Class 12 practice
Class 12 maths questions.
Maths questions for Class 12, at the level of students aged 15–17, each with its answer and a short explanation. Try each one before opening the answer.
Aarav and Bina can complete a landscaping project together in 12 days. Aarav works alone for the first 4 days and leaves. Bina then finishes the remaining work alone in 20 days. In how many days could Bina complete the entire project on her own?
Level 4 of 5Show the answer
Answer: 24 days
Four days of joint work completes 4/12 = 1/3 of the task. Bina finishes the remaining 2/3 in 16 days alone, meaning she needs 24 days for the full job.
The line segment joining the points P(-2, 3) and Q(6, -5) forms the diameter of a circle. What is the area of this circle in square units?
Level 4 of 5Show the answer
Answer: 32π
The diameter length is √((6 - (-2))² + (-5 - 3)²) = 8√2. The radius is 4√2, so the area equals π(4√2)² = 32π square units.
In a triangle ABC, the angle bisector of angle A meets side BC at point D. If AB = 12 cm, AC = 18 cm, and BC = 15 cm, what is the length of segment BD?
Level 4 of 5Show the answer
Answer: 6 cm
By the Angle Bisector Theorem, BD/DC = AB/AC = 12/18 = 2/3. Since BC = BD + DC = 15 cm, BD = (2/5) * 15 cm = 6 cm.
A bag contains 5 red balls, 4 blue balls, and 3 green balls. If two balls are drawn at random one after another without replacement, what is the probability that both balls have the same colour?
Level 4 of 5Show the answer
Answer: 19/66
Total ways to choose 2 from 12 balls is 66. Favourable outcomes for identical colours are C(5,2) + C(4,2) + C(3,2) = 10 + 6 + 3 = 19. Hence, the probability is 19/66.
If alpha and beta are the roots of the quadratic equation x^2 - 7x + 9 = 0, what is the numerical value of (alpha / beta) + (beta / alpha)?
Level 4 of 5Show the answer
Answer: 31/9
From Vieta's relations, alpha + beta = 7 and alpha * beta = 9. Then (alpha/beta) + (beta/alpha) = (alpha^2 + beta^2)/(alpha*beta) = ((alpha + beta)^2 - 2*alpha*beta)/(alpha*beta) = (49 - 18)/9 = 31/9.
Rohan drops a metallic sphere of radius 6 cm into a cylindrical vessel partially filled with water. The internal diameter of the cylinder is 16 cm. If the sphere is fully submerged, by how many centimetres does the water level rise?
Level 4 of 5Show the answer
Answer: 4.5 cm
The volume displaced equals the sphere's volume: (4/3)*pi*(6^3) = 288*pi cm^3. Vessel radius is 8 cm. Equating pi*(8^2)*h = 288*pi gives 64*h = 288, so h = 4.5 cm.
- True or false?
For any real angle theta, the expression sin(theta) + cos(theta) can attain a maximum value strictly greater than the square root of 2.
Level 4 of 5Show the answer
Answer: False
The expression can be rewritten as sqrt(2) * sin(theta + pi/4). Because the maximum value of the sine function is 1, the upper bound of the entire expression is exactly sqrt(2).
- True or false?
If two events A and B both have positive probabilities and are mutually exclusive, they are also statistically independent.
Level 4 of 5Show the answer
Answer: False
Mutually exclusive events cannot occur together, so P(A ∩ B) = 0. However, independent events with positive probabilities require P(A ∩ B) = P(A)P(B) > 0.
- True or false?
Two distinct circles with radii r₁ and r₂ touch each other externally on a flat plane. The length of their direct common tangent is equal to 2√(r₁r₂).
Level 4 of 5Show the answer
Answer: True
The distance between the centres is r₁ + r₂. Applying Pythagoras' theorem gives the tangent length as √((r₁ + r₂)² - (r₁ - r₂)²), which simplifies to 2√(r₁r₂).
- Fill in the blank.
In a Cartesian plane, if the distance between points P(2, -3) and Q(10, y) is 10 units, where y is positive, then y equals ___.
Level 4 of 5Show the answer
Answer: 3
By the distance formula, (10 - 2)^2 + (y + 3)^2 = 100, which gives 64 + (y + 3)^2 = 100. Thus, (y + 3)^2 = 36, so y + 3 = 6, yielding y = 3.
- Fill in the blank.
If a jar has 5 red marbles and the probability of picking a blue marble is twice that of a red marble, the count of blue marbles is ___.
Level 4 of 5Show the answer
Answer: 10
Because probability is proportional to the count of each colour, the number of blue marbles is twice the number of red marbles: 2 × 5 = 10.
Which of the following quadratic equations is the odd one out?
Level 4 of 5Show the answer
Answer: 2x² - 7x + 3 = 0
The other three quadratic equations have negative discriminants (b² - 4ac < 0) and no real roots, whereas 2x² - 7x + 3 = 0 has a positive discriminant and distinct real roots.
Which of the following equations of planes in three-dimensional space is the odd one out?
Level 4 of 5Show the answer
Answer: x + y + z = 3
The other three planes are each at a perpendicular distance of exactly 3 units from the origin (|d|/√(a² + b² + c²) = 9/3 = 3), whereas this plane is at distance √3.
Ananya lists all distinct six-digit numbers formed by permuting the digits 1, 2, 3, 4, 5, and 6. If she selects one of these numbers at random, what is the probability that it is divisible by 4?
Level 5 of 5Show the answer
Answer: 4/15
A number is divisible by 4 if its last two digits form a multiple of 4. There are 8 such pairs from the given digits, giving 8 * 24 = 192 favorable numbers out of 720, or 4/15.
The straight line 3x + 4y - k = 0 is tangent to the circle x^2 + y^2 - 4x - 6y - 12 = 0. What is the sum of all possible real values of k?
Level 5 of 5Show the answer
Answer: 36
The circle has center (2, 3) and radius 5. Setting the perpendicular distance from the center to the line equal to 5 gives |18 - k| = 25. Thus k can be 43 or -7, whose sum is 36.
Kabir evaluates a limit in calculus. As x approaches 0, what is the limiting value of (1 - cos 4x) / (x sin 2x)?
Level 5 of 5Show the answer
Answer: 4
Using 1 - cos 4x = 2 sin^2(2x), the expression becomes 2 sin(2x) / x. Rewriting this as 4 * (sin 2x / 2x) shows that the limit as x approaches 0 is 4 * 1 = 4.
Pradeep determines the shortest distance between two straight lines in three-dimensional space: (x - 1)/2 = (y - 2)/3 = (z - 3)/4 and (x - 2)/3 = (y - 4)/4 = (z - 5)/5. What is this distance?
Level 5 of 5Show the answer
Answer: 1 / sqrt(6)
The displacement vector is (1, 2, 2) and the cross product of the direction vectors is (-1, 2, -1). The scalar projection yields |1(-1) + 2(2) + 2(-1)| / sqrt(6) = 1 / sqrt(6).
In the binomial expansion of (x² + 2/x)⁹ in powers of x, Rohan needs to find the constant term. What is the value of the term that is completely independent of x?
Level 5 of 5Show the answer
Answer: 5376
The general term is ⁹Cᵣ (x²)^(9-r) (2/x)ʳ = ⁹Cᵣ 2ʳ x^(18 - 3r). For the constant term, 18 - 3r = 0, so r = 6. This gives ⁹C₆ × 2⁶ = 84 × 64 = 5376.
Kabir arranges seven distinct textbooks on a study shelf: four mathematics books and three physics books. If all three physics books must remain together, how many distinct arrangements can Kabir make?
Level 5 of 5Show the answer
Answer: 720
Treat the three physics books as one unit, giving five units arranged in 5! = 120 ways. The physics books can be arranged internally in 3! = 6 ways, giving 120 × 6 = 720.
- Fill in the blank.
The maximum possible value attained by the function f(theta) = 3 sin(theta) + 4 cos(theta) + 7 across all real angles theta is ___.
Level 5 of 5Show the answer
Answer: 12
The expression a sin(theta) + b cos(theta) reaches a maximum of sqrt(a^2 + b^2). Thus, the maximum of 3 sin(theta) + 4 cos(theta) + 7 is sqrt(9 + 16) + 7 = 12.
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