Class 12 Applied Mathematics Half-Yearly Exam 2026: Important Questions & Complete Revision

A wide range of mathematical concepts, formula-based problems and numerical applications can make the Class 12 Applied Mathematics Half-Yearly Exam 2026 difficult to revise for. Topics such as calculus, matrices, modular arithmetic and time-speed-distance problems require conceptual clarity, regular practice and accurate calculations.

The Class 12 Applied Mathematics Half-Yearly Important Questions 2026 bring together 15 MCQs with correct answers and step-by-step solutions. Working through these questions can help reinforce key formulas, identify common calculation errors and improve readiness for the half-yearly examination.

Class 12 Applied Mathematics Half-Yearly Exam 2026: Important Topics

Key areas for the Class 12 Applied Mathematics Half-Yearly Exam 2026 include differentiation, applications of derivatives, exponential functions, modular arithmetic, time-speed-distance problems and matrix operations.

For additional practice across Class 12 mathematics topics, explore Important Questions for Class 12 Maths.

Class 12 Applied Mathematics Half-Yearly 2026: Important Questions with Answers

These MCQs cover important concepts from calculus, applications of derivatives, modular arithmetic and matrices. Each question includes four options, the correct answer and a step-by-step explanation.

Question 1. Finding the Derivative of a Curve

For the curve (\sqrt{y}+\sqrt{x}=1), find the value of (\dfrac{dy}{dx}) at the point (\left(\dfrac14,\dfrac14\right)).

Options:

• A) (\dfrac12)

• B) (-1)

• C) (1)

• D) (2)

Correct Answer: B) (-1)

Question 2. Finding the Second Derivative

If (f(x)=\log(x^2+1)), find the value of (f''(2)).

Options:

• A) (-\dfrac{6}{25})

• B) (\dfrac{3}{25})

• C) (\dfrac{6}{25})

• D) (-\dfrac{4}{25})

Correct Answer: A) (-\dfrac{6}{25})

Question 3. Derivative of a Logarithmic Function

If (f(x)=\log_a x), find (f'(e)).

Options:

• A) (\dfrac{1}{e\ln a})

• B) (\dfrac{1}{e})

• C) (\dfrac{1}{a\ln a})

• D) None of these

Correct Answer: A) (\dfrac{1}{e\ln a})

Question 4. Differentiation of an Infinite Radical

Given

[
y=\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}},
]

find (\dfrac{dy}{dx}).

Options:

• A) (2y-1)

• B) (2y+1)

• C) (\dfrac{1}{2y-1})

• D) (\dfrac{1}{2y+1})

Correct Answer: C) (\dfrac{1}{2y-1})

Question 5. Second Derivative of an Exponential Function

If (y=e^{-bx}), find (y'').

Options:

• A) (b^2e^{bx})

• B) (b^2e^{-bx})

• C) (be^{bx})

• D) None of these

Correct Answer: B) (b^2e^{-bx})

Question 6. Rate of Change of a Cube's Volume

The edge of a cube is increasing at (3\text{ cm/s}). How quickly is its volume increasing when the edge is (10\text{ cm}) long?

Options:

• A) (900\text{ cm}^3/\text{s})

• B) (700\text{ cm}^3/\text{s})

• C) (500\text{ cm}^3/\text{s})

• D) None of these

Correct Answer: A) (900\text{ cm}^3/\text{s})

For further revision of derivatives and related concepts, Class 12 Maths Revision Notes provide additional topic-wise study material.

Question 7. Increasing Exponential Functions

For which value of (a) is the function (f(x)=a^x) increasing over all real numbers?

Options:

• A) (a>0)

• B) (a>1)

• C) (a<0)

• D) (0<a<1)

Correct Answer: B) (a>1)

Question 8. Calculating Marginal Cost

A toy manufacturer has a variable cost of (x(30+x)), where (x) is the number of toys produced. The storage cost is (1500). Find the marginal cost when (20) toys are produced.

Options:

• A) (60)

• B) (70)

• C) (40)

• D) None of these

Correct Answer: B) (70)

Question 9. Addition Using Modular Arithmetic

If (a\equiv7\pmod{10}) and (b\equiv3\pmod{10}), find the least non-negative value of (x) for which

[
a+b\equiv x\pmod{10}
]

Options:

• A) (0)

• B) (1)

• C) (3)

• D) (10)

Correct Answer: A) (0)

Derivatives and continuity are important calculus concepts covered in the Class 12 Maths Chapter 5 Solutions, along with step-by-step explanations and solved questions.

Question 10. Finding the Remainder of (7^6)

Find the remainder when (7^6) is divided by (3).

Options:

• A) (1)

• B) (2)

• C) (3)

• D) (4)

Correct Answer: A) (1)

Question 11. Speed of a Boat in Still Water

A boat travels (12\text{ km}) upstream in (48) minutes. If the stream flows at (2\text{ km/h}), find the speed of the boat in still water.

Options:

• A) (6.5\text{ km/h})

• B) (17\text{ km/h})

• C) (13\text{ km/h})

• D) (8.5\text{ km/h})

Correct Answer: B) (17\text{ km/h})

Question 12. Winning Distance in a Race

In a (500\text{ m}) race, the speed ratio of contestants A and B is (3:4). If A receives a head start of (140\text{ m}), by how many metres does A win?

Options:

• A) (10\text{ m})

• B) (20\text{ m})

• C) (60\text{ m})

• D) (40\text{ m})

Correct Answer: B) (20\text{ m})

Question 13. Adjoint of a Singular Matrix

If (A) is a singular matrix, what is the value of (A(\operatorname{adj}A))?

Options:

• A) Null matrix

• B) Scalar matrix

• C) Identity matrix

• D) None of these

Correct Answer: A) Null matrix

Question 14. Finding an Unknown in a Symmetric Matrix

The matrix

[
A=
\begin{bmatrix}
x+7 & 2\
2x-3 & x+8
\end{bmatrix}
]

is symmetric. Find the value of (x).

Options:

• A) (10)

• B) (-\dfrac43)

• C) (-10)

• D) (\dfrac52)

Correct Answer: D) (\dfrac52)

Question 15. Determinant of a Matrix

If (A) is a non-singular matrix of order (3) and (A^2=2A), find (|2A|).

Options:

• A) (4)

• B) (8)

• C) (64)

• D) (16)

Correct Answer: C) (64)

For additional matrix practice, NCERT Solutions for Class 12 Maths Chapter 3, Exercise 3.1 provide worked examples and step-by-step solutions.

Class 12 Applied Mathematics Half-Yearly Exam 2026 Important Questions PDF 

Keeping the questions and solutions available in one place can make revision easier. Use the resource below to access the Class 12 Applied Mathematics Half-Yearly Important Questions PDF.

How to Revise for the Class 12 Applied Mathematics Half-Yearly Exam 2026

A focused revision plan can help improve accuracy and strengthen understanding of important mathematical concepts.

• Revise formulas: Review differentiation rules, logarithmic derivatives, matrix identities and determinant properties.

• Practise step-by-step calculations: Write each step clearly, especially when differentiating functions or solving numerical problems.

• Understand applications: Practise related-rate questions, marginal cost problems and time-speed-distance calculations.

• Review modular arithmetic: Work through congruences and remainders to understand how modular operations work.

• Check matrix properties: Remember the conditions for symmetric and singular matrices and the rules for calculating determinants.

• Solve timed MCQs: Practise answering questions within a fixed time while checking calculations and signs.

• Learn from mistakes: Review incorrect answers and identify whether the error came from a formula, calculation or misunderstanding of the question.

A focused revision of key concepts is essential for the Class 12 Applied Mathematics Half-Yearly Exam 2026, especially when solving formula-based questions and numerical problems. These 15 important MCQs include answers and step-by-step solutions covering calculus, modular arithmetic, speed and distance, matrices and determinants. Practising them can help improve accuracy, reinforce mathematical concepts and support effective revision before the examination.