All Questions of. JEE-Main 2016 Included. Joint Entrance Examination (Main) (All India Common Engineering Entrance Test)

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2 All Questions of JEE-Main 06 Included PART II Joint Entrance Eamination (Main) (All India Common Engineering Entrance Test) By J.P. Mohindru (Author of Modern's abc + of Mathematics Series) & Bharat Mohindru THOROUGHLY REVISED AND UPDATED EDITION 07 (Producers of Quality Tet & Competition Books) Price of Part - I & II : ` 80.00

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4 CONTENTS PART II CHAPTERS PAGES MODULE IV (INVERSE TRIGONOMETRIC FUNCTION, DETERMINANTS & MATRICES). Inverse Trigonometric Functions. Determinants. Matrices Assertion-Reason & Column Matching Type Questions IV Unit Test Paper No. / /7 / /7 / /9 A-IV/ A-IV/ U-IV/ U-IV/ Real Numbers Limit and Continuity Differentiability and Differentiation Application of Derivatives Indefinite Integrals Definite Integrals Areas under Curves Differential Equations Assertion-Reason & Column Matching Type Questions V / /6 / / 6/ 6/ 7/ 7/9 8/ 8/9 9/ 9/8 0/ 0/ / / A-V/ A-V/ Unit Test Paper No. U-V/ U-V/ MODULE V (CALCULUS) MODULE VI (THREE DIMENSIONAL GEOMETRY, VECTORS AND PROBABILITIES) / /8. Three Dimensional Geometry / /8. Vector Algebra / /. Probability A-VI/ A-VI/6 Assertion-Reason & Column Matching Type Questions VI Unit Test Paper No. 6 U-VI/ U-VI/ MT/ MT/ MOCK TESTS Mock Tests (For Revision)

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8 INVERSE TRIGONOMETRIC FUNCTIONS IMPORTANT TERMS, FACTS AND FORMULAE. DOMAIN AND RANGE OF INVERSE FUNCTIONS Function Domain Range y = sin y y = cos 0 y y = tan R < y < y = cot R 0 < y < y = sec y = cosec < or < < or <. PROPERTIES OF INVERSE FUNCTIONS sin (sin ) = sin (sin ) = ; etc. sin ( ) = cosec ; etc. sin ( ) = sin, cos ( ) = cos, tan ( ) = tan ; etc. 0 < y <, y < y <, y 0. sin + cos = tan + cot = sec + cosec =.. FORMULAE OF SUMS AND DIFFERENCES cos ( ) = cos sin + cos = sin ± sin y = sin [ y ± y ] cos ± cos y = cos [ y + y ] tan + tan y = R S T tan + tan + y y + y tan tan y = tan + y y y if if y < y > + y + z yz (E) tan + tan y + tan z = tan. y yz z L N M O Q P /

9 / MODERN S abc OF OBJECTIVE MATHEMATICS. MORE RESULTS (i) sin = sin (ii) cos = cos ( ) (iii) tan sin cos tan = = = + + (i) sin = sin ( ) (ii) cos = cos ( ) (iii) tan = tan.. IMPORTANT SUBSTITUTIONS Epression Substitution a a + or a + = a sin θ <θ< = a tan θ <θ< a = a sec θ (0 < θ < ) a and a + = a cos θ <θ<. sin (sin ) = sin (sin ) = ; etc.. sin ( ) = sin, cos ( ) = cos ; etc.. sin + sin y = sin ( y + y ). cos + cos y= cos ( y y ). tan + tan y = + y tan if y < y + y tan if y + y > 6. tan sin cos tan = = =. + + Select the correct answer :. The value of sin cos The value of sin cos

10 INVERSE TRIGONOMETRIC FUNCTIONS / 7. The value of cot cos 7.. The principal value of the epression cos [ cos( 680 )] If tan + cot =, then equals : The domain of the function sin [, ] [, ] [0, ] 7. If cos sin + cos = 0, then is equal to : The greatest and least values of (sin ) + (cos ) are respectively : and 8 and and and The domain of the function defined by : f() = sin + cos [, ] [, + ] (, ) φ. 0. The value of sin ( sin ( 6)) 8 96 sin.. If. If sin + sin =, 0 y then the value of cos + cos y tan + tan =,. y then cot + cot y equals :.. The value of tan cos + tan The value of the epression ( ( sin cot cos tan ) ) 0. The equation. tan cot = tan has : no solution unique solution infinite number of solutions two solutions. 6. If α sin + cos β, then : α=, β= α = 0, β = α=, β= α = 0, β =.

11 / MODERN S abc OF OBJECTIVE MATHEMATICS 7. The value of tan (sec ) + cot (cosec ). 8. The value of the epression sec + sin 9. If a a sin cos tan, + = + a + a where a, (0, ), then the value of 0 a a a. a 0. If cos > sin, then : < 0 < < > 0.. Considering only the principal values, if tan (cot ) = sin cot, then. The value of sin (cot ) +. ( + ) / ( + ) /.. The principal value of sin.. If = sin k, y = cos k, k, then the correct relationship + y = y = + y = y =.. Two angles of a triangle are cot and cot. Then the third angle Solution set of cos sin = sin ( ) [, ] [, 0] 0, 7. If 0 and θ = sin + cos tan, then : θ = θ θ 8. Principal value of : θ. 7 sin + cos cos Value of : cos sin + cos tan ( ) equals : 7 F 0. If sin sin + cos I HG K J = 0., then is equal to :.

12 INVERSE TRIGONOMETRIC FUNCTIONS /. The value of cos cos + sin sin 0 0. (Roorkee 000). The trigonometric equation sin = sin a has a solution for : all real values a < a < a <. (A.I.E.E.E. 00). The value of for which : sin (cot ( + )) = cos (tan ) is 0. (I.I.T. (Screening) 00). If sin + cosec =, then the value of. (A.I.E.E.E. 007). The value of cot cosec + tan (A.I.E.E.E. 008) 7 6. The value of tan cos + tan If cos p + cos q + cos r =, then : p + q + r + pqr 8. If sin + sin + sin = y z, then : y + z + y + z 0 9. The number of real solutions of : tan ( + ) + sin + + = zero one two infinite. y 0. If cos + cos = α, then : a b cot α cos α tan α sin α.. The principal value of : 9 9 cos cos sin n n. If sin i = n, then i is equal to : i = i = n n n( n + ). sin cot tan cos is equal to :. Value of cos tan tan a y y cos α + equals : ab b

13 /6 MODERN S abc OF OBJECTIVE MATHEMATICS. If cos + cos y + cos z =, then : 6. If y + yz + z equals : 0. n sin i = n i = n n( n + ), then n n i i = equals : 7. The number of positive integral solutions of : tan cos y + = sin + y Value of tan + tan + tan + tan If tan =, then equals : tan tan tan 6 tan The integral solution of : tan tan + = tan y (, ) (, ) (, ) 8. If sin + cos ( ) = sin ( ), then satisfies :. If + + = 0 = 0 + = = 0. + y = sec + sin + and F HG I z =cosec + + KJ + cos +, then y + z + equals : F HG I KJ 0. If (tan ) + (cot ) =, then equals : n m tan equals : m = m + m + tan (n n + ) tan (n + n + ) tan n + n n + n +. If A = tan F HG k I KJ and then the value of (A B) k B= tan k, y 6. If sin + sin =, then the value of 6 y y If sin + for 0< <, then equals : 6 + cos + =. (I.I.T. (Screening) 00) 8. cot ( cos α) tan ( cos α) =, then sin = tan α cot α tan α α cot. (A.I.E.E.E. 00)

14 INVERSE TRIGONOMETRIC FUNCTIONS /7 9. The value of : tan a y a a + tan ay+ + aa F HG I + + KJ + + a a a n an tan tan aa + an an + tan a n tan y The value of : tan F HG y tan sin + cos + + if 0 if < not finite if > 6. If tan + tan + + ( )( ) + tan tan + ()() = tan, then equals : n n + n n + + n( n + ) n n + n + n A.M. of the non-zero solutions of the equation : tan tan tan = 7 6. I KJ 6. If,,, are the roots of the equation : sin θ + cos θ cos θ sin θ = 0, then tan + tan + tan + tan equals : θ θ θ θ. 6. If α<, then the number of solutions of (sin ) + (cos ) = α 0 infinite. 6. y If cos + cos = and a b y sin sin =, then a b the value of y + equals : a b 66. If θ, θ, θ be the roots of + m + + m = 0, then the general value of tan θ + tan θ + tan θ n n ( n + ) 67. If cos cos y = α, then y cos α + y is equal to : sin α sin α sin α. (A.I.E.E.E. 00) 68. If 0 < <, then : / + { cos (cot ) + sin (cot ) } is equal to : (I.I.T. 008). The value of cot cot n + k n= k=. (I.I.T. (Advanced) 0)

15 /8 MODERN S abc OF OBJECTIVE MATHEMATICS. If tan y = tan + tan, Then a value of y when < (J.E.E (Main) 0) (A,C) sin cos = sin cos cot cos = 7 cot cot 6 9. = sin cos = sin sin = sin sin 0 = 0. sin cos = sin cos 6+ = sin cos = sin sin = sin sin 0 =. 0 = =. cos [cos ( 680 )] = cos [cos 680 ] = cos [cos (70 0 )] = cos [cos ( 0 )] = cos [cos (0 )] = 0 =. 9. tan + cot = tan + (tan + cot ) = = tan + = tan = tan Hence, =. = tan = tan.

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