HBSE CLASS 12 MATHEMATICS 2026 | PYQs & MOST IMPORTANT QUESTIONS

Table of Contents

🟢 SECTION A – 1 MARK QUESTIONS (2021)

Q1. Write the condition for a relation to be equivalence relation.

✅ Solution:

A relation R is equivalence if it satisfies:

1️⃣ Reflexive ⇒ (a, a) ∈ R
2️⃣ Symmetric ⇒ If (a, b) ∈ R then (b, a) ∈ R
3️⃣ Transitive ⇒ If (a, b) and (b, c) ∈ R then (a, c) ∈ R

👉 Therefore, relation must be Reflexive + Symmetric + Transitive

Q2. Find f⁻¹(x) if f(x) = x² + 4 (x ≥ 0)

✅ Solution:

Let y = x² + 4

y − 4 = x²

x = √(y − 4)

Since x ≥ 0

👉 f⁻¹(x) = √(x − 4)

Q3. If a*b = LCM(a, b), then binary operation is:

✅ Solution:

LCM(a, b) = LCM(b, a)

So operation is Commutative

Q4. Evaluate:

tan⁻¹(√3) − cos⁻¹(-1/2)

✅ Solution:

tan⁻¹(√3) = π/3
cos⁻¹(-1/2) = 2π/3

So,

π/3 − 2π/3 = -π/3

👉 Answer = -π/3

Q5. Evaluate:

tan⁻¹(1/√(x²−1))

✅ Solution:

We know identity:

tan⁻¹(1/√(x²−1)) = sec⁻¹x

👉 Answer = sec⁻¹x

Q6. Condition for matrix multiplication AB to exist?

✅ Solution:

Number of columns of A = Number of rows of B

Q7. (AB)⁻¹ = ?

✅ Solution:

(AB)⁻¹ = B⁻¹A⁻¹

Q8. If determinant equation given, find x

(Example form: | x 1 |
| 2 3 | = 0)

Solve:

3x − 2 = 0
x = 2/3

Q9. Find k for continuity at x = 3

Method:

LHL = RHL = f(3)

Solve equation

👉 Value of k obtained accordingly

Q10. Differentiate e^(3x+5)²

Let u = (3x+5)²

d/dx e^u = e^u × du/dx

du/dx = 2(3x+5)×3

So answer:

6(3x+5)e^(3x+5)²

Q11. Slope of tangent of y² = 4ax

Differentiate:

2y dy/dx = 4a

dy/dx = 2a/y

Q12. ∫ 2x sin(x²) dx

Let u = x²
du = 2x dx

= ∫ sin u du

= -cos u + C

= -cos(x²) + C

Q13. ∫₀¹ dx/(1+x²)

= tan⁻¹x [0 to1]

= π/4 − 0

= π/4

Q14. Area bounded by x² = y from x=0 to1

Area = ∫₀¹ x² dx

= x³/3 [0 to1]

= 1/3

Q15. Differential equation of family y = mx

Differentiate:

dy/dx = m

Eliminate m

m = y/x

So:

dy/dx = y/x

Q16. If events independent, which statement not true?

Independent ⇒

P(A∩B) = P(A)P(B)

Other incorrect option selected accordingly

Q17. Find λ if vectors perpendicular

Use:

a·b = 0

Solve λ

Q18. Angle made with z-axis

Use:

cosθ = z/|a|

Q19. If x=4t, y=4/t find dy/dx

dy/dx = (dy/dt)/(dx/dt)

dx/dt=4
dy/dt=-4/t²

So:

dy/dx = (-4/t²)/4

= -1/t²

Q20. ∫ tan²x dx

tan²x = sec²x −1

= ∫(sec²x −1)dx

= tan x − x + C

Q21. Degree of differential equation

Highest power of highest order derivative

Answer accordingly

Q22. Type of relation on set {1,2,3}

If R = {(1,1),(2,2),(3,3)}

👉 Identity relation

Q23. Principal value of tan⁻¹(-1)

= -π/4

Q24. Range of sin⁻¹x

= [-π/2, π/2]

Q25. If A given, find AAᵀ

Multiply matrix

Q26. |2A| (3×3)

= 8|A|

Q27. Check continuity of sin2x/x at 0

Use limit:

lim x→0 sin2x/x

So continuous

Q28. ∫ e^x(sin x + cos x) dx

= e^x sin x + C

Q29. Probability coin tossed 8 times

Use binomial formula

Q30. Relation R in N

Check reflexive/symmetric/transitive

Q31. sin⁻¹(1/√2)

= π/4

Q32. |Adj A|

= |A|²

Q33. Implicit differentiation

Apply differentiation both sides

Q34. Maximum of sin x + cos x

= √2

Q35. ∫ sin²x/(1+cos x) dx

Simplify identity

Answer accordingly

Q36. ∫ dx/(x²−9)

= (1/6) ln|(x−3)/(x+3)| + C

Q37. Area of quadrant of ellipse

= πab/4

Q38. P(A∪B)

= P(A)+P(B)-P(A∩B)

Q39. Cross product formula

Q40. Angle between vectors

cosθ = (a·b)/|a||b|

🟢 SECTION A – 1 MARK QUESTIONS (2023)

Q1. Nature of function

$f(x)=x−2x−3f(x)=\dfrac{x-2}{x-3}$

✅ Solution:

Let

$y=x−2x−3y=\frac{x-2}{x-3}$

For injective test:

Assume

$f(x_1)=f(x_2)$ $x1−2×1−3=x2−2×2−3\frac{x_1-2}{x_1-3}=\frac{x_2-2}{x_2-3}$

Cross multiply:

$x_1-2)(x_2-3)=(x_2-2)(x_1-3)$

Simplifying gives:

$x_1=x_2$

👉 Hence function is One-One (Injective)

Q2. Evaluate:

$tan⁡−1(3)−cot⁡−1(−3)\tan^{-1}(\sqrt3)-\cot^{-1}(-\sqrt3)$

✅ Solution:

$tan⁡−1(3)=π3\tan^{-1}(\sqrt3)=\frac{\pi}{3}$ $cot⁡−1(−3)=−π6\cot^{-1}(-\sqrt3)= -\frac{\pi}{6}$

So,

$π3−(−π6)=π3+π6=π2\frac{\pi}{3}-\left(-\frac{\pi}{6}\right) = \frac{\pi}{3}+\frac{\pi}{6} = \frac{\pi}{2}$

👉 Answer = π/2

Q3. If matrix A is given, find $A^{'} A$

✅ Solution:

Transpose A ⇒ $A^{'}$

Multiply:

$A^{'} A$

Perform normal matrix multiplication.

(Exact multiplication depends on matrix given in paper)

Q4. Evaluate determinant

Example form:

$∣1234∣\begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix}$

= (1×4 − 2×3)

= 4 − 6

= −2

Q5. Find a, b for continuity

Condition:

$LHL=RHL=f(a)\text{LHL}=\text{RHL}=f(a)$

Substitute values and solve simultaneous equations.

Q6. Related rates (Area of circle)

If area $A=πr2A=\pi r^2$

Differentiate:

$dAdt=2πrdrdt\frac{dA}{dt}=2\pi r \frac{dr}{dt}$

Substitute values accordingly.

Q7. Slope of tangent

Given y=f(x)

$Slope=dydx\text{Slope}=\frac{dy}{dx}$

Differentiate and substitute point.

Q8.

$∫sec⁡2xcsc⁡2xdx\int \frac{\sec^2 x}{\csc^2 x} dx$

✅ Solution:

$sec⁡2xcsc⁡2x=1/cos⁡2×1/sin⁡2x=sin⁡2xcos⁡2x=tan⁡2x\frac{\sec^2 x}{\csc^2 x} = \frac{1/\cos^2 x}{1/\sin^2 x} = \frac{\sin^2 x}{\cos^2 x} = \tan^2 x$

So,

$∫tan⁡2xdx=∫(sec⁡2x−1)dx=tan⁡x−x+C\int \tan^2 x dx = \int (\sec^2 x -1) dx = \tan x – x + C$

Q9.

$∫xcos⁡2xdx\int x \cos 2x dx$

✅ Solution:

Use Integration by Parts:

Let
u=x
dv=cos2x dx

Then:

du=dx
v= (sin2x)/2

So:

$x\frac{\sin2x}{2}-\int \frac{\sin2x}{2}dx$ $\frac{x\sin2x}{2}+\frac{\cos2x}{4}+C$

Q10. Differential equation of

$y=asin⁡(x+b)y=a\sin(x+b)$

Differentiate:

$dydx=acos⁡(x+b)\frac{dy}{dx}=a\cos(x+b)$

Eliminate constants →

Final differential equation:

$d2ydx2+y=0\frac{d^2y}{dx^2}+y=0$

Q11. Solve

$dydx=ytan⁡x\frac{dy}{dx}=y\tan x$

✅ Solution:

Separate variables:

$dyy=tan⁡xdx\frac{dy}{y}=\tan x dx$

Integrate:

$ln⁡y=−ln⁡∣cos⁡x∣+C\ln y = -\ln|\cos x|+C$ $y=Csec⁡xy=C\sec x$

Q12. Independent events – find union

Formula:

$P(A∪B)=P(A)+P(B)−P(A)P(B)P(A\cup B)=P(A)+P(B)-P(A)P(B)$

Q13. Die thrown 6 times

Total outcomes:

$6^6$

Use binomial formula if required.

Q14. Formula of variance

$Var(X)=E(X^2)-(E(X))^2$

Q15. Angle made by unit vector

If vector is unit,

$ratio\cos\theta = \text{direction ratio}$

Q16. Angle between planes

Formula:

$cos⁡θ=∣a1a2+b1b2+c1c2∣a12+b12+c12a22+b22+c22\cos\theta= \frac{|a_1a_2+b_1b_2+c_1c_2|} {\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}$

🟢 SECTION A – 1 MARK QUESTIONS (2024)

Q1. Nature of function

$f (n) = 2 n + 3$ on N

✅ Solution:

Assume
f(n₁)=f(n₂)

2n₁+3 = 2n₂+3

2n₁ = 2n₂

n₁ = n₂

👉 Function is One-One (Injective)

Q2. Domain of sin⁻¹(2x)

Condition:

-1 ≤ 2x ≤ 1

Divide by 2:

-1/2 ≤ x ≤ 1/2

👉 Domain = [-1/2, 1/2]

Q3. Total number of 3×3 matrices with entries 0 or 2

Each entry has 2 choices

Total entries = 9

Total matrices = 2⁹

= 512

Q4. If

$∣x123∣=0\begin{vmatrix} x & 1 \\ 2 & 3 \end{vmatrix} = 0$

✅ Solution:

Determinant:

3x − 2 = 0

3x = 2

x = 2/3

Q5. |Adj A| when |A| = k (3×3 matrix)

Formula:

|Adj A| = |A|ⁿ⁻¹

Here n=3

= k²

Q6. Differentiate sin(log x)

✅ Solution:

Let u = log x

d/dx sin u = cos u × du/dx

du/dx = 1/x

Answer:

= (cos(log x))/x

Q7. Derivative of sin x with respect to cos x

✅ Solution:

$dd(cos⁡x)(sin⁡x)=d(sin⁡x)/dxd(cos⁡x)/dx\frac{d}{d(\cos x)}(\sin x) = \frac{d(\sin x)/dx}{d(\cos x)/dx}$

Q8.

$∫exsec⁡x(1+tan⁡x)dx\int e^x \sec x (1+\tan x) dx$

✅ Solution:

Notice:

Derivative of (e^x sec x) = e^x sec x (1+tan x)

So integral =

$e^x \sec x + C$

Q9.

$∫−π/2π/2sin⁡5xdx\int_{-\pi/2}^{\pi/2} \sin^5 x dx$

✅ Solution:

sin⁵x is odd function

Integral from -a to a of odd function = 0

👉 Answer = 0

Q10. Area enclosed by circle

$x^2+y^2=2$

Radius:

r²=2

r=√2

Area = πr²

= π(2)

= 2π

Q11. Order of differential equation

$d2ydx2+y=0\frac{d^2y}{dx^2}+y=0$

Highest derivative = 2

👉 Order = 2

Q12. Integrating factor of

$dydx+Py=Q\frac{dy}{dx}+Py=Q$

Integrating factor =

$e∫Pdxe^{\int P dx}$

Q13. Angle between two vectors

Formula:

$cos⁡θ=a⋅b∣a∣∣b∣\cos\theta= \frac{a·b}{|a||b|}$

Q14. If vectors perpendicular, find λ

Use:

a·b = 0

Solve for λ

Q15. Equation of x-axis in space

On x-axis:

y = 0
z = 0

Q16. Probability of even prime number (die)

Even prime = 2

Probability = 1/6

Q17. Red balls without replacement

Use:

$P=favourabletotalP=\frac{favourable}{total}$

Multiply successive probabilities

Q18. Two queens with replacement

Probability of queen:

4/52

Multiply accordingly

Q19. Assertion-Reason (Relation reflexive)

Correct answer selected as per property:

Reflexive ⇒ (a,a) ∈ R

Q20. Assertion-Reason (Angle between lines)

Use dot product formula

✅ CLASS 12 MATHEMATICS – ALL 1 MARK QUESTIONS TOGETHER

📘 2021 PAPER (Code 5631 – Set A)

🔹 Questions 1 to 40 (All are 1 Mark each)

Condition for equivalence relation
Find $f^{-1}(x)$ if $f(x)=x^2+4$
Binary operation $a * b = L CM (a, b)$ is?
$tan⁡−13−cos⁡−1(−1/2)\tan^{-1}\sqrt3 – \cos^{-1}(-1/2)$
$tan⁡−1(1×2−1)\tan^{-1}\left(\frac{1}{\sqrt{x^2-1}}\right)$
Condition for matrix multiplication $A B$
Inverse of product of matrices
Determinant equality solve for x
Continuity at x=3 find k
Differentiate $e^{(3x+5)^2}$
Slope of tangent of $y^2=4ax$
$∫2xsin⁡(x2)dx\int 2x \sin(x^2) dx$
$∫01dx1+x2\int_0^1 \frac{dx}{1+x^2}$
Area bounded by $x^2=y$ , x=0 to 1
Differential equation of family $y = m x$
Independent events – NOT true statement
Find λ (perpendicular vectors)
Angle made with z-axis
If $x = 4 t, y = 4/ t$ , find dy/dx
$∫tan⁡2xdx\int \tan^2 x dx$
Degree of differential equation
Type of relation on set {1,2,3}
Principal value of $tan^{-1}(-1)$
Range of $sin^{-1}x$
If matrix A given, find $A A^{'}$
Determinant property $∣2 A ∣$
Continuity of $sin⁡2xx\frac{\sin2x}{x}$
$∫ex(sin⁡x+cos⁡x)dx\int e^x(\sin x + \cos x)dx$
Probability – coin tossed 8 times
Relation R in N
Principal value of $sin⁡−1(1/2)\sin^{-1}(1/\sqrt2)$
Value of $∣ A d j A ∣$
Differentiate implicit function
Maximum of $sin⁡x+cos⁡x\sin x + \cos x$
Evaluate $∫sin⁡2×1+cos⁡xdx\int \frac{\sin^2 x}{1+\cos x} dx$
$∫1×2−9dx\int \frac{1}{x^2-9} dx$
Area of quadrant of ellipse
Find $P (A \cup B)$ (Independent events)
Vector cross product
Angle between vectors

📘 2023 PAPER (Code 231 – Set A)

🔹 Section A: Questions 1 to 16 (1 Mark each)

Nature of function $f(x)=x−2x−3f(x)=\frac{x-2}{x-3}$
Principal value $tan⁡−13−cot⁡−1(−3)\tan^{-1}\sqrt3 – \cot^{-1}(-\sqrt3)$
If matrix A given, find $A^{'} A$
Determinant value
Continuity – find a, b
Related rates – area of circle
Slope of tangent
$∫sec⁡2xcsc⁡2xdx\int \frac{\sec^2 x}{\csc^2 x} dx$
Evaluate $∫xcos⁡2xdx\int x \cos2x dx$
Differential equation of $y=asin⁡(x+b)y=a\sin(x+b)$
Solve $dydx=ytan⁡x\frac{dy}{dx}=y\tan x$
Independent events – find union
Probability – die thrown 6 times
Formula of variance
Angle made by unit vector
Angle between planes

📘 2024 PAPER (Code 1232 – Set A, GRAPH)

🔹 Section A: Questions 1 to 20 (1 Mark each)

Nature of function $f (n) = 2 n + 3$
Domain of $sin^{-1}(2x)$
Total matrices (3×3 entries 0 or 2)
Determinant solve for x
Value of $∣ a d j A ∣$
Differentiate $sin⁡(log⁡x)\sin(\log x)$
Derivative of sin x w.r.t cos x
$∫exsec⁡x(1+tan⁡x)dx\int e^x \sec x(1+\tan x)dx$
Evaluate definite integral $∫−π/2π/2sin⁡5xdx\int_{-\pi/2}^{\pi/2} \sin^5x dx$
Area enclosed by circle $x^2+y^2=2$
Order and degree of differential equation
Integrating factor
Angle between vectors
Orthogonal vectors find λ
Equation of x-axis in space
Probability – even prime number
Probability – red balls without replacement
Probability – two queens with replacement
Assertion-Reason (Relation reflexive)
Assertion-Reason (Angle between lines)

🎯 TOTAL 1 MARK QUESTIONS

Year	Number of 1 Mark Questions
2021	40
2023	16
2024	20
✅ Total	76 Questions

📊 CLASS 12 MATHEMATICS

🔥 MOST REPEATED TOPICS (2021–2024)

🥇 1. Probability (Highest Repeated Topic)

✅ Appeared in:

2021 → Coin, Independent events, Union
2023 → Independent events, Die probability
2024 → Even prime, Balls, Cards, Assertion-based probability

🔢 Total Questions: 9–10 Questions

🔥 Subtopics Repeated:

Independent Events
$\cup B)$
Binomial Probability
Without Replacement
With Replacement
Conditional Probability

👉 Very High Exam Weightage

🥈 2. Integration

✅ Appeared in:

2021 → 6+ integrals
2023 → 2 integrals
2024 → 3 integrals

🔢 Total Questions: 10–11 Questions

🔥 Repeated Forms:

Trigonometric integrals
Substitution type
Definite integrals
Standard forms
Exponential + Trig combinations

👉 Most Favourite 1 Mark Area

🥉 3. Differentiation / Derivatives

✅ Appeared in:

2021 → dy/dx, implicit differentiation
2023 → Related rates, derivative
2024 → Direct derivative, order & degree

🔢 Total Questions: 7–8 Questions

🔥 Common Types:

Chain rule
Implicit differentiation
Related rates
Order & Degree

🏅 4. Vectors & 3D Geometry

✅ Appeared in:

2021 → Angle between vectors
2023 → Angle between planes
2024 → Orthogonality, unit vector, axis equation

🔢 Total Questions: 7 Questions

🔥 Repeated Types:

Angle between vectors
Perpendicular condition
Cross product
Direction ratios

🏅 5. Matrices & Determinants

✅ Appeared in:

2021 → Inverse property, determinant
2023 → Determinant value
2024 → adj A, determinant equation

🔢 Total Questions: 7–8 Questions

🔥 Common Repeats:

|2A| property
|adj A|
Determinant solving for x
Inverse property

🏅 6. Continuity & Differentiability

✅ Appeared in:

2021 → Continuity at point
2023 → Continuity
2024 → Domain questions

🔢 Total Questions: 5 Questions

🏅 7. Relations & Functions

✅ Appeared in:

2021 → Equivalence relation
2023 → Nature of function
2024 → Injective / Surjective

🔢 Total Questions: 5 Questions

📊 FINAL RANKING (MOST REPEATED)

Rank	Topic	Frequency	Prediction 2026
🥇 1	Probability	🔥🔥🔥🔥🔥	Guaranteed
🥈 2	Integration	🔥🔥🔥🔥	Very High
🥉 3	Derivatives	🔥🔥🔥🔥	Very High
4	Vectors & 3D	🔥🔥🔥	High
5	Matrices	🔥🔥🔥	High
6	Continuity	🔥🔥	Medium
7	Relations	🔥🔥	Medium

🎯 STRONG PREDICTION FOR 2026 (1 MARK SECTION)

Most Expected:

1 Probability Question (100%)
1 Integration Question (100%)
1 Vector Question
1 Determinant Property
1 Derivative Question
1 Domain / Range Question

📌 Smart Strategy for Students (Principal Sir Style 😎)

👉 Master Standard Integrals
👉 Practice Independent Probability Cases
👉 Memorize Determinant Properties
👉 Revise Vector Angle Formula
👉 Practice 5 Continuity Questions

📘 CLASS 12 MATHEMATICS

🔥 100 MOST IMPORTANT MCQs (Based on 2021–2024)

✅ SECTION A – RELATIONS & FUNCTIONS (1–12)

A relation R is reflexive if:
(A) (a, b) ∈ R
(B) (a, a) ∈ R for all a
(C) (b, a) ∈ R
(D) None
If f(x)=2x+3 on N→N, then f is:
(A) Injective
(B) Surjective
(C) Bijective
(D) None
Domain of sin⁻¹(2x) is:
(A) [-1,1]
(B) [-½,½]
(C) [-2,2]
(D) [0,1]
Principal value of tan⁻¹(-1):
(A) π/4
(B) -π/4
(C) π/2
(D) 0
Range of sin⁻¹x is:
(A) [0,π]
(B) [-π/2,π/2]
(C) [0,2π]
(D) (-π,π)
If R={(1,1),(2,2),(3,3)}, then R is:
(A) Reflexive
(B) Symmetric
(C) Transitive
(D) All
f(x)=x² is:
(A) Injective
(B) Surjective
(C) Neither
(D) One-one
Domain of log x:
(A) x>0
(B) x≥0
(C) All real
(D) x≠0
cos⁻¹(-1)=
(A) 0
(B) π
(C) π/2
(D) -π
sin⁻¹(1/√2)=
(A) π/6
(B) π/4
(C) π/3
(D) π/2
A relation is equivalence if:
(A) Reflexive
(B) Symmetric
(C) Transitive
(D) All
tan⁻¹√3=
(A) π/6
(B) π/4
(C) π/3
(D) π/2

✅ MATRICES & DETERMINANTS (13–25)

|2A| (3×3) equals:
(A) 2|A|
(B) 4|A|
(C) 8|A|
(D) |A|
|Adj A| (3×3, |A|=5) =
(A) 5
(B) 25
(C) 125
(D) 1
(AB)⁻¹ =
(A) A⁻¹B⁻¹
(B) B⁻¹A⁻¹
(C) A⁻¹+B⁻¹
(D) BA
Determinant of identity matrix =
(A) 0
(B) 1
(C) n
(D) -1
If |A|=0 then A is:
(A) Singular
(B) Non-singular
(C) Identity
(D) Zero
Number of 3×3 matrices with entries 0 or 2:
(A) 27
(B) 81
(C) 512
(D) 9
If A is skew symmetric, then diagonal elements are:
(A) 1
(B) -1
(C) 0
(D) Any
If A is invertible, then |A| ≠
(A) 1
(B) 0
(C) -1
(D) 5
Determinant changes sign if:
(A) Rows interchanged
(B) Multiply row by k
(C) Add row
(D) None
|I|=
(A) 0
(B) 1
(C) n
(D) -1
If A²=I, then A⁻¹=
(A) A
(B) I
(C) -A
(D) 0
If two rows equal, determinant=
(A) 1
(B) 0
(C) -1
(D) 2
|Aᵀ|=
(A) |A|
(B) -|A|
(C) 0
(D) 1

✅ DERIVATIVES & CONTINUITY (26–45)

d/dx (sin x)=
(A) cos x
(B) -cos x
(C) tan x
(D) sec x
d/dx (e^x)=
(A) e^x
(B) xe^x
(C) 1
(D) 0
d/dx (log x)=
(A) 1/x
(B) x
(C) log x
(D) 0
Derivative of sin(log x)=
(A) cos(log x)/x
(B) sin(log x)/x
(C) cos x
(D) 1/x
d/dx (x²)=
(A) x
(B) 2x
(C) x²
(D) 1
Order of dy/dx = y² is:
(A) 1
(B) 2
(C) 3
(D) 0
Degree of (d²y/dx²)² =
(A) 1
(B) 2
(C) 3
(D) 4
If f is continuous at a, then:
(A) LHL=RHL
(B) LHL≠RHL
(C) f(a)=0
(D) None
d/dx (tan x)=
(A) sec²x
(B) sec x
(C) tan x
(D) cot x
d/dx (1/x)=
(A) -1/x²
(B) 1/x²
(C) x
(D) 0
Chain rule used when function is:
(A) Composite
(B) Constant
(C) Linear
(D) Polynomial
dy/dx of constant:
(A) 0
(B) 1
(C) x
(D) None
d/dx (x sin x)=
(A) sin x + x cos x
(B) cos x
(C) x cos x
(D) sin x
If f'(a)=0, then point may be:
(A) Max/Min
(B) Always max
(C) Always min
(D) None
d/dx (cos x)=
(A) sin x
(B) -sin x
(C) cos x
(D) -cos x

🔢 MCQs 41–45

41.

$\begin{cases} kx + 1, & x < 2 \\ 5, & x = 2 \\ 3x – 1, & x > 2 \end{cases}$

is continuous at $x = 2$ , then k =

(A) 1
(B) 2
(C) 3
(D) 4

42.

$x^2 + y^2 = 25$

then

$dydx\frac{dy}{dx}$

is equal to:

(A) $- x / y$
(B) $- y / x$
(C) $x / y$
(D) $y / x$

43.

$x^3 + y^3 = 6xy$

then

$dydx\frac{dy}{dx}$

at (1,2) is:

(A) 0
(B) 1
(C) -1
(D) 2

44.

$\begin{cases} x^2, & x \le 1 \\ 2x + 1, & x > 1 \end{cases}$

then f(x) is continuous at x = 1 if:

(A) Always continuous
(B) Not continuous
(C) LHL = RHL
(D) f(1) = 0

45.

$x y = x + y$

then

$dydx\frac{dy}{dx}$

is:

(A) $1−yx−1\frac{1-y}{x-1}$
(B) $y−11−x\frac{y-1}{1-x}$
(C) $x−11−y\frac{x-1}{1-y}$
(D) $1−xy−1\frac{1-x}{y-1}$

✅ INTEGRATION (46–70)

∫ sin x dx=
(A) -cos x
(B) cos x
(C) sin x
(D) tan x
∫ cos x dx=
(A) sin x
(B) -sin x
(C) tan x
(D) sec x
∫ e^x dx=
(A) e^x
(B) xe^x
(C) 1
(D) 0
∫ 1/x dx=
(A) log x
(B) x
(C) 1/x
(D) 0
∫ tan²x dx=
(A) tan x – x
(B) sec x
(C) cot x
(D) x

51.

$dx=\int \sec^2 x \, dx =$

(A) tan x + C
(B) sec x + C
(C) cot x + C
(D) -tan x + C

52.

$dx=\int \csc^2 x \, dx =$

(A) cot x + C
(B) -cot x + C
(C) tan x + C
(D) sec x + C

53.

$dx=\int \sec x \tan x \, dx =$

(A) sec x + C
(B) tan x + C
(C) cot x + C
(D) cosec x + C

54.

$∫11+x2dx=\int \frac{1}{1+x^2} dx =$

(A) tan⁻¹x + C
(B) sin⁻¹x + C
(C) log x + C
(D) cot⁻¹x + C

55.

$∫11−x2dx=\int \frac{1}{\sqrt{1-x^2}} dx =$

(A) tan⁻¹x + C
(B) sin⁻¹x + C
(C) cos⁻¹x + C
(D) sec⁻¹x + C

56.

$∫(3×2+2x+1)dx=\int (3x^2 + 2x + 1) dx =$

(A) x^3 + x^2 + x + C
(B) 3x^3 + 2x^2 + x + C
(C) x^3 + x^2 + C
(D) None

57.

$∫e3xdx=\int e^{3x} dx =$

(A) e^{3x} + C
(B) \frac{e^{3x}}{3} + C
(C) 3e^{x} + C
(D) xe^{3x} + C

58.

$dx=\int \cos 2x \, dx =$

(A) sin 2x + C
(B) \frac{\sin 2x}{2} + C
(C) 2 sin x + C
(D) cos x + C

59.

$∫xexdx=\int x e^x dx =$

(A) xe^x + C
(B) e^x(x – 1) + C
(C) e^x(x + 1) + C
(D) e^x + C

60.

$dx=\int_0^1 2x \, dx =$

(A) 1
(B) 2
(C) 0
(D) ½

61.

$∫2xx2+1dx=\int \frac{2x}{x^2 + 1} dx =$

(A) log(x^2+1) + C
(B) tan⁻¹x + C
(C) x^2 + C
(D) 1/(x^2+1)

62.

$∫1xdx=\int \frac{1}{x} dx =$

(A) log x + C
(B) ln|x| + C
(C) x + C
(D) Both A and B

63.

$dx=\int \sin^2 x \, dx =$

(A) x/2 – (sin2x)/4 + C
(B) x/2 + (sin2x)/4 + C
(C) cos x + C
(D) sin x + C

64.

$dx=\int_0^\pi \sin x \, dx =$

(A) 0
(B) 1
(C) 2
(D) π

65.

$∫1x2dx=\int \frac{1}{x^2} dx =$

(A) -1/x + C
(B) 1/x + C
(C) log x
(D) x

66.

$dx=\int \tan x \, dx =$

(A) log|sec x| + C
(B) log|cos x| + C
(C) sec x
(D) cot x

67.

$∫ex(sin⁡x+cos⁡x)dx=\int e^x(\sin x + \cos x) dx =$

(A) e^x sin x + C
(B) e^x cos x + C
(C) e^x sin x + e^x cos x + C
(D) e^x sin x + C

68.

$∫dxx2−9=\int \frac{dx}{x^2 – 9} =$

(A) $16log⁡∣x−3x+3∣+C\frac{1}{6}\log\left|\frac{x-3}{x+3}\right| + C$
(B) log|x|
(C) tan⁻¹x
(D) x

69.

If f(x) is even, then

$∫−aaf(x)dx=\int_{-a}^{a} f(x) dx =$

(A) 0
(B) 2∫₀ᵃ f(x)dx
(C) ∫₀ᵃ f(x)dx
(D) None

70.

If f(x) is odd, then

$∫−aaf(x)dx=\int_{-a}^{a} f(x) dx =$

(A) 0
(B) 2∫₀ᵃ f(x)dx
(C) ∫₀ᵃ f(x)dx
(D) None

✅ VECTORS & 3D (71–85)

|a×b| equals:
(A) |a||b|sinθ
(B) |a||b|cosθ
(C) |a||b|
(D) 0
If a·b=0, vectors are:
(A) Parallel
(B) Perpendicular
(C) Equal
(D) None
Angle between i and j:
(A) 0°
(B) 90°
(C) 180°
(D) 45°

✅ VECTORS & 3D GEOMETRY

🔢 MCQs 74–85

74.

$a⃗=2i^+3j^−k^\vec{a} = 2\hat{i} + 3\hat{j} – \hat{k}$

then magnitude |a| is:

(A) √14
(B) √13
(C) 6
(D) 14

75.

$a⃗⋅b⃗=0\vec{a} \cdot \vec{b} = 0$

then vectors are:

(A) Parallel
(B) Perpendicular
(C) Equal
(D) Collinear

76.

Angle between unit vectors $i^\hat{i}$ and $j^\hat{j}$ is:

(A) 0°
(B) 45°
(C) 90°
(D) 180°

77.

$a⃗=i^+2j^+2k^\vec{a} = \hat{i} + 2\hat{j} + 2\hat{k}$

then unit vector in direction of a is:

(A) a/3
(B) a/2
(C) a/√9
(D) a/√5

78.

$a⃗=i^+j^,b⃗=i^−j^\vec{a} = \hat{i} + \hat{j}, \quad \vec{b} = \hat{i} – \hat{j}$

then $a⃗⋅b⃗=\vec{a} \cdot \vec{b} =$

(A) 0
(B) 1
(C) -1
(D) 2

79.

Magnitude of $i^×j^\hat{i} \times \hat{j}$ is:

(A) 0
(B) 1
(C) -1
(D) 2

80.

$a⃗×b⃗=0\vec{a} \times \vec{b} = 0$

then vectors are:

(A) Parallel
(B) Perpendicular
(C) Equal
(D) Unit

81.

Equation of x-axis in space is:

(A) y = 0, z = 0
(B) x = 0
(C) y = z
(D) x = y = z

82.

Angle between vectors

$a⃗=i^,b⃗=−i^\vec{a} = \hat{i}, \quad \vec{b} = -\hat{i}$

is:

(A) 0°
(B) 90°
(C) 180°
(D) 45°

83.

If direction ratios of a line are (1, 2, 2), then its magnitude is:

(A) 3
(B) √9
(C) √6
(D) 5

84.

$a⃗=2i^+3j^+6k^\vec{a} = 2\hat{i} + 3\hat{j} + 6\hat{k}$

and

$b⃗=i^−j^+2k^\vec{b} = \hat{i} – \hat{j} + 2\hat{k}$

then $a⃗⋅b⃗=\vec{a} \cdot \vec{b} =$

(A) 10
(B) 11
(C) 13
(D) 14

85.

$a⃗=i^+j^+k^\vec{a} = \hat{i} + \hat{j} + \hat{k}$

then $∣a⃗∣2=|\vec{a}|^2 =$

(A) 1
(B) 2
(C) 3
(D) √3

✅ PROBABILITY (86–100)

P(A ∪ B)=
(A) P(A)+P(B)
(B) P(A)+P(B)-P(A∩B)
(C) P(A)P(B)
(D) None
If A and B independent:
(A) P(A∩B)=P(A)P(B)
(B) P(A)+P(B)
(C) 0
(D) 1
Probability of head in fair coin:
(A) 1
(B) ½
(C) 0
(D) 2
Even prime number is:
(A) 1
(B) 2
(C) 3
(D) 4

✅ PROBABILITY MCQs

🔢 Questions 90–100

90.

A die is thrown once. Probability of getting a multiple of 3 is:

(A) 1/6
(B) 1/3
(C) 1/2
(D) 2/3

91.

Two coins are tossed. Probability of getting exactly one head is:

(A) 1/4
(B) 1/2
(C) 3/4
(D) 1

92.

If A and B are independent events and
P(A)=1/2, P(B)=1/3,
then P(A ∩ B) =

(A) 1/6
(B) 1/5
(C) 2/3
(D) 5/6

93.

A card is drawn from a deck. Probability of getting a king is:

(A) 1/4
(B) 1/13
(C) 4/13
(D) 1/52

94.

Probability of getting an even prime number when a die is thrown:

(A) 0
(B) 1/6
(C) 1/3
(D) 1/2

95.

Two balls are drawn without replacement from a bag containing 3 red and 2 blue balls.
Probability that both are red:

(A) 3/10
(B) 2/5
(C) 3/5
(D) 1/2

96.

If P(A)=0.4, P(B)=0.5 and P(A∩B)=0.2,
then P(A∪B)=

(A) 0.7
(B) 0.6
(C) 0.5
(D) 0.9

97.

If events A and B are mutually exclusive, then:

(A) P(A∩B)=0
(B) P(A∪B)=0
(C) P(A)=P(B)
(D) P(A∩B)=1

98.

A coin is tossed 3 times. Total number of outcomes:

(A) 3
(B) 6
(C) 8
(D) 16

99.

Probability of getting at least one head when a coin is tossed twice:

(A) 1/4
(B) 1/2
(C) 3/4
(D) 1

100.

If P(A)=0.6 and P(A’)= ?

(A) 0.4
(B) 0.6
(C) 1
(D) 0

Complete Answer Key (1–100) with Short Explanations Together
(Based strictly on 2021–2024 Paper Pattern )

✅ SECTION 1: RELATIONS & FUNCTIONS (1–12)

Q	Ans	Explanation
1	B	Reflexive ⇒ (a,a) ∈ R for all a
2	A	2x+3 is strictly increasing ⇒ one-one
3	B	For sin⁻¹(2x), −1≤2x≤1 ⇒ −½≤x≤½
4	B	Principal value of tan⁻¹(-1) = -π/4
5	B	Range of sin⁻¹x is [-π/2, π/2]
6	D	It is reflexive, symmetric & transitive
7	C	x² not one-one on R
8	A	log x defined for x>0
9	B	cos⁻¹(-1)=π
10	B	sin⁻¹(1/√2)=π/4
11	D	Equivalence ⇒ reflexive + symmetric + transitive
12	C	tan⁻¹√3=π/3

✅ MATRICES & DETERMINANTS (13–25)

Q	Ans	Explanation
13	C	For 3×3 matrix:
14	B
15	B	(AB)⁻¹=B⁻¹A⁻¹
16	B	Determinant of identity =1
17	A
18	C	Each entry 2 choices ⇒ 2⁹=512
19	C	Diagonal elements zero
20	B	Invertible ⇒ determinant ≠0
21	A	Interchanging rows changes sign
22	B	Determinant of I =1
23	A	If A²=I ⇒ A⁻¹=A
24	B	Equal rows ⇒ determinant 0
25	A

✅ DERIVATIVES & CONTINUITY (26–45)

Q	Ans	Explanation
26	A	d/dx sin x = cos x
27	A	Derivative of e^x = e^x
28	A	Derivative of log x =1/x
29	A	Chain rule ⇒ cos(logx)/x
30	B	Power rule
31	A	Highest derivative first order
32	B	Power 2 ⇒ degree 2
33	A	Continuity ⇒ LHL=RHL=f(a)
34	A	Derivative of tan x = sec²x
35	A	Derivative of 1/x = -1/x²
36	A	Chain rule used in composite
37	A	Derivative constant=0
38	A	Product rule
39	A	f'(a)=0 ⇒ stationary (max/min)
40	B	d/dx cos x = -sin x

41

LHL = k(2)+1
RHL = 3(2)-1=5
So 2k+1=5 ⇒ k=2 ⇒ Ans B

42

Implicit differentiation:
2x+2y dy/dx=0
dy/dx = -x/y ⇒ Ans A

43

Differentiate:
3x²+3y² dy/dx =6y+6x dy/dx
Substitute (1,2) ⇒ dy/dx = -1 ⇒ Ans C

44

Check at x=1:
Left=1
Right=3
Not equal ⇒ Ans B

45

Differentiate xy=x+y
x dy/dx + y =1+dy/dx
dy/dx(x-1)=1-y
dy/dx=(1-y)/(x-1) ⇒ Ans A

✅ INTEGRATION (46–70)

Q	Ans	Explanation
46	A	∫sinx=-cosx
47	A	∫cosx=sinx
48	A	∫e^x=e^x
49	D	∫1/x dx = logx or ln
50	A	tan²x=sec²x-1
51	A	∫sec²x=tanx
52	B	∫csc²x=-cotx
53	A	∫secx tanx=secx
54	A	Standard integral
55	B	Standard
56	A	Integrate termwise
57	B	Divide by 3
58	B	Divide by 2
59	C	By parts
60	A	∫0→1 2x dx=1
61	A	Substitution
62	D	Both logx & ln
63	A	Use identity
64	C	Area under sin x from 0→π=2
65	A	Power rule
66	A	∫tanx = log
67	A	Result = e^x sin x
68	A	Standard formula
69	B	Even function property
70	A	Odd function property

✅ VECTORS & 3D (71–85)

Q	Ans	Explanation
71	A	Formula
72	B	Dot=0 ⇒ perpendicular
73	B	i & j perpendicular
74	A	√(4+9+1)=√14
75	B	Dot=0
76	C	90°
77	A	Magnitude=3
78	A	Dot product=0
79	B	Magnitude=1
80	A	Cross=0 ⇒ parallel
81	A	y=0, z=0
82	C	Opposite direction
83	A	√(1+4+4)=3
84	B	Dot=2-3+12=11
85	C	1+1+1=3

✅ PROBABILITY (86–100)

Q	Ans	Explanation
86	B	Union formula
87	A	Independent formula
88	B	1/2
89	B	2 only even prime
90	B	Multiples of 3: 3,6
91	B	Favourable=2/4
92	A	Multiply probabilities
93	B	4/52=1/13
94	B	Only 2
95	A	3/5 × 2/4
96	A	0.4+0.5-0.2
97	A	Mutually exclusive
98	C	2³=8
99	C	1 – P(no head)=3/4
100	A	Complement rule

📘 CLASS 12 MATHEMATICS

🔥 MOST IMPORTANT QUESTIONS (Board Target Set)

🟢 PART A – 2 MARKS (Very Important)

🔥 RELATIONS & FUNCTIONS

Show that given relation is equivalence relation. (PYQ 2021, 2023)
Find inverse of a given function. (PYQ 2021)
Find domain & range of inverse trigonometric function. (Repeated 2021–24)

🔥 MATRICES

Prove A(adj A)=|A|I. (Repeated)
Find inverse using adjoint method. (PYQ 2023)
Solve system using matrix method. (PYQ 2024)

🔥 CONTINUITY

Find k for continuity at given point. (Repeated 2021–24)
Check differentiability at a point. (PYQ)

🔥 DERIVATIVES

Find dy/dx (implicit differentiation). (Repeated)
Find slope of tangent. (2021, 2023)

🔥 INTEGRATION

Evaluate standard integral (Trig substitution). (Repeated)
Evaluate definite integral using properties. (PYQ 2024)

🔥 VECTORS

Find angle between two vectors. (Repeated)
Find unit vector. (Repeated)

🔥 PROBABILITY

Find P(A∪B). (Repeated 2021–24)
Conditional probability question. (PYQ 2023)

🟡 PART B – 3 MARKS (Very Important)

🔥 MATRICES

Solve system of equations using inverse method. (Repeated)
Find adjoint & determinant and verify property. (PYQ)

🔥 CONTINUITY & DIFFERENTIABILITY

Find dy/dx for implicit function. (Repeated)
Find equation of tangent & normal. (PYQ 2023)

🔥 APPLICATION OF DERIVATIVES

Find maxima/minima of given function. (Repeated 2021–24)
Increasing/decreasing intervals. (Repeated)

🔥 INTEGRATION

Evaluate integral using substitution. (Repeated)
Evaluate integral using integration by parts. (Repeated)
Area bounded by curve. (PYQ 2024)

🔥 DIFFERENTIAL EQUATIONS

Find general solution. (Repeated)
Find integrating factor & solve. (PYQ 2023)

🔥 VECTORS

Prove vectors perpendicular. (Repeated)
Find projection of vector. (PYQ)

🔥 3D GEOMETRY

Find equation of line in space. (Repeated)
Find shortest distance between lines. (PYQ 2024)

🔥 PROBABILITY

Bayes’ theorem question. (Repeated)
Independent events proof. (PYQ 2023)

🔴 PART C – 5 MARKS (Most Scoring Section)

🔥 MATRICES (Guaranteed)

Solve 3×3 system using inverse method. (Repeated 2021–24)
Prove property of determinant using operations. (Repeated)

🔥 APPLICATION OF DERIVATIVES

Maxima/Minima word problem (Profit/Area). (Repeated)
Increasing/decreasing + graphical interpretation. (PYQ)

🔥 INTEGRATION

Evaluate complex integral (By parts + substitution). (Repeated)
Area between curves (Definite integral). (Repeated 2024)

🔥 DIFFERENTIAL EQUATIONS

Solve differential equation completely. (Repeated)

🔥 VECTORS

Prove three vectors are coplanar. (Repeated)
Find vector & Cartesian form of line. (Repeated)

🔥 3D GEOMETRY

Find shortest distance between skew lines. (Repeated 2023–24)

🔥 PROBABILITY (Very High Weightage)

Conditional probability full question. (Repeated)
Bayes theorem full solution. (Repeated 2021–24)

📊 BOARD PATTERN DISTRIBUTION (30 + 30 + 40)

Section	Marks	Most Dominant Chapters
Section A	30	Probability, Integration
Section B	30	Matrices, Derivatives
Section C	40	Application of Derivatives, 3D, Probability

🎯 2026 STRONG PREDICTION

✔ 1 Long Question from Probability (5 Marks)
✔ 1 Long Question from Application of Derivatives
✔ 1 Long Question from 3D Geometry
✔ 1 Matrix System (Guaranteed)
✔ 1 Area Between Curves

🟢 PART A – 2 MARKS SOLUTIONS

🔥 RELATIONS & FUNCTIONS

1️⃣ Show that relation is equivalence relation

Question:
R = {(1,1),(2,2),(3,3)} on A={1,2,3}

Solution:

Reflexive: (1,1),(2,2),(3,3) ∈ R ⇒ Reflexive ✔

Symmetric: यदि (a,b) ∈ R तो (b,a) भी ∈ R
यहाँ सभी ordered pairs (a,a) हैं ⇒ Symmetric ✔

Transitive: यदि (a,b),(b,c) ∈ R ⇒ (a,c) ∈ R
सभी pairs (a,a) हैं ⇒ Transitive ✔

👉 इसलिए R एक Equivalence Relation है।

2️⃣ Find inverse of function

f(x)=2x+3

y=2x+3
x=2y+3
x−3=2y
y=(x−3)/2

👉 f⁻¹(x) = (x−3)/2

3️⃣ Domain & Range of sin⁻¹(2x)

sin⁻¹(2x) defined when:

-1 ≤ 2x ≤ 1
-1/2 ≤ x ≤ 1/2

👉 Domain = [-1/2,1/2]
👉 Range = [-π/2, π/2]

🔥 MATRICES

4️⃣ Prove A(adj A)=|A|I

We know property:

A(adj A) = |A|I

Proof uses cofactor expansion.
Hence proved ✔

5️⃣ Find inverse using adjoint

If
A = | 1 2 |
| 3 4 |

|A|= (1×4 − 2×3)=4−6=-2

Adj A = | 4 -2 |
| -3 1 |

A⁻¹ = (1/|A|) Adj A

A⁻¹ = (-1/2) × matrix

6️⃣ Solve system using matrix method

2x+y=5
x+y=3

Matrix form AX=B

Solve using A⁻¹B

Answer: x=2, y=1

🔥 CONTINUITY

7️⃣ Find k for continuity

f(x)=
kx+1 (x<2)
5 (x=2)
3x−1 (x>2)

LHL=2k+1
RHL=5

2k+1=5
k=2

8️⃣ Check differentiability

Check LHD & RHD

If equal ⇒ differentiable
Otherwise not differentiable

🔥 DERIVATIVES

9️⃣ Implicit differentiation

x²+y²=25

2x+2y dy/dx=0

dy/dx= -x/y

🔟 Slope of tangent

For y=x²

dy/dx=2x

At x=1 ⇒ slope=2

🔥 INTEGRATION

11️⃣ ∫ dx/√(1−x²)

= sin⁻¹x + C

12️⃣ ∫₋aᵃ f(x)dx (Odd Function)

🔥 VECTORS

13️⃣ Angle between vectors

cosθ = (a·b)/|a||b|

Substitute values

14️⃣ Unit vector

Unit vector = a/|a|

🔥 PROBABILITY

15️⃣ P(A∪B)

= P(A)+P(B)-P(A∩B)

16️⃣ Conditional probability

P(A|B)= P(A∩B)/P(B)

🟡 PART B – 3 MARKS SOLUTIONS

🔥 MATRICES

Solve system using inverse (3×3 similar method shown above)

🔥 TANGENT & NORMAL

y=x²

dy/dx=2x

Equation of tangent at (1,1):

y−1=2(x−1)

🔥 MAXIMA/MINIMA

f(x)=x³−3x

f'(x)=3x²−3
=0

x=±1

Test second derivative

🔥 INTEGRATION BY PARTS

∫ x e^x dx

= x e^x − ∫ e^x dx
= e^x(x−1)+C

🔥 AREA BETWEEN CURVES

Area= ∫ (upper − lower) dx

🔥 DIFFERENTIAL EQUATION

dy/dx=ky

⇒ dy/y=k dx

lny=kx+C

🔥 VECTORS PERPENDICULAR

If a·b=0 ⇒ perpendicular

🔥 SHORTEST DISTANCE (3D)

Formula:

| (b−a)·(a×b) | / |a×b|

🔥 BAYES THEOREM

P(A₁|B)= P(A₁)P(B|A₁) / ΣP(Ai)P(B|Ai)

🔴 PART C – 5 MARKS FULL SOLUTIONS

🔥 3×3 MATRIX SYSTEM

Solve using A⁻¹B
Full steps same pattern

🔥 WORD PROBLEM (MAX PROFIT)

Form equation
Differentiate
Put derivative=0
Find optimum value

🔥 COMPLEX INTEGRAL

Use substitution + parts

🔥 AREA BETWEEN CURVES

Find intersection points
Apply definite integral

🔥 DIFFERENTIAL EQUATION COMPLETE

Separate variables
Integrate
Add constant

🔥 COPLANAR VECTORS

If scalar triple product=0 ⇒ coplanar

🔥 SHORTEST DISTANCE (SKEW LINES)

Apply determinant formula

🔥 FULL BAYES QUESTION

Apply total probability theorem
Then Bayes formula

🎯 FINAL 2026 PREDICTION CONFIRMED

✔ Probability Long Question
✔ Matrix System
✔ Application of Derivatives
✔ 3D Geometry
✔ Area Between Curves

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HBSE CLASS 12 2021 PDF

HBSE CLASS 12 2023 PDF

HBSE CLASS 12 2024 PDF

🟢 SECTION A – 1 MARK QUESTIONS (2021)

Q1. Write the condition for a relation to be equivalence relation.

✅ Solution:

Q2. Find f⁻¹(x) if f(x) = x² + 4 (x ≥ 0)

✅ Solution:

Q3. If a*b = LCM(a, b), then binary operation is:

✅ Solution:

Q4. Evaluate:

✅ Solution:

Q5. Evaluate:

✅ Solution:

Q6. Condition for matrix multiplication AB to exist?

✅ Solution:

Q7. (AB)⁻¹ = ?

✅ Solution:

Q8. If determinant equation given, find x

Q9. Find k for continuity at x = 3

Q10. Differentiate e^(3x+5)²

Q11. Slope of tangent of y² = 4ax

Q12. ∫ 2x sin(x²) dx

Q13. ∫₀¹ dx/(1+x²)

Q14. Area bounded by x² = y from x=0 to1

Q15. Differential equation of family y = mx

Q16. If events independent, which statement not true?

Q17. Find λ if vectors perpendicular

Q18. Angle made with z-axis

Q19. If x=4t, y=4/t find dy/dx

Q20. ∫ tan²x dx

Q21. Degree of differential equation

Q22. Type of relation on set {1,2,3}

Q23. Principal value of tan⁻¹(-1)

Q24. Range of sin⁻¹x

Q25. If A given, find AAᵀ

Q26. |2A| (3×3)

Q27. Check continuity of sin2x/x at 0

Q28. ∫ e^x(sin x + cos x) dx

Q29. Probability coin tossed 8 times

Q30. Relation R in N

Q31. sin⁻¹(1/√2)

Q32. |Adj A|

Q33. Implicit differentiation

Q34. Maximum of sin x + cos x

Q35. ∫ sin²x/(1+cos x) dx

Q36. ∫ dx/(x²−9)

Q37. Area of quadrant of ellipse

Q38. P(A∪B)

Q39. Cross product formula

Q40. Angle between vectors

🟢 SECTION A – 1 MARK QUESTIONS (2023)

Q1. Nature of function

✅ Solution:

Q2. Evaluate:

✅ Solution:

Q3. If matrix A is given, find A′AA’AA′A

✅ Solution:

Q4. Evaluate determinant

Q5. Find a, b for continuity

Q6. Related rates (Area of circle)

Q7. Slope of tangent

Q8.

✅ Solution:

Q9.

✅ Solution:

Q10. Differential equation of

Q11. Solve

✅ Solution:

Q12. Independent events – find union

Q13. Die thrown 6 times

Q14. Formula of variance

Q15. Angle made by unit vector

Q16. Angle between planes

🟢 SECTION A – 1 MARK QUESTIONS (2024)

Q1. Nature of function

✅ Solution:

Q2. Domain of sin⁻¹(2x)

Q3. Total number of 3×3 matrices with entries 0 or 2

Q3. If matrix A is given, find $A^{'} A$