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Assignment (chapter 1 8) maths
1.
MATHEMATICS ASSIGNMENT (CHAPTER :-
1 – 8) CLASS – XII 4 MARKS QUESTIONS: 1. y = (𝐭𝐚𝐧−𝟏 𝒙) 𝟐 , show (x2 + 1)2 y2 + 2x (x2 + 1)2 y1 = 2. 2. Prove using properties :- | 𝒃 + 𝒄 𝒒 + 𝒓 𝒚 + 𝒛 𝒄 + 𝒂 𝒓 + 𝒑 𝒛 + 𝒙 𝒂 + 𝒃 𝒑 + 𝒒 𝒙 + 𝒚 | = 𝟐 | 𝒂 𝒑 𝒙 𝒃 𝒒 𝒚 𝒄 𝒓 𝒛 | . 3. Evaluate :- ∫ 𝟐 (𝟏+𝒙)(𝟏+ 𝒙 𝟐) dx 4. If (cos x)y = (cos y)x , find 𝒅𝒚 𝒅𝒙 . 5. Prove :- 𝐭𝐚𝐧−𝟏 ( 𝐜𝐨𝐬 𝒙 𝟏+𝐬𝐢𝐧 𝒙 ) = 𝝅 𝟒 − 𝒙 𝟐 . 6. A = R – {3}, B = R – {1}. f : A → B. f(x) = 𝒙−𝟐 𝒙−𝟑 . Show that f is one – one & onto. 7. Prove :- ∫ (√ 𝒕𝒂𝒏 𝒙 + √ 𝒄𝒐𝒕 𝒙 )𝒅𝒙 = √𝟐 𝝅 𝟐 𝝅 𝟐 𝟎 . 8. If a + b + c ≠ 0 & | 𝒂 𝒃 𝒄 𝒃 𝒄 𝒂 𝒄 𝒂 𝒃 | = 0. Prove a = b = c 9. Evaluate :- ∫(𝟐 𝒔𝒊𝒏 𝟐𝒙 − 𝒄𝒐𝒔 𝒙)√𝟔 − 𝒄𝒐𝒔 𝟐 𝒙 − 𝟒 𝒔𝒊𝒏 𝒙 dx. 10. Differentiate :- log (xsinx + cot2 x), w.r.t. x 11. Evaluate :- ∫ 𝟓𝒙 (𝒙+𝟏)( 𝒙 𝟐+ 𝟗) dx 12. Prove :- y = 𝟒 𝒔𝒊𝒏 𝒙 (𝟐+𝒄𝒐𝒔 𝒙) - x is an increasing function of x in [0, 𝝅 𝟐 ] 13. Evaluate :- ∫ 𝑠𝑒𝑐3 𝑥 𝑑𝑥 . 14. Differentiate :- tan−1 ( 𝑎 sin 𝑥+𝑏 cos 𝑥 𝑎 cos 𝑥−𝑏 sin 𝑥 ) w.r.t. x 15. If x √ 𝟏 + 𝒚 + 𝒚 √ 𝟏 + 𝒙 = 𝟎. 𝐏𝐫𝐨𝐯𝐞 𝒅𝒚 𝒅𝒙 = 𝟏 (𝟏+𝒙) 𝟐 . 16. If A = [ 𝟏 𝟎 −𝟏 𝟕 ], find k : A2 = 8A + kI . 17. If f(x) = { |𝒙| 𝒙 𝒙 ≠ 𝟎 𝟎 𝒙 = 𝟎 . check the continuity at x = 0. 18. If y = ecos x , prove y2 + sin x y1+ y cos x = 0. 19. Evaluate :- ∫ 𝟏 𝒙 𝟒+ 𝟏 dx. 20. Prove :- | (𝒃 + 𝒄) 𝟐 𝒃𝒂 𝒄𝒂 𝒂𝒃 (𝒄 + 𝒂) 𝟐 𝒄𝒃 𝒂𝒄 𝒃𝒄 (𝒂 + 𝒃) 𝟐 | = 2abc (a + b + c)3 . 21. If 𝐜𝐨𝐬−𝟏 𝒙 + 𝐜𝐨𝐬−𝟏 𝒚 + 𝐜𝐨𝐬−𝟏 𝒛 = 𝝅 , Prove x2 + y2 + z2 + 2xyz = 1.
2.
22. Find equation
of tangent to the curve x2 + 3y = 3 which is PARALLEL to line y – 4x + 5 = 0. 23. Find the matrix X : [ 𝟐 −𝟏 𝟎 𝟏 −𝟐 𝟒 ] 𝑿 = [ −𝟏 −𝟖 −𝟏𝟎 𝟑 𝟒 𝟎 𝟏𝟎 𝟐𝟎 𝟏𝟎 ] 24. Evaluate :- ∫ sin−1 𝑥 𝑥2 dx. 25. Prove : ∫ 𝒙 𝒅𝒙 𝒂 𝟐 𝒄𝒐𝒔 𝟐 𝒙+ 𝒃 𝟐 𝒔𝒊𝒏 𝟐 𝒙 𝝅 𝟎 = 𝝅 𝟐 𝟐𝒂𝒃 . 26. Using property of determinants, Prove : | 𝒂 + 𝒃 + 𝒄 −𝒄 −𝒃 −𝒄 𝒂 + 𝒃 + 𝒄 −𝒂 −𝒃 −𝒂 𝒂 + 𝒃 + 𝒄 | = 2 (a + b) (b + c) ( c + a) 27. Solve for x : 𝐬𝐢𝐧−𝟏 ( 𝟐𝒂 𝟏+ 𝒂 𝟐) + 𝐬𝐢𝐧−𝟏 ( 𝟐𝒃 𝟏+ 𝒃 𝟐) = 𝟐 𝐭𝐚𝐧−𝟏 𝒙 28. Evaluate :- ∫ 𝒅𝒙 𝟓+𝟒 𝒄𝒐𝒔 𝒙 𝝅 𝟎 . 29. Prove : 𝑑 𝑑𝑥 ( 𝑥 2 √𝑎2 − 𝑥2 + 𝑎2 2 sin−1 𝑥 2 ) = √𝑎2 − 𝑥2 . 30. Show that the relation R in set R of real numbers, defined as R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive. 31. Using elementary operation find A-1 , A = [ 𝟐 𝟑 𝟒 𝟓 ] . 32. Show that the curve 2x = y2 & 2xy = k cut at right angles if k2 = 8. 33. Find 𝒅𝒚 𝒅𝒙 if y = 𝐭𝐚𝐧−𝟏 [ √ 𝟏+ 𝒙 𝟐 + √ 𝟏− 𝒙 𝟐 √ 𝟏+ 𝒙 𝟐 – √ 𝟏− 𝒙 𝟐 ] . 34. Evaluate : ∫ 𝒇(𝒙)𝒅𝒙, 𝒇(𝒙) = |𝒙| + |𝒙 + 𝟐| + |𝒙 + 𝟓| 𝟎 −𝟓 . 35. Find the intervals in which f(x) = sin x + cos x, 0 ≤ 𝒙 ≤ 𝟐𝝅 is strictly increasing or decreasing. 36. Show that the relation R on z defined by (a, b) ∈ R a – b is divisible by 5 is an equivalence relation. 37. Solve by matrix method :- x + 2y + 5z = 10, x – y – z = -2, 2x + 3y – z = -11. 38. Evaluate :- ∫ 𝒙 𝒅𝒙 𝟒− 𝒄𝒐𝒔 𝟐 𝒙 𝝅 𝟎 . 39. If y = Aemx + Benx , show that y2 – (m + n) y1 + mny = 0. 40. Evaluate : ∫ ( 𝒙 𝟐 + 𝟓𝒙)𝒅𝒙 𝟑 𝟏 as limit of sum.
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