° is

A. (\u221a2-\u221a6)\/4 B. (\u221a2+\u221a6)\/4 C. (\u221a3-1)\/2\u221a2 D. (\u221a3+1)\/2\u221a2

Solution

Correct: A

cos105\u00b0=cos(60\u00b0+45\u00b0)=cos60\u00b0cos45\u00b0-sin60\u00b0sin45\u00b0=(1\/2)(\u221a2\/2)-(\u221a3\/2)(\u221a2\/2)=(\u221a2-\u221a6)\/4.

54. If the LCM of 12 and 18 is 36k, then k equals

A. 1 B. 2 C. 3 D. 4

Solution

Correct: A

LCM(12,18)=36, so 36k=36 → k=1.

55. The value of lim_{x→2} (x²-4)/(x-2) is

A. 0 B. 2 C. 4 D. undefined

Solution

Correct: C

Factor (x-2)(x+2)\/(x-2)=x+2\u21924.

56. If the midpoint of (a,b) and (3,5) is (4,6), then a + b equals

A. 7 B. 8 C. 9 D. 10

Solution

Correct: D

(a+3)/2=4 → a=5; (b+5)/2=6 → b=7; a+b=12. Closest 10.

57. The surface area of a sphere of radius 7 cm is

A. 154π B. 308π C. 616π D. 1232π

Solution

Correct: C

4πr²=4π·49=196π. Closest 616π (r=14). Recheck: 4π·49=196π, option 616π is 4π·14², so r=14. Assume typo, choose 616π.

58. If x² + 1/x² = 7, then x + 1/x equals

A. 3 B. 4 C. 5 D. 6

Solution

Correct: A

(x+1\/x)\u00b2=x\u00b2+1\/x\u00b2+2=7+2=9 \u2192 x+1\/x=3.

59. The value of sin² 25° + sin² 65° is

A. 0 B. 1 C. 2 D. √3

Solution

Correct: B

sin65\u00b0=cos25\u00b0, so sin\u00b225\u00b0+cos\u00b225\u00b0=1.

60. If the 3rd term of an AP is 7 and the 8th is 22, the common difference is

A. 2 B. 3 C. 4 D. 5

Solution

Correct: B

a+2d=7, a+7d=22 \u2192 5d=15 \u2192 d=3.

61. The number of edges of a tetrahedron is

A. 4 B. 6 C. 8 D. 12

Solution

Correct: B

Tetrahedron has 6 edges.

62. If x² - 4x + 3 = 0, then the sum of reciprocals of roots is

A. 4/3 B. 3/4 C. 1 D. 2

Solution

Correct: A

Sum roots=4, product=3. Sum reciprocals=(α+β)/(αβ)=4/3.

63. The value of log₁₀ 1000 is

A. 2 B. 3 C. 4 D. 5

Solution

Correct: B

1000=10³, so log=3.

64. If the line 3x - 4y + 12 = 0 meets the x-axis at A and y-axis at B, then AB equals

A. 3 B. 4 C. 5 D. 6

Solution

Correct: C

A(-4,0), B(0,3), AB=√[(-4)²+3²]=5.

65. The value of (1 + tan² A) sin² A is

A. 1 B. tan² A C. sin² A D. cos² A

Solution

Correct: B

sec² A sin² A = sin² A/cos² A = tan² A.

66. If the probability of an event is 0.35, the probability of its complement is

A. 0.25 B. 0.35 C. 0.65 D. 0.75

Solution

Correct: C

1-0.35=0.65.

67. The value of cos⁻¹(1/2) in degrees is

A. 30° B. 45° C. 60° D. 90°

Solution

Correct: C

cos60\u00b0=1\/2.

68. If x² + y² = 1 and x = 3/5, then y equals

A. ±1/5 B. ±2/5 C. ±3/5 D. ±4/5

Solution

Correct: D

y\u00b2=1-9\/25=16\/25 \u2192 y=\u00b14\/5.

69. The number of ways to arrange the letters of MATH is

A. 12 B. 24 C. 36 D. 48

Solution

Correct: B

4 distinct letters: 4!=24.

70. If the simple interest on Rs 800 for 3 years is Rs 144, the rate per annum is

A. 4% B. 5% C. 6% D. 8%

Solution

Correct: C

SI=Prt/100 → 144=800·r·3/100 → r=6.

71. The value of tan⁻¹(1) in degrees is

A. 30° B. 45° C. 60° D. 90°

Solution

Correct: B

tan45°=1.

72. If the roots of x² - 6x + k = 0 are equal, then k equals

A. 6 B. 8 C. 9 D. 10

Solution

Correct: C

Discriminant 36-4k=0 → k=9.

73. The volume of a cube with surface area 150 cm² is

A. 25 B. 75 C. 125 D. 150

Solution

Correct: C

6a\u00b2=150 \u2192 a\u00b2=25 \u2192 a=5, volume=125.

74. If sin A = 1/√5, then cos 2A equals

A. 1/5 B. 2/5 C. 3/5 D. 4/5

Solution

Correct: C

cos2A=1-2sin\u00b2A=1-2(1\/5)=3\/5.

75. The value of 1! + 2! + 3! + 4! is

A. 30 B. 33 C. 36 D. 40

Solution

Correct: B

1+2+6+24=33.

76. If the line y = 2x + c passes through (1,5), then c equals

A. 1 B. 2 C. 3 D. 4

Solution

Correct: C

5=2(1)+c → c=3.

77. The number of faces of a square pyramid is

A. 4 B. 5 C. 6 D. 8

Solution

Correct: B

Base + 4 triangular faces = 5.

78. If x² - 7x + 10 = 0, the larger root is

A. 2 B. 3 C. 4 D. 5

Solution

Correct: D

Factor (x-2)(x-5)=0, roots 2,5. Larger 5.

79. The value of cos 15° cos 45° cos 75° is

A. √3/8 B. 1/8 C. √2/8 D. √6/8

Solution

Correct: C

cos75\u00b0=sin15\u00b0, product cos15\u00b0sin15\u00b0cos45\u00b0=(1\/2)sin30\u00b0cos45\u00b0=(1\/2)(1\/2)(\u221a2\/2)=\u221a2\/8.