Top CBSE Class 12 Maths Questions for Board Exam 2026 – Prepare Smartly

 CBSE Class 12 Mathematics Important Questions 2026 – Most Expected Board Exam Questions

                  CHAPTER 1: RELATIONS AND FUNCTIONS

Mock Test: Functions and Relations

Q1: If f: {1,3, 4} → {1, 2, 5} and g: {1,2, 5} → {1, 3} given by f = {(1,2), (3, 5), (4,1)} and g = {(1,3), (2, 3), (5,1)}. Write down g∘f. (All India 2014C)

Show Hint/Answer

g∘f = {(1,3), (3,1), (4,3)}

Q2: Let R be the equivalence relation in the set A = {0,1,2,3,4,5} given by R = {(a, b) : 2 divides (a – b)}. Write the equivalence class [0]. (Delhi 2014C)

Show Hint/Answer

[0] = {0, 2, 4}

Q3: 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. (All India 2011)

Show Hint/Answer

f is one-one (injective), but not onto (not all elements of S are covered)

Q4: If f : R → R is defined by f(x) = 3x + 2, then define f[f(x)]. (Foreign 2011; Delhi 2010)

Show Hint/Answer

f[f(x)] = 3(3x+2) + 2 = 9x + 8

Q5: State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2,1)} not to be transitive. (Delhi 2011)

Show Hint/Answer

Because (1,2) ∈ R and (2,1) ∈ R but (1,1) ∉ R, so R is not transitive

Q6: What is the range of the function f(x) = |x−1|/(x−1), x ≠ 1? (Delhi 2010)

Show Hint/Answer

Range = {−1, 1}

Q7: If f : R → R and g:R → R are given by f(x) = sin x and g(x) = 5x², then find g∘f(x). (Foreign 2010)

Show Hint/Answer

g∘f(x) = g(f(x)) = 5(sin x)²

Q8: If the function f:R → R defined by f(x) = 3x – 4 is invertible, then find f⁻¹. (All India 2010C)

Show Hint/Answer

f⁻¹(y) = (y + 4)/3

Q9: Let f : N → Y be a function defined as f(x) = 4x + 3, where Y = {y ∈ N : y = 4x + 3, for some x ∈ N}. Show that f is invertible. Find its inverse. (All India 2019)

Show Hint/Answer

f⁻¹(y) = (y−3)/4

📘 Inverse Trigonometric Functions – IMP Quiz

Class 12 Mathematics | Chapter 2

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1. IMP. Find the principal value of:
   tan^-1(√3) – sec^-1(−2)
   
   Solution: _______________________________
   Answer (Check): ___________________________

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2. IMP. If
   sin ( sin^-1(1/5) + cos^-1x ) = 1,
   find the value of x.
   
   Solution: _______________________________
   Answer (Check): ___________________________

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3. IMP. Evaluate:
   cos^-1(−1/2) + 2 sin^-1(1/2)
   
   Solution: _______________________________
   Answer (Check): ___________________________

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4. IMP. Find the value of:
   cot ( π/2 − 2 cot^-1√3 )
   
   Solution: _______________________________
   Answer (Check): ___________________________

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5. IMP. Find the value of:
   tan ( 2 tan^-15 )
   
   Solution: _______________________________
   Answer (Check): ___________________________

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6. IMP. Evaluate:
   tan^-1 [ 2 sin ( 2 cos^-1(3/√2) ) ]
   
   Solution: _______________________________
   Answer (Check): ___________________________

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7. IMP. Find the value of:
   sin [ π/3 − sin^-1(−1/2) ]
   
   Solution: _______________________________
   Answer (Check): ___________________________

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8. IMP. Find the principal value of:
   cos^-1(cos 2π/3) + sin^-1(sin 2π/3)
   
   Solution: _______________________________
   Answer (Check): ___________________________

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9. IMP. Prove that:
   3 sin^-1x = sin^-1(3x − 4x³),
   where x ∈ [ −1/2 , 1/2 ].
   
   Proof: _______________________________
   Result (Check): ___________________________

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10. IMP. Solve for x:
    tan^-1(x + 1) + tan^-1(x − 1) = tan^-1(8/31)
    
    Solution: _______________________________
    Answer (Check): ___________________________

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11. IMP. Find the value of:
    sin ( cos^-1(4/5) + tan^-1(3/4) )
    
    Solution: _______________________________
    Answer (Check): ___________________________

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12. IMP. If
    tan^-1((x−3)/(x−4)) + tan^-1((x+3)/(x+4)) = π/4,
    find the value of x.
    
    Solution: _______________________________
    Answer (Check): ___________________________

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13. IMP. Prove that:
    tan ( π/4 + ½ cos^-1(a/b) ) + tan ( π/4 − ½ cos^-1(a/b) ) = 2b/a
    
    Proof: _______________________________
    Result (Check): ___________________________

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14. IMP. Solve for x:
    tan^-1(x−1) + tan^-1x + tan^-1(x+1) = tan^-1(3x)
    
    Solution: _______________________________
    Answer (Check): ___________________________

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15. IMP. Prove that:
    cot^-1 [ (√(1+sin x) + √(1−sin x)) / (√(1+sin x) − √(1−sin x)) ] = x/2,
    for 0 < x < π/2.
    
    Proof: _______________________________
    Result (Check): ___________________________

Class 12 Maths – Matrices Quiz (CBSE Previous Year Questions)

Q1. If \(3A - B = \begin{bmatrix}5 & 1 \\ 0 & 1\end{bmatrix}\) and \(B = \begin{bmatrix}4 & 2 \\ 3 & 5\end{bmatrix}\), find matrix A.

\(\begin{bmatrix}3 & 1 \\ 1 & 2\end{bmatrix}\)
\(\begin{bmatrix}3 & 1 \\ 1 & 2\end{bmatrix}\)
\(\begin{bmatrix}2 & 1 \\ 1 & 3\end{bmatrix}\)

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Q2. Find the value of \(x - y\), if \(2\begin{bmatrix}1 & 0 \\ 3 & x\end{bmatrix} + \begin{bmatrix}y & 1 \\ 0 & 2\end{bmatrix} = \begin{bmatrix}5 & 1 \\ 6 & 8\end{bmatrix}\)

1
0
2

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Q3. If A is a square matrix such that \(A^2 = I\), find the value of \((A - I)^3 + (A + I)^3 - 7A\).

0
I
A

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Q4. Write the element \(a_{ij}\) of a 3 × 3 matrix if \(a_{ij} = |i - j|^2\).

\(i - j\)
\((i - j)^2\)
\(|i + j|\)

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Q5. If A is a square matrix such that \(A^2 = A\), find \(7A - (I + A)^3\).

A
−I
0

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📌 Tip: These questions are extremely important for board examinations.

Class 12 Maths – Determinants Quiz (CBSE Previous Year Questions)

Q1. If A = ⎡⎣5 −4 −4
6 3 −7
−3 2 3⎤⎦, find the cofactor of element a₂₁.

2
−20
20

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Q2. If | 2x 2(x+1) x+3 | | x+1 3 5 | = | 1 3 5 | | 3 x 3 | | 3 5 3 | find the value of x.

1
2
−1

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Q3. If Δ = | 5 2 1 | | 3 0 2 | | 8 1 3 |, find the cofactor of a₂₃.

−3
−11
11

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Q4. For what value of x is the matrix [ 2x+4 x+5 ] [ 4 3 ] singular?

−2
−1
1

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Q5. For what value of x is the matrix [ 5−x 2 ] [ x+1 4 ] singular?

1
3
−3

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Q6. Find the value of Δ = | x+y z−3 | | y+z x−3 |.

x² + y² + z² − 9
(x+y)(x−3) − (y+z)(z−3)
0

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Q7. Find the value of the determinant | 1 0 2 | | 1 1 7 | | 1 8 3 |.

10
−36
36

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Class 12 Maths – Determinants Quiz (CBSE)

Q1. If the matrix A is:

5	-4	-4
6	3	-7
-3	2	3

Find the cofactor of element a₂₁.

20
-20
10

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Q2. If the determinant of the following matrix is equal to that of the given matrix, find x:

2x	2(x+1)	x+3
x+1	3	5
3	x	3

and

1	3	5
3	5	3
5	3	1

1
2
-1

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Q3. If Δ is the determinant of the matrix:

5	2	1
3	0	2
8	1	3

Find the cofactor of a₂₃.

11
-11
3

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Q4. For what value of x is the matrix singular?

2x+4	x+5
4	3

-2
-1
1

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Q5. For what value of x is the matrix singular?

5 - x	2
x + 1	4

1
3
-3

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© 2026 Board Exam Prep - Chapter 4 Determinants Important Questions

Determinants – CBSE Board Questions (Quiz)

1. Find the maximum value of
   | 1 1 1 |
   | 1 1+sinθ 1 |
   | 1 1 1+cosθ |
   (Delhi 2016)
2. If
   | x −sinθ cosθ |
   | sinθ −x 1 |
   | cosθ 1 x | = 8
   Find the value of x.
   (Foreign 2016)
3. In the interval π/2 < x < π, find the value of x for which the matrix
   | 2sinx 1 |
   | 3 2sinx |
   is singular.
   (All India 2015)
4. Find the value of the determinant
   | p p−1 |
   | p+1 p |
   (Delhi 2014)
5. For what value of x is the matrix
   | 2 x |
   | x+2 4 |
   singular?
   (Delhi 2011)
6. Find the value of the determinant
   | 0 2 4 |
   | 2 3 5 |
   | 0 4 6 |
   (Delhi 2010)
7. Find the value of
   | 2 3 5 |
   | 7 8 9 |
   | 6 5 8 |
   (All India 2014)
8. Find the value of
   | 4 4 4 |
   | a b c |
   | b+c c+a a+b |
9. Using properties of determinants, prove that
   | a+b+c −c −b |
   | −c a+b+c −a |
   | −b −a a+b+c |
   = 2(a+b)(b+c)(c+a)
   (Delhi 2019)
10. Using properties of determinants, prove that
    | 1 1+3y 1 |
    | 1 1 1+3z |
    | 1+3x 1 1 |
    = 9(3xyz + xy + yz + zx)
    (CBSE 2018)
11. Using properties of determinants, prove that
    | a² a²+ab ab |
    | ab b² b²+bc |
    | ac+c² ac c² |
    = 4a²b²c²
    (All India 2015, Foreign 2014)
12. Using properties of determinants, prove that
    | a+1 a+2 a+2 |
    | a+2 a+3 a+3 |
    | a+2 a+3 a+4 |
    | 1 1 1 |
    = 2
    (Delhi 2015)
13. Using properties of determinants, prove that
    | x²+1 xy xz |
    | xy y²+1 yz |
    | xz yz z²+1 |
    = 1 + x² + y² + z²
    (Delhi 2014)
14. Using properties of determinants, prove that
    | 2y 2z x |
    | −y−z −x 2z |
    | 2x 2y −x−y |
    = (x + y + z)³
    (Delhi 2014)
15. Using properties of determinants, prove that
    | 1+a 1 1 |
    | 1 1+b 1 |
    | 1 1 1+c |
    = abc + ab + bc + ca
    (All India 2014, 2009)
16. Using properties of determinants, prove that
    | α α² β+γ |
    | β β² γ+α |
    | γ γ² α+β |
    = (α−β)(β−γ)(γ−α)(α+β+γ)
    (Delhi 2012, 2010, 2008)

Continuity & Differentiability - Class 12 Board Exam Quiz

1. Verify whether the function
   \( f(x)=\begin{cases} x\sin\left(\frac{1}{x}\right), & x\neq 0 \\ 0, & x=0 \end{cases} \)
   is continuous at \(x=0\). :contentReference[oaicite:1]{index=1}
2. Check for differentiability of the function \(f(x)=|x-5|\) at the point
   \(x=5\). :contentReference[oaicite:2]{index=2}
3. Find the value of \(\lambda\), if the function
   \( f(x)=\begin{cases} \frac{\sin^2(\lambda x)}{x^2}, & x\neq 0 \\ 1, & x=0 \end{cases} \)
   is continuous at \(x=0\). :contentReference[oaicite:3]{index=3}
4. Find the values of \(p\) and \(q\) such that a piecewise function is differentiable at \(x=1\). (Board Style) :contentReference[oaicite:4]{index=4}
5. If \( f(x)=\frac{\sin 3x}{2x} \) for \(x\neq 0\) and \(f(0)=k+1\), find the value of \(k\) so that \(f(x)\) is continuous at 0. :contentReference[oaicite:5]{index=5}
6. Discuss the continuity of the greatest integer function \(f(x)=[x]\) in the interval \(-3 < x < 3\). :contentReference[oaicite:6]{index=6}
7. Examine the continuity of \(f(x)=x^3+2x^2-1\) at \(x=1\). :contentReference[oaicite:7]{index=7}
8. Find all points of discontinuity of the function
   \(f(x)=\begin{cases}2x+3, &x\le 2 \\ 2x-3, &x > 2\end{cases}\). :contentReference[oaicite:8]{index=8}
9. Differentiate \(y=x^{\sin x}+(\sin x)^{\cos x}\) with respect to \(x\). (Higher–order differentiation) :contentReference[oaicite:9]{index=9}
10. If \(\log(x^2+y^2)=2\tan^{-1}\left(\frac{y}{x}\right)\), find \(\frac{dy}{dx}\). :contentReference[oaicite:10]{index=10}
11. If \( (x^2+y^2)^2=xy\), find \(\frac{dy}{dx}\). :contentReference[oaicite:11]{index=11}
12. If \(x=A\cos 4t+B\sin 4t\), find \(\frac{d^2x}{dt^2}\). (Derivative in parametric form) :contentReference[oaicite:12]{index=12}

Continuity & Differentiability – Class 12 Important Board Questions

1. Verify whether the function
   \( f(x) = \begin{cases} x\sin(1/x), & x \neq 0 \\ 0, & x = 0 \end{cases} \)
   is continuous at \(x = 0\).
   (CBSE Subjective – Previous Year) :contentReference[oaicite:1]{index=1}
2. Check if the function
   \( f(x)=|x-5| \)
   is differentiable at \(x=5\).
   (CBSE Subjective – Previous Year) :contentReference[oaicite:2]{index=2}
3. Find the value of \(k\) such that the function
   \( f(x)=\frac{\sin^2(kx)}{x^2} \) for \(x\neq 0\) and \(f(0)=1\)
   is continuous at \(x=0\).
   (Previous Boards) :contentReference[oaicite:3]{index=3}
4. Determine the value of constants \(p\) and \(q\) such that the piecewise function is differentiable at \(x=1\):
   \[ f(x)=\begin{cases} px+2, & x\leq1\\ xq-1, & x>1 \end{cases} \]
   (Board-style subjective) :contentReference[oaicite:4]{index=4}
5. Find the value of \(k\) so that the function
   \( f(x)=\frac{\sin 3x}{2x} \) for \(x\neq 0\) and \(f(0)=k+1\)
   is continuous at 0.
   (CBSE PYQs) :contentReference[oaicite:5]{index=5}
6. Discuss the continuity and points of discontinuity of the greatest integer function \(f(x)=[x]\) in the interval \(-3 < x < 3\).
   (Board model question) :contentReference[oaicite:6]{index=6}
7. Find all points of discontinuity of
   \[ f(x)=\begin{cases} 2x+3, & x\le 2\\ 2x-3, & x>2 \end{cases}. \]
   (Subjective continuity) :contentReference[oaicite:7]{index=7}
8. Use Rolle’s theorem to find a value \(c\) in the interval \([-√3,0]\) such that
   \(f(x)=x^3-3x\) satisfies \(f'(c)=0\).
   (Board-style long question) :contentReference[oaicite:8]{index=8}
9. If \(\log(x^2+y^2)=2\tan^{-1}\left(\frac{y}{x}\right)\), find \(\frac{dy}{dx}\).
   (Subjective differentiability) :contentReference[oaicite:9]{index=9}
10. If \((x^2 + y^2)^2 = xy\), find \(\frac{dy}{dx}\).
    (Previous board question) :contentReference[oaicite:10]{index=10}
11. Differentiate \(y=x^{\sin x}+(\sin x)^{\cos x}\) with respect to \(x\).
    (Board long answer) :contentReference[oaicite:11]{index=11}
12. Find the value(s) of \(c\) such that the Mean Value Theorem applies to the function
    \(f(x)=e^x\sin x\), on the interval \([0,\pi]\).
    (Subjective/MVT-based) :contentReference[oaicite:12]{index=12}

Application of Derivatives – Class 12 Important Questions (Board Pattern)

1. The total cost \(C(x)=0.005x^3-0.02x^2+30x+5000\). Find the marginal cost when 3 units are produced. (CBSE 2018) :contentReference[oaicite:1]{index=1}
2. The volume of a cube is increasing at 8 cm³/s. How fast is the surface area increasing when edge = 12 cm? (All India 2019) :contentReference[oaicite:2]{index=2}
3. Find the intervals in which
   \(f(x)=\frac{x^4}{4}-x^3-5x^2+24x+12\) is increasing and decreasing. (CBSE 2018) :contentReference[oaicite:3]{index=3}
4. Find the intervals where
   \(f(x)=3x^4-4x^3-12x^2+5\) is increasing or decreasing. (Delhi 2014) :contentReference[oaicite:4]{index=4}
5. Find the intervals where
   \(f(x)=2x^3-9x^2+12x+15\) is increasing or decreasing. (All India 2010) :contentReference[oaicite:5]{index=5}
6. Prove \(y=\frac{4\sin\theta}{2+\cos\theta}-\theta\) is increasing in \((0,\frac{\pi}{2})\). (All India 2016, 2011) :contentReference[oaicite:6]{index=6}
7. For \(f(x)=\sin3x-\cos3x\), \(0
8. Find the angle of intersection of curves \(x^2+y^2=4\) and \((x-2)^2+y^2=4\) in the first quadrant. (CBSE 2018) :contentReference[oaicite:8]{index=8}
9. Show that the normal at any point \(t\) on
   \(x=3\cos t-\cos^3t, \; y=3\sin t-\sin^3t\)
   satisfies \(4(y\cos^3t - x\sin^3t)=3\sin4t\). (Delhi 2016) :contentReference[oaicite:9]{index=9}
10. Find the points on \(x^2+y^2-2x-3=0\) where the tangent is parallel to the X-axis. (Delhi 2011) :contentReference[oaicite:10]{index=10}
11. The sum of perimeters of a circle and a square is \(k\). Prove that the sum of their areas is least when the side of the square equals the circle’s diameter. (Delhi 2014) :contentReference[oaicite:11]{index=11}
12. A tank with rectangular base and open top has volume 8 m³ and depth 2 m. Find the least cost of construction if base costs ₹70/m² and sides ₹45/m². (Delhi 2019) :contentReference[oaicite:12]{index=12}
13. Find the point \(P(x,y)\) on the curve \(y^2=4ax\) nearest to \((11a,0)\). (All India 2014) :contentReference[oaicite:13]{index=13}
14. Show that a right circular cone of given volume has least curved surface area when its semi-vertical angle is \(\cot^{-1}\sqrt{2}\). (Delhi 2014) :contentReference[oaicite:14]{index=14}
15. Prove that a cylinder with a given volume and open top has minimum surface area when height equals radius of base. (Foreign 2014; Delhi 2011) :contentReference[oaicite:15]{index=15}
16. Find the area of the greatest rectangle that can be inscribed in an ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\). (All India 2013) :contentReference[oaicite:16]{index=16}
17. Sketch the curve \(y=x^3+6x^2-9x+1\) using derivatives (increasing/decreasing, concavity & points of inflection). :contentReference[oaicite:17]{index=17}
18. If an object’s position is \(s(t)=t^3-6t^2+9t\), find velocity & acceleration at \(t=2\). :contentReference[oaicite:18]{index=18}
19. The displacement of a car is \(s(t)=4t^2-2t+7\). Find velocity and acceleration at \(t=3\). :contentReference[oaicite:19]{index=19}
20. Find the abscissa of the point on the curve \(3y=6x-5x^3\) whose normal passes through the origin. (MCQ style) :contentReference[oaicite:20]{index=20}

Integrals – Class 12 Important Board Questions (Chapter 7)

1. Evaluate: ∫\(\frac{\sin^2x-\cos^2x}{\sin x\cos x}\) dx. (All India 2017) :contentReference[oaicite:1]{index=1}
2. Find: ∫\(\frac{\sin^2x-\cos^2x}{\sin^2x\cos^2x}\) dx. (Delhi 2014) :contentReference[oaicite:2]{index=2}
3. Evaluate: ∫\(\frac{\sin^6x}{\cos^8x}\) dx. (All India 2014) :contentReference[oaicite:3]{index=3}
4. Compute: ∫\(\frac{1}{\sin^2x\cos^2x}\) dx. (Delhi 2014; Foreign 2014) :contentReference[oaicite:4]{index=4}
5. Find ∫cos⁻¹(sin x) dx. (Delhi 2014) :contentReference[oaicite:5]{index=5}
6. Write ∫(3√x + 1/√x) dx. (Delhi 2014) :contentReference[oaicite:6]{index=6}
7. Evaluate ∫(1−x)√x dx. (Delhi 2012) :contentReference[oaicite:7]{index=7}
8. Given ∫e^x(tan x+1) sec x dx = e^x f(x) + C. Find f(x). (All India 2012) :contentReference[oaicite:8]{index=8}
9. Evaluate: ∫\(\frac{2}{1+\cos 2x}\) dx. (Foreign 2012) :contentReference[oaicite:9]{index=9}
10. Find: ∫\(\frac{x+\cos 6x}{3x^2+\sin 6x}\) dx. (All India 2012) :contentReference[oaicite:10]{index=10}
11. Evaluate: ∫\(\frac{dx}{x^2+16}\) dx. (Delhi 2011) :contentReference[oaicite:11]{index=11}
12. Evaluate: ∫\(\frac{2-3\sin x}{\cos^2x}\) dx. (Delhi 2011) :contentReference[oaicite:12]{index=12}
13. Compute: ∫\(\frac{x^3-x^2+x-1}{x-1}\) dx. (Delhi 2011C) :contentReference[oaicite:13]{index=13}
14. Evaluate definite integral: ∫₀¹ x e^{x²} dx. (Foreign 2014) :contentReference[oaicite:14]{index=14}
15. Compute: ∫₀^(π/4) sin 2x dx. (Foreign 2014) :contentReference[oaicite:15]{index=15}
16. Evaluate: ∫₀¹ \(\frac{1}{\sqrt{1-x²}}\) dx. (All India 2014) :contentReference[oaicite:16]{index=16}
17. If ∫₀^a 1/(4+x²) dx = π/8, find a. (All India 2014) :contentReference[oaicite:17]{index=17}
18. Evaluate: ∫₀¹ 2x/(1+x²) dx. (All India 2011) :contentReference[oaicite:18]{index=18}
19. Compute: ∫₀^(−π/4)^(π/4) sin³x dx. (Delhi 2010) :contentReference[oaicite:19]{index=19}
20. Evaluate: ∫₀^(−π/2)^(π/2) sin⁵x dx. (All India 2010) :contentReference[oaicite:20]{index=20}

Application of Integrals – Class 12 Important Questions (CBSE)

1. Find the area bounded by the curve
   y = x², the x-axis and the ordinates x = 1 and x = 3.
   (Board Pattern)
2. Find the area enclosed between the curve
   y = x² − 4x + 3 and the x-axis.
   (CBSE Previous Year)
3. Find the area bounded by the curve
   y² = 4ax and the line x = a.
   (All India Board)
4. Find the area enclosed between the curves
   y = x² and y = 4x − x².
   (Delhi Board)
5. Find the area of the region bounded by the curve
   y = sin x, the x-axis, x = 0 and x = π.
   (CBSE PYQ)
6. Find the area enclosed between the curves
   y = x² and y = |x|.
   (Board-style question)
7. Find the area bounded by the curve
   y = √x, the x-axis and the line x = 4.
   (Previous Year)
8. Find the area enclosed between the curves
   x = y² and x = 4.
   (All India Board)
9. Find the area of the region bounded by
   y = x³ and y = x.
   (Delhi Board)
10. Find the area bounded by the curve
    y = cos x, the x-axis and x = 0 to x = π/2.
    (Board Pattern)
11. Find the area enclosed between the curves
    y = x² and y = 2 − x².
    (CBSE PYQ)
12. Find the area of the region bounded by the curve
    x = a cos θ, y = a sin θ (parametric form).
    (Board-style)
13. Find the area enclosed between the curves
    y = sin x and y = cos x in the interval [0, π/2].
    (All India Board)
14. Find the area of the region bounded by
    y² = x and x + y = 2.
    (Delhi Board)
15. Find the area bounded by the curve
    y = eˣ, y = 0, x = 0 and x = 1.
    (Board Pattern)
16. Find the area enclosed between the curves
    x = y² − 1 and x = 1 − y².
    (CBSE PYQ)
17. Find the area bounded by
    y = tan x and y = sec x between x = 0 and x = π/4.
    (Previous Year)
18. Find the area of the region enclosed by the ellipse
    x²/a² + y²/b² = 1.
    (Board-style long question)
19. Find the area bounded by the curve
    y = x² + 1 and y = 2x + 3.
    (CBSE PYQ)
20. Find the area enclosed between the curves
    y = |x| and y = x².
    (Board Pattern – High Frequency)

Differential Equations – Class 12 Important Questions (CBSE)

1. Form the differential equation representing the family of curves
   y = Ae^x + Be^−x.
   (CBSE Previous Year)
2. Form the differential equation of the family of curves
   y = Ax + B.
   (Board Pattern)
3. Form the differential equation of the family of circles having centre on y-axis and radius equal to the distance of the centre from the x-axis.
   (All India Board)
4. Solve the differential equation
   dy/dx = 3x² − 4x + 1.
   (Board Pattern)
5. Solve the differential equation
   dy/dx = sin x + cos x.
   (CBSE PYQ)
6. Solve:
   dy/dx = (x + y)².
   (Board Style)
7. Solve the differential equation
   dy/dx = (x − y)/(x + y).
   (Delhi Board)
8. Solve the differential equation
   (x + y) dx + (x − y) dy = 0.
   (All India Board)
9. Solve:
   (x² + y²) dx = 2xy dy.
   (CBSE PYQ)
10. Solve the differential equation
    dy/dx + y cot x = sin x.
    (Board Pattern – Linear DE)
11. Solve:
    dy/dx + y tan x = sec x.
    (Previous Year)
12. Solve the differential equation
    dy/dx − y/x = x².
    (CBSE PYQ)
13. Solve:
    dy/dx + y = e^x.
    (Board Pattern)
14. Solve the differential equation
    dy/dx + y/x = sin x.
    (All India Board)
15. Solve the differential equation
    dy/dx + 2y = e^−x.
    (Board Style)
16. Find the particular solution of
    dy/dx + y tan x = sin x,
    given y = 1 when x = 0.
    (CBSE PYQ)
17. Solve the differential equation
    dy/dx = (y² − x²)/(2xy).
    (Board Pattern)
18. Solve:
    (x + y) dy = (x − y) dx.
    (Delhi Board)
19. Solve the differential equation
    dy/dx = (x + y + 1)/(x + y − 1).
    (CBSE PYQ)
20. Solve the differential equation
    dy/dx + y/x = x²e^x.
    (Board Pattern – Long Question)

Vector Algebra – Class 12 Important Questions (CBSE)

1. Find the magnitude of the vector
   \(\vec{a} = 2\hat{i} - 3\hat{j} + 6\hat{k}\).
   (Board Pattern)
2. Find the unit vector in the direction of
   \(\vec{a} = 3\hat{i} - 4\hat{j} + 12\hat{k}\).
   (CBSE PYQ)
3. Find the angle between the vectors
   \(\vec{a} = \hat{i} + 2\hat{j} - 2\hat{k}\) and \(\vec{b} = 2\hat{i} - \hat{j} + \hat{k}\).
   (All India Board)
4. Find the projection of the vector
   \(\vec{a} = 4\hat{i} + 3\hat{j}\) on \(\vec{b} = 2\hat{i} + 2\hat{j}\).
   (Board Pattern)
5. Find the value of \( \lambda \) if the vectors
   \(\vec{a} = \hat{i} + 2\hat{j} + \lambda\hat{k}\) and \(\vec{b} = 2\hat{i} - \hat{j} + 3\hat{k}\)
   are perpendicular.
   (CBSE PYQ)
6. Find the value of \(k\) if the vectors
   \(\vec{a} = k\hat{i} + 3\hat{j} - 4\hat{k}\) and \(\vec{b} = 2\hat{i} - \hat{j} + k\hat{k}\)
   are parallel.
   (Delhi Board)
7. Find the position vector of a point which divides the line joining
   \(\vec{a} = 2\hat{i} - 2\hat{j} + \hat{k}\) and \(\vec{b} = \hat{i} + 2\hat{j} - 3\hat{k}\)
   internally in the ratio 2:1.
   (CBSE PYQ)
8. Find the area of the triangle whose vertices are
   A(1, 2, 3), B(2, −1, 1) and C(3, 1, −2).
   (All India Board)
9. Find the area of the parallelogram whose adjacent sides are represented by
   \(\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}\) and \(\vec{b} = 2\hat{i} - \hat{j} + \hat{k}\).
   (CBSE PYQ)
10. Find the volume of the parallelepiped whose edges are
    \(\vec{a} = \hat{i} + \hat{j} + \hat{k}\), \(\vec{b} = 2\hat{i} - \hat{j} + \hat{k}\) and \(\vec{c} = \hat{i} + 2\hat{j} - \hat{k}\).
    (Delhi Board)
11. Find the value of \( \lambda \) if the vectors
    \(\vec{a} = 3\hat{i} + 2\hat{j} + \hat{k}\), \(\vec{b} = \hat{i} + \lambda\hat{j} + 2\hat{k}\) and \(\vec{c} = 2\hat{i} + \hat{j} + 3\hat{k}\)
    are coplanar.
    (CBSE PYQ)
12. Find the vector equation of the line passing through the point
    with position vector \(2\hat{i} - \hat{j} + 3\hat{k}\)
    and parallel to the vector \(\hat{i} + 2\hat{j} - \hat{k}\).
    (Board Pattern)
13. Find the equation of the line passing through the points
    A(1, 2, 3) and B(3, 4, 5) in vector form.
    (All India Board)
14. Find the position vector of the foot of the perpendicular from the origin to the line
    \(\vec{r} = (\hat{i} + \hat{j}) + \lambda(2\hat{i} - \hat{j} + 2\hat{k})\).
    (Board Pattern)
15. Find the value of \(k\) if the vectors
    \(\vec{a} = \hat{i} + k\hat{j} + 2\hat{k}\) and \(\vec{b} = 2\hat{i} + \hat{j} + k\hat{k}\)
    are orthogonal.
    (CBSE PYQ)
16. Show that the vectors
    \(\vec{a} = 2\hat{i} - \hat{j} + \hat{k}\), \(\vec{b} = \hat{i} + \hat{j} - \hat{k}\) and \(\vec{c} = 3\hat{i} - \hat{j} + 2\hat{k}\)
    are coplanar.
    (Delhi Board)
17. Find the cosine of the angle between the vectors
    \(\vec{a} = \hat{i} + 3\hat{j} + 5\hat{k}\) and \(\vec{b} = 2\hat{i} - \hat{j} + \hat{k}\).
    (Board Pattern)
18. Find the vector equation of the line passing through the point
    A(1, −1, 2) and perpendicular to the plane
    \(\vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 5\).
    (CBSE PYQ)
19. Find the area of the triangle formed by the vectors
    \(\vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}\) and \(\vec{b} = \hat{i} + 2\hat{j} - \hat{k}\).
    (All India Board)
20. Find the volume of the tetrahedron formed by the vectors
    \(\vec{a} = \hat{i} + \hat{j} + \hat{k}\), \(\vec{b} = 2\hat{i} - \hat{j} + \hat{k}\) and \(\vec{c} = \hat{i} + 2\hat{j} - \hat{k}\).
    (Board Pattern)

Three Dimensional Geometry – Class 12 Important Questions (CBSE)

1. Find the direction cosines of the line joining the points
   A(2, −1, 3) and B(4, 2, −1).
   (Board Pattern)
2. Find the direction ratios of the line whose direction cosines are
   \(\frac{1}{3}, \frac{2}{3}, \frac{2}{3}\).
   (CBSE PYQ)
3. Find the angle between the two lines having direction ratios
   (1, −2, 2) and (2, 1, −2).
   (All India Board)
4. Find the equation of the line passing through the point
   (1, −2, 3) and parallel to the line
   \(\frac{x−1}{2}=\frac{y+1}{−1}=\frac{z}{3}\).
   (CBSE PYQ)
5. Find the equation of the line passing through the points
   A(2, 1, −1) and B(3, 4, 2).
   (Board Pattern)
6. Find the equation of the plane passing through the point
   (1, 2, 3) and perpendicular to the vector
   \(\hat{i} − 2\hat{j} + 3\hat{k}\).
   (CBSE PYQ)
7. Find the equation of the plane passing through the points
   (1, 0, −1), (2, 1, 0) and (0, −1, 2).
   (All India Board)
8. Find the equation of the plane which cuts the axes at intercepts
   3, 4 and 5.
   (Board Pattern)
9. Find the angle between the planes
   2x − y + 2z = 5 and x + 2y − z = 3.
   (CBSE PYQ)
10. Find the distance of the point
    (2, −1, 4) from the plane 2x + y − 2z + 5 = 0.
    (All India Board)
11. Find the distance between the parallel planes
    3x − 4y + 5z + 6 = 0 and 6x − 8y + 10z − 3 = 0.
    (CBSE PYQ)
12. Find the foot of the perpendicular from the point
    (1, 2, 3) to the plane x − y + z − 2 = 0.
    (Board Pattern)
13. Find the image of the point
    (3, −2, 1) in the plane 2x − y + 2z = 4.
    (CBSE PYQ)
14. Find the equation of the plane passing through the intersection of the planes
    x + y + z = 1 and 2x − y + z = 3
    and perpendicular to the plane
    x − 2y + 3z = 4.
    (Delhi Board – Long Question)
15. Find the shortest distance between the lines
    \(\frac{x−1}{2}=\frac{y−2}{3}=\frac{z}{−1}\)
    and
    \(\frac{x}{1}=\frac{y+1}{−2}=\frac{z−3}{2}\).
    (All India Board)
16. Find the equation of the plane passing through the line of intersection of the planes
    x − y + z = 1 and 2x + y − z = 3
    and at a distance 2 units from the origin.
    (Board Pattern – HOTS)
17. Find the coordinates of the point where the line
    \(\frac{x−1}{2}=\frac{y−2}{3}=\frac{z−3}{4}\)
    meets the plane x + y + z = 6.
    (CBSE PYQ)
18. Find the angle between the line
    \(\frac{x−1}{2}=\frac{y}{−1}=\frac{z+1}{2}\)
    and the plane 2x + y + 2z = 5.
    (All India Board)
19. Find the equation of the plane passing through the points
    A(1, 2, 3), B(2, 3, 1) and C(3, 1, 2).
    (Board Pattern)
20. Find the distance between the skew lines
    \(\frac{x}{1}=\frac{y−1}{2}=\frac{z+1}{3}\)
    and
    \(\frac{x−2}{2}=\frac{y+1}{−1}=\frac{z}{1}\).
    (CBSE PYQ – Long Question)

Linear Programming – Class 12 Important Questions (CBSE)

1. Define a Linear Programming Problem (LPP). State its basic components.
   (CBSE – Theory Question)
2. Write the standard form of a Linear Programming Problem.
   (Board Exam Pattern)
3. What is meant by a feasible solution and an optimal solution of an LPP?
   (CBSE PYQ)
4. Solve the following LPP graphically:
   Maximize Z = 3x + 5y
   subject to
   x + y ≤ 4
   x ≤ 2
   y ≤ 3
   x, y ≥ 0
   (Delhi Board)
5. Solve the following LPP graphically:
   Minimize Z = 2x + y
   subject to
   x + 2y ≥ 6
   x + y ≥ 4
   x, y ≥ 0
   (All India Board)
6. A manufacturer produces two types of goods A and B. Each unit of A requires 2 hours of labour and each unit of B requires 1 hour. Total labour available is 60 hours. Profit on A is ₹40 per unit and on B is ₹30 per unit. Formulate the LPP and find the maximum profit graphically.
   (CBSE PYQ – Case Study)
7. Solve the LPP:
   Maximize Z = 4x + 3y
   subject to
   x + y ≤ 10
   x + 3y ≤ 24
   x, y ≥ 0
   (Board Pattern)
8. Solve the LPP:
   Minimize Z = x + 2y
   subject to
   2x + y ≥ 8
   x + y ≥ 6
   x, y ≥ 0
   (CBSE PYQ)
9. A dietician wants to prepare a diet using foods P and Q. Food P contains 3 units of protein and 2 units of fat. Food Q contains 4 units of protein and 1 unit of fat. The diet should contain at least 12 units of protein and 6 units of fat. Formulate the LPP and find the optimal solution graphically.
   (Board Exam)
10. Solve the LPP:
    Maximize Z = 2x + 3y
    subject to
    x + y ≤ 8
    2x + y ≤ 10
    x, y ≥ 0
    (CBSE PYQ)
11. Explain the term “unbounded solution” of an LPP.
    (Board Theory Question)
12. Solve the following LPP graphically:
    Maximize Z = 5x + 4y
    subject to
    x + y ≤ 5
    3x + y ≤ 9
    x, y ≥ 0
    (Delhi Board)
13. What is a bounded feasible region? How does it affect the solution of an LPP?
    (CBSE Theory)
14. Solve the LPP:
    Minimize Z = 3x + 5y
    subject to
    x + y ≥ 6
    x + 3y ≥ 9
    x, y ≥ 0
    (All India Board)
15. A farmer wants to grow two crops A and B on a land of 100 hectares. Crop A requires 2 labour units per hectare and crop B requires 1 labour unit per hectare. Total labour available is 120 units. Profit from A is ₹2000 per hectare and from B is ₹3000 per hectare. Formulate the LPP and find the maximum profit graphically.
    (CBSE PYQ – Long Question)
16. Solve the LPP graphically:
    Maximize Z = 6x + 5y
    subject to
    x + y ≤ 10
    x + 2y ≤ 16
    x, y ≥ 0
    (Board Pattern)
17. Explain the importance of corner points in solving an LPP graphically.
    (CBSE Theory Question)
18. Solve the LPP:
    Minimize Z = 4x + y
    subject to
    x + y ≥ 5
    2x + y ≥ 8
    x, y ≥ 0
    (CBSE PYQ)
19. State any two limitations of Linear Programming.
    (Board Exam)
20. Solve the LPP:
    Maximize Z = 7x + 3y
    subject to
    x + y ≤ 6
    3x + y ≤ 12
    x, y ≥ 0
    (CBSE PYQ)

Probability – Class 12 Important Questions (CBSE)

1. If A and B are two events such that P(A) = 1/2, P(B) = 1/3 and P(A ∪ B) = 2/3, find P(A ∩ B).
   (CBSE PYQ)
2. Two dice are thrown at the same time. Find the probability that the sum of numbers on the top is 8.
   (Board Pattern)
3. A card is drawn at random from a well-shuffled pack of 52 cards. Find the probability that the card drawn is neither an ace nor a king.
   (CBSE PYQ)
4. Two coins are tossed simultaneously. Find the probability of getting at least one head.
   (Board Exam)
5. Find the probability that a number selected at random from the numbers 1, 2, 3, …, 20 is divisible by 3 or 5.
   (CBSE PYQ)
6. In a random experiment, P(A) = 0.4, P(B) = 0.3 and A and B are independent events. Find P(A ∩ B).
   (Board Pattern)
7. A die is thrown twice. Find the probability that the number obtained on the first throw is greater than the number obtained on the second throw.
   (CBSE PYQ)
8. Two cards are drawn at random from a well-shuffled pack of cards. Find the probability that both the cards are red.
   (All India Board)
9. If A and B are mutually exclusive events such that P(A) = 0.35 and P(B) = 0.25, find P(A ∪ B).
   (Board Exam)
10. Find the probability that a leap year selected at random will have 53 Sundays.
    (CBSE PYQ)
11. A bag contains 5 red and 7 blue balls. Two balls are drawn at random without replacement. Find the probability that both balls are blue.
    (Board Pattern)
12. Define conditional probability. Find P(A | B) if P(A ∩ B) = 1/5 and P(B) = 2/5.
    (CBSE PYQ)
13. If events A and B are independent and P(A) = 1/4, P(B) = 1/5, find P(A ∪ B).
    (Board Exam)
14. A die is thrown. Let A be the event “number obtained is a multiple of 3” and B be the event “number obtained is even”. Find P(A | B).
    (CBSE PYQ)
15. State and explain Bayes’ Theorem.
    (Board Theory Question)
16. Three boxes B₁, B₂ and B₃ contain balls as follows:
    B₁: 2 red, 3 blue
    B₂: 4 red, 1 blue
    B₃: 1 red, 5 blue
    One box is selected at random and a ball is drawn. If the ball drawn is red, find the probability that it was drawn from box B₂.
    (CBSE PYQ – Bayes Theorem)
17. Two events A and B are such that P(A) = 0.6, P(B) = 0.5 and P(A ∩ B) = 0.3. Find P(A | B).
    (Board Exam)
18. From a lot of 10 bulbs containing 4 defective ones, a bulb is drawn at random. Find the probability that the bulb drawn is defective.
    (Board Pattern)
19. If P(A) = 2/3 and P(B) = 3/5, find P(A ∩ B) when A and B are independent events.
    (CBSE PYQ)
20. A coin is tossed three times. Find the probability of getting exactly two heads.
    (All India Board)