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

August 28, 2017 | Author: varunkohliin | Category: Sine, Triangle, Trigonometric Functions, Trigonometry, Classical Geometry

#### Short Description

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

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Vidyamandir Classes DATE :

  MARKS :    10 

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : INVTG & PROP  [1]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

TIME TAKEN:

PARENT’S SIGNATURE :

 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 x  (A)

2.

4.

5.

6.

7.

8.

24 25

(B)

(D)

1 5

10  3

(D)

None of these

 6

(D)

x 1

(D)

all x  R

 4

(D)

    4 2

(D)

None of these

(D)

1:2:3

(D)

None of these

24 25

(C)

3  10

(C)

1 5

The value of sin 1  sin10  is : (A)

3.

1 , the value of cos cos 1 x  2 sin 1 x is : 5

10

(B)

2 If sin 1 x  sin 1 y  , then cos 1 x  cos 1 y is equal to : 3 2  (A) (B) (C) 3 3

 2x sin 1   1  x2 (A) x

 1   2 tan x for :  (B) 1

x0

If 0  x  1 and   sin 1 x  cos 1 x  tan 1 x , then :   (A)  (B)  2 2     3  7 The principal value of sin 1   cos 1  cos   2   6    5  (A) (B) 6 2

(C)

(C)



   , is : 

(C)

3 2

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 If cot (A)

A bc  then the ABC is : 2 a isosceles (B)

equilateral

(C)

right angled

9.

Which of the following pieces of data does not uniquely determine an acute-angled triangle ABC (R being the circumradius) ? (A) (B) a, b, c (C) (D) a, sin A, sin B a, sin B, R 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 : (A)

1:1  3

(B)

2:3

VMC/Inverse Trigonometry & Properties of Triangle

(C)

3 :2 3

46

(D)

1: 2  3

HWT/Mathematics

Vidyamandir Classes DATE :

  MARKS :    10 

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : INVTG & PROP  [2]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

TIME TAKEN:

PARENT’S SIGNATURE :

 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.

tan cos 1 x is equal to :

1  x2 x

(A) 2.

3.

1 x

1  x2 x

(C)

2

If x  y  z  xyz , then tan 1 x  tan 1 y  tan 1 z   (A) (B)  2 n r 1   2 tan 1   is equal to :  1  22 r  1    r 1

(C)

1

(C)

tan 1 2n  1

(D)

1  x2

(D)

None of these

(D)

tan 1 2n  1 

(D)

   0, 2   

(A) 4.

x

(B)

 

tan 1 2n

 

tan 1 2n 

(B)

 4

 4

Range of the function f  x   cos 1   x , where { } is fractional part function, is : (A)

   2 ,   

(B)

   2 ,   

(C)

   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 (A) 2 (B) 3 (C) 4 (D) 5

6.

The sides a, b, c of ABC are in G.P. where log a  log 2b, log 2b  log 3c and log 3c  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   and the side a = 2, then area of triangle is : a b c

(A) 8.

(B)

2

 AC  2 cos    2 

(B)

3 2

(C)

ac a  ac  c 2 2

(D)

3

 AC  2 sin    2 

(C)

 AC  2 cos    2 

(D)

 AC  2 sin    2 

x

(C)

2  x

(D)

2  x

(C)

a

(D)

None of these

If   x  2 then cos 1  cos x   (A)

10.

1

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

9.

 3 

x

(B)

If a  sin 1 x  cos 1 x  tan 1 x  b , then : (A)

a  0, b  

(B)

a  0, b 

 2

VMC/Inverse Trigonometry & Properties of Triangle

47

 ,b  2

HWT/Mathematics

Vidyamandir Classes DATE :

  MARKS :    10 

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : INVTG & PROP  [3]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

TIME TAKEN:

PARENT’S SIGNATURE :

 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.

2.

For which value of x, sin cot 1  x  1   cos tan 1 x   1 (A) (B) 0 2

x

7.

8.

9.



2

1

(D)

1 2

12

  1 

x

(C)

x 1  x2

(D)

1  x2

0.96

(C)

0.6

(D)

None of these

0.48

(B)

If cos 1     cos 1     cos 1     3 , then the value of      is : 0

(B)

3

(C)

3

(D)

1

(C)

0

(D)

None of these

In a triangle ABC, a  b cos C  ccosB  is : (A)

6.

(B)

1  x2

(A) 5.

(C)

The value of cos 2 cos 1 0.8 is : (A)

4.

 1  x 2  x cos cot 1 x  sin cot 1 x 

If 0 < x < 1, then (A)

3.

 

a2

(B)

b2  c2

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) The in-radius of the triangle formed by the axes and the line 4 x  3 y  12  0 is : 1 (A) (B) r (C) r2 r 1 2 2 2 5 2 If tan 1 x  cot 1 x  , then x = 8 (A) (B) 1 (C) 0 1

 

7

1 4

(D)

r

(D)

None of these

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)

10.

100 3

(B)

The area of an equilateral triangle is 1 (A) (B) 3

200 3 3

(C)

100 3 3

(D)

3 square units. The circumradius of equilateral triangle is : 2 (C) 1 (D) 3

VMC/Inverse Trigonometry & Properties of Triangle

48

200 3

3

HWT/Mathematics

Vidyamandir Classes DATE :

  MARKS :    10 

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : INVTG & PROP  [4]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

TIME TAKEN:

PARENT’S SIGNATURE :

 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.

 2a  1  2b  1 If sin 1    sin    2 tan x , then x is equal to :  1  a2   1  b2  a b b (A) (B) (C) ab 1  ab 2n

2.

If

sin 1 xi  n then find the value of

i 1

(A) 3.

4.

5.

6.

7.

9.

x  cos

1

x , then tan sin

(D)

None of these

(D)

1/3

(D)

1

(D)

4

x . i

2n

(C)

1

(C)

n  n  1 2

2/3

x =

3 1 (C) 2 2 In a ABC ,  a  b  c   b  c  a    bc , where   I , the greatest value of  is : 0

(B)

(A)

1

(B)

2

(C)

3

 3  If x   , 2  , then the value of the expression sin 1 cos 1  cos x   sin 1  sin x   is :    2    (A) (B) (C) 0 (D) 2 2

 

In ABC a 2 cos 2 B  cos 2 C  b 2 cos 2 C  cos 2 A  c 2 cos 2 A  cos 2 B is equal to :

1

(C)

a  b2  c2

(D)

2 a 2  b2  c2

B C If b  c  3a , the cot   . cot    2   2 (A) 1 (B) 2

(C)

3

(D)

4

0

(B)

2

In any triangle the ratio of angles is 1 : 2 : 3, the ratio of corresponding sides is : (A)

10.

(B)

(A)

(A) 8.

ab 1  ab

2n

  2 If sin  sin 1    cos 1 x   1 , then x = 3     (A) 1 (B) 0 If sin

(D)

i 1

n

1

b 1  ab

1:2:3

(B)

1: 2 : 2

(C)

1: 3 : 2

(D)

1: 2 : 3

(C)

5 4

(D)

 2

tan 1 1  tan 1  2   tan 1  3  (A)

3 4

(B)

VMC/Inverse Trigonometry & Properties of Triangle

49

HWT/Mathematics

Vidyamandir Classes DATE :

  MARKS :    10 

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : INVTG & PROP  [5]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

TIME TAKEN:

PARENT’S SIGNATURE :

 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.

2.

 , then cot 1 x  cot 1 y  4  3 (A) (B) (C) 2 4  The domain of the function f  x   sin 1 x   log 1  x  is : 4

If tan 1 x  tan 1 y 

(A) 3.

 1  x , 1  2 

(B)

 1  x , 1  2 

(C)

(D)

5 4

 1  x , 1  2 

(D)

 1  x , 1  2 

If x1 , x2 , x3 , x4 are roots of the equation x 4  x 3 sin 2   x 2 cos 2   sin   x cos   0 , then

tan 1 x1  tan 1 x2  tan 1 x3  tan 1 x4 

4.

5.

 

(D)

–

(C)

2

(D)

3

19/35

(D)

None of these

If in a ABC, 8 R  a  b  c , then the triangle ABC is : (A) right angled (B) isosceles (C)

equilateral

(D)

None of these

sin B If in a ABC , cos A  then the ABC is : 2 sinC (A) equilateral (B) isosceles

7.

right angled

(D)

None of these

*8.

(D)

  nR 2 tan   n

(D)

None of these

1/5 2

bc ca ab   then cos A is equal to : 11 12 13 (B) 5/7 (C) 2

2

2

(C)

The area of a regular polygon of n sides is : nR 2  2  sin   2  n 

(B)

  nr 2 tan   n

Angles A, B and C of a triangle ABC are in A.P. If (A)

10.

(B)

If in a triangle ABC ,

(A) 9.

In a right angled ABC, sin 2 A  sin 2 B  sin 2 C  (A) 0 (B) 1

(A) 6.

  2

(C)

(A)

 6

(B)

1 1 cos 1    2 sin 1   is equal to : 2   2  (A) (B) 4

b  c

 4

 6

VMC/Inverse Trigonometry & Properties of Triangle

(C)

nr 2  2  sin   2  n 

3 , the angle A is equal to : 2  (C) 3

(C)

 3

50

(D)

2 3

HWT/Mathematics

Vidyamandir Classes DATE :

IITJEE :

NAME :

  MARKS :    10 

TIME : 25 MINUTES

TEST CODE : INVTG & PROP  [6]

ROLL NO.

START TIME :

STUDENT’S SIGNATURE :

END TIME :

TIME TAKEN:

PARENT’S SIGNATURE :

 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.

2.

1 1 The value of tan 1    tan 1   is : 2 3 (A) 0 (B)  3

5.

6.

7.

8.

9.

 4

(D)

3 9 , the value of x100  y100  z100  101 101 101 is : 2 x y z (B) 1 (C) 2 (D) 5    is : 3  

3 5 2

(B)

3 5

1   The numerical value of tan  2 tan 1   is : 5 4  (A) 1 (B) 0 1 1 If x   2 , the principal value of sin x is : x   (A) (B) 4 2  3 The equation sin 1 x  cos 1 x  cos 1  has :  2    (A) no solution (B) unique solution

3

(C)

3 5 2

(D)

None of these

(C)

7/17

(D)

–7/17

(C)

(D)

3 2

(C)

infinite no. of solution

(D)

None of these

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

B C If in a triangle ABC, b + c = 3a then cot   . cot   is equal to : 2   2 (A) 1 (B) –1 (C) 2

(D)

–2

(D)

None of these

If in a ABC ,  sin A  sin B  sin C   sin A  sin B  sin C   3 sin A . sin B then : (A)

10.

0

1  The value of tan  cos 1    2  (A)

4.

 6

If sin 1 x  sin 1 y  sin 1 z  (A)

3.

(C)

A  60

(B)

B  60

(C)

C  60 2

2

2

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 : (A)

AP

(B)

GP

VMC/Inverse Trigonometry & Properties of Triangle

(C)

HP

51

(D)

None of these

HWT/Mathematics

Vidyamandir Classes DATE :

  MARKS :    10 

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : INVTG & PROP  [7]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

TIME TAKEN:

PARENT’S SIGNATURE :

 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.

2.

3.

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

If in a ABC, sin 2 A  sin 2 B  sin 2 C then the triangle is : (A) acute angled (B) right angled (C)

(D)

None of these

c 2  3c  7  0

(D)

c 2  3c  7  0

(C)

4

(D)

24

(C)

equilateral

(D)

None of these

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

c 2  3c  7  0

(A) 4.

67 8

If sin (A)

7.

65 4

isosceles 1

(B)

right angled

 x  , for some x   1, 1 , then the value of cos 1 x is : 5 3 10

(B)

5 10

(C)

7 10

(D)

9 10

1 3 cos 1   2 5

(B)

1 1  3  sin   2 5

(C)

1 3 tan 1   2 5

(D)

1 tan 1   2

0

1

(D)

2

2

(D)

3

(B)

1 2

(C)

If cos 1 x  cos 1 y  cos 1 z   , then x 2  y 2  z 2  2 xyz  (A)

10.

(B)

If tan 1  x  1  tan 1  x   tan 1  x  1  tan 1  3x  , then x = (A)

9.

(C)

1 2 tan 1    tan 1    4 9 (A)

*8.

c 2  3c  7  0

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

6.

(B)

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

5.

obtuse angled

0

(B)

1

(C)

3  The value of cos  tan 1  is : 4  (A)

3 5

(B)

4 5

VMC/Inverse Trigonometry & Properties of Triangle

(C)

1 2

52

(D)

2 3

HWT/Mathematics

Vidyamandir Classes DATE :

  MARKS :    10 

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : INVTG & PROP  [8]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

TIME TAKEN:

PARENT’S SIGNATURE :

 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.

1 1  sin  sin 1  cos 1   2 2  (A)

2.

 2 tan   5 (A)

4.

5.

7.

4tan 1 x

(B)

0

(C)

1

(D)

None of these

(C)

 2

(D)

1

(D)

(C)

17/6

(D)

None of these

(C)

 tan 1  0.3

(D)

 1  tan 1    18 

(D)

None of these

(D)

30

    2      tan  15   3 tan  5  . tan  15  is equal to :        1  3 (B) (C) 3

4 2  The value of tan  cos 1  tan 1  : 5 3  (A) 6/17 (B) 7/16

3

If tan  x  y   33 and x  tan 1 3 , then y will be : (A)

6.

1 2

(B)

 2x  If x  1 , then 2 tan 1 x  sin 1   is equal to :  1  x2 

(A) 3.

0

tan 1  2.3

(B)

tan 1 1.3

     If tan  tan      tan      k tan 3 , then the value of k is : 3   3  1 (A) 1 (B) (C) 3 3  x 3   2x  k  If A  tan 1  and B  tan 1   , then the value of A  B is :  2k  x   k 3    (A) 0 (B) 45 (C) 60

8.

If the radius of the circumcircle of an isosceles triangle PQR is equal to PQ(= PR), then the angle P is : (A) (B) (C) (D)  6  3  2 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)

10.

1 km

(B)

2 km

(B)

 s  b  tan

(C)

4 km

(C)

 s  b  tan

(D)

7 km

In a triangle ABC, r = (A)

 s  a  tan

B 2

B 2

VMC/Inverse Trigonometry & Properties of Triangle

53

C 2

(D)

None of these

HWT/Mathematics

Vidyamandir Classes DATE :

  MARKS :    10 

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : INVTG & PROP  [9]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

TIME TAKEN:

PARENT’S SIGNATURE :

 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.

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

r  4R

(A) 2.

s4

(D)

None of these

b2c2

(C)

c2a2

(D)

a 2b 2

2

(C)

3

(D)

4

(B)

1

(C)

tan  nA  . tan  nB  . tan  nC  (D)

None of these

2, 3

(B)

3, 4

A 5 C 2  , tan  , then : 2 6 2 5 a, c, b are in AP (B) a, b, c are in AP

(C)

4, 3

(D)

6, 8

(C)

b, a, c are in AP

(D)

a, b, c are in GP

(C)

x3

(D)

x4

(C)

tan 1 n

(D)

tan 1 n  1

(B) (D)

Geometric mean of a and b None of these

(C)

0,

6 sin 1 x 2  6 x  8.5   if :

 sin r 1

(A)

10.

R 3

In a ABC , tan

n

9.

r

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

(A) 8.

(B)

0

(A) *7.

(C)

If A  B  C   , n  Z , then tan  nA   tan  nB   tan  nC  is equal to :

(A) 6.

(B)

(A) 5.

R 2

In any ABC, b 2 sin 2C  c 2 sin 2 B  (A)

4.

r

If c 2  a 2  b 2 , 2s  a  b  c , then 4s  s  a  s  b  s  c   : (A)

3.

(B)

x 1

(B)

x2

 r  r 1  is equal to :  r  r  1   

1 

tan 1 n 

 4

(B)

tan 1 n  1 

 4

a b  If tan 1    tan 1    , then x is : x    x 2 (A) Arithmetic mean of a and b (C) Harmonic mean of a and b If sin 1 x  sin 1 1  x   cos 1 x , then x equals : (A)

1,  1

(B)

1, 0

VMC/Inverse Trigonometry & Properties of Triangle

1 2

54

(D)

None of these

HWT/Mathematics

Vidyamandir Classes DATE :

  MARKS :    10 

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : INVTG & PROP  [10]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

TIME TAKEN:

PARENT’S SIGNATURE :

 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.

 3 The principal value of sin 1  is :  2   

(A) 2.

(B)

 4

(B)

(A)

7.

3 4

 n  1 d a1  an

 d 1    tan  1  a 2 a3   (B)



x 2

(B)

0

(B)

  cot   2 cot 1 3  is equal to : 4  (A) 1 (B)

5 3

(C)

(D)

 3

nd 1  a1an

(D)

an  a1 a1  an

x 2

(D)

2 

 6

  d 1   . . . . .  tan  1  a n 1an  

 n  1 d 1  a1an

(C)

    

2  x

(C)

x 2

 2

(C)

(D)

2

7

(C)

–1

(D)

None of these

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

8.

(D)

 ab  1  1  bc  1  1  ca  1  If a  b  c  0 , then cot 1    cot  b  c   cot  c  a    a b      (A)

6.

4 3

 1  sin x  1  sin x    The value of tan 1   is :   x    2   1  sin x  1  sin x 

(A)

5.

(C)

If a1 , a2 , a3 , . . . . ., an is an AP with common difference d, then :

  d tan tan 1    1  a1a2

*4.

 3

Two angles of a triangle are cot 1  2  and cot 1  3 . Then the third angle is : (A)

3.

2 3

420 17

(B)

84 5

 , then x  y  z  2 x  yz (B)

(C)

15

(D)

None of these

(C)

xz  y

(D)

xyz

If cot 1 x  cot 1 y  cot 1 z  (A)

xy  z

VMC/Inverse Trigonometry & Properties of Triangle

55

HWT/Mathematics

Vidyamandir Classes Paragraph for Q.9 - 10 The vertices of a ABC are A(1, 1), B(5, 1) and C(5, –4). 9.

The orthocenter of ABC has coordinates : (A)

10.

(5, 1)

(B)

3   3,  2   

(C)

3   5,  2   

(D)

(3, 1)

(C)

3   3,  2   

(D)

 11 2   3 , 3  

The circumcentre of ABC has coordinates : (A)

(3, 1)

(B)

3   5,  2   

VMC/Inverse Trigonometry & Properties of Triangle

56

HWT/Mathematics