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 markshard
Differential Equations
Solve the linear differential equation dy/dx + (1/x) y = x.
Reveal model answer + marking pointsHide answer▾
Here P = 1/x and Q = x. The integrating factor IF = e^(integral of (1/x) dx) = e^(ln x) = x. Multiply through by x (or use the solution formula): y x (IF) = integral of [Q x (IF)] dx, that is, y x = integral of (x times x) dx = integral of x^2 dx = x^3/3 + C. Dividing by x gives the general solution y = x^2/3 + C/x.
y (IF) = integral of Q (IF) dx + C
Marking-scheme points
- ✓P = 1/x, Q = x; IF = e^(integral 1/x dx) = x
- ✓y x = integral of x^2 dx = x^3/3 + C
- ✓y = x^2/3 + C/x
MathsClass 123 marksmedium
Vector Algebra
If a = i + 2 j + 3 k and b = 2 i - j + k, find the scalar (dot) product a . b.
Reveal model answer + marking pointsHide answer▾
The scalar (dot) product of two vectors is the sum of the products of their corresponding components: a . b = (1)(2) + (2)(-1) + (3)(1) = 2 - 2 + 3 = 3. The dot product is a scalar quantity and also equals |a||b| cos theta, where theta is the angle between the vectors.
a . b = a1 b1 + a2 b2 + a3 b3
Marking-scheme points
- ✓a . b = sum of products of corresponding components
- ✓= (1)(2) + (2)(-1) + (3)(1)
- ✓a . b = 3
MathsClass 123 marksmedium
Vector Algebra
Find the angle between the vectors a = i + j + k and b = i - j + k.
Reveal model answer + marking pointsHide answer▾
The angle theta satisfies cos theta = (a . b)/(|a| |b|). Here a . b = (1)(1) + (1)(-1) + (1)(1) = 1 - 1 + 1 = 1. Also |a| = sqrt(1 + 1 + 1) = sqrt(3) and |b| = sqrt(1 + 1 + 1) = sqrt(3). So cos theta = 1/(sqrt(3) x sqrt(3)) = 1/3. Therefore theta = cos inverse (1/3).
cos theta = (a . b)/(|a| |b|)
Marking-scheme points
- ✓cos theta = (a . b)/(|a| |b|)
- ✓a . b = 1, |a| = |b| = sqrt(3)
- ✓theta = cos inverse (1/3)
MathsClass 123 marksmedium
Vector Algebra
Find the cross product a x b and its magnitude, where a = i + j + k and b = i - j + k.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
Three Dimensional Geometry
Write the vector and Cartesian equations of a line passing through a point and parallel to a given direction.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
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.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
Three Dimensional Geometry
Find the distance of the point (0, 0, 0) from the plane 3x + 4y + 12z - 26 = 0.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
Linear Programming
State the corner point method for solving a linear programming problem.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
Linear Programming
Maximise Z = 3x + 4y subject to the constraints x + y <= 4, x >= 0 and y >= 0.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
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).
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
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).
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 markshard
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.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
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.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 123 marksmedium
Probability
A fair coin is tossed 3 times. Using the binomial distribution, find the probability of getting exactly 2 heads.
🔒 Model answer + marking pointsUnlock with Premium →BiologyClass 123 marksmedium
Sexual Reproduction in Flowering Plants
What is microsporogenesis? Describe the structure of a mature pollen grain.
🔒 Model answer + marking pointsUnlock with Premium →BiologyClass 123 marksmedium
Sexual Reproduction in Flowering Plants
What is double fertilization in angiosperms?
🔒 Model answer + marking pointsUnlock with Premium →BiologyClass 123 marksmedium
Human Reproduction
What is spermatogenesis? Briefly describe the process.
🔒 Model answer + marking pointsUnlock with Premium →BiologyClass 123 marksmedium
Human Reproduction
Describe the phases of the human menstrual cycle.
🔒 Model answer + marking pointsUnlock with Premium →BiologyClass 123 marksmedium
Principles of Inheritance and Variation
Explain a monohybrid cross between a pure tall (TT) and a pure dwarf (tt) pea plant up to the F2 generation.
🔒 Model answer + marking pointsUnlock with Premium →BiologyClass 123 marksmedium
Principles of Inheritance and Variation
What phenotypic ratio is obtained in the F2 generation of a dihybrid cross? What does it illustrate?
🔒 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.