Maths Musing was started in January 2003 issue of Mathematics Today. The aim of Maths Musing ...

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Vol. XXXV No. 5 May 2017
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mathematics today | MAY ‘17 7
Maths Musing was started in January 2003 issue of Mathematics Today. The aim of Maths Musing is to augment the chances of bright
students seeking admission into IITs with additional study material.
During the last 10 years there have been several changes in JEE pattern. To suit these changes Maths Musing also adopted the new
pattern by changing the style of problems. Some of the Maths Musing problems have been adapted in JEE benefitting thousand of our
readers. It is heartening that we receive solutions of Maths Musing problems from all over India.
Maths Musing has been receiving tremendous response from candidates preparing for JEE and teachers coaching them. We do hope
that students will continue to use Maths Musing to boost up their ranks in JEE Main and Advanced.
173
JEE MAIN COMPREHENSION
ABC is a triangle right angled at A. Points D and E are
1. A jar has 6 red marbles and 6 blue marbles. Anil picks
on the side AC such that DC = 20, ED = 8, ∠ACB = a,
two marbles at random, then Balu picks two of the
∠ADB = 2a, ∠AEB = 3a.
remaining marbles at random. The probability that
they get the same colour combination, irrespective 7. cos2a =
of order, is 2 3 4 5
(a) (b) (c) (d)
2 4 4 5 3 4 5 6
(a) (b) (c) (d) 8. AE =
7 9 11 11
(a) 4 (b) 5 (c) 6 (d) 7
2. The locus of the middle points of chords of the
circle x2 + y2 = a2 passes through a fixed point (a, b)
INTEGER MATCH
is
9. If a + b + c = 0, a3 + b3 + c3 = 3 and a5 + b5 + c5 = 10,
(a) x2 + y2 = ax – by (b) x2 + y2 = ax + by then a4 + b4 + c4 is
(c) x2 – y2 = ax – by (d) none of these
MATRIX MATCH
π/2 dx 10. Match the following.
3. ∫ =
0 cos6x+sin6x List-I List-II
3
(a) (b) p (c) (d) 2p P. When 2323 is divided by 53, the 1. 55
2 2
remainder is
16 24
4. ∑ (−1)r  =
 r  Q. The number of integer solutions 2. 30
r=0
of the equation
24 24 23 23 x + y + z + u = 3, x ≥ – 2,
(a)  8  (b)  7  (c)  8  (d)  7  y ≥ – 1, z ≥ 0, u ≥ 1, is
R. The coefficient of x2y in the 3. 56
5. If a , a , a , .... and b , b , b , ... are two A.P.s,
1 2 3 1 2 3 expansion of (1 + x + 2y)5 is
a b = 120, a b = 143, a b = 154, then a b =
1 1 2 2 3 3 8 8
(a) 29 (b) 129 10
(c) 229 (d) 329 S. If Cr =(1r0), then ∑r.Cr is 4. 60
C
r=1 r−1
JEE ADVANCED
P Q R S
6. The function f (x) = log [x3+ x6+1]is an
e (a) 1 2 3 4
(a) even function (b) 2 3 4 1
(b) odd function (c) 3 4 1 2
(c) increasing function (d) 4 1 2 3
(d) decreasing function
See Solution Set of Maths Musing 172 on page no. 51
8 MATHEMATICS TODAY | MAY ‘17

*ALOK KUMAR, B.Tech, IIT Kanpur
Single option correct tYpe (a) 1 (b) 5
(c) 8 (d) none of these
1. The number of solution(s) of equation
sin sin–1([x]) + cos–1cosx = 1 (where [⋅] denotes the 6. If f(x) = sinx + cosax is periodic, then a is
greatest integer function) is (a) 2 (b) p (c) p/2 (d) 2
(a) one (b) two 7. The line y = 2x + 4 is shifted 2 units along +y axis,
(c) three (d) none of these keeping parallel to itself and then 1 unit along
2. If P(x) is a polynomial with integer coefficients such +x axis direction in the same manner, then equation
of the line in its new position is,
that for 4 distinct integers a, b, c, d; P(a) = P(b) =
(a) y = 2x + 6 (b) y = 2x + 5
P(c) = P(d) = 3, if P(e) = 5, (e is an integer) then
(c) y = 2x + 4 (d) none of these
(a) e = 1 (b) e = 3
(c) e = 4 (d) no real value of e 8. The number of real root(s) of the equation
 x2tanx = 1 lie(s) between 0 and 2p is/are
3. A non-zero vector a is parallel to the line of
(a) 1 (b) 2 (c) 3 (d) 4
intersection of the plane P determined by
1
^ ^ ^ ^ 9. If A = (p, q, r) and B = (p′, q′, r′) are two points on
i +j and i −2j and plane P determined by
2
^ ^ ^ ^ the line lx = my = gz, such that OA = 3, OB = 4, then
vector 2i +j and 3i +2k, then angle between
pp′ + qq′ + rr′ is equal to
 ^ ^ ^
a and vector i −2j+2k is (a) 7 (b) 12
(a) p/4 (b) p/2 (c) 5 (d) none of these
(c) p/3 (d) none of these
cotA
10. In DABC, if b2 + c2 = 2a2, then value of
4. The differential equation of the system of circles cotB+cotC
touching the x-axis at origin is is
dy (a) 1/2 (b) 3/2 (c) 5/2 (d) 5/3
(a) (x2−y2) −2xy=0
dx 11. Solution of the differential the equation
(b) (x2−y2)dy +2xy=0 y(2x4+y)dy =(1−4xy2)x2 is given by
dx dx
dy (a) 3(x2y)2 + y3 – x3 = c
(c) (x2+y2) −2xy=0
dx y3 x3
(b) xy2+ − +c=0
dy
(d) (x2+y2) +2xy=0 3 3
dx 3 3 3
2 y x 4xy
5. The remainder on dividing 1234567 + 891011 by 12 (c) yx5+ = − +c
5 3 3 3
is
(d) none of these
* Alok Kumar is a winner of INDIAN NAtIoNAl MAtheMAtIcs olyMpIAD (INMo-91).
he trains IIt and olympiad aspirants.
10 mathematics today | MAY ‘17

12. The value of x which satisfies the equation (a) a3x3+3a2cx2+3acx2+3ac2x+b3=0
2tan−12x=sin−11+44xx2 is (b) a3x3−3a2cx2+3acx2+3ac2x−c3=0
1   1 (c) a3x3−3a2cx2−3ac2x+b3x+c3=0
(a) 2, ∞ (b) −∞, −2 (d) None of these
20. If a2 + b2 + c2 = 1 where a, b, c ∈ R, then the
 1 1
(c) [–1, 1] (d) − ,  maximum value of (4a – 3b)2 + (5b – 4c)2 +
 2 2
(3c – 5a)2 is
13. The number of solutions of the equation (a) 25 (b) 50
cos−11+x2−cos−1x= p+sin−1x is given by (c) 144 (d) none of these
 2x  2 ∞cosx p ∞1
21. If ∫ dx= , then ∫ (1−sin2x)3/2dx is
(a) 0 (b) 1 (c) 2 (d) 3 x 2 x
0 0
14. Let f(x) be a continuous and differentiable function equal to
and f(y) f(x + y) = f(x) " R. If f(5) = 3 and f ′(3) = 7, (a) p/2 (b) p/4
then the value of f ′(8) is (c) p/6 (d) 3p/2
(a) 0 (b) 1/7
22. A cubical die faces marked 1, 2, 3, ..., 6 is toaded
(c) 7/3 (d) 7
such that the probability of throwing the number t is
15. Consider 26 tangent lines to an ellipse. The lines proportional to t2. The probability that the number
separate the plane into several regions, some 5 has appeared given that when the die is rolled the
enclosed and others unbounded then numbers of number turned up is not even, is
unbounded regions are (a) 1/7 (b) 3/7
(a) 50 (b) 52 (c) 5/7 (d) 2/3
(c) 26C (d) none of these
2 23. There is a point inside an equilateral triangle ABC
16. A plane 2x + 3y + 5z = 1 has point P which is at of side d whose distance from the vertices is 3, 4, 5.
minimum distance from line joining A (1, 0, –3) Rotate the triangle and P through 60° about C. Let
and B (1, –5, 7), then distance AP is equal to A go to A′ and P to P′. The area of triangle PAP′ is
(a) 8 (b) 12
(a) 3 5 (b) 2 5
(c) 6 (d) none of these
(c) 4 5 (d) none of these
one or more than one option
17. For the curve x2y3 = c (where c is a constant) the correct tYpe
portion of the tangent between the axes is divided
24. If f(x) = 0 is a polynomial whose coefficients all ±1
in the ratio
and whose roots are all real, then the degree of f(x)
(a) 3 : 5 (b) 2 : 5
can be equal to
(c) 3 : 2 (d) 1 : 5
(a) 1 (b) 2
18. In a triangle OAB, E is the mid point of OB and D is (c) 3 (d) 4
a point on AB such that AD : DB = 2 : 1. If OD and
25. If cosecx+1dx=k fog(x)+c, where k is a real
OP
AE intersect at P, then ratio of is equal to constant, then
PD
(a) k=−2, f(x)=cot−1x,g(x)= cosecx−1
(a) 3 : 2 (b) 2 : 3
(c) 3 : 4 (d) 4 : 3 (b) k=−2, f(x)=tan−1x, g(x)= cosecx−1
19. If a1, a2 and a3 are the roots of the equation (c) k=2, f(x)=tan−1x, g(x)= cotx
ax3 + bx + c = 0, then the equation whose roots cosecx−1
a14 , a24 , a34 is (d) k=2, f(x)=cot−1x, g(x)= cotx
a2+a3 a3+a1 a2+a1 cosecx+1
12 mathematics today | MAY ‘17

26. A complex number z is rotated in anticlockwise 33. For the quadratic equation x2 + 2(a + 1)x + 9a – 5 = 0
direction by an angle a and we get z′ and if the which of the following are true?
same complex number z is rotated by an angle a in (a) If 2 < a < 5 then roots are of opposite sign
clockwise direction and we get z′′, then (b) If a < 0, then roots are of opposite sign
(a) z′, z, z′′ are in G.P. (c) If a > 7, then both roots are negative
(b) z′2+z′′2=2z2cos2a (d) If 2 ≤ a ≤ 5 roots are unreal
(c) z′ + z′′ = 2zcosa comprehenSion tYpe
(d) z′, z, z′′ are in H.P. Passage for Q. No. 34 to 36
27. f(x) is defined for x ≥ 0 and has a continuous In an argand plane z1, z2 and z3 are respectively the
derivative. It satisfies f(0) = 1, f ′(0) = 0 and vertices of an isosceles triangle ABC with AC = BC and
(1 + f(x)) f ′′(x) = 1 + x. The values f(1) can’t take ∠CAB = q. If z4 is incentre of triangle. Then
is (are) 2
AB AC
(a) 2 (b) 1.75 34. The value of    
 IA  AB
(c) 1.50 (d) 1.35
(z −z )(z −z ) (z −z )(z −z )
28. If z=sec−1x+1x+sec−1y+ 1y where xy < 0, (a) 2(z41−z11)2 3 (b) 2 (z41−z31) 1
(z −z )(z −z )
then the values of z which is (are) possible (c) 2 1 3 1 (d) none of these
(z −z )2
8p 7p 4 1
(a) (b)
10 10 35. The value (z – z )2 (1 + cosq)secq is
4 1
(c) 9p (d) none of these (a) (z – z )(z – z ) (b) (z2−z1)(z3−z1)
2 1 3 1
10 (z −z )(z −z ) (z4−z1)
(c) 2 1 3 1 (d) (z – z )(z – z )2
29. Let f(x)=[b2+(a−1)b+2]x−∫(sin2x+cos2x)dx (z −z )2 2 1 3 1
4 1
be an increasing function of x ∈ R and b ∈ R, then
q
a can take value (s) 36. The value of (z −z )2tanq⋅tan is
2 1
2
(a) 0 (b) 1
(a) (z + z – z )(z + z – 2z )
(c) 2 (d) 4 1 2 3 1 2 4
(b) (z + z – z )(z + z – z )
1 2 3 1 2 4
x−2 y−1 z−1
30. The line = = intersects the curve (c) –(z1 + z2 – z3)(z1 + z2 – 2z4)
3 2 −1 (d) none of these
x2 – y2 = a2, z = 0 if a is equal to
matriX match tYpe
(a) 4 (b) 5
37. Match the following :
(c) –4 (d) none of these
Column-I Column-II
31. p is the fundamental period of
(a) Number of solutions of the (1) 1
(a) |sinx| + |cosx|
equation
(b) cos(sinx) + cos(cosx)
(c) sin2x + cos2x sin−1x+cos−1x2= p is
(d) none of these 2
32. The locus of the point of intersection of the tangents (b) The number of ordered (2) 2
at the extremities of a chord of the circle x2 + y2 =b2 pairs (x, y) satisfying
which touches the circle x2 + y2 – 2by = 0 passes −1 −1
sin x sin y
through the point + =2 is
x y
 b
(a) 0, 2 (b) (0, b) (c) Number of solutions of (3) 0
b  the equation cos(cosx) =
(c) (b, 0) (d)  , 0
2  |sin (sinx)| is
14 mathematics today | MAY ‘17
mathematics today | MAY ‘17 15