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 a1' a2' , a m are the roots of the equation xm - 1 = 0, then what is the equation whose roots are ß1 = a2 +a3 + +am - (m - 1) a1 ß2 = al + a3 + + am - (m - 1)a2 ßj=al + +ai-I +aj+1 +...+am-(m-l)aj 10. ßm = a1 + + am-1 - (m - 1) am ? (a) xm + mm = 0 (b) xm - (-m)m = 0 (c) xm + (m-l)m = 0 (d) xm - (m- I)m = 0 7.If a, ß, ? are the roots of the equation x3 - px2 + qx - r = 0, then what is the value of ?a2ß ? (a) pq + 3r (b) pq + r (c) pq - 3r (d) q2/r 8.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 9.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 10.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,?), where R * = R {0} and (a, b) ? (c, 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) Non-abelian groups may have abelian subgroups (c) Cyclic groups may have non-cyclic subgroups (d) Non-cyclic groups cannot have cyclic subgroups 14.Let s = (1 3 5 7 11) (2 4 6) € S11 What is the smallest positive integer n such that sn = s37 ? (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. 17.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 18.Let F be a finite field with n elements. What is the possible value ofn ? . (a) I (b) 36 (c) 37 (d)125 19.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 20.If Q, R, ( are respectively the fields of rational numbers, real 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 21.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 22.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 field F with the relation xn = 0, for a fixed n € N. 23.What is the dimension of F[x] over F ? (a) 1 (b) n - 1 (c) n (d) Infinite 24.Which one of the following is correct? The set S = {a + ib, c + id} is a basis for the vector space C over R iff (a) ad - be = 0 (b) ad + be = 0 (c) ad + be ? 0 (d) ad - be ?0 25.Let V be the vector space of all 2 x 2 matrices over the field R of real numbers and B = [1 2]. 1f : V --> V is .linear transformation defined by T(A) = AB - BA, then what is the dimension of the kernel of T ? (a) 1 (b) 2 (c) 3 (d) 4 26.What is the rank of the linear transformation T : R3 --> R3 defined by T(x, y, z) = (y, 0, z) ? (a) 3 (b) 2 (c) 1 (d) 0 Consider the vector space Cover R and let T : C?C 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 (b)1 (c) 2 (d) 3 [COSO Sin.O] 28. If n is a positive integer and A = . . then -sinO cosO what is. An equal to ?? [ coS.O -sin .0] [ -cos.O sin.O ] (a) sin nO cas nO (b) sin nO cos nO [ COS .0 sin nO] [cosnO sin .0 ] (c) sin nO - cas nO (d) - sin nO cos nO 29.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.
3 -2 ] 1 b+c b2 +C2 1 c+a C2 +a2 30. What is the value of the detenninant ? 1 a+b a2 +b2 (a) (a-b)(b-c)(c-a) (b) (a+b)(b+c)(c+a) (c) abc (d) a + b + c 31. Under which one of the following conditions does the system of equations [
1 2 4 ][ x ] [ 6 ] 2 1 2 y = 4 1 2 a-4 z a have a unique solution? (a) For all a € R (b) a = 8 (c) For all a € Z (d) a ? 8 32 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) 33. Which one of the following is correct ? For different values of a and b, the straight line given by 30. If A = satisfies the matrix equation A 2 - kA + 4 -2 21 = 0, then what is the value of k ? (a) 0 (b) I (c:) 2 (d) 3 34.x(a + 2b) + y(a - 3b) = a - b passes through (a) a conjugate point. (b) a fixed point. (c) the origin. (d) None of the above
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. 37.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 38.What is the equation of the plane which passes through the z-axis 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 39.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 > 40.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 41.~ ~ ~ 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 42. Consider the following two statements : S1 : ~ ~ ~ a, b, c are non-zero, non-coplanar vectors S2 : ~ ~ ~ ~ ~ ~ ~ bxc ~ b ' cxa ~ axb a= ,= ,c= ~~~ ~~~ - ¬ (a b c) (a b c) (a b c) are non-coplanar. 43.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 ~ 44.If r satisfies the equation
~ A A A A A r x (i + 2j + k) = i - k," then for any scalar m, what ~ is r equal to ? (a) i + m(i + 2j + k) (c) k + m (i + 2j + k) (b) j + m (i + 2j + k)(d) i - k + m(i + 2j + k) 45.For the triangle OBC, one vertex 0 is the origin and the . ~ .position vectors of the other vertices B and C are b and 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 -. -. (a) b + c c b+c -. -. (b) b + C a+b+c -. -. (c) c b + b c a+b+c -. -. (d) b b + c c a+b+c 46.What is the range of the function f(x) = log2 {(sin x - cos x + 3 .fi )/.fi} ? (a) [I, 2J (b) [0, I] (c) (1, 2) (d) (0, I) f I 47. (x+3sinx- x3 -k sinh x) . h h . h 48. I. 1m 2 3 . eXIsts, t en w at IS t e x~o I-cosx + X -3x value of k ? (a) -I (c) 3 (b) 2 (d) 4 { {Sin(a + 2)x + sin x} / x x < 0 49. Iff(x) = b, X = 0 {(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 (c) 2, I (d) -2, I 50. Let f{x) = x"[xJ for real x. f(x) is differentiable at the origin if n is equal to which one of the following? (a) -J< (b) 0 (c) any real number (d) any positive integer What is the maximum value of y =" sin3 x cos x, 0 < x < 1t ?
51Match List - I with List - II and select the correct answer using the code given below the lists: List,.. I A. The function x3 - 6x2 - 36x + 7 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 52.ABC DAB C D (a) 4 2 1 3 (b) 3 1 2 4 (c) 3 2 I 4 ( d) 4 1 ~ 3 53.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 54.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 ? (a) m ~ -1/2 (b) m> 0 (c) m > 1/2 (d) m ~ 1/2 j 55.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)) on ' (a) -3.,fj /16 (c) -3/16 (b) 3.,fj / 4 (d) 3.,fj /16 List - II 1. x = -2 2 x= 6 3. x < - 2 or x>6 4. -2 < x < 6 56.the curve, then which one of the following is correct? (a) a, c, b are in A.P. (b) a, c, b are in G.P. (c) a, c, b are in H.P. (d) a, c, b do not follow definite sequence 57.What is the maximum area of the rectangle whose sides pass through the angular points of a given rectangle of sides 'a' and 'b' ? (a) (a + b)2/2 (c) (a2 + b2)/2 (b) (a + b)2 (d) (a2 + b2) 58.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 59.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 60.Which one of the following is correct? The inclined asymptotes of the curve x3 - xy2 - 2xy + 2x - y = 0 are themselves (a) perpendicular (b) parallel (c) inclined at an angle 1t/3 (d) inclined at an angle 1t/4 61.Which one of the properties pertaining to the tangent at any point on the curve x2/3 + y2/3 = a2/3 is correct? . (a) Sum of its intercepts made with the coordinate axes is constant (b) It encloses a triangle of constant area with the coordinate axes (c) Length of its portion intercepted between the --=coordinate axes is constant (d) It always passes through the origin 62.What is the least absolute value of the radius of curvature for the curve y = In x ? (a) 3..[j (b) 2..[j (c) ..[jl.,fi (d) 3..[j 12 1t/2 f 5sinx + 3cosx dx 63.. What is the value of sin x + cosx ? 0 (a) 0 (b) n/2 (c) 4n ( d) 2n 64. x T~e maximum value off(x), where f(x) = J sin {x(l- x)} dx 0 occurs at which one of the following points? (a) x = 0 (b) x = I (c) x = - I (d) None of the above 65.What is the volume of solid generated, when the area of the ellipse (x2/9) + (y2/4) = I (in the first quadrant) is revolved about y-axis? (a) 16n (b) 12n (c) 8n (d) 6n 66.If J xm(1- x)n dx = J xn(l- x)P dx, then what is p equal 0 0 to ? (a) 2n (b) m (c) m + n (d) mln 67.What is the area of the region bounded by the curve 2y =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 68.X3 sin x cosxl If f(x) = .6 p -1 p2 0 p3 , where p is a constant, then 70. d3 what is the value of -r {f(x)} at x = 0 ? dx (a) P (b) p + p2 (c) p + p3 (d) Independent of p 69.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 70.What is,the solution of the differential equationd; = 2dy < x x+ 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 71.What is the solution of the differential equation (1 + ex/y) dx + ex/Y (1- ~) dy = 0 ? (a) x + y eX/Y =c (b) y + x eX/Y = c (c) x - y eX/Y = c (d) None of the above 72.The singular solution of the differential equation y = px +f(p) will be obtained by eliminating p between the equation y = px + f(p) and which one of the following equations? df dv df (a) x + = 0 (b) = X + ¬ dp dp dp dy . (c) - = P dx dy df (d) - = p + ¬ dx dp 73.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 a family of circles 3. The differential equation of its orthogonal trajectories . dy 2xy IS - = 2 2 dx x-y 74.Which of the statements given above are correct? (a) 1 and 2 only (b) 1 and 3 only (c) 2 and 3 only (d) 1,2 and 3 75.What are the orthogonal trajectories of the system of curves 'I (:Y = ~ ? (a) 9a (y + c)l = :J:2 x3/2 (b) 9a (y + c)l = :J:2 xll3 (c) 9a (y + c)3 = 4x2 (d) 9a (y + c)2 = 4 x3 76.From a square lamina ABCD whose diagonals meet at 0, the triangle AOB is cut and the remaining part is hung up at D. In the position of equilibrium, how much angle does DC make with the vertical? (a) tan-l(7/9) (c) 45° (b) tan-l (5/9) (d) 30° 77.A pillar OD is to be pulled down by tying a rope of length 1= 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 moment about 0 when OB equals to which one of the following? (a) J2 I (b) I / J2 (c) fi I (d) I / fi 78.A force F, having magnitude of 10 dyne, is applied on the comer C of a rectangular plate ABCD, as shown in the figure above. If AB = 8 em, AD = 12 em, then what is the moment -+ of F about A ? (a) 20 (-2 + 3.fi) x 1O-7Nm (b) 20 (-2 + 3.fi) x 10-5 Nm (c) 20 (2. + 3 .fi) x 10-7 Nm (d) 20 (2 + 3.fi) x 10-5 Nm 79.A heavy spherical ball of weight W is on a smooth inclined plane (a. = angle of inclination of the plane to the horizontal). A force of magnitude P is applied through the centre of the ball in order to maintain the ball at rest. What is the value ofP? (a) P = W JI + cos2 a. (c) P = W sin a. (b) P = W cos a. (d) P = W ~I+sin2a. 80.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 gravity divides the median through A in the ratio 3 : 4 ? (a) 2 g (b) 3 g (c) 4 g (d) 5 g 81.Two spheres of radii 6 em, 3 em are firmly united. The two spheres are solid and of the same material. What is the distance of the centre of gravity of the whole body from the centre of the larger sphere? (a) 1 em (~) 2 cm (c) 3 em (d) 4 em 82.If the angle of friction is A, then what is the greatest height at which a particle can rest inside a hollow sphere of radius a? (a) a sin A (b) a (1 - cos A) (c) a tan A. (d) a (1 - sin A.) 83.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 to B ? (a) (ul cos al - u2 cos a2)/d (b) (ul cos al + u2 COS a2)/d (c) (ul sin al - u2 sin a2)/d (d) (ul sin al + u2 sin a2)/d 84.Two particles are projected vertically upwards from a place at an interval of 2 seconds. If the first and the second particle attain the respective greatest heights HI and H2 simultaneously, then which one of the following is correct? (a) ~ = (.jH; + fii) (c) ~HIH2 = 2g (b) .jH; = (JI-G + fii) (d) ~HI / H2 = 2 85.A particle of unit mass is constrained to move in a smooth circular path of radius a with constant speed. If now an additional radial force of magnitude P acts on the particle, how does the kinetic energy (E) of the particle change? (a) E changes by Pa/2 (b) E changes by Ji Pa (c) E changes by Pa/4 (d) E changes by 2 Pa 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 87.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? (a) 2RW (c) 4RW/3 (b) 2RW/3 (d) 3RW/2 88. . xY - yx What is the value of hm x y x-+y X -y ? I+In y (a) I-Iny(b) 1 + In y -I-In y I+Iny (c) I+In y 1- In y (d) 1 -In y 89.. A floppy with 1.44 MB capacity can store the infonnation equivalent to which one of the following? (a) 1-44 x 26 bytes (b) 1.44 x 210 bytes (c) 1.44 x 220 bytes (d) 1.44 x 1024 bytes 90. Under what conditions of the inputs A and B, will the output in the gates for operations OR and XOR be different? (a) A = I, 8 = 0 (b) A = 0, 8 = I (c) A = 0, B = 0 (d) A = 1, B = 1 91. Step I : get A, B Comment: A (i, j) and B(i, j) are m x nand n x p matrices For i = 1 to m do for j = I to P do C(i,j) ~ 0 For k = 1 to n do C(i,j) ~ X Step 2 : Step 3 : Comment: 92.Output C C = C(i, j) is the product matrixAB of the order m x p What is X in the above algorithm ? (a} C(i, j) + ,A(i, k) . B(k, j) (b) C(i, j) + A(i, k) . BU, k) (c) A(i, k) + B(k, j) (d) C(i, j) + A(i, j) . B(i, j) 93.What is the decimal equivalent of the hexadecimal number FF? (a) 225 (c) 255 (b) 245 (d) 256 94. Which one is called "coincidence detector" ? (a) OR gate (b) NAND gate (c) NOT gate' (d) AND gate Directions: The following 5 (Five) items consist of two statements : one labelled as the 'Assertion (A)' and the other as 'Reason (R)'. You are to examine these two statements carefully and select the answers to these items using the codes given below: (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 95.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 96.Reason (R) : 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. 97.xAssertion (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 dy dy of dx - x dx + y = 0 . Reason (R) 98The general solution of the given equation is y = cx - c2 and the given solution cannot be obtained by assigning a definite value to c in the general solution. . .12 99. Assertion (A): I 005. X dx = 2 I co;. x dx 0 0 Reason (R) : The integrand is an even function. |