INVERSE TRIGONOMETRIC FUNCTION. Contents. Theory Exercise Exercise Exercise Exercise

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1 INVERSE TRIGONOMETRIC FUNCTION Toic Contents Page No. Theory 0-06 Eercise Eercise Eercise Eercise Answer Key 0 - Syllabus Inverse Trigonometric Function (ITF) Name : Contact No. ARRIDE LEARNING ONLINE E-LEARNING ACADEMY A-79 indra Vihar, Kota Rajasthan 00 Contact No

2 INVERSE TRIGONOMETRIC FUNCTION. Princial Values & Domains of Inverse Trigonometric/Circular Functions: Function Domain Range (i) y = sin where - y (ii) y = cos where 0 y (iii) y = tan where Î R - < y < (iv) y = cosec where - or ³ - y, y ¹ 0 (v) y = sec where or ³ 0 y ; y ¹ (vi) y = cot where Î R 0 < y < NOTE: (a) st quadrant is common to the range of all the inverse functions. (b) rd quadrant is not used in inverse functions. (c) th quadrant is used in the clockwise direction i.e. - y 0. (d) No inverse function is eriodic. (See the grahs on age ). Proerties of Inverse Trigonometric Functions: A (i) sin (sin ) =, (ii) cos (cos ) =, (iii) tan (tan ) =, Î R (iv) cot (cot ) =, Î R (v) sec (sec ) =,, ³ (vi) cosec (cosec ) =,, ³ These functions are equal to identity function in their whole domain which may or may not be R.(See the grahs on age ) B (i) sin (sin ) =, - (ii) cos (cos ) = ; 0 (iii) tan (tan ) = ; - < < (iv) cot (cot ) = ; 0 < < (v) sec (sec ) = ; 0, ¹ (vi) cosec (cosec ) = ; ¹ 0, - These functions are defined on R, but they are equal to identity function for a short interval of only. (See the grahs on age 6) C (i) sin (-) = - sin, (ii) tan (-) = - tan, Î R (iii) cos (-) = - cos, (iv) cot (-) = - cot, Î R The functions sin, tan and cosec are odd functions and rest are neither even nor odd. D (i) cosec = sin ;, ³ (ii) sec = cos ;, ³ (iii) cot ì ï tan = í ï + tan î ; > 0 ; < 0 A-79 Indra Vihar, Kota Rajasthan 00 Page No. #

3 E (i) sin + cos =, (ii) tan + cot =, Î R (iii) cosec + sec =, ½½ ³ F (i) sin (cos ) = cos (sin ) = -, (ii) tan (cot ) = cot (tan ) =, Î R, ¹ 0 (iii) cosec (sec ) = sec (cosec ) =, ½½ > -. Identities of Addition and Substraction: A (i) sin + sin y = sin - y + y - ê, ³ 0, y ³ 0 & ( + y ) = - sin - y + y - ê, ³ 0, y ³ 0 & + y > Note that: + y Þ 0 sin + sin y + y > Þ < sin + sin y < (ii) cos + cos y = cos y y, ³ 0, y ³ 0 ê (iii) tan + tan y = tan + y, > 0, y > 0 & y < - y = + tan + y, > 0, y > 0 & y > - y =, > 0, y > 0 & y = Note that : y < Þ 0 < tan + tan y < ;y > Þ < tan + tan y < B (i) sin - sin y = sin - y - y -, ³ 0, y ³ 0 ê (ii) cos - cos y = cos y y, ³ 0, y ³ 0, y ê - y (iii) tan - tan y = tan, ³ 0, y ³ 0 + y Note: For < 0 and y < 0 these identities can be used with the hel of roerties (C) i.e. change and y to - and - y which are ositive. A-79 Indra Vihar, Kota Rajasthan 00 Page No. #

4 ê sin ê C (i) sin æ - ê - sin = ê ê- ê ( + sin ) if if if > < - (ii) cos ( - ) = ê ê cos - cos if if ³ 0 < 0 (iii) tan - = tan ê ê + tan ê- ( - tan ) if if if < < > (iv) sin + = tan ê ê - tan ê- ( + tan ) if if if > < - (v) cos + tan = ê - tan if if ³ 0 < 0 + y + z - yz D If tan + tan y + tan z = tan ê - y - yz - z if, > 0, y > 0, z > 0 & (y + yz + z) < NOTE: (i) If tan + tan y + tan z = then + y + z = yz (ii) If tan + tan y + tan z = then y + yz + z = (iii) tan + tan + tan = (iv) tan + tan + tan = A-79 Indra Vihar, Kota Rajasthan 00 Page No. #

5 Inverse Trigonometric Functions Some Useful Grahs. (i) y = sin, ½½, y yî ê -, (ii) y = cos, ½½, y Î [0, ] Ù y O - - O - (iii) æ y = tan, Î R, y Î-, è, ø y (iv) - y = cot, Î R, y Î (0, ) y O O - - (v) æ y = sec, ½½ ³, yî (vi) ê 0, U, y = cosec ø è, ½½ ³, - y - O - y æ yî ê -,0 U 0, ø è - O - A-79 Indra Vihar, Kota Rajasthan 00 Page No. #

6 . (i) y = sin (sin ) = cos (cos ), Î [-, ], y Î [-, ] = ; y is aeriodic y y = )º O + (ii) y = tan (tan ) = cot (cot ) =, Î R, y Î R; y is aeriodic y )º O y = ¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾ (iii) y = cosec (cosec ) = sec (sec ), ½½ ³, ½y½ ³, = ; y is aeriodic y y = ¾¾¾¾¾ - O y = ¾¾¾¾¾ - A-79 Indra Vihar, Kota Rajasthan 00 Page No. #

7 6. (i) y = sin (sin ), Î R, yî -, ê, is eriodic with eriod Ùy y =- ( + ) - )º O y = + y = y = - y = - - (ii) y = cos (cos ), Î R, y Î [0, ], is eriodic with eriod y y = + y = - y = y = O - (iii) ì ü æ y = tan (tan ), Î R - í( n ) nîiý, y Î-, î þ è is eriodic with eriod ø Ùy - y = + - y = + y = - O - y = - y = - - (iv) ì ü æ y = sec (sec ), y is eriodic with eriod ; Î R - í( n ) nîiý, yî 0, î þ ê U, ø è y y = + y = - y = y = O - - A-79 Indra Vihar, Kota Rajasthan 00 Page No. # 6

8 PART - I : OBJECTIVE QUESTIONS * Marked Questions are having more than one correct otion. Section (A) : Fundamentals of ITF æ A. The value of sin è æ + sin ø è ø is equal to : (A) 7 (B) 0 (C) (D) L NM A-. sin - sin - = F HG IO KJ QP (A) (B) (C) (D) A-. The rincial value of F I HG K J F + sin sin I HG K J is - cos cos (A) (B) / (C) / (D) / A-. cosec (cos ) is real if : (A) Î [, ] (C) is an odd multile of (B) Î R (D) is a multile of A-. If cos [tan{sin(cot- )}] = y, then : (A) y = (B) y = (C) y = - (D) y = 0 A-6.* If a satisfies the inequation > 0, then a value eists for : (A) sin a (B) cos a (C) sec a (D) cosec a Section (B) : Inter Conversion and Proerties of ITF B. The value of sin æ - cos + cos æ - sin è ø è ø 7 9 (A) (B) 6 6 is- 6 (C) 9 (D) None of these A-79 Indra Vihar, Kota Rajasthan 00 Page No. # 7

9 F B-. The value of sin cos I HG K J is - (A) (B) 7 (C) 0 (D) 0 B-. The value of ïì æ tan í cos ïî è æ ïü - - / ý è 7 ø ø ïþ is - (A) (B) (C) (D) B-. cos[tan {sin (cot )}] is equal to- (A) + + (B) + + (C) + + (D) None of these B-. tandcos i is equal to : (A) - (B) + (C) + (D) - æ B-6. If = tan cos - + sin ; æ æ y = cos cos è è8øø, then : (A) = y (B) y = (C) tan = -(/)y (D) tan = (/)y B-7. If sin + sin y =, then cos + cos y is equal to : (A) (B) (C) 6 (D) B-8. If sin cos tan, [0,] q = + - Î, then the interval in which q lies is given by : (A) ê0, (B) ê, (C) ê0, (D) ê, æ B-9. If = cos + æ sin statements holds good? + tan ( ) and y = cos æ æ sin sin è ø,then which of the following (A) y = cos (B) 6 y = cos (C) 6 cos y = (D) None of these ì æ ü B0.* The value of cos ê cos ícos - ý î þ is : æ 7 (A) cos - (B) sin æ 0 æ (C) cos æ (D) - cos A-79 Indra Vihar, Kota Rajasthan 00 Page No. # 8

10 B.* If 0 < <, then tan - + is equal to : (A) + cos (B) cos (C) sin - (D) tan + - B.* If cos = tan, then : (A) = æ (B) = æ + (C) sin (cos ) = æ (D) tan (cos ) = æ Section (C) : Addition of ITF æ C. If < 0 then value of tan () + tan (A) (B) is equal to : (C) 0 (D) none of these C-. The value of tan sin - æ tan - æ ê + is : (A) 6 7 (B) 7 6 (C) 7 (D) 7 6 C-. cos æ + cos æ is equal to : (A) cos æ 6 (B) cos æ - 6 æ6 (C) cos 6 (D) none of these æ æ C-. tan + tan is equal to : è ø è ø (A) (B) (C) (D) none of these C-. tan + tan = cosec, the is equal to : (A) (B) (C) - (D) none of these C-6. If q = cot 7 + cot 8 + cot 8, then cotq is equal to : (A) (B) (C) (D) Section (D) : ITF Equations æ D. The solution of the equation sin tan - æ sin - = 0 is : 6 (A) = (B) = - (C) = (D) none of these A-79 Indra Vihar, Kota Rajasthan 00 Page No. # 9

11 D-. If sin + cot æ =, then is equal to : (A) 0 (B) D-. ( sin ) ( sin y) ( sin )( sin y) (C) + + =, then +y is equal to : (A) (B) / (C) (D) / (D) D-. The equation sin = sin a has a solution for : (A) all real values of a (B) a < (C) a > (D) - a D-. If n å i= cos - a i = 0, then n å i= a i is equal to : (A) n (B) n (C) 0 (D) none of these D-6. The value of a for which + a + sin ( + ) + cos ( + ) = 0, is : æ (A) + (B) + (C) - + D-7.* sin > cos holds for : æ (D) + (A) all values of (B) Î æ 0, è ø (C) Î æ, è ø (D) = 0.7 D-8. If cot n >, n Î N, then the maimum value of 6 n is : (A) (B) (C) 9 (D) none of these D-9. The solution of the inequality - (tan ) - tan + ³ 0 is : (A) (, tan ] È [tan, ) (B) (, tan ] (C) (, tan] È [tan, ) (D) [tan, ) æ D0.* If 6 sin 6 + =, then : 7 (A) = (B) = (C) = (D) = D.* If sin + sin y + sin z =, then : 9 (A) 00 + y 00 + z y + z = 0 (B) + y + z 6 0 y 0 z 60 = y + z (C) 0 + y + z = 0 (D) = (yz ) D.* The sum n å tan is equal to : n= n - n + (A) tan + tan (B) tan (C) (D) sec (- ) A-79 Indra Vihar, Kota Rajasthan 00 Page No. # 0

12 PART - II : MISCELLANEOUS OBJECTIVE QUESTIONS Comrehension : Comrehension # A young mathematician while redefining the inverse trigonometric functions chose the range of sin as ê, and of cos as [, ], i.e. ê f :[-,], f() = sin [ ] g:[ -,], g() = cos In his scheme of things he remodelled the whole eressions for sum, difference of these inverse functions, their derivatives & anti-derivatives. Solve the following roblems based on this new range of these inverse functions.. Identify the correct statement. (A) sin is an increasing function. (C) sin is a decreasing function. (B) cos is an increasing function. (D) sin and cos both are increasing function.. Which of the following function is constant function? (A) cos + sin (B) cos sin (C) cos + sin (D) cos + sin. Solution set of the equation sin + cos = is : (A) {0, } (B) {, } (C) (0, ) (D) none of these Comrehension # ì ì ï+q - <q<- ï--q - q<- ï tan ï ï (tan q) = í q - <q<, sin (sin q ) = í q - q ï ï ï ï ï-+q <q< ï -q <q î î ì -q, - q< 0 ï q = í q q ï î -q, < q cos (cos ), 0 Based on the above results, answer each of the following :. cos is equal to : (A) sin - if < < (B) sin - if < < 0 (C) sin - if < < 0 (D) sin - if 0 < <. sin is equal to : (A) cos - if < < (B) cos - if < < (C) cos - if 0 < < (D) cos - if 0 < < 6. cos is equal to : - - (A) tan if < < 0 (B) tan if < < 0 - (C) tan if 0 < < (D) + tan - if < < 0 A-79 Indra Vihar, Kota Rajasthan 00 Page No. #

13 Match the Column : 7. Match the column Column I Column II [.] and {.} reresent the greatest integer and fractional art functions resectively. (A) Number of solutions of [] = cos () (B) Number of solutions of sin = sgn() (q) (C) Number of solutions of {} = e (r) (D) Number of solutions of 8. Match the column Column - I + sin cos = {} (s) 0 Column - II (A) If >, then sec (cosec ) is equal to : () - - (B) If <, then sec (cosec ) is equal to : (q) - (C) If >, then cosec (sec ) is equal to : (r) - (D) If <, then cosec (sec ) is equal to : (s) not defined Assertion / Reason Tye Direction : Each question has choices (A), (B), (C), (D) and (E) out of which ONLY ONE is correct. (A) Statement is True, Statement- is True; Statement- is a correct elanation for Statement. (B) Statement is True, Statement- is True; Statement- is NOT a correct elanation for Statement. (C) Statement is True, Statement- is False. (D) Statement is False, Statement- is True. (E) Statement and Statement- both are False. 9. Statement : If a, b are roots of = 0 then cos a eist but not cos b, (a > b). Statement- : Domain of cos is [, ]. 0. Statement : tan (sec ) + cot (coses ) =. Statement- : tan q + sec q = = cot q + cosec q. A-79 Indra Vihar, Kota Rajasthan 00 Page No. #

14 PART - I : OBJECTIVE QUESTIONS * Marked Questions are having more than one correct otion.. If æ sin q sin =, then tan q is equal to è + cosq ø (A) / (B) (C) (D) -. If cos l + cos m + cos v = then lm + mv + vl is equal to (A) (B) 0 (C) (D) n å i i =. If sin - n å = n then i i = is equal to (A) n (B) n (C) nn ( +) (D) n (n - ). If =, the value of cos (cos + sin ) is : (A) - (B) (C) - (D) +. If tan =, then : (A) = tan (B) = tan (C) = tan (/) (D) = tan 8 6.* a, b and g are three angles given by a = tan ( - ), b = sin + æ - sin è ø and g = cos. Then (A) a > b (B) b > g (C) a < g (D) a > g 7. If X = tan () + tan () + tan () ; æ æ æ Y = tan + tan è + tan ø è ø è ø æ (A) 0 (B) - - tan 8 6 è ø then (X - Y) equals to: (C) tan (D) none of these A-79 Indra Vihar, Kota Rajasthan 00 Page No. #

15 8. Number of integral value(s) of satisfying - ( - ) ( ) tan - tan - 0, is : (A) (B) (C) (D) 9. Domain of the function (A) ên,n +,n ÎI f() = sin (sin) + cos (cos) is : (B) [(n + ), (n + )],nîi (C) [n, (n + )],nîi (D) ên +,n +,n ÎI 0. The function f() = cot ( + ) + cos + + is defined on the set S, where S = (A) {0,} (B) (0,) (C) {0, } (D) [,0]. Solution set of the inequality + > sin (sin) + cos (cos) is : (A) R (B) R {} (C) R {} (D) R { }. sin æ - - = sin is true if : (A) Î [0, ] (B) ê-, (C) ê-, (D) ê- ê,. The value of ì æaü ì æaü êtan í + sin ý+ tan í - sin ý î èbøþ î èbøþ, where ( 0 < a < b), is : (A) b a (B) a b (C) b - a b (D) b - a a. Which of the following is the solution set of the equation sin = cos + sin ( )? (A) ì ü í, ý î þ (B) ê, (C) ê, (D) ì ü í, ý î þ. Value of k for which the oint (a, sin a) (a>0) lies inside the triangle formed by + y = k with co-ordinate aes is : æ (A) +, æ (C) -,+ è ø æ æ æ (B) - +, + è ø (D) ( sin, +sin) ì ï 6. cos í îï ü ï ý þï = cos - cos holds for (A) (B) Î R (C) 0 (D) 0 A-79 Indra Vihar, Kota Rajasthan 00 Page No. #

16 ìï - sin + + sin üï 7. The value of cot í ý, where < <, is : ïî - sin - + sin ïþ (A) - (B) + (C) (D) - (sin ) + (cos ) 8.* If =l, then l Î [a,b] : (A) (a + b) is a rime number. (C) If l is an integer, it can only be 0 or. (D) b a = (B) l cannot be an integer. 7 9.* If f () = ì cos + cos í + - ü ý î þ then : (A) f æ = è ø æ (B) f è ø æ = cos (C) f è ø = æ (D) f è ø = cos PART - II : SUBJECTIVE QUESTIONS. Evaluate the following : (i) tan êcos æ + tan è ø - - (ii) sin æ ê -sin è ø (iii) cos (tan ) (iv) tan tan æ (v) cos æ sin è ø (vi) æ tan sin è + cot ø (vii) sin æ- ê -sin ê (viii) cos êcos ê æ- + 6 (i) tan - êtan () cos - êcos (i) sin êcos. Find sin (sin q), cos (cos q), tan (tan q) and cot (cot q) for q Î ê,. Evaluate each of the following : æ (i) sin sin + cos è ø æ (ii) sin (tan + tan - ) (iii) tan cos è ø. Prove each of the following : (i) tan = + cot = sin + = cos + when < 0. (ii) cos = sec = sin - = + tan - = cost - when << 0 A-79 Indra Vihar, Kota Rajasthan 00 Page No. #

17 . Find the value of sin (sin) + cos (cos0) + tan [ tan (-6)] + cot [ cot (0)]. 6. Solve the following inequalities: (i) cos > cos (ii) tan > cot. (iii) arccot - arccot + 6 > 0 7. If X = cosec tan cos cot sec sin a & Y = sec cot sin tan cosec cos a; where 0 a <. Find the relation between X & Y. Eress them in terms of 'a'. 8. Solve the following equation : sec a - sec b = sec b - sec a a ³ ; b ³, a ¹ b. 9. (i) Find all ositive integral solutions of the equation, tan + cot y = tan. (ii) If 'k' be a ositive integer, then show that the equation: tan + tan y = tan k has no non-zero integral solution. 0. If y cos cos a b + =a, then rove that. y y cos sin a b a - a+ b = a.. Prove that : (i) æ- æ- sin ê cot + cos ê cot = è ø è ø (ii) sin + cos + cot = 7 6. In a D ABC if Ð A = 90º, then rove that b c tan - + tan - = c + a a+ b.. If a sin b cos = c, then find the value of a sin + b cos.. (i) Prove that if 0 < A < æ + + = tan tana tan (cot A) tan (cot A) tan. (ii) Prove that : æ tan + sin - cos =-+ cot 0 -. æ æ æ +. Solve each of the following for : (i) sin + sin = (iii) tan + tan = (ii) tan + tan = (iv) sin + sin = sin. æ 6. Prove that sin + sin y = sin - y + y - when either y < 0 or è ø + y. 7. Find the sum of series : (i) tan + tan tan 9 n + n +... (ii) tan tan tan tan to n terms. (iii) sin + sin n - n- sin +... n (n + ) (iv) cot 7 + cot + cot + cot +... to n terms. A-79 Indra Vihar, Kota Rajasthan 00 Page No. # 6

18 PART-I IIT-JEE (PREVIOUS YEARS PROBLEMS) * Marked Questions are having more than one correct otion.. The number of real solutions of tan ( + ) + sin + + = is: [IIT-JEE 999, Part, (, 0), 80] (A) zero (B) one (C) two (D) infinite. 6 æ æ If sin cos è ø è ø = for 0 < <, then equals : [IITJEE-00, Scr. (, 0), ] (A) / (B) (C) / (D). Prove that, cos tan sin cot = + +. [IIT-JEE-00, Main (, 0), 60]. Domain of f () = sin () + is : [JEE 00 (screening)] 6 (A) æ - è, (B) ê -, ø (C) ê-, (D) ê-,. The value of for which sin ( cot ( + )) = cos (tan ) is : [IIT-JEE-00, Scr. (, ), 8] (A) / (B) (C) 0 (D) / 6. Match the column [IIT-JEE-007, Paer-, (6, 0), 8] Let (, y) be such that : sin (a) + cos (y) + cos (b y) = Column I Column II (A) If a = and b = 0, then (, y) () lies on the circle + y = (B) If a = and b =, then (, y) (q) lies on ( ) (y ) = 0 (C) If a = and b =, then (, y) (r) lies on y = (D) If a = and b =, then (, y) (s) lies on ( ) (y ) = 0 7. If 0 < <, then + [{ cos (cot ) + sin (cot )} ] / = [IIT-JEE 008, Paer, (, ), 8] (A) + (B) (C) + (D) + 8. Values of which satisfies the equation [IIT-JEE 00] tan ( + ) tan ( ) = sin (/) are : (A) ± (B) ± (C) ± (D) ± A-79 Indra Vihar, Kota Rajasthan 00 Page No. # 7

19 PART-II AIEEE (PREVIOUS YEARS PROBLEMS) æ.(a) tan + tan æ 9 is equal to : [AIEEE-00] () cos - æ () æ sin èø æ () tan.(b) cot ( ) cosa tan ( ) cosa =. then sin is equal to : () tan [AIEEE-00] æa () tan æa () cot è ø () tan a () cot æa. The trigonometric equation sin = sin a, has a solution for : [AIEEE-00] () < a < () all real values of a () a () a ³. If cos cos y = a, then y cos a + y is equal to : [AIEEE-00] () sin a () () sin a () sin a. æ If sin + æ cosec = then a value of is : [AIEEE-007] () () () (). æ The value of cot cos ec + tan is : [AIEEE-008] () 7 () 7 () 7 () If, y, z are in A.P. and tan, tan y and tan z are also in A.P., then : [JEE Mains_0] () = y = z () = y = 6z () 6 = y = z () 6 = y = z SUBJECTIVE QUESTIONS Write the rincial value of the following :.. cos æ - æ sin - è ø [ Marks] [ Marks]. tan (- ) [ Marks]. cos - è ø æ [ Marks] A-79 Indra Vihar, Kota Rajasthan 00 Page No. # 8

20 æ æ cos cos + sin sin sin sin æ cos cos 6 æ 7 [ Marks] [ Marks] [ Marks] 8. Evaluate : cot [tan a + cot a] [ Marks] 9. Find if sec ( ) + cos ec = [ Marks] Prove : sin = sin ( ) [ Marks]. Write the following in simlest form : æ + tan, ¹ 0 [ Marks] Prove that : sin + sin = tan [ Marks] 7 6. Prove that : tan + tan + tan + tan = [ Marks] 7 8 æ æ æ. Prove that : tan tan tan + = 7 7 [ Marks] æ 8 æ æ6. Prove that : sin sin cos 7 + = 8 [ Marks] sin sin 6. Prove that : cot æ æ =, Î 0, sin sin è ø [6 Marks] 7. Prove that : tan æ = - cos è ø [6 Marks] 8. Solve : tan + tan = / [6 Marks] 9. Solve : tan ( + ) + tan ( - ) = tan [6 Marks] 8 0. Solve : tan tan = - + [6 Marks]. Prove that : cos tan - æ æ,, = - Î - + sin [6 Marks] A-79 Indra Vihar, Kota Rajasthan 00 Page No. # 9

21 Eercise # PART - I A. (B) A-. (C) A-. (A) A-. (D) A-. (B) A-6.* (CD) B. (B) B-. (D) B-. (A) B-. (C) B-. (A) B-6. (C) B-7. (B) B-8. (B) B-9. (A) B0.* (BCD) B.* (ABC) B.* (AC) C. (B) C-. (D) C-. (B) C-. (A) C-. (D) C-6. (C) D. (C) D-. (B) D-. (C) D-. (D) D-. (A) D-6. (D) D-7.* (CD) D-8. (B) D-9. (B) D0.* (BD) D.* (AB) D.* (AD) PART - II. (C). (B). (D). (D). (C) 6. (D) 7. (A) (S), (B) (P), (C) (S), (D) (Q) 8. (A) (), (B) (q), (C) (), (D) (q) 9. (A) 0. (C) Eercise # PART - I. (B). (C). (B). (C). (D) 6.* (BC) 7. (C) 8. (B) 9. (C) 0. (C). (C). (B). (C). (A). (A) 6. (C) 7. (B) 8.* (AB) 9.* (AD) PART - II. (i) (ii) (iii) (iv) - (v) (vi) 7 6 (vii) (viii) (i) - () (i). êq -, ê ê - q, q < q ; ê ê - q, q -, q < q q -, ê ê ; ê q -, < q < < q ; ê ê q -, q -, q < < q < A-79 Indra Vihar, Kota Rajasthan 00 Page No. # 0

22 . (i) 8 + (ii) 7 70 (iii) (i) [-, 0) (ii) > (iii) (-, cot ) U (cot, ) 7. X = Y = -a 8. = ab 9. (i) Two solutions (, ) (, 7). ab + c (a - b) a + b. (i) (ii) (iii) ± (iv), 0, 7. (i) (ii) tan ( + n) - tan (iii) (iv) arc cot n + ê n Eercise # PART - I. (C). (B). (D). (D) 6. (A) (), (B) (q), (C) (), (D) (s) 7. (C) 8. (D) PART - II.(a) ().(b) (). (). (). (). () 6. () Eercise # tan ± A-79 Indra Vihar, Kota Rajasthan 00 Page No. #

METHODS OF DIFFERENTIATION. Contents. Theory Objective Question Subjective Question 10. NCERT Board Questions

METHODS OF DIFFERENTIATION. Contents. Theory Objective Question Subjective Question 10. NCERT Board Questions METHODS OF DIFFERENTIATION Contents Topic Page No. Theor 0 0 Objective Question 0 09 Subjective Question 0 NCERT Board Questions Answer Ke 4 Sllabus Derivative of a function, derivative of the sum, difference,