Class 11 maths questions.

1. Ananya observes the top of a transmission tower from a point on level ground. The angle of elevation is 30°. Walking 40 metres closer to the base along the ground, the angle becomes 60°. What is the height of the tower?

   • 20√3 m
   • 40√3 m
   • 20 m
   • 30√3 m
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   Answer: 20√3 m

   Let height be h. The horizontal difference gives h√3 - h/√3 = 40. Solving 2h/√3 = 40 yields h = 20√3 metres.

   Level 3 of 5

2. A point P lies on the x-axis and is equidistant from the points (2, -5) and (-2, 9). What are the coordinates of point P?

   • (-7, 0)
   • (7, 0)
   • (0, -7)
   • (-6, 0)
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   Answer: (-7, 0)

   Let P be (x, 0). Equating squared distances gives (x - 2)² + 25 = (x + 2)² + 81. Expanding and simplifying yields -4x + 29 = 4x + 85, so 8x = -56, meaning x = -7.

   Level 3 of 5

3. An observer stands on horizontal ground 30 m away from the foot of a vertical transmission tower. The angle of elevation to the top of the tower is 30°. The angle of elevation to the top of a mast mounted on it is 45°. What is the height of the mast?

   • 10(3 - √3) m
   • 10(√3 - 1) m
   • 30(√3 - 1) m
   • 15(√3 - 1) m
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   Answer: 10(3 - √3) m

   Tower height is 30 tan 30° = 10√3 m. Total height is 30 tan 45° = 30 m. Mast height is 30 - 10√3 = 10(3 - √3) m.

   Level 3 of 5

4. Kavita buys 3 science journals and 4 mathematics workbooks for ₹430. Rohan buys 2 science journals and 5 mathematics workbooks of the same type for ₹450. What is the price of one mathematics workbook?

   • ₹70
   • ₹50
   • ₹65
   • ₹80
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   Answer: ₹70

   Subtracting 2 × (3j + 4w = 430) from 3 × (2j + 5w = 450) eliminates j, leaving 7w = 490, so w = ₹70.

   Level 3 of 5

5. The line segment joining point A(-3, 5) and point B(6, -4) is divided internally by point P in the ratio 1 : 2. What are the coordinates of point P?

   • (0, 2)
   • (1, 1)
   • (0, 3)
   • (3, -1)
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   Answer: (0, 2)

   By the section formula, x = (1×6 + 2×(-3)) / (1 + 2) = 0/3 = 0. Similarly, y = (1×(-4) + 2×5) / (1 + 2) = 6/3 = 2. Thus, P is (0, 2).

   Level 3 of 5

6. Two tangents PA and PB are drawn to a circle with centre O from an external point P. If the angle APB measures 70°, what is the measure of angle OAB?

   • 35°
   • 40°
   • 55°
   • 20°
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   Answer: 35°

   Because radii meet tangents at 90°, angle AOB = 180° - 70° = 110°. Triangle OAB is isosceles with OA = OB, making angle OAB = (180° - 110°) / 2 = 35°.

   Level 3 of 5

7. A metallic solid sphere of radius 6 cm is melted and recast into three smaller solid spheres. If the radii of two of these spheres are 3 cm and 4 cm, what is the radius of the third sphere?

   • 5 cm
   • 4.5 cm
   • 5.2 cm
   • 2 cm
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   Answer: 5 cm

   Equating total volumes gives R³ = r1³ + r2³ + r3³. Here, 6³ = 3³ + 4³ + r3³, so 216 = 27 + 64 + r3³. Thus r3³ = 125, meaning r3 = 5 cm.

   Level 3 of 5

8. In a school plantation drive, 25 students planted an average of 14 saplings each. Later, five additional students joined, raising the combined average of all 30 students to 15 saplings. How many saplings did the five new students plant?

   • 100
   • 75
   • 120
   • 90
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   Answer: 100

   The original group planted 25 × 14 = 350 saplings. The full group planted 30 × 15 = 450 saplings. The new students contributed 450 - 350 = 100 saplings.

   Level 3 of 5

9. A line segment connects points A(-3, 5) and B(6, -1). The y-axis intersects this line segment at point P. In what internal ratio does the y-axis divide segment AB?

   • 1 : 2
   • 2 : 1
   • 1 : 3
   • 3 : 2
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   Answer: 1 : 2

   Any point on the y-axis has an x-coordinate of 0. Applying the section formula gives (6k - 3)/(k + 1) = 0, which yields k = 1/2, meaning the ratio is 1 : 2.

   Level 3 of 5

10. Preeti has a bag containing 5 red, 4 blue, and 3 green marbles. She draws two marbles one after the other without replacement. What is the probability that both selected marbles have different colours?

    • 47/66
    • 19/66
    • 23/33
    • 35/66
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    Answer: 47/66

    Total pairs equal 12 × 11 = 132. Matching pairs equal 5(4) + 4(3) + 3(2) = 38. Probability of different colours is 1 - 38/132 = 47/66.

    Level 3 of 5

11. A quadratic equation is given by (k - 1)x² + 2(k - 1)x + 1 = 0, where k does not equal 1. For which value of k does the equation have two real and equal roots?

    • k = 2
    • k = 3
    • k = 0
    • k = -1
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    Answer: k = 2

    Equal roots require discriminant b² - 4ac = 0. Thus 4(k - 1)² - 4(k - 1) = 4(k - 1)(k - 2) = 0. Since k ≠ 1, k = 2.

    Level 3 of 5

12. If the roots of the quadratic equation x² - px + q = 0 differ by 1, which relation between p and q must hold true?

    • p² = 4q + 1
    • p² = 4q - 1
    • p² = 2q + 1
    • p² = q² + 1
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    Answer: p² = 4q + 1

    Let the roots be α and β. Then α + β = p and αβ = q. Since (α - β)² = 1, expanding gives (α + β)² - 4αβ = 1, so p² - 4q = 1.

    Level 3 of 5
13. True or false?

    Doubling the radius of a solid sphere increases its total surface area by a factor of eight.

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    Answer: False

    Surface area is proportional to the square of the radius. Doubling the radius multiplies the surface area by two squared, which is four, not eight.

    Level 3 of 5
14. True or false?

    The sum of any rational number and any irrational number is an irrational number.

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    Answer: True

    If the sum were rational, subtracting the original rational number would yield a rational result. This contradicts the fact that the added number is irrational.

    Level 3 of 5
15. True or false?

    If two events in a random experiment have non-zero probabilities and are mutually exclusive, they are independent events.

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    Answer: False

    For mutually exclusive events, P(A ∩ B) = 0. For independent events with positive probabilities, P(A ∩ B) = P(A)P(B) > 0. Therefore, they cannot be independent.

    Level 3 of 5
16. Fill in the blank.

    In an arithmetic progression studied by Aarav, the first term is 7 and the common difference is 4, making the 11th term ___.

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    Answer: 47

    The nth term of an arithmetic progression is a + (n - 1)d. Substituting the values gives 7 + (11 - 1) × 4 = 7 + 40 = 47.

    Level 3 of 5
17. Fill in the blank.

    On a survey grid of Jaipur, the straight-line distance between the landmark at coordinates (0, 0) and a post at (9, 12) is ___ units.

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    Answer: 15

    By applying the distance formula, the distance equals √(9² + 12²) = √(81 + 144) = √225 = 15 units.

    Level 3 of 5
18. Fill in the blank.

    For the quadratic equation x^2 - 6x + k = 0 to possess two equal real roots, the value of k must be ___.

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    Answer: 9

    For equal real roots, the discriminant b^2 - 4ac must equal zero, so (-6)^2 - 4(1)(k) = 0, giving k = 9.

    Level 3 of 5

19. Which real-valued function is the odd one out regarding its monotonicity across all real numbers?

    • f(x) = 2x + 5
    • f(x) = eˣ
    • f(x) = x³
    • f(x) = x²
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    Answer: f(x) = x²

    The other three functions are strictly increasing over the entire real line, whereas f(x) = x² decreases for negative inputs and increases for positive inputs.

    Level 3 of 5

20. Which three-dimensional vector is the odd one out?

    • (1/√3)i + (1/√3)j + (1/√3)k
    • (3/5)i + (4/5)j
    • (1/√2)i - (1/√2)k
    • i + j + k
    Show the answer

    Answer: i + j + k

    The first three are unit vectors with a magnitude equal to 1. The vector i + j + k has a magnitude of √3.

    Level 3 of 5

Olympiads for Class 11 · Maths olympiad prep