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
Determinants
Using determinants, find the area of the triangle whose vertices are (1, 0), (6, 0) and (4, 3).
Reveal model answer + marking pointsHide answer▾
The area of a triangle with vertices (x1, y1), (x2, y2), (x3, y3) is (1/2)|x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|. Substituting: area = (1/2)|1(0 - 3) + 6(3 - 0) + 4(0 - 0)| = (1/2)|-3 + 18 + 0| = (1/2)(15) = 7.5 square units.
Area = (1/2)|x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|
Marking-scheme points
- ✓Area = (1/2)|x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|
- ✓= (1/2)|-3 + 18 + 0|
- ✓= 7.5 square units
MathsClass 123 marksmedium
Determinants
Find the inverse of the matrix A = [[2, 3],[1, 4]] using the adjoint method.
Reveal model answer + marking pointsHide answer▾
First find the determinant: |A| = (2)(4) - (3)(1) = 8 - 3 = 5, which is non-zero, so the inverse exists. For a 2 x 2 matrix [[a, b],[c, d]], the adjoint is [[d, -b],[-c, a]], so adj A = [[4, -3],[-1, 2]]. Then the inverse is A inverse = (1/|A|) adj A = (1/5)[[4, -3],[-1, 2]] = [[4/5, -3/5],[-1/5, 2/5]].
A inverse = (1/|A|) adj A
Marking-scheme points
- ✓|A| = 8 - 3 = 5 (non-zero, so inverse exists)
- ✓adj A = [[4, -3],[-1, 2]]
- ✓A inverse = (1/5)[[4, -3],[-1, 2]]
MathsClass 123 markshard
Determinants
Solve the system of equations 2x + 3y = 8 and x + 2y = 5 using determinants (Cramer's rule).
Reveal model answer + marking pointsHide answer▾
Write D = determinant of coefficients = |[[2, 3],[1, 2]]| = (2)(2) - (3)(1) = 4 - 3 = 1. Dx (replace first column by constants) = |[[8, 3],[5, 2]]| = (8)(2) - (3)(5) = 16 - 15 = 1. Dy (replace second column by constants) = |[[2, 8],[1, 5]]| = (2)(5) - (8)(1) = 10 - 8 = 2. By Cramer's rule, x = Dx/D = 1/1 = 1 and y = Dy/D = 2/1 = 2. So x = 1, y = 2.
x = Dx/D, y = Dy/D
Marking-scheme points
- ✓D = 1, Dx = 1, Dy = 2
- ✓Cramer's rule: x = Dx/D, y = Dy/D
- ✓x = 1, y = 2
MathsClass 123 marksmedium
Continuity and Differentiability
Find the value of k for which the function f(x) = kx + 1 for x <= 5 and f(x) = 3x - 5 for x > 5 is continuous at x = 5.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
Continuity and Differentiability
If x^2 + y^2 = 25, find dy/dx by implicit differentiation.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 markshard
Continuity and Differentiability
Differentiate y = x^x with respect to x using logarithmic differentiation.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
Continuity and Differentiability
If y = e^x sin x, find the second order derivative d2y/dx2.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
Application of Derivatives
The radius of a circle is increasing at the rate of 3 cm/s. Find the rate at which the area is increasing when the radius is 10 cm.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
Application of Derivatives
Find the intervals in which the function f(x) = x^2 - 4x + 3 is increasing or decreasing.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
Application of Derivatives
Find the equation of the tangent to the curve y = x^2 at the point (1, 1).
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 markshard
Application of Derivatives
Find the local maximum and local minimum values of the function f(x) = x^3 - 3x + 2.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
Application of Derivatives
Find two positive numbers whose sum is 16 and whose product is maximum.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
Integrals
Evaluate the integral of 2x/(1 + x^2) dx.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
Integrals
Evaluate the integral of x e^x dx using integration by parts.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 markshard
Integrals
Evaluate the integral of 1/((x - 1)(x - 2)) dx using partial fractions.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
Integrals
Evaluate the definite integral from 0 to 1 of x^2 dx.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
Integrals
Evaluate the definite integral from 0 to pi/2 of sin x dx.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
Integrals
Evaluate the definite integral from -1 to 1 of x^3 dx using a property of definite integrals.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
Application of Integrals
Find the area bounded by the curve y = x^2, the x-axis and the lines x = 0 and x = 3.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
Application of Integrals
Find the area bounded by the line y = 2x, the x-axis and the lines x = 0 and x = 3.
🔒 Model answer + marking pointsUnlock with Premium →You are more ready than you feel.
One question at a time is how every topper started. Bookmark this, revise a few each day, and watch the fear shrink. And if a friend is stressing about boards — send this their way. You both win.