JEE-MAIN -2016-PHASE-1-PCM-13

Mathematics
1. The points (1, 3) and (5, 1) are two opposite vertices of a rectangle the other two vertices lie on
the line y = 2x + c. Then joint equation of both the diagonals is
(A) 2 y 2 − 2 x 2 + 18 x + y − 28 = 0 (B) 2 y 2 + 2 x 2 + 18 x + y − 28 = 0
2 2
(C) 2 y − 2 x + 18 x + y + 28 = 0 (D) None of these

2 2
2. If the equation 2x + 6xy − py − 4x + 2qy + 1 = 0 represents two perpendicular lines then q is
(A) integer only (B) rational number only
(C) irrational number only (D) None of these

3. The difference of the slopes of the lines x 2 ( sec 2 θ − sin 2 θ ) − 2xy tan θ + y 2 sin 2 θ = 0 is
(A) 1 (B) - 1 (C) 2 (D) None of these

 3
4. The set of real values of a for which the line segment AB where A  a 2 + 1, a +  and
 7
B ( 5 − a, a 2 + 2 ) is divided into two segments by the line 2x + 7y = 9 is
 1  1  1
(A) a ∈  −5,  (B) a ∈  −4,  (C) a ∈  −7,  (D) None of these
 2  2  2

5. Sum of slopes of the legs of a right isosceles triangle if hypotenuse is x − 2y − 3 = 0 and vertex
at right angle is (1, 6), is
8 8 3
(A) (B) − (C) (D) None of these
3 3 8

4 3  2π   4π 
6.  cos α + cos3  + α  + cos3  + α  =
3  3   3 
(A) − cos 3α (B) cos 3α (C) cos 2α (D) − cos 2α

 π  π
7. Minimum value of the expression 4 cos  θ +  − 3sin  θ −  is
 3  6
7 7
(A) 7 (B) − (C) (D) −7
2 2

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o 2 2 2
8. If A + B + C = 180 then sin A − sin B + sin c is
(A) 2sin A sin Bsin C (B) 2 sin A cos Bsin C
(C) 2 cos A sin Bsin C (D) 4sin A sin Bsin C

2 π 3π 5π 7π
9. Value of sec + sec2 + sec 2 + sec 2 is
16 16 16 16
(A) 16 (B) 36 (C) 32 (D) 28

2 2
10. Minimum value of ( sin α + cos ecα ) + ( cos α + sec α ) is
(A) 2 (B) 4 (C) 7 (D) 9

11.
2 2
The circles x + y + 2ux + 2vy = 0 and x 2 + y 2 + 2u1x + 2v1 y = 0 touch each other if
(A) uu1 = vv1 (B) uv1 = u1v (C) uv = u1v1 (D) None of these

12. If l1 , l2 are lengths of circumference of circle x 2 + y 2 − 2x = 0 fall outside and inside of the circle
l
3 ( x 2 + y 2 ) − 2 3x + 2y + 1 = 0 then 1 is
l2
(A) 1 (B) 2 (C) 3 (D) None of these

13. Equation of the circle described on the common chord of circles x 2 + y 2 − 4x − 5 = 0 and
x 2 + y 2 + 8y + 7 = 0 as a diameter
(A) x 2 + y 2 − 2x + 4y + 1 = 0 (B) x 2 + y 2 + 2x + 4y + 1 = 0
(C) x 2 + y 2 + 2x − 4y + 1 = 0 (D) None of these

14. Equation of the circle passing through the origin and cutting the circles
x 2 + y 2 − 4x + 6y + 10 = 0 and x 2 + y 2 + 12y + 6 = 0 orthogonally, is
(A) 2 ( x 2 + y 2 ) + 7x + 2y = 0 (B) x 2 + y 2 + 7x + 2y = 0
(C) 2 ( x 2 + y 2 ) − 7x + 2y = 0 (D) None of these

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15. If vertex of an equilateral triangle is (2 )

2, −1 and its centroid is (0, 0) then the equation of
incircle is
(A) 4 ( x 2 + y 2 ) = 81 (B) x 2 + y2 = 9 (C) x 2 + y2 = 4 (D) x 2 + y 2 = 36

2 x + 21− x − 1
16. Minimum value of , ( x ∈ R ) is
2− x + 1
1
(A) (B) 0 (C) 1 (D) 2
2

17. If two straight lines meet the coordinate axes in four concyclic points and m1 , m 2 are their slopes
then
(A) m1 − m 2 = 1 (B) m1 + m 2 = 1 (C) m1m 2 = 1 (D) m1 = m 2

3 2
18. If sin A + sin B = and cos A + cos B = then sin ( A + B ) is
10 5

24 13 12
(A) (B) (C) (D) None of these
25 25 13
3
2
19. If  1 + sin xf ( x ) dx =
3
(1 + sin x ) 2 + c then
(A) f ( x ) = cos x (B) f ( x ) = cos ecx
(C) f ( x ) is constant function (D) None of these
x −2
20. The complete set of values of x satisfying the inequality ≤ 0 is
x −3
(A) x ∈ [ 2,3] (B) x ∈ ( −3, −2] ∪ [ 2,3)
(C) x ∈ ( 2,3] (D) None of these

21. Sum of all the values of x satisfying log 2 ( 9 − 2 x ) = 3 − x

(A) -3 (B) 3 (C) 9 (D) None of these

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22. If largest value of x satisfying ( log 2 x )( log3 x ) = log3 x 3 + log 2 x 2 − 6 is p, then p is

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

23. If y = 1 + x( 1/4
)(1 + x )(1 − x ) then dy
1/ 2

dx
1/4
=

(A) 1 (B) -1 (C) x (D) x

sec 2 ( log x )
24.  x
dx is equal to

(A) tan ( log x ) + c (B) cot ( log x ) + c (C) sin ( log x ) + c (D) None of these

π /2

 cos x.e
sin x
25. If dx = e − k , then k =
0
(A) 1 (B) 0 (C) -1 (D) None of these

2 2
sin x − sin β
26. lim 2 2
=
x→β
x −β
1 sin β cos β
(A) − sin 2β (B) − (C) (D) None of these
cos ec 2β β

27. Which of the following is not true?

(A) ( A − B) ∪ B = A (B) ( A − B) ∩ B = φ (C) ( A − B) ∪ A = A (D)
A ⊆ B  A∪ B = B

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3 x3 + cos x + 2
28. lim is equal to
x →∞ 2 x 3 + 2 x 2 + 1

2 3
(A) (B) (C) 2 (D) None of these
3 2

dy
29. If x = at2 and y = 2at then is not equal to
dx
1 2a
(A) (B) t (C) (D) None of these
t y

1 − cos x π 
30. If f ( x ) = , then f '   + 1 is equal to
1 + cos x 2
(A) 1 (B) 2 (C) 3 (D) 4

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MATHEMATICS
1. D 2. C 3. C 4. B
5. A 6. B 7. D 8. B
9. C 10. D 11. B 12. B
13. A 14. C 15. * 16. C
17. C 18. A 19. A 20. B
21. B 22. C 23. B 24. A
25. A 26. C 27. A 28. B
29. B 30. C

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