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): ___________________________

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