Relations & Functions (8 marks)
- Show f(x)=x² is not one-one.
- Find domain/range of f(x)=√(x-2)+1.
- Prove equivalence relation on N: a~b if divides.
- Inverse of f(x)=(x-1)/(x+1), x≠-1.
- Binary operation: closure for + mod 5.
Inverse Trigonometric Functions (5 marks)
- Evaluate tan⁻¹(1)+cos⁻¹(1/2).
- sin⁻¹((2x+1)/(2x²+1)) principal value.
- Prove tan⁻¹a + tan⁻¹b = tan⁻¹((a+b)/(1-ab)).
- Range of principal value branch.
- Simplify cos(tan⁻¹(3/4)).
Matrices (10 marks, Algebra)
- Product AB where A=diag(1,2,3), B=[,].
- Adjoint of [,[-1,3,0],].
- Solve system: 2x+y=7, x+2y=5 using matrix inverse.
- Determinant |A| if A transpose = 2A .
- Eigenvalues of [,].
Determinants (Algebra)
- Area triangle vertices (1,0),(6,0),(0,3) using det.
- Minor/cofactor expansion 3×3.
- det(A+B)=detA+detB? True/false justify.
- Solve 3x-4y+z=5 etc. Cramer’s rule.
- Property: det(kA)=kⁿ detA.
Continuity & Differentiability (8 marks)
- Check continuity f(x)=|x|/x at x=0 .
- Differentiate sin(x²) chain rule.
- Rolle’s theorem verify x²-5x+6 on.
- dy/dx: xy=log(xy) implicit.
- Second derivative test extrema.
Applications of Derivatives (16 marks, Calculus)
- Rate: ladder slips 0.5m/s base 2m/s find length.
- Tangent normal to y=√(25-x²) at (3,4).
- Max area rectangle in semicircle.
- Increasing/decreasing f(x)=x³-3x+2.
- Approximation √36.1 using differentials.
Integrals (16 marks, Calculus)
- ∫(x+1)/(x+2) dx partial fractions.
- ∫sin²x dx trig identity.
- Area bounded y=sinx, x-axis [0,π].
- ∫dx/√(1-x²) standard form.
- Definite ∫₀¹ x e^x dx integration by parts.
Differential Equations (8 marks)
- Solve dy/dx + y tanx = sin²x.
- Order/degree: (d²y/dx²) + (dy/dx)³ = x.
- Orthogonal trajectories xy=ce^x.
- Variable separable: dy/y = dx/x.
- Homogeneous: (x-y)dy=(x+ y)dx.
Vector Algebra & 3D Geometry (14 marks)
- Scalar triple product [a b c]=6, find volume.
- Equation line through (1,0,1) parallel i+2j-k.
- Distance point (1,2,3) to plane x+2y+z=7.
- Angle between vectors a·b=3, |a|=√10, |b|=1 .
- Shortest distance skew lines.
Application of Integrals, Probability (Remaining)
- Area between y=x², y=√x.
- Bayes theorem P(A|B)=? Given priors .
- Binomial P(X=5) n=10,p=0.4.
- Variance discrete RV.
- Random variable expectation.
These are the most important ques for class 12th boards exam 2025-2026 for cbsc boards and state boards as well .