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JEE MATHEMATICS PRACTICE MODULE Complex Numbers | Progressions | Quadratic Equations CHAPTER 01: COMPLEX NUMBERS (Q1-Q10) ani? 21= 2740 +4) gina, Sol: Let 2 — % +, Viera ry = (e+4)+i(y+ 4) im part: yt4=0> y= —4 peal part: Ve 2 tt = 2 +401 pig. -4i Q2. Find min value of [22 —6 + 5a) given |z— 3 23] < - Sol: 22-3 — 2.58) |. Dist from center (3,2) to (3,-2.5) is 45. Min dist to circle =45-2= 2.5. Result = 2 36 Bl = 5 za ver Q3. If —2—, find (gl0L + 4103) 105 2 — et7/8 gl0l — eil0In/6 _ ei(16a+50/6) _ ei5n/6 5103 — Sol —* Calculate sum using Euler form. ; — w)7 Q4. Let” be a cube root of unity. Find value of (lw = wi y Sol: Uapaa = —4? Expression becomes (-w? — w?)7 — (- 2w?)7 — — 128wl4 — 1 28u?- Q5. Solve Je] 2=14 24 Sol: frre x iy—1t 2 y= 2 Vit xilo2e- 1 pyre 1.5 — 2% Q6. If 7 ar a — 7 arg(=*) +, find the locus of *. Sol: Locus is a segment of a circle passing through (, 0) and (-1,0). 7. if !@1) — 4s 1221 = 21231 = 3 ang 21 + 22 + 23] — 1 fina [92,2 + 42,2, + 2q2\_ Sol: Use lel? me 1 oo . lex2ozg(8 + 4 + 1)| = leqzoegll2g + % +2%|=1-2-3-1=6 Q8. Find the area of polygon with vertices as roots of we 1 — ngin( 20 Sol: Regular "gon inscribed in unit circle. Area gsin(*). ;)10 Q9. Find real part of (1+) F sol; (L ~ #)? = 2%, (24) = 322° = 32% peal part — 0 2 — 2023 2024 ~2025 aio.if 2 +1=9 ging 27 + 2 a Sol: “is or &, sum — #°S(L + w@ + w?) = w?05(0) — 0 CHAPTER 02: QUADRATIC EQUATIONS (Q11-Q20) 2 6p 9 = _ _é¢ ) Q11. if ® B are roots of ” a iy and n = 0" pr find “a. Sol: Newton's method; 710 ~ 649 — 24g = 0 = 449 — 205 = Bag ang, 6/2 = 3. . et — 72 Q12. Find the number of real roots of 7 2 Sol: Plot ¥ — © and ¥ — 2”. intersect at? < 9 (one point). Ans: 1. CHAPTER 03: PROGRESSIONS (Q21-Q30) Q21. Find sum of all 3-digit numbers divisible by 7. ol: ap; 105, 112, ...,994 a = 105,1 = 994,d—7 n= 128. S = 138 (105 + 994) = 70336 Ts = 24,T; =192 pS Q22. In GP, . Find “10, Sol: 07? = 24, ar5 = 192 > r8 = 8 > r= 2,a=6 Sy) = 6(2!0 — 1) = 6138 1/be, 1/ca, 1/ab Q23. if” b, © are in AP, show are in AP. Sol: Divide & 8 py °C, Resutt is 1/be 1/ac, 1/ab, otal Q24. Find sum of | ata te ®, Sol: Infinite GP sum: @/(1— 7) = 1/(1— 0.5) = 2 25. Find” if oan ** — 441, sole 2 = 44d > MD = 21 > n? +n -42=0n=6 Q26. AM of two numbers is 10, GM is 8. Find numbers. sor. @ +5 = 20, ab = 64 2? —20r + 64=0 = 16,4 Q27. Sum of series | ° 2+2-34-- b(n 1). 2p) — rint1)Qn—-1) 4 n(n+1) _ n(n+1)(n+2) sol; (”? — 2) 6 + $ Q28. If 3 S, = 3n?+ nm find a. Sol: @n — §,,= 8,1 = a= 2 d— ad y—a, —10—-4— 6 Q29. Sum of 9-6 + 0.66 + 0.666+... to” terms. sol: 31% — 9"). Q30. Solve / 3+ oped @ = 625. Sol: Sum of first odds is 7,” = 625 > n = 25 @ = 2(25) —1= 49. 
