MATHEMATICS - 2007 (PRELIMINARY) Time Allowed: 2 hours Maximum Marks: 300 1. Let X = {n : n is a positive integer, n≤ 50}. If A = {n E X : n is even} and B = {n EX: n is a multiple of 7}, then what is the number of elements in the smallest subset of X containing both A and B? (a) 28 (b) 29 (c) 32 (d) 35 2. Let A = {t €N: 12 and t are relatively prime} and B = {t €N: t ≤ 24}. What is the number of elements in A n B? (a) 10 (b) 8 (C)7 (d) 4 3. Let z = cos (n/8) + i sin (n/8) and A = {zn : n €N}. Which one of the following is correct? (a) A is not a finite set (b) A contains 12 non-real complex numbers (c) The number of elements in A is 16 (d) A contains no integers 4. Which one of the following is correct? The equation x3 - 266x2 - (266)2 x + (266)3 = 0 (a) has no multiple roots (b) has exactly one real root (c) has no non-real roots (d) has no integral roots 5. What is the sum of the roots of the equation {(x - 2)2 + 9} {(x - 3)2 + 4} = 0 (a) 5 (b) 10 (c) 13 (d) 18
6.
Let m be a positive integer, m ≥ 2. If α1' α2'......., α m are the roots of the equation xm - 1 = 0, then what is the equation whose roots are β1 = α2 +α3 +............+αm - (m - 1) α1 β2 =
αl + α3 +.......+ αm - (m - 1)α2
βj=αl +............+αi-I +αj+1 +...+αm-(m-l)αj βm =
7.
8.
9.
10.
α1 +........+ αm-1 - (m - 1) αm ?
(a) xm + mm = 0 (b) xm - (-m)m = 0 (c) xm + (m-l)m = 0 (d) xm - (m- I)m = 0 If α, β, γ are the roots of the equation x3 - px2 + qx - r = 0, then what is the value of ∑α2β ? (a) pq + 3r (b) pq + r (c) pq - 3r (d) q2/r Let G be an infinite cyclic group and H is its subgroup. Which one of the following is correct? (a) H is not necessarily cyclic (b) H is finite (c) H is infinite (d) H is not necessarily abelian Let G ≠ {e} be a group with no subgroup other than {e} and G. Then which one of the following is correct? (a) G is an infinite cyclic group (b) G is a finite cyclic group (c) G is an abelian non-cyclic group (d) G is neither abelian nor cyclic Which one of the following groups is cyclic? (a) Z12 x Z9 (b) ZIO x Z85 (c) Z4 x Z25 x Z6 (d) Z22 x Z21 x Z65
11.
Which one of the following is a group? (a) (N, *), where a * b = a for all a, b € N (b) (Z, *), where a * b = a - b for all a, b € Z (c) (Q, *), where a * b = ab/2 for all a, b €; Q (d) (R, *), where a * b = a + b + 1 for all a, b € R
12.
Consider the group (R * x R,)סּ, where R * = R \{0} and (a, b) ( סּc, d) = (ac, bc + d). What are the identity element and the a-1 of (a, b) respectively? (a) (1, 0) and (a-1 a-1, b a-1) (b) (0, 1) and (a a-1, b a-1) (c) (0, I) and (a a-1, - b a-1) (d) (1, 0) and (a a-1, -b a-1)
13.
Which' one of the following statements is correct? (a) Abelian groups may have non-abelian subgroups (b) Nonabelian groups may have abelian subgroups (c) Cyclic groups may have non-cyclic subgroups 14. (d) Non-cyclic groups cannot have cyclic subgroups Let σ = (1 3 5 7 11) (2 4 6) € S11 What is the smallest positive integer n such that σn = σ37 ? (a) 3 (b) 5 (c) 7 (d) 11 15. Let (R, +) be an abelian group. If multiplication '.' is defined on R by setting a . b = 0 for all a, b € R, then which one of the following statements is correct? (a) (R, +, .) is not a ring (b) (R, +, .) is a ring, but not commutative (c) (R, +, .) is a commutative ring, but has no unity (d) (R, +, .) is a field 16. Consider the following assertions: 1. The characteristic of the ring (Z, +, .) is zero. 2. For every composite number n, Z n' the ring of
.
residue classes modulo n, is a field. 3. Z 5' the ring of residue classes modulo 5, is an integral domain. 4. The ring of all complex numbers is a field. Which of the above assertions are correct? (a) 1, 3 and 4 (b) 1, 2 and 3 (c) 1, 2 and 4 (d) 2, 3 and 4 17.
Let F be a finite field with n elements. What is the possible
value ofn ? (a) I (c) 37 18.
.
(b) 36 (d) 125
If R is a fmite integral domain with n elements, then what is the number of invertible elements under multiplication in R ? (a) 1 (b) n (c) n - 1 (d) [n/2] where [.] is the bracket function
19. If Q, R, ( are respectively the fields of rational numbers, real
20.
numbers and complex numbers then which one of the following algebraic structures is not a vector space? (a) R over the field Q (b) R over the field R (c) Q over the field R (d) C over the field C Let x = (3, 2, -1), Y = (2, 4, 1), z = (4, 0, -3) and w = (10, 4, -5) be vectors in R3, a real vector space. Which one of the following is correct? (a) 2x + Z = w, y + Z = w (b) 2x - y = z, Y +. 2z = w (c) x + Z = w, 2x + Y = z (d) y + 2z = w, x - y = z
21.
If V is the real vector space of all mappings from R to R, V1 = {f € V1 | f(-x) = f(x)} and V2 = {f € V | f(-x) = -f(x)}, then which one of the following is correct? a) Neither V1 nor V2 is a subspace of V (b) V1 is a subspace of V, but V2 is not a subspace of V (c) V1 is not a subspace of V, but V2 is a subspace of V (d) Both V1 and V2 are subspaces of V Let F[x] be the ring of polynomials in one variable x over a 22. field F with the relation xn = 0, for a fixed n € N. What is the dimension of F[x] over F ? (a) 1 (b) n - 1 (c) n (d) Infinite Which one of the following is correct? 23. The set S = {a + ib, c + id} is a basis for the vector space
over R iff (a) ad - be = 0 (c) ad + be ≠ 0 C
24.
(b) ad + be = 0 (d) ad - be ≠0
Let V be the vector space of all 2 x 2 matrices over the field R of real numbers and B =
[1 2]. 1f
transformation defined by T(A) = AB
-
: V --> V is .linear BA, then what is the
dimension of the kernel of T ? (a) 1 (b) 2 (c) 3 (d) 4 25. What is the rank of the linear transformation T : R3 --> R3 defined by T(x, y, z) = (y, 0, z) ?
(a) 3 (c) 1
(b) 2 (d) 0
26. Consider the vector space Cover R and let T : CàC be a linear transfonnation given by T(z) = Z. Then :which one of the following is correct? (a) T is one-one, but not onto. (b) T is onto. but not one-one (c) T is one-one as well as onto. (d) T is neither one-one nor onto. 27. If T is a linear transformation ftom a real vector. space R2 to a real vector space R3 such that T(x, y) = (x - y, y - x, -x), then what is the nullity of T ?
(a) 0
.
(c) 2
(b) 1 (d) 3 [COSO
28. If n is a positive integer and A =
Sin.O] .
.
-sinO
then
cosO
what is. A equal to ?؛ n
29.
[ coS.O
-sin .0]
[ -cos.O
sin.O ]
(a) sin nO
cas nO
sin nO
cos nO
[ COS .0
sin nO]
[cosnO
sin .0 ]
-
-
(b)
(c) sin nO (d) cas nO sin nO cos nO If A and B are symmetric matrices of the same order, then which one of the following is not correct? (a) A + B is a symmetric matrix. (b) AB - BA is a symmetric matrix. (c) AB + BA. is a symmetric matrix. (d) A + AT and B + B T are symmetric: matrices.
30. If A = 3 -2 4 -2
satisfies the matrix equation A 2
- kA +
21 = 0, then [ what]is the value of k ? (a) 0 (b) I (c:) 2 (d) 3
1 b+c b2 +C2 31.
1 c+a C2 +a2 1 a+b a2 +b2
What is the value of the detenninant
?
(a) (a-b)(b-c)(c-a) (b) (a+b)(b+c)(c+a) (c) abc (d) a + b + c 32. Under which one of the following conditions does the
system of equations
.
[ unique solution? (a) For all a € R (c) For all a € Z
12 4
x
212
y
1 2 a-4
][
6
= ]
4
have a
[ z]
a
(b) a = 8 (d) a ≠ 8 33. Consider the equations 2x + 2y = 1 and 2x - y = lover Z 3. What is the solution (x, y) ? (a) (1, 1) but not (2, 0) (b)(2, 0) but not (1, 1) (c) Both (1, 1) and (2,0) (d) (1/2, 0) 34. Which one of the following is correct ? For different values of a and b, the straight line given by through (a) a conjugate point. (b) a fixed point. (c) the origin. (d) None of the above
x(a + 2b) + y(a - 3b) =
a - b passes
35. The line 3x + 2y = 24 meets the y-axis at A and the x-axis at S, and perpendicular bisector of AS meets the line through (0,
-1) parallel to the x-axis at C. What is the area of the triangle ABC ? (a) 91 square unit (b) 81 square unit (c) 61 square unit (d) 41 square unit 36. Consider the following statements :
.
S 1 : The equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 S2 S2 S1 S1 S2 S2 S2 S2 S1
represents a pair of straight lines.
S2: The equation ax2 + 2hxy + by2 = 0 always represents
a pair of straight lines passing through the origin. Which one of the following is correct? (a) If S1 is true, S2 is always true. (b) If S1 is not true, then S2 is also not true. (c) S2 is always true and S1 implies S2 if c = O. (d) Both S1 and S2 imply each other.
40.
What is the equation of the plane which bisects the line joining the points (3, -2, 1) and (1, 4, -3) at right angles? (a) x - 3y + 2z + 3 = 0 (b) 3x - 2y + Z + 3 = 0 (c) x + 4y - 3z + 2 = 0 (d) x - 3y + 2z + 2 = 0 What is the equation of the plane which passes through the zaxis and is perpendicular to the line x-I y+2 z-3 -=-=- ? cos9 sin 9 0 . (a) x + y tan 9= 0 (b) y + x tan 9 = 0 (c) x cos 9 y sin 9 = 0 (d) x sin 9 -
-
y cos e = 0
42. A straight lin; LOn the XY.plane bisects-the angie between ox and OY. What are the direction cosines of L ? (a) < 1/.,/2, 1/.,/2, 0 > (b) < 1/2, .fi/2, 0 > (c) < 0, 0, 1 >
(d) < 2/3, 2/3, 1/3 >
43. What is the equation of the cone with vertex at origin and passing through the circle x2 + y2 = 4, z = 2 ? (a) x2 + y2 + z2 = 4
(b) x2 + y2 - z2 = 0
(c) x2 + y2 - z2 = 2
(d) x2 + y2 + z2 = 2
~ 44. If a,~ ~b, c are non-zero vectors such that
~~~4~~
(a x b)x c = a x (b x c), then which one of the following is correct? ~~
(a) a and b are collinear ~~
(b) a and c are collinear
(c)
b and ~ are collinear
(d) None of the above 45. Consider the following two statements : S1 :
~~~
a, b, c are non-zero, non-coplanar vectors ~
S2 :
46.
~
~a=bxc ,= ~,c=
b
~~~
(a b c)
~ ' cxa ~~ axb ~~~
.
(a b c)
~
~ -
(a b c)
are non-coplanar Which one of the following is correct? (a) 81 implies 82 but 82 does not im~!y 8} (b) 8} does noUmply-S2 but 82 implies 81 (c) 8-1 imPlies 82 and 82 implies 81 (d) 81 does not imply 82 and 82 does not imply 8} What is the volume of the tetrahedron with vertices at (0,0,0), (I, I, I), (2, I, 1) and (1, 2, 1)? (a) 1/6 (b) Ii3 (c) 1/2 (d) 1 ~
47. If r satisfies the equation ~ AAAAA
r x (i + 2j + k) = i ~
- k," then for any scalar m, what
is r equal to ? (a)
i + m(i + 2j + k) (c)
(b) j + m (i + 2j + k)
k + m (i + 2j + k) (d) i -
k + m(i + 2j + k)
48. For the triangle OBC, one vertex 0 is the origin and the and C are b and
.
~
.position vectors of the other vertices B
(a) -J< (c) any real number
(b) 0 (d) any positive integer
What is the maximum value of
y =" sin3 x cos x, 0 < x < 1t ? c respectively and a, b, c are the lengths of the sides BC, OB and OC respectively. What is the position vector of the incentre of the triangle OBC ? b
c
bc 49.
-. -. (a) b + c C
-.b+c -. (c) c b +
cc
a+b+c
What is the range of the function
-. -. (b) b + a+b+c -. -. (d) b b + a+b+c
f(x) = log2 {(sin x - cos x + 3 .fi )/.fi} ? (a) [I, 2J (b) [0, I] (c) (1, 2) (d) (0, I) f I. (x+3sinx- x3 -k sinh . 50. I. 1m 2 x)3 .. eXIsts, then w at h IS t eh x~o I-cosx + X -3x value of k ? (a) -I (b) 2 (c) 3 (d) 4
51. Iff(x) =
{Sin(a + 2)x + sin x} / x x < 0 X= 0
b,
{ {(x + 3x2)1/3
- x1/3} / x4/3, x> 0
is continuous at x = 0, then what are the values of a and b respectively? (a)
- I, - I
-
(b) 1,-1
Match and the correct answer (c) 2, I List I with List - II(d) -2,select I using the code given below the lists: 52. Let f{x) = x"[xJ List,.. for realI x. f(x) is differentiable at the origin function x3 -one 6x2of- 36x +7 ifA. n isThe equal to which the following? increases when B. The function x3 - 6x2 - 36x + 7 is maximum at C. The function x3 - 6x2 - 36x + 7 is minimum at D. The function x3 - 6x2 - 36x + 7 decreases when ABC DAB C D (a) 4 2 1 3 (b) 3 1 2 4 (c) 3 2 I 4 ( d) 4 1 ~ 3 If 4a + 2b + c = 0 , then the equation 3ax2 + 2bx + c = 0 has at least one real root lying between which of the following? (a) 0 and 1 (b) 1 and 2 (c) 0 and 2 (d) None of the above Under which one of the folJowing conditions does the function f(x) = {(x2)m sin (x-2)n} x * 0, n > 0 and f(O) = 0
have a derivative at x = 0 ?
53.
(a) m ~ -1/2 (c) m > 1/2
(b) m> 0 (d) m ~ 1/2
j
If the tangent to the curve f(x) = x2 at any point (e, fee)) is
I
paralJel to the line joining the points (a, f(a)) and (b, f(b)) (a) (b) 3.,fj / 4 (d) 3. x < - 2 or on ' -3.,fj /16 - 6List 4. -2 54. 56. 57. 55.
(c) -3/16
3.,fj /16
x =6 -2 6 21.x=
What is the abscissa of the point at which the tangent to the curve y = eX is parallel to the chord joining the extremities of the curve in the interval [0, I] ? (a) 1/2 (b) in (lie) (c) in (e I) (d) lie What is the subnormal at x = 1t/2 on the curve y = x sin x? (a) I (b) 2/1t (c) 1t I 2 (d) 2 Which one of the following is correct? The inclined asymptotes of the curve x3 - xy2 - 2xy + 2x - y = 0 are themselves (a) perpendicular (b) curve, parallelthen which one of the following is correct? (a) a, c, the (c) inclined b are in A.P. at an angle 1t/3 (d) a, inclined angle 1t/4 (b) c, b areatinanG.P. Which one of the properties pertaining to the tangent at any (c) a, c, b are in H.P. point onbthe x2/3 +definite y2/3 =sequence a2/3 is correct? . (a) Sum of (d) a, c, do curve not follow its intercepts made with theof coordinate axeswhose sides pass What is the maximum area the rectangle is constant through the angular points of a given rectangle of sides 'a' and (b)? It encloses a triangle of constant area with the 'b' 58. (a) (acoordinate + b)2/2 axes (b) (a + b)2 (d) (c)(a2 Length of its portion intercepted between the + b2)/2 (a2 + b2) 62: (c) 59. 61. 60. ~
--=
x
T~e maximum value off(x), where f(x) =
(b) (d)
(a) 3..[j x)} dx
(c) ..[jl.,fi
J sin {x(l-
2..[j 3..[j 120
1 following points? occurs at whichaxes one isofconstant the coordinate t/2 the (d) passes through (a) xIt1 =1always 0 (b)origin x=I If x = xm(1x)n dx = xn(lx)P dx, what is p equal for (c) I least (d) of the above 63. What is- the absolute value ofNone thethen radius of curvature 5sinx + 3cosx dx What 0 0is the the curve y =volume In x ? of solid generated, when the area of the 64. to What the value of = I (in thesin x +quadrant) cosx ? is revolved ellipse ? is(x2/9) + (y2/4) first 0 (b) m about (a) 2n y-axis? (a) m 0 +n (b) mln n/212n (a) (c) 16n (d) (b) (c) 4n ( d) 2n by the curve 2y = (c) 8n is the area of the region (d) bounded 6n 66. 67. What 65. 68.
J
Jf
2 - 3x - 2x2 and the x-axis?
(a) 125/48 square unit (b) 4 square unit (c) 3 square unit (d) 125/24 square unit X3 sin x cosxl 69.
If f(x) = .6 p
-1 p2
0
, where p is a constant, then
p3 d3
what is the value of -r {f(x)} at x = 0 ? dx
(a) P (c) p + p3 70.
(b) p + p2 (d) Independent of p
What are the order and degree respectively of the differential equation of the family of curves y2 = 2c (x + ~), where c is an' arbitrary constant? (a) 1, 1 (b) 1, 2 (c) I, 3 (d) 2, 1 d2 d What is,they dx = 2dx
;
71.
solution of the differential equation + 2y = 0, with the given conditions y(O) = 0 and y'(O) = I ? (a) y = e-X cos x (b) y = e-X sin x (c) y = (cos x + sin x) e-X (d) y = sin x <
72.
What is the solution of the differential equation (1 + ex/y) dx + ex/Y (1- ~) dy = 0 ? (a) x + y eX/Y =c
73.
(b) y + x eX/Y = c
(c) x - y eX/Y = c
(d) None of the above
The singular solution of the differential equation y = px +
:
Consider the following statements in respect of the differential equation 2xy
= yl - xl.
1. The differential equation is a homogeneous equation 2. The curve represented by the differential equation is = ~ of ? circles a family 3. differential equation (a)The 9a (y + c)l = :J:2 x3/2 (b)of9aits(yorthogonal + c)l = :J:2trajectories xll3
(:Y
(c) 9a + c)3 = 4x2 by (d)eliminating 9a (y + c)2 p= between 4 x3 f(p) be obtained the equation y .will dy(y2xy =From pxIS+a-f(p) square one of the whose following diagonals equations? meet at 0, the =andlamina 2which 2 ABCD dx x-y triangle AOB is cut and the remaining part is hung up at D. In df dv df Which given above are.... correct? the xposition equilibrium, how much angle (a) + of=the 0 ofstatements (b) ................. = X does + DC make dp dp dp (a) 1 and 2 only (b) 1 and 3 only with the vertical? (c) 2 and 3 only (d) anddy 3(5/9) dy df (b) 1,2tan-l (a)(c) -tan-l(7/9) =P (d) - = p + What of curves dxare the orthogonal trajectories (d) 30°of the (c) 45° dx system dp .
74. 76. 75.
'I
77 .
D B
78.
(b) I / J2 (a) J2 I A pillar OD is to be pulled down by tying a rope of length 1= (c) fi I (d) I / fi AB to some point B of the pillar and then puIling the rope with a force F ~as shown in the above figure. F will have maximum A force F,Bhaving moment magnitude about 0 when of 10 dyne, OB equals is applied to which on the one of comer CAof a rectangular plateDABCD, 0as shown in the figure the following?
ck;
The weight of a triangular lamina ABC is 9 g. What is the additional weight to be placed at A so that the new centre of -+ divides the median through A in the ratio 3 : 4 ? gravity of F about A? (a) 2 g (b) 3 g (a) (b) 5 g (c) 420g (-2 + 3.fi) x 1O-7Nm (d) 20 + 3.fi) of x radii 10-5 6Nm Two(-2spheres em,(c)3 20 em are firmly united. The two spheres are solid and of the same material. What is the (2. + 3 .fi) x 10-7 Nm distanceIf of of =gravity the what wholeis body from the above. ABthe = 8centre em, AD 12 em,ofthen the moment (d) 20 (2 3.fi) x 10-5 Nm centre of +the larger sphere? (a)heavy 1 em spherical ball of weight (~) 2W cmis on a smooth inclined A (c) 3 em (d) 4 em plane (a. = angle of inclination of the plane to the horizontal).
80. 79. 81. 82.
(b) what P =through Wiscos (d)centre P =height A If force the angle of magnitude of friction Pis is A,applied then thea.the greatest of the at ball which in aorder particle to maintain can rest inside the ball a hollow at rest. sphere Whatofisradius the value a? W + cos2 a. ofP? (c) aP sin (a) = WAJI sin a. (b)~I+sin2a. a (1 - cos A)
Two points A and B have velocities ul and u2 as shown in the figure above. If AB = d, what is the angular velocity of A relative (c) a tan to A. B ? (d) a (1 - sin A.) (a) (ul cos al - u2 cos a2)/d 83. (b) (ul cos al + u2 COS a2)/d UI A particle mass is constrained to moveU2 in a smooth (c) (ul sin alof- unit u2 sin a2)/d circular radius (d) (ul sinpath al + of u2 sin a2)/da with constant speed. If now an additional radial force of magnitude P acts on thea particle, Two particles are projected vertically upwards from place at howinterval does the kinetic energy (E) of the particle change? an of 2 seconds. If the first and the second particle (a) = (.jH; + fii) (c) (b) .jH; = (JI-G + fii) attain the respective greatest heights (a) E changes by Pa/2 (b) E changes by HIPa and (c) E H2 (d) / H2 = 2is correct? ~HIH2 = 2g one of~HI changes by Pa/4 (d) Ewhich changes by 2the PaBfollowing 84. 85. simultaneously, A then
~
Ji
86.
A particle is executing simple harmonic motion and its displacement from its mean position is given by x = a cos (nt + k), where t denotes the time and a, Ii, k are positive constants. Under what condition will the speed of the particle be maximum? (a) t = (2p + 1) 1t 12n, p being an integer (b) t = (2p + 1) 1t/2n - (kin), p being an integer (c) t = (2p + 1) 1t/2n + (kin), p being an integer (d) t = 'p1t/n - (kin), p being an integer
88. 89.
A particle whose weight on the surface of the earth is W, falls JQ the surface of the earth from a height equal to the
diameter 2R of the earth. What is the work done by the earth's attraction? 2RWwith 1.44 MB capacity(b) 2RW/3 91. A (a) floppy can store the infonnation (c) 4RW/3 (d) 3RW/2 equivalent to which one of the following? (a) 1-44 x 26 bytes . xY (b) - yx1.44 x 210 bytes (c) 1.44 x 220 bytes (d) What is the value of hm1.44 x y x ?1024 bytes 90. x-+y X -y 92. Under what conditions of the inputs A and B, will the output in
93.
the gates beydifferent? I+In fory operations OR and XOR 1- In (a) I-Iny A = I, 8 = 0 (b) A = 0, 8 = I (a) (b) 1 + In y (c) A = 0, B = 0 (d) A = 1, B = 1 StepI+Iny I: get A, B -I-In y Comment: A (i, j) and B(i,(d) j) 1are-Inmyx nand n x p (c) I+In y matrices For i = 1 to m do for j = I to P do C(i,j) ~ 0 For k = 1 to n do Step 2 : C(i,j) ~ X
Step 3 : Comment:
Output C C = C(i, j) is the product matrix AB of the order m x p What is X in the above algorithm ? (a} C(i, j) + ,A(i, k) . B(k, j) 95. (b) Which called detector" ? C(i,one j) + isA(i, k) . "coincidence BU, k) (a) A(i, OR gate (b) NAND gate (c) k) + B(k, j) (c) NOT (d) C(i, j)gate' + A(i, j) . B(i, j) (d) AND gate Directions: 5 (Five) items consist of two statements 94. What The is thefollowing decimal equivalent of the hexadecimal number : one labelled FF? as the 'Assertion (A)' and the other as 'Reason (R)'. You are to (a) examine carefully 225 these two statements(b) 245 and select the answers to these items using the codes given below: (c) 255 (d) 256 (a) Both A and R are individually true and R is the correct explanation of A (b) Both A and R are individually true but R is not the correct explanation of A (c) A is true but R is false (d) A is false but R is true 96. Let n ~ 3, n be odd Assertion (A) : For any i = 1,2,.............., n - 1; if aI' a2'.............., an are the roots of the equation xn xi - 1 = 0, then (1 + at) (1 + a2) .... (1 + an) = 1
-
Reason (R)
97.
98.
99.
: If at,..........., an are the roots of the equation, xn - x-I = 0, then (1 + at) (l + a2) .... (1 + an) = 1. Assertion (A) : There is at least one cyclic group of order 100 which has only 5 subgroups. Reason (R) : A fmite cyclic group of order m has a unique subgroup of order n, where n is a divisor of m.
x
Assertion (A) : The function f(x) = 1 + Ixl is not differentiable at x = O. Reason (R) : I x I and hence (1 + I x I) is not differentiable at x = O. Assertion (A) : The function y = x2/4 is a singular solution
2 : The general . .12dy dy solution of the given equation is y = cx - c2 and the given solution cannot be Note: The publisher is not responsible any we have x xdx + mistake, y = value 100. Assertion (A): 005. X dx =of 2 dx co;.for dx obtained by assigning a definiteAll to c in tried our best to collect the correct data/answers. disputes 0the 0 .0 general Reasonto(R) : The integrand is anofeven function. solution. are subject the exclusive jurisdiction Delhi courts only.
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