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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 113 markshard
Conic Sections
Find the eccentricity of an ellipse whose latus rectum is half of its major axis.
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Latus rectum = 2b^2/a and major axis = 2a. Given 2b^2/a = (1/2)(2a) = a, so 2b^2 = a^2. Since b^2 = a^2(1 - e^2), we get 2a^2(1 - e^2) = a^2, so 2(1 - e^2) = 1, giving 1 - e^2 = 1/2, e^2 = 1/2 and e = 1/sqrt2.
b^2 = a^2(1 - e^2)
Marking-scheme points
- ✓2b^2/a = a -> 2b^2 = a^2
- ✓Use b^2 = a^2(1 - e^2)
- ✓e^2 = 1/2 -> e = 1/sqrt2
MathsClass 113 marksmedium
Introduction to Three Dimensional Geometry
Find the coordinates of the point which divides the line segment joining (1, -2, 3) and (3, 4, -5) internally in the ratio 2 : 3.
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By the section formula for internal division in ratio m : n, the coordinates are ((m x2 + n x1)/(m + n), (m y2 + n y1)/(m + n), (m z2 + n z1)/(m + n)) with m = 2, n = 3. x = (2(3) + 3(1))/5 = 9/5; y = (2(4) + 3(-2))/5 = 2/5; z = (2(-5) + 3(3))/5 = -1/5. The point is (9/5, 2/5, -1/5).
((m x2 + n x1)/(m+n), ...)
Marking-scheme points
- ✓Use internal section formula with m = 2, n = 3
- ✓x = 9/5, y = 2/5, z = -1/5
- ✓Point = (9/5, 2/5, -1/5)
MathsClass 113 marksmedium
Introduction to Three Dimensional Geometry
Find the centroid of the triangle whose vertices are (1, 2, 3), (3, -1, 5) and (4, 0, -2).
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The centroid of a triangle with vertices (x1,y1,z1), (x2,y2,z2), (x3,y3,z3) is ((x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3). x = (1+3+4)/3 = 8/3; y = (2-1+0)/3 = 1/3; z = (3+5-2)/3 = 6/3 = 2. Centroid = (8/3, 1/3, 2).
G = ((x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3)
Marking-scheme points
- ✓Centroid = average of the three vertices
- ✓x = 8/3, y = 1/3, z = 2
- ✓Centroid = (8/3, 1/3, 2)
MathsClass 113 marksmedium
Limits and Derivatives
Evaluate lim as x approaches 0 of (sin 3x)/(sin 5x).
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Limits and Derivatives
Evaluate lim as x approaches 3 of (x^2 - 9)/(x^2 - x - 6).
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Limits and Derivatives
Evaluate lim as x approaches 0 of (sqrt(1 + x) - 1)/x.
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Limits and Derivatives
Differentiate y = (x^2 + 1)(x - 3) using the product rule.
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Limits and Derivatives
Differentiate y = (x + 1)/(x - 1) using the quotient rule.
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Limits and Derivatives
Find the derivative of sin x from first principles.
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Statistics
Find the mean deviation about the mean for the data 6, 7, 10, 12, 13, 4, 8, 12.
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Statistics
Find the variance and standard deviation of the data 2, 4, 6, 8, 10.
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Statistics
The mean of a data set is 6 and its standard deviation is 2 sqrt2. Find the coefficient of variation.
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Probability
Two fair dice are thrown together. Find the probability that the sum of the numbers on them is 8.
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Probability
One card is drawn at random from a well-shuffled pack of 52 cards. Find the probability that it is a king or a heart.
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Relations and Functions
Show that the relation R on the set of integers Z defined by aRb if (a - b) is divisible by 3 is an equivalence relation.
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Relations and Functions
Show that the function f: R -> R given by f(x) = 2x + 3 is a bijection.
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Relations and Functions
If f: R -> R is given by f(x) = 2x + 3, find its inverse function.
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Inverse Trigonometric Functions
Prove that tan inverse (1) + tan inverse (2) + tan inverse (3) = pi.
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Matrices
If A = [[1, 2],[3, 4]] and B = [[2, 0],[1, 3]], find the product AB.
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Matrices
Show that any square matrix A can be written as the sum of a symmetric and a skew-symmetric matrix.
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