CBSE Class 12 Maths Exam Pattern 2025

The table below shows the exam pattern and the marking scheme for CBSE Class 12 exams.

Important Questions to Expect in CBSE Board Class 12 Maths Exam 2025

1. If $A=\{1,2,3\},S=\{4,5,6,7\}$ and $f=\{(1,4),(2,5),(3,6)\}$ is a function from $A$ to $B$. State whether $f$ is one-one or not.
2. If $f:R\to R$ is defined by $f(x)=3x+2$, then define $f[f(x)]$.
3. Write fog, if $f:R\to R$ and $g:R\to R$ are given by $f(x)=|x|$ and $g(x)=|5x-2|$.
4. Prove that the relation $R$ on the set $A=\{1,2,3,4\}$, defined as $R=\{(a,b):a-$ $b$ is even $\}$, is an equivalence relation.
5. Find the inverse of $f(x)=\frac{2x+3}{x-4},x\ne 4$.
6. Show that the function $f:R\to R$ defined as $f(x)=x^3$ is one-one and onto.
7. Let $f(x)=|x|+|x-1|$. Prove its continuity at all points.
8. Solve for $x\cdot {\tan }^{-1}\left(\frac{x}{2}\right)+{\tan }^{-1}\left(\frac{1}{x}\right)=\frac{\pi }{4}$.
9. Simplify ${\sin }^{-1}\left(\frac{3}{5}\right)+{\cos }^{-1}\left(\frac{4}{5}\right)$.
10. Prove that ${\tan }^{-1}x+{\tan }^{-1}y={\tan }^{-1}\left(\frac{x+y}{1-xy}\right)$ if $xy<1$.
11. Write the principal value of $\cos -1[\cos (680{)}^{\circ }]$.
12. Write the value of ${\cos }^{-1}(1/2)-2\sin -1(1/2)$
13. Find the domain of $f(x)={\sin }^{-1}(\sqrt{x})$.
14. If $A$ is a square matrix such that $A2=A$, then write the value of $7A-(I+A)3$, where I is an identity matrix.
15. Solve for $x\cdot {\tan }^{-1}\left(\frac{x}{2}\right)+{\tan }^{-1}\left(\frac{1}{x}\right)=\frac{\pi }{4}$.
16. Write the principal value of ${\sin }^{-1}(-1/2)$.
17. Solve: 2 x+y-3 z=5,3 x-2 y+4 z=7, x+y+z=6
18. If $A=\left[\begin{array}{cc}2& -1\\ 3& 4\end{array}\right]$, find $A^{-1}$.
19. Prove that $\det (AB)=\det (A)\cdot \det (B)$.
20. If a matrix has 5 elements, then write all possible orders it can have.
21. Verify if $A=\left[\begin{array}{ll}1& 2\\ 3& 4\end{array}\right]$ is invertible and find $A^{-1}$ if invertible.
22. Prove that $f(x)=|x-1|$ is continuous but not differentiable at $x=1$.
23. Differentiate $f(x)=e^{x^2}\sin x$ using the chain rule.
24. Find $\frac{dy}{dx}$ if $y={\tan }^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$.
25. Show that $f(x)=x^3-3x+2$ has exactly one root in $(0,1)$.
26. Find the equation of the tangent to the curve $y=x^3-3x^2+2x$ at $x=1$.
27. Prove that the function $f(x)=x^3-6x^2+9x+5$ has a local minimum at $x=2$.
28. Solve: Find the intervals where $f(x)=x^4-4x^3+6x^2$ is increasing or decreasing.
29. A ladder 10 m long leans against a wall. The top of the ladder slides down the wall at a rate of 2 $\mathrm{m}/\mathrm{s}$. How fast is the bottom moving away from the wall when the top is 6 m above the ground?
30. The volume of a cube is increasing at the rate of $8{\mathrm{cm}}^3/\mathrm{s}$. How fast is the surface area increasing when the length of its edge is 12 cm?
31. Evaluate: $\int \frac{x^2}{{(1+x^2)}^2}dx$.
32. Solve: $\int \frac{1}{1+x^2}dx$.
33. Evaluate: $\int e^x(\sin x+\cos x)dx$.
34. Solve using substitution: $\int \frac{1}{\sqrt{1-x^2}}dx$.
35. Find: $\int \ln (x)dx$.
36. Find the area enclosed by $y=\sqrt{x}$, the $x$-axis, and $x=4$.
37. Determine the area between the curves $y=x^2$ and $y=2x$.
38. Solve: Find the area of the region enclosed by the parabola $y^2=4ax$ and its latus rectum.
39. Solve: $\frac{dy}{dx}+y=e^{2x}$.
40. Form the differential equation of the family of curves $y=c_1{e}^{2x}+c_2{e}^{-2x}$.
41. Solve: $(x^2+y^2)dx+2xydy=0$.
42. Find the particular solution of $\frac{dy}{dx}=\frac{x+1}{y+1}$, given $y=0$ when $x=0$.
43. Find the angle between the vectors $\mathbf{a}=\hat{i}+\hat{j}+\hat{k}$ and $\mathbf{b}=\hat{i}-\hat{j}+\hat{k}$.
44. Find the projection of $\mathbf{a}=2\hat{i}-\hat{j}+3\hat{k}$ on $\mathbf{b}=\hat{i}+\hat{j}$.
45. Find the shortest distance between the lines:

$\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z}{1},\frac{x+1}{3}=\frac{y-2}{4}=\frac{z-1}{5}$.

46. Find the equation of the plane passing through the points $(1,0,0),(0,1,0)$, and $(0,0,1)$.
47. A factory manufactures two products A and B. Each unit of A requires 1 hour of labour and 2 hours of machine time, and each unit of $B$ requires 3 hours of labour and 1 hour of machine time. The factory has 12 labour hours and 8 machine hours. Find the production mix to maximize profit if the profit per unit of $A$ is ₹ 40 and per unit of $B$ is ₹ 60.
48. Solve graphically: Maximize $z=3x+4y$, subject to:

$x+2y\le 10,2x+y\le 8,x\ge 0,y\ge 0$

49. A die is thrown twice. Find the probability of getting a sum of 8.
50. Two cards are drawn from a pack of 52 cards. Find the probability that one is a king and the other is a queen.
51. If $A$ and $B$ are independent events, prove that $P(A\cap B)=P(A)\cdot P(B)$.
52. Find the mean and variance of a random variable $X$ where $P(X=x_i)$ is uniform for $i=$ $1,2,3$.
53. Prove Rolle's theorem for $f(x)=x^2-4x$ in $[0,4]$.
54. Solve: $\int \frac{dx}{x^2+1}$.
55. If $f(x)=e^{x^2},g(x)=e^{-x^2}$, find $f(x)\cdot g(x)$.
56. Evaluate: ${\int }_0^{\pi /2}{\sin }^{2}xdx$.
57. Prove that $f(x)=x^3-3x+1$ is strictly increasing in $R$.
58. Verify that $y=A\sin x+B\cos x$ satisfies the equation $\frac{d^{2}y}{d{x}^2}+y=0$.
59. Show that the mean and variance of $X$ are given by $\mu =np$ and ${\sigma }^2=npq$.
60. A bag contains 6 white balls and 4 black balls. Find the probability of selecting 2 white balls in 3 draws without replacement.

Preparation Tips for Excel CBSE Board Class 12 Maths Exam

• Analyze the CBSE Maths syllabus and pay special emphasis on high-weightage topics like Calculus, Algebra, Probability, Vector Geometry, etc.
• Solve all the examples and exercises contained in the NCERT as many questions in the exam are based on this.
• Use the standard methods to solve problems to maintain step-by-step accuracy because the solutions are marked in detail in the exam.
• Students are advised to solve a number of questions every day to build better concept knowledge.
• Prepare a list of formulas and theorems review them from time to time and get a grasp of their uses.
• To familiarize with the format of the questions, students should solve CBSE sample papers.
• Solve sample/model papers of the previous years to know about the repeated questions and subjects. It assists you in managing your time well.
• Make sure to leave some time to cross-check your answers at the end of the exam.
• Have a balanced diet, drink enough water, and ensure you take enough rest before exams.
• It is useful to take a break from time to time in order to prevent getting burned with studies.
• Continue to keep a positive attitude and confidence in your preparation do not worry yourself unnecessarily.

Additional Resources For the CBSE Class 12 Board Exam

Apart from the sample paper and marking scheme students can focus on other resources as well like NCERT exemplar, solution notes and more to achieve good marks. Check the links below for relevant resources.