Board Boosters
The questions your board exam loves to ask
800 most-asked Class 11 & 12 (+1 / +2) questions across Physics, Chemistry, Maths and Biology — each with a model answer and the exact marking-scheme points examiners reward. Revise smart, walk in calm.
MathsClass 123 marksmedium
Application of Integrals
Find the area enclosed by the circle x^2 + y^2 = 16.
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The equation x^2 + y^2 = 16 represents a circle centred at the origin with radius r = 4 (since r^2 = 16). The area enclosed by a circle of radius r is pi r^2. So the area = pi (4)^2 = 16 pi square units. (By integration, the area = 4 times the integral from 0 to 4 of sqrt(16 - x^2) dx = 16 pi.)
Area of circle = pi r^2
Marking-scheme points
- ✓Circle x^2 + y^2 = 16 has radius r = 4
- ✓Area of circle = pi r^2
- ✓Area = pi (4)^2 = 16 pi square units
MathsClass 123 marksmedium
Differential Equations
Solve the differential equation dy/dx = (1 + y^2)/(1 + x^2).
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Separate the variables: dy/(1 + y^2) = dx/(1 + x^2). Now integrate both sides: the integral of dy/(1 + y^2) = the integral of dx/(1 + x^2). This gives tan inverse (y) = tan inverse (x) + C, where C is the constant of integration. This is the general solution of the differential equation.
integral dy/(1 + y^2) = tan inverse (y)
Marking-scheme points
- ✓Separate variables: dy/(1 + y^2) = dx/(1 + x^2)
- ✓Integrate both sides
- ✓tan inverse (y) = tan inverse (x) + C
MathsClass 123 marksmedium
Differential Equations
Solve the differential equation dy/dx = x/y.
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Separate the variables by cross-multiplying: y dy = x dx. Integrate both sides: the integral of y dy = the integral of x dx, giving y^2/2 = x^2/2 + C1. Multiplying by 2: y^2 = x^2 + C (where C = 2 C1). So the general solution is y^2 - x^2 = C, which represents a family of hyperbolas.
Marking-scheme points
- ✓Separate variables: y dy = x dx
- ✓Integrate: y^2/2 = x^2/2 + C1
- ✓General solution: y^2 - x^2 = C
MathsClass 123 marksmedium
Differential Equations
Write the standard form of a linear differential equation and its integrating factor.
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Differential Equations
Solve the linear differential equation dy/dx + (1/x) y = x.
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Vector Algebra
If a = i + 2 j + 3 k and b = 2 i - j + k, find the scalar (dot) product a . b.
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Vector Algebra
Find the angle between the vectors a = i + j + k and b = i - j + k.
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Vector Algebra
Find the cross product a x b and its magnitude, where a = i + j + k and b = i - j + k.
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Three Dimensional Geometry
Write the vector and Cartesian equations of a line passing through a point and parallel to a given direction.
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Three Dimensional Geometry
Write the formula for the angle between two lines in terms of their direction ratios, and the condition for the lines to be perpendicular and parallel.
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Three Dimensional Geometry
Find the distance of the point (0, 0, 0) from the plane 3x + 4y + 12z - 26 = 0.
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Linear Programming
State the corner point method for solving a linear programming problem.
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Linear Programming
Maximise Z = 3x + 4y subject to the constraints x + y <= 4, x >= 0 and y >= 0.
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Probability
If P(A) = 0.5, P(B) = 0.6 and P(A intersection B) = 0.3, find P(A | B) and P(B | A).
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Probability
If A and B are independent events with P(A) = 0.3 and P(B) = 0.4, find P(A intersection B) and P(A union B).
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Probability
Bag I contains 3 red and 4 black balls, and Bag II contains 5 red and 6 black balls. A bag is chosen at random and a ball drawn from it is found to be red. Find the probability that it was drawn from Bag I.
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Probability
A random variable X takes the values 0, 1 and 2 with probabilities 1/4, 1/2 and 1/4 respectively. Find the mean (expected value) of X.
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Probability
A fair coin is tossed 3 times. Using the binomial distribution, find the probability of getting exactly 2 heads.
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