Jee-2014-Booklet4-Hwt-Inverse Trigo & Prop of Triangle

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NAME : TEST CODE : INVTG & PROP  [1]

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 This test contains a total of 10 Objective Type Questions.  Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct.

1. If 1

5

x , the value ofcos cos



1x2sin1x



is :

(A) 24 25  (B) 24 25 (C) 1 5  (D) 1 5

2. The value of sin1



sin10



is :

(A) 10 (B) 3 10 (C) 10 3 (D) None of these

3. If 1 1 2

3

sin xsin y  , thencos1xcos1y is equal to :

(A) 2 3  _(B) 3  _(C) 6  _(D) _ 4. 1 2 ₂ 2 1 1 x sin tan x x   _   _    for : (A) x 1 (B) x0 (C) x 1 (D) all xR

5. If 0  andx 1 sin1xcos1x tan 1x, then :

(A) 2    (B) 2    (C) 4    (D) 4 2  _{ }_ 

6. The principal value of 1 3 1 7

2 6

sin _ _cos cos_ _

    , is : (A) 5 6  (B) 2  (C) 3 2  (D) None of these

7. Sides of a triangle are in ratio 1 : 3 : 2 , then angles of triangle are in ratio :

(A) 1 : 3 : 5 (B) 2 : 3 : 4 (C) 3 : 2 : 1 (D) 1 : 2 : 3 8. If 2 A b c cot a 

 then the ABC is :

(A) isosceles (B) equilateral (C) right angled (D) None of these

9. Which of the following pieces of data does not uniquely determine an acute-angled triangle ABC (R being the circumradius) ?

(A) a, sin A, sin B (B) a, b, c (C) a, sin B, R (D) a, sin A, R 10. The angles of a  are in the ratio 4 : 1 : 1, then the ratio of the largest side to the perimeter is :

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1. tan cos



1x



is equal to :

(A) 2 1 x x  (B) 2 1 x x  (C) 2 1 x x  (D) 1 x 2

2. If x  y z xyz, then tan1x tan 1ytan1z

(A)  (B) 2  _(C) ₁ _(D) _{None of these} 3. 1 1 2 1 1 2 1 2 n _r r r tan          _   



is equal to : (A) tan1

 

2n (B) 1

 

2 4 n tan  (C) tan1

 

2n1 (D) 1

 

2 1 4 n tan   4. Range of the function f x

 

cos1





 

x



, where { } is fractional part function, is :

(A) 2,         (B)  2,       (C)  2,      (D) 0,2       

5. Incircle of radius 4 of aABC touches the side BC at D. If BD = 6, DC = 8 and  be the area of triangle then   3

(A) 2 (B) 3 (C) 4 (D) 5

6. The sides a, b, c ofABC are in G.P. where log alog b, log b2 2 log c3 andlog c3 log a are in A.P., thenABC is :

(A) acute angled (B) right angled (C) obtuse angled (D) None of these

7. If cos A cos B cos C

a  b  c and the side a = 2, then area of triangle is :

(A) 1 (B) 2 (C) 3

2 (D) 3

8. If the angles A, B, C of the triangle ABC be in A.P., then

2 2 a c a ac c  _   (A) 2 2 A C cos_  _   (B) 2 2 A C sin_  _   (C) 2 2 A C cos_  _   (D) 2 2 A C sin_  _  

9. If x 2 thencos1



cos x





(A) x (B) x (C) 2 x (D) 2 x

10. If asin1xcos1x tan 1x , then :b (A) a0, b (B) 0

2

a , b (C)

2

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1. For which value of x, sin cot_ 1



x1



_cos tan



1x



(A) 1 2 (B) 0 (C) 1 (D) 1 2  2. If 0 < x < 1, then





 





1 2 2 2 1 1

1x _ x cos cot x sin cot x 1_ 

  (A) 2 1 x x  (B) x (C) 2 1 x x (D) 1 x 2

3. The value of cos



2cos10 8.



is :

(A) 0.48 (B) 0.96 (C) 0.6 (D) None of these

4. If cos1

 

 cos1

 

 cos1

 

 3 , then the value of     is :

(A) 0 (B) 3 (C) 3 (D) 1

5. In a triangle ABC,a b cos C



ccosB



is :

(A) a2 (B) b2c2 (C) 0 (D) None of these

6. If the lengths of the sides of a triangle are 5, 12 and 13 units, the circumradius of triangle is :

(A) 2.5 (B) 6 (C) 6.5 (D) 7

7. The in-radius of the triangle formed by the axes and the line 4x3y12 is :0

(A) r2 (B) 1 2 r (C) r1 (D) 1 4 r 8. If



 



2 2 2 1 1 5 8

tan x  cot x   , then x =

(A) 1 (B) 1 (C) 0 (D) None of these

9. A man from the top of a 100m high tower sees car moving towards the tower at an angle of depression of 30 . After some time, the angle of depression becomes 60 . The distance (in metres) travelled by the car during this time is :

(A) 100 3 (B) 200 3

3 (C)

100 3

3 (D) 200 3

10. The area of an equilateral triangle is 3 square units. The circumradius of equilateral triangle is :

(A) 1

3 (B)

2

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1. If 1 1 1 2 2 2 2 2 1 1 a b

sin sin tan x

a b   _   _   _   _      , then x is equal to : (A) a b a b   (B) 1 b ab  (C) 1 b ab  (D) 1 a b ab   2. If 2 1 1 n i i sin x n  



then find the value of

2 1 n i i x 



. (A) n (B) 2n (C)



1



2 n n (D) None of these 3. If 1 2 1 1 3

sin sin_    _{ } cos x_

 

  , then x =

(A) 1 (B) 0 (C) 2/3 (D) 1/3

4. If sin1xcos1x, thentan sin



1x



=

(A) 0 (B) 1

2 (C)

3

2 (D) 1

5. In aABC , a



 b c

 

b c a



bc, where   , the greatest value of  is :I

(A) 1 (B) 2 (C) 3 (D) 4

6. If 3 2

2

x   , _

 , then the value of the expression









1 1 1

sin _cos cos x sin sin x _{ is :} (A) 2   (B) 2  (C) 0 (D) 

7. In ABC a2



cos B cos C2  2

 

b2 cos C2 cos A2

 

c2 cos A cos B2  2



is equal to :

(A) 0 (B) 1 (C) a2b2c2 (D) 2 a



2b2c2



8. Ifb c 3a, the 2 2 B C cot _{ }. cot  _{ }     (A) 1 (B) 2 (C) 3 (D) 4

9. In any triangle the ratio of angles is 1 : 2 : 3, the ratio of corresponding sides is :

(A) 1 : 2 : 3 (B) 1: 2 : 2 (C) 1: 3 : 2 (D) 1: 2 : 3

10. tan1

 

1 tan1

 

2 tan1

 

3 

(A) 3 4  _(B) _ _(C) 5 4  _(D) 2 

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TEST CODE : INVTG & PROP  [5]

 This test contains a total of 10 Objective Type Questions.  Each question carries 1 mark. There is NO NEGATIVE marking.

Choose the correct alternative. Only one choice is correct. However, questions marked with '*' may have more than correct option.

1. If 1 1

4

tan xtan y , thencot1xcot1y (A) 2  (B) 3 4  (C)  (D) 5 4 

2. The domain of the function

 

1



1



4 f x  sin x log x is : (A) 1 1 2 x  , _   (B) 1 1 2 x  , _   (C) 1 1 2 x_ , _   (D) 1 1 2 x  , _  

3. If x , x , x , x are roots of the equation1 2 3 4 x4x sin3 2x cos2 2 sin x cos , then0

1 1 1 1

1 2 3 4

tan x tan x tan x tan x 

(A)  (B)

2

  (C)   (D) –

4. In a right angled ABC , sin A sin B2  2 sin C2 

(A) 0 (B) 1 (C) 2 (D) 3

5. If in a triangle

11 12 13

b c c a a b

ABC ,      then cos A is equal to :

(A) 1/5 (B) 5/7 (C) 19/35 (D) None of these

6. If in a ABC , R8 2a2b2c2, then the triangle ABC is :

(A) right angled (B) isosceles (C) equilateral (D) None of these

7. If in a

2

sin B ABC , cos A

sin C

  then theABC is :

(A) equilateral (B) isosceles (C) right angled (D) None of these

*8. The area of a regular polygon of n sides is :

(A) 2 2 2 nR sin n        (B) 2 nr tan n        (C) 2 2 2 nr sin n        (D) 2 nR tan n       

9. Angles A, B and C of a triangle ABC are in A.P. If 3 2

b

c , the angle A is equal to : (A) 6  (B) 4  (C) 3  (D) None of these 10. 1 1 2 1 1 2 2 cos  _{ } sin  _{ }     is equal to : (A) 4  (B) 6  (C) 3  (D) 2 3 

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1. The value of 1 1 1 1 2 3 tan  _{ }tan  _{ }     is : (A) 0 (B)  3 (C)  6 (D)  4 2. If 1 1 1 3 2

sin xsin ysin z  , the value of x100 y100 z100 ₁₀₁ ₁₀₁9 ₁₀₁

x y z      is : (A) 0 (B) 1 (C) 2 (D) 3 3. The value of 1 1 5 2 3 tan cos _ _      is : (A) 3 5 2  (B) 3 5 (C) 3 5 2  (D) None of these

4. The numerical value of 2 11

5 4 tan_ tan _   is : (A) 1 (B) 0 (C) 7/17 (D) –7/17 5. If x 1 2 x

  , the principal value of _sin1_x_{is :} (A) 4  (B) 2  (C)  (D) 3 2  6. The equation 1 1 1 3 2

sin x cos  xcos _ _

  has :

(A) no solution (B) unique solution (C) infinite no. of solution (D) None of these

7. The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60 . If the third side is 3, the remaining fourth side is :

(A) 2 (B) 3 (C) 4 (D) 5

8. If in a triangle ABC, b + c = 3a then

2 2

B C

cot _{ }. cot _{ }

    is equal to :

(A) 1 (B) –1 (C) 2 (D) –2

9. If in a ABC , sin A



sin Bsin C

 

sin Asin Bsin C



3sin A . sin B then :

(A) A 60 (B) B 60 (C) C 60 (D) None of these

10. If the angles A, B, C of a triangle ABC are in AP and sides a, b, c are in G.P. then a , b , c are in :2 2 2

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1. If the sides of a triangle are in the ratio 3 : 7 : 8, the R : r is equal to :

(A) 2 : 7 (B) 7 : 2 (C) 3 : 7 (D) 7 : 3

2. If in aABC, sin A sin B2  2 sin C2 then the triangle is :

(A) acute angled (B) right angled (C) obtuse angled (D) None of these

3. In aABC , a4, b3, A 60 . Then c is the root of the equation :

(A) c23c 7 0 (B) c23c 7 0 (C) c23c 7 0 (D) c23c 7 0

4. The sides of a triangle are 13, 14, 15. The radius of its incircle is :

(A) 67

8 (B)

65

4 (C) 4 (D) 24

5. If in a ABC , a sin Ab sin B, then the triangle is :

(A) isosceles (B) right angled (C) equilateral (D) None of these

6. If 1

5

sin x , for some x 



1 1,



, then the value of cos1x is :

(A) 3 10  (B) 5 10  (C) 7 10  (D) 9 10  7. 1 1 1 2 4 9 tan  _{ }tan  _{ }     (A) 1 1 3 2cos 5        (B) 1 1 3 2sin 5        (C) 1 1 3 2tan 5        (D) 1 1 2 tan  _{ }  

*8. Iftan1



x 1



tan1

 

x tan1



x 1



tan1

 

3x , then x =

(A) 0 (B) 1

2 (C) 1 (D) 2

9. If cos1xcos1ycos1z , then x2y2z22xyz

(A) 0 (B) 1 (C) 2 (D) 3 10. The value of 13 4 cos tan_  _   is : (A) 3 5 (B) 4 5 (C) 1 2 (D) 2 3

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1. 11 11

2 2

sin sin  cos 

  (A) 0 (B) 1 2 (C) 1 (D) None of these 2. If x , then1 2 1 1 2 ₂ 1 x tan x sin x  _     _    is equal to : (A) 4tan1x (B) 0 (C) 2  (D)  3. 2 3 2 5 15 5 15

tan_ _tan_ _ tan_ _ . tan_ _

        is equal to :

(A)  3 (B) 1

3 (C) 1 (D) 3

4. The value of 14 12

5 3

tan cos_  tan _

  :

(A) 6/17 (B) 7/16 (C) 17/6 (D) None of these

5. Iftan x



y



33 and xtan13, then y will be :

(A) tan1

 

2 3. (B) tan1

 

1 3. (C) tan1

 

0 3. (D) 1 1

18

tan _ _

 

6. If 3

3 3

tantan_ _tan_ _k tan 

    , then the value of k is :

(A) 1 (B) 1 3 (C) 3 (D) None of these 7. If 1 3 2 x A tan k x     _ _ _   and 1 2 3 x k B tan k      _ _

 , then the value of A B is :

(A) 0 (B) 45 (C) 60 (D) 30

8. If the radius of the circumcircle of an isosceles triangle PQR is equal to PQ(= PR), then the angle P is :

(A)  6 (B)  3 (C)  2 (D) 2 3

9. Two straight roads intersect at an angle of 60. A bus on one road is 2km away from the intersection and a car on the other road is 3km away from the intersection. Then the direct distance between the two vehicles is :

(A) 1 km (B) 2 km (C) 4 km (D) 7 km 10. In a triangle ABC, r = (A)





2 B sa tan (B)





2 B s b tan (C)





2 C

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1. In an equilateral triangle the in-radius and the circum-radius are connected by :

(A) r4R (B) 2 R r (C) 3 R r (D) None of these 2. If c2a2b , s2 2    , thena b c 4s s



a



s b s



  :c



(A) s4 (B) b c2 2 (C) c a2 2 (D) a b2 2

3. In any ABC , b sin C2 2 c sin B2 2 

(A)  (B) 2 (C) 3 (D) 4

4. If A  B C , n , thenZ tan nA

 

tan nB

 

tan nC

 

is equal to :

(A) 0 (B) 1 (C) tan nA . tan nB . tan nC

 

(D) None of these

5. If in aABC, right angled at B, s a 3, s  , then the values of a and c are respectively :c 2

(A) 2, 3 (B) 3, 4 (C) 4, 3 (D) 6, 8

6. In a 5 2

2 6 2 5

A C

ABC , tan , tan

   , then :

(A) a, c, b are in AP (B) a, b, c are in AP (C) b, a, c are in AP (D) a, b, c are in GP *7. 6sin1



x26x8 5.



 if : (A) x1 (B) x2 (C) x3 (D) x4 8.





1 1 1 1 n r r r sin r r    _ _     _   



is equal to : (A) 1 4 tan n (B) 1 1 4

tan n  (C) tan1 n (D) tan1 n1

9. If 1 1 2 a b tan tan x x    _   _         , then x is :

(A) Arithmetic mean of a and b (B) Geometric mean of a and b

(C) Harmonic mean of a and b (D) None of these

10. If sin1xsin1



1x



cos1x, then x equals :

(A) 1,1 (B) 1, 0 (C) 0 1

2

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1. The principal value of 1 3

2 sin _ _   is : (A) 2 3   (B) 3   (C) 4 3  (D) 5 3 

2. Two angles of a triangle are cot1

 

2 and cot1

 

3 . Then the third angle is :

(A) 4  (B) 3 4  (C) 6  (D) 3 

3. If a , a , a , . . . . ., a is an AP with common difference d, then :₁ ₂ ₃ _n

1 1 1

1 2 2 3 1

1 1 1 _n _n

d d d

tan tan tan . . . tan

a a a a a a              _   _   _ _ _  _ _ _ _ _ _   (A)





1 1 n n d a a   (B)





1 1 1 _n n d a a   (C) 1 1 n nd a a  (D) 1 1 n n a a a a   *4. The value of 1 1 1 1 1 sin x sin x tan sin x sin x  _    _        is : 2 x  _  _{ }      (A) 2 x   (B) 2 x (C) 2 x  (D) 2 2 x  

5. If a   , thenb c 0 cot 1 ab 1 cot 1 bc 1 cot 1 ca 1

a b b c c a    _    _    _  _   _   _        (A) 0 (B) 2  (C)  (D) 2 6. 2 13 4 cot_  cot _   is equal to :

(A) 1 (B) 7 (C) –1 (D) None of these

7. The sides of a triangle are 17, 25, 28. The greatest altitude is of length :

(A) 420 17 (B) 84 5 (C) 15 (D) None of these 8. If 1 1 1 2

cot xcot ycot z , then x  y z

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(A) (5, 1) (B) 3 3 2 ,  _      (C) 3 5 2 ,  _      (D) (3, 1)

10. The circumcentre ofABC has coordinates :

(A) (3, 1) (B) 5 3 2 ,  _      (C) 3 3 2 ,  _      (D) 11 2 3, 3  _     