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⚡ Surprise Test — Inverse Trig Functions

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⚡ Surprise Test · Inverse Trig Functions
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takeatest.net · ISC Class XII
Chapter 3 — Inverse Trigonometric Functions
4 Days · 56 Questions · 30 Mar – 7 Apr 2026 | ISC Mathematics
Teacher: Ravishankar
Campus: CAMP / KLN / UNDRI
Board Weightage: ~10 marks
Year: 2026–27
Class
Division
Subject
Class XII · Mathematics
Chapter 01
Days 4
Periods 6 × 40 min
QR Slots / Day 4
Solution Mode Manual Click · Auto Play
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Overview
Day 1 · Types
Day 2 · RST
Day 3 · Equivalence
Day 4 · ℕ×ℕ & Review
Lesson Overview
Subject
Mathematics
Chapter
1 — Relations
Reference
NCERT
Dates
16–19 Mar 2026
How to Use Each Question
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⚠ Common Misconceptions
✗Confusing symmetric with antisymmetric
✗Empty relation is vacuously S & T — but not Reflexive unless A = ∅
✗Transitivity — only pairs already in R need checking
✗Equivalence class [a] = set of elements, not ordered pairs
DAY 1 · MON 30 MAR 2026
Day 1 — Types of Relations
Period 1–2 · 80 min
Binary · Empty · Universal · Identity
Day 1 — Domain & Range
sin⁻¹Domain: [−1,1] · Range: [−π/2, π/2]
cos⁻¹Domain: [−1,1] · Range: [0, π]
tan⁻¹Domain: ℝ · Range: (−π/2, π/2)
KeyPrincipal value = unique angle in principal range

Day 1 — Principal Value Ranges (Must Memorise)

FunctionDomainRange (Principal Values)
sin⁻¹ x[−1, 1][−π/2, π/2]
cos⁻¹ x[−1, 1][0, π]
tan⁻¹ xℝ (all reals)(−π/2, π/2)
cosec⁻¹ xℝ − (−1, 1)[−π/2, π/2] − {0}
sec⁻¹ xℝ − (−1, 1)[0, π] − {π/2}
cot⁻¹ xℝ (all reals)(0, π)
📌 The principal value is the UNIQUE angle in the principal range that satisfies the equation. Always reduce to this range.
✎Concept Board — Day 1Click to open · Draw diagrams & explain concepts▶
Q1 \(Find the principal value of each: (i) \sin^{-1}\left(\frac{1}{2}\right) (ii) \sec^{-1}(2) (iii) \cot^{-1}(\sqrt3) (iv) \cos^{-1}\left(\frac{\sqrt3}{2}\right) (v) \tan^{-1}(1) (vi) \sin^{-1}\left(\frac{1}{\sqrt2}\right)\)
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Q2 \(Which of the following are NOT defined? Give reason. (i) co\sec^{-1}\left(\frac{1}{2}\right) (ii) \cos^{-1}\left(\frac{−3}{2}\right) (iii) \sec^{-1}\left(\frac{4}{5}\right) (iv) \tan^{-1}(1) (v) \cot^{-1}(7) (vi) \sin^{-1}\left(\frac{1.7}{1.8}\right)\)
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Q3 \(Find the domain of f(x) = \sin^{-1}(\sqrt{x − 1}). State the domain in interval notation.\)
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Q4 \(Find the domain AND range of f(x) = \sin^{-1}(x^{2} − 4). Express domain as a union of intervals.\)
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Q5 \(Find the domain of each: (i) \cos^{-1}(x^{2} − 4) (ii) \sec^{-1}(3x − 1)\)
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Q6 \(Find the domain of f(x) = 2 \cos^{-1}(2x) + sin x\)
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Q7 \(Find the domain of y = \cos^{-1}(|x − 1|)\)
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Q8 \(Write the domain and range of f(x) = \tan^{-1} x. Also find domain of f(x) = \sin^{-1} x + cos x\)
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Q9 \(Find the value of: 2 \sec^{-1}(2) + \sin^{-1}\left(\frac{1}{2}\right)\)
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Q10 \(Choose the correct answer: sin\left[\left(\frac{\pi}{3 + \sin^{-1}\frac{1}{2}}\right)\right] = ? (a) 1 (b) \left(\frac{1}{2}\right) (c) \left(\frac{1}{3}\right) (d) \left(\frac{1}{4}\right)\)
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Day 1 — Homework
HW 1
Write Universal relation on A={1,3,5}; verify using a−b<5.
HW 2
How many relations possible from A={1,2} to B={a,b,c}?
📎 Assignments — Day 1 (Optional · Viewable by students)
Day 1 — Video Resources
Exit Ticket
1Write one example each: empty relation and universal relation on A={1,2,3}
2Is Identity relation on A={a,b} an Equivalence Relation?
ISC Board Tags — Day 1
DAY 2 · TUE 01 APR 2026
Day 2 — Reflexive, Symmetric & Transitive
Period 3–4 · 80 min
RST · Matrix · Arrow Diagrams
Day 2 — Theorems 1–4
T1f⁻¹(f(θ)) = θ in principal range
T2\(cosec⁻¹x = sin⁻¹\dfrac{1}{x}, sec⁻¹x = cos⁻¹\dfrac{1}{x}\)
T3\(sin⁻¹(−x) = −sin⁻¹x; cos⁻¹(−x) = π−cos⁻¹x\)
T4\(sin⁻¹x + cos⁻¹x = π/2; tan⁻¹x + cot⁻¹x = π/2\)

Day 2 — Theorems 1–4: Properties

Theorem 1 — Self-Adjusting Property
sin⁻¹(sin θ) = θ   θ ∈ [−π/2, π/2]    sin(sin⁻¹ x) = x   x ∈ [−1, 1]
cos⁻¹(cos θ) = θ   θ ∈ [0, π]    tan⁻¹(tan θ) = θ   θ ∈ (−π/2, π/2)
Theorem 2 — Reciprocal Property
cosec⁻¹ x = sin⁻¹(1/x)  |  sec⁻¹ x = cos⁻¹(1/x)  |  cot⁻¹ x = tan⁻¹(1/x) if x > 0
Theorem 3 — Odd/Even Property
sin⁻¹(−x) = −sin⁻¹ x  |  cos⁻¹(−x) = π − cos⁻¹ x  |  tan⁻¹(−x) = −tan⁻¹ x
Theorem 4 — Complementary Property
sin⁻¹ x + cos⁻¹ x = π/2  |  tan⁻¹ x + cot⁻¹ x = π/2  |  sec⁻¹ x + cosec⁻¹ x = π/2
✎Concept Board — Day 2Click to open · Draw diagrams & explain concepts▶
Q11 \(Evaluate (reduce to principal range): (i) \sin^{-1}\left(\frac{sin 5\pi}{6}\right) (ii) \cos^{-1}\left(\frac{cos 2\pi}{3}\right) (iii) \tan^{-1}\left(\frac{tan 5\pi}{4}\right) (iv) \sin^{-1}\left(\frac{sin 2\pi}{3}\right)\)
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Q12 \(Evaluate: (v) \cos^{-1}\left(\frac{cos 9\pi}{8}\right) (vi) \tan^{-1}\left(\frac{tan 5\pi}{6}\right) + \cos^{-1}\left(\frac{cos 13\pi}{6}\right) (vii) \cos^{-1}(cos 680°)\)
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Q13 \(Evaluate: \sin^{-1}\left(\frac{sin 3\pi}{4}\right) + \cos^{-1}\left(\frac{cos 3\pi}{4}\right) + \tan^{-1}(1)\)
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Q14 \(Evaluate each: (i) sin\left(\frac{\sin^{-1} 2}{5}\right) (ii) sin\left(\frac{\cos^{-1} 3}{5}\right) (iii) cos\left(\frac{\tan^{-1} 3}{4}\right)\)
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Q15 \(Find the principal values of: (i) \sin^{-1}\left(\frac{−\sqrt3}{2}\right) (ii) \cos^{-1}\left(\frac{−1}{2}\right) (iii) \tan^{-1}(−1) (iv) \sec^{-1}(−\sqrt2) (v) \cot^{-1}(−\sqrt3)\)
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Q16 \(Evaluate: \tan^{-1}\left(\frac{−1}{\sqrt3}\right) + \cot^{-1}\left(\frac{1}{\sqrt3}\right) + \tan^{-1}[sin\left(\frac{−\pi}{2}\right)]\)
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Q17 \(Prove that: sec^{2}(\tan^{-1} 2) + cosec^{2}(\cot^{-1} 3) = 15\)
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Q18 \(If 4 \sin^{-1} x + \cos^{-1} x = \pi, find x.\)
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Q19 \(Evaluate: \cos^{-1}\left(\frac{cos 13\pi}{8}\right)\)
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Q20 \(Prove that: tan^{2}(\sec^{-1} 2) + cot^{2}(co\sec^{-1} 3) = 11\)
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Day 2 Homework
HW 1
R={(a,b): a≤b²} on ℝ. Show R is neither R, S, nor T.
HW 2
Write a relation which is: (a) only T (b) only S (c) only R and T (d) only S and R.
📎 Assignments — Day 2 (Optional · Viewable by students)
Day 2 — Video Resources
Exit Ticket
1Symmetric but neither R nor T — give one example.
2R={(1,2),(2,1)} on {1,2,3}: Is R symmetric? Reflexive? Transitive?
Mark Scheme Note
State property → assume (a,b)∈R → show consequence → conclude. Every step earns marks.
DAY 3 · WED 02 APR 2026
Day 3 — Equivalence Relations & Classes
Period 5 · 40 min
ER · Equivalence Classes · Partition
Equivalence Relation
DefinitionR is ER ⟺ Reflexive AND Symmetric AND Transitive
Class [a]{b ∈ A : (b,a) ∈ R} — set of all elements related to a
PartitionClasses are mutually disjoint; their union = A
ISC NoteAlways write conclusion sentence — 1 mark awarded by ISC for it

Day 3 — Theorems 5–8: Identities & Addition Formulae

Theorem 5 — Conversion Property
sin⁻¹ x = cos⁻¹(√(1−x²)) = tan⁻¹(x/√(1−x²)) = cot⁻¹(√(1−x²)/x)
tan⁻¹ x = sin⁻¹(x/√(1+x²)) = cos⁻¹(1/√(1+x²)) = cot⁻¹(1/x)
Theorem 6 — tan⁻¹ Addition
tan⁻¹ x + tan⁻¹ y = tan⁻¹[(x+y)/(1−xy)]   if xy < 1
tan⁻¹ x + tan⁻¹ y = π + tan⁻¹[(x+y)/(1−xy)]   if x>0, y>0, xy>1
Theorem 7 — tan⁻¹ Subtraction
tan⁻¹ x − tan⁻¹ y = tan⁻¹[(x−y)/(1+xy)]   if xy > −1
Theorem 8 — Double Angle Formulae (|x| < 1)
2 tan⁻¹ x = tan⁻¹[2x/(1−x²)] = sin⁻¹[2x/(1+x²)] = cos⁻¹[(1−x²)/(1+x²)]
✎Concept Board — Day 3Click to open · Draw diagrams & explain concepts▶
Q21 \(Express in simplest form (−3\pi/2 < x < \pi/2): \tan^{-1}(cos x / (1 − sin x))\)
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Q22 \(Prove that: \tan^{-1} x + \cot^{-1}(x+1) = \tan^{-1}(x^{2} + x + 1)\)
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Q23 \(Prove that: \tan^{-1}\left(\frac{2}{11}\right) + \tan^{-1}\left(\frac{7}{24}\right) = \tan^{-1}\left(\frac{1}{2}\right)\)
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Q24 \(Prove that: \tan^{-1} 1 + \tan^{-1} 2 + \tan^{-1} 3 = \pi\)
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Q25 \(Prove that: \tan^{-1}\left(\frac{1}{5}\right) + \tan^{-1}\left(\frac{1}{7}\right) + \tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1}\left(\frac{1}{8}\right) = \pi/4\)
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Q26 \(Prove that: \tan^{-1}[\frac{\sqrt{1+x^{2}} + \sqrt{1−x^{2}}}{\sqrt{1+x^{2}} − \sqrt{1−x^{2}}}] = \pi/4 + \left(\frac{1}{2}\right)\cos^{-1}(x^{2})\)
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Q27 \(Prove that: \tan^{-1}\left(\frac{1}{4}\right) + \tan^{-1}\left(\frac{2}{9}\right) = \sin^{-1}\left(\frac{1}{\sqrt5}\right)\)
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Q28 \(Prove that: \cot^{-1} 7 + \cot^{-1} 8 + \cot^{-1} 18 = \cot^{-1} 3\)
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Q29 \(Prove that: \sin^{-1}\left(\frac{4}{5}\right) + \tan^{-1}\left(\frac{5}{12}\right) + \cos^{-1}\left(\frac{63}{65}\right) = \pi/2\)
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Q30 \(Prove that: \sin^{-1}\left(\frac{5}{13}\right) + \cos^{-1}\left(\frac{3}{5}\right) = \tan^{-1}\left(\frac{63}{16}\right)\)
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Q31 \(Prove that: \sin^{-1}\left(\frac{8}{17}\right) + \sin^{-1}\left(\frac{3}{5}\right) = \sin^{-1}\left(\frac{77}{85}\right)\)
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Q32 \(If \tan^{-1}x + \tan^{-1}y + \tan^{-1}z = \pi/2, show that: xy + yz + zx = 1\)
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Day 3 Homework
HW 1
Prove R={(L₁,L₂): L₁∥L₂} on all lines in XY-plane is an ER. Find all lines related to y=2x+4.
HW 2
aRb iff (a+b) is even, a,b∈ℤ. Prove R is an ER.
📎 Assignments — Day 3 (Optional · Viewable by students)
Day 3 — Video Resources
Exit Ticket
1What is [3] under R={(a,b): 3|(a−b)} on ℤ?
2Can two equivalence classes overlap? Prove or disprove.
6M Mark Scheme
1M Reflexive + 1M Symmetric + 1M Transitive + 1M Conclusion + 2M for Equivalence Class
DAY 4 · THU 03 APR 2026
Day 4 — ℕ×ℕ Equivalence & Review
Period 6 · 40 min
ℕ×ℕ · VSA · True/False
Equivalence on ℕ×ℕ
Standard(a,b)R(c,d) ⟺ ad=bc (cross-multiplication form of a/b=c/d)
ISC 2024(a,b)R(c,d) iff a−c=b−d → same as a−b=c−d
Class[(2,6)]={(1,3),(2,6),(3,9),…} all pairs in ratio 1:3

Day 4 — Theorems 9–10: Multiple Angle & sin⁻¹/cos⁻¹ Addition

Theorem 9 — Multiple Angle Formulae
2 sin⁻¹ x = sin⁻¹(2x√(1−x²))  |  2 cos⁻¹ x = cos⁻¹(2x² − 1)
3 sin⁻¹ x = sin⁻¹(3x − 4x³)  |  3 cos⁻¹ x = cos⁻¹(4x³ − 3x)
3 tan⁻¹ x = tan⁻¹[(3x − x³)/(1 − 3x²)]
Theorem 10 — sin⁻¹ x ± sin⁻¹ y
sin⁻¹ x + sin⁻¹ y = sin⁻¹[x√(1−y²) + y√(1−x²)]   if x²+y² ≤ 1
sin⁻¹ x + sin⁻¹ y = π − sin⁻¹[x√(1−y²)+y√(1−x²)]   if x,y>0, x²+y²>1
Theorem 10 — cos⁻¹ x ± cos⁻¹ y
cos⁻¹ x + cos⁻¹ y = cos⁻¹[xy − √(1−x²)·√(1−y²)]   if x+y ≥ 0
cos⁻¹ x − cos⁻¹ y = cos⁻¹[xy + √(1−x²)·√(1−y²)]   if x ≤ y
✎Concept Board — Day 4Click to open · Draw diagrams & explain concepts▶
Q33 \(Solve for x (x ≠ 0): \sin^{-1}\left(\frac{5}{x}\right) + \sin^{-1}\left(\frac{12}{x}\right) = \pi/2\)
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Q34 \(Solve for x (x > 0): \sin^{-1} x + \sin^{-1} 2x = \pi/3\)
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Q35 \(Solve: cos(\tan^{-1} x) = sin\left(\frac{\cot^{-1} 3}{4}\right)\)
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Q36 \(Solve: \sin^{-1}(1 − x) + \sin^{-1} x = \cos^{-1} x\)
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Q37 \(Find x if: (\tan^{-1} x)^{2} + (\cot^{-1} x)^{2} = 5\pi^{2}/8\)
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Q38 \(Solve: \tan^{-1}[\left(\frac{x−1}{x−2}\right)] + \tan^{-1}[\left(\frac{x+1}{x+2}\right)] = \pi/4\)
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Q39 \(Solve: \tan^{-1}(x+1) + \tan^{-1}(x−1) = \tan^{-1}\left(\frac{8}{31}\right)\)
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Q40 \(Solve: 5 \tan^{-1} x + 3 \cot^{-1} x = 2\pi\)
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Q41 \(If \tan^{-1} a + \tan^{-1} b + \tan^{-1} c = \pi, prove that: a + b + c = abc\)
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Q42 \(If \cos^{-1}x + \cos^{-1}y + \cos^{-1}z = \pi, prove that: x^{2} + y^{2} + z^{2} + 2xyz = 1\)
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Q43 \(Prove that: \sin^{-1}\left(\frac{12}{13}\right) + \cos^{-1}\left(\frac{4}{5}\right) + \tan^{-1}\left(\frac{63}{16}\right) = \pi\)
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Q44 \(If \cos^{-1}\left(\frac{x}{a}\right) + \cos^{-1}\left(\frac{y}{b}\right) = \alpha, prove that: x^{2}/a^{2} − 2xy cos\alpha/ab + y^{2}/b^{2} = sin^{2}\alpha\)
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Q45 \(Solve: \cos^{-1}[sin(\cos^{-1} x)] = \pi/3, x > 0\)
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Q46 \(Solve: \cos^{-1} x + \sin^{-1}\left(\frac{x}{2}\right) = \pi/6\)
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Q47 \(Solve: \tan^{-1} 2x + \tan^{-1} 3x = \pi/4\)
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Q48 \(Solve: \tan^{-1}(x+2) + \tan^{-1}(2−x) = \tan^{-1}\left(\frac{2}{3}\right)\)
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Q49 \(Find the value of: (i) \cos^{-1}(−1) − \sin^{-1}(1) (ii) cot\left(\frac{\pi}{2 − 2\cot^{-1}\sqrt3}\right) (iii) \tan^{-1}\left(\frac{1}{7}\right) + \tan^{-1}\left(\frac{1}{13}\right)\)
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Q50 \(True or False - justify: (i) 2\tan^{-1}\left(\frac{2}{3}\right) = \tan^{-1}\left(\frac{12}{5}\right) (ii) (\cos^{-1}x)^{2} = sec^{2}x (iii) \sin^{-1}[cos(\sin^{-1}\left(\frac{1}{2}\right))] = \pi/3\)
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Q51 \(Choose correct: \cot^{-1}\left(\frac{1}{\sqrt3}\right) + 2\sin^{-1}\left(\frac{1}{2}\right) = ? (a) \pi/4 (b) \pi/6 (c) \pi/3 (d) 2\pi/3\)
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Q52 \(Choose correct: \cos^{-1}\left(\frac{−1}{2}\right) − 2\sin^{-1}\left(\frac{1}{2}\right) + 3\cos^{-1}\left(\frac{−1}{\sqrt2}\right) − 4\tan^{-1}(−1) = ? (a)19\pi/12 (b)13\pi/12 (c)47\pi/12 (d)43\pi/12\)
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Q53 \(Choose: If \tan^{-1}x + \tan^{-1}y = 4\pi/5, then \cot^{-1}x + \cot^{-1}y = ? (a)\pi/5 (b)2\pi/5 (c)3\pi/5 (d)\pi\)
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Q54 \(Choose: If \tan^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(\frac{x}{3}\right) = \pi/4, 0<x<6, find x. (a)1 (b)2 (c)3 (d)5\)
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Q55 \(A-R: Assertion: Domain of \sec^{-1}(2x) is (−\infty,−\frac{1}{2}]∪[\frac{1}{2},\infty). Reason: \sec^{-1}(−2) = −\pi/4. Choose (a/b/c/d).\)
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Q56 \(A-R: Assertion: sin(2\tan^{-1}\left(\frac{1}{3}\right)) = \left(\frac{3}{5}\right). Reason: 2\tan^{-1}x = \tan^{-1}(2x/(1−x^{2})), if −1<x<1. Choose (a/b/c/d).\)
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Day 4 Homework
HW 1
Complete all pending Ex1(A). NCERT Exemplar: aRb iff (a−b) divisible by n on ℤ — show R is ER.
HW 2
Attempt Assertion–Reason Q21(i)–(vi) from textbook.
📎 Assignments — Day 4 (Optional · Viewable by students)
HW 3
Read Case Study Q24 (General Elections context) and answer all parts.
Day 4 — Video Resources
Final Exit Ticket
1In one sentence: what makes a relation an equivalence relation?
2Write [(1,2)] under (a,b)R(c,d) iff a+d=b+c on ℕ×ℕ.
3How many equivalence relations on {1,2,3}? (Bell number)
Chapter Summary
R(a,a)∈R ∀a
S(a,b)∈R ⟹ (b,a)∈R
T(a,b),(b,c)∈R ⟹ (a,c)∈R
ER+S+T simultaneously
[a]Disjoint partition of A
🎬 Inverse Trig — Interactive Animation
ISC Class XII Mathematics · Chapter 1: Relations · 16–19 Mar 2026