s special centum coaching team class: xii centum … 12, 2018 · question paper 2017-2018 class:...

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Padasalai.Net’s Special Centum Coaching Team

Question Paper 2017-2018

CLASS: XII CENTUM SPECIAL TEST TIME: 3 Hrs

SUBJECT: MATHEMATICS MARKS: 200

I. Choose the correct answer: - 40×1=40

1. If A = [2 0 1], then rank of AAT is a) 1 b) 2 c) 3 d) 0

2. If A and B are any two matrices such that AB = O and A is non-singular, then

a) B = O b) B is singular c) B is non-singular d) B = A

3. In a system of 3 linear non-homogeneous equation with three unknowns, if

= 0 and x = 0, y 0 and z = 0 then the system has

a) unique solution b) two solutions c) infinitely many solutions d) no

solution

4. (AT)

-1 is equal to a) A

-1 b) A

T c) A d) (A

-1)

T

5. The vectors kji 432 and kcjbia are perpendicular when

a) a = 2, b = 3, c = -4 b) a = 4, b = 4, c = 5

c) a = 4, b = 4, c = -5 d) a = -2, b = 3, c = 4

6. If qp, and qp are vector of magnitude then the magnitude of qp is

a) 2 b) 3 c) 2 d) 1

7. If accbba ,, = 8 then cba ,, a) 4 b) 16 c) 32 d) -4

8. The following two lines are 11

1

2

1 zyx

and

2

1

5

1

3

2

zyx

a) parallel b) intersecting c) skew d) perpendicular

9. The centre and radius of the sphere given by x2 + y

2 + z

2 – 6x + 8y–10z+1 = 0

is

a) (-3, 4, -5),49 b) (-6, 8, -10),1 c) (3, -4, 5),7 d) (6, -8, 10),7

10. The angle between the vectors ji

and kj

is

a) 3

b)

3

2 c)

3

d)

3

2

11. If the amplitude of a complex number is /2 then the number is

a) purely imaginary b) purely real c) 0 d) neither real nor imaginary

12.

i

i

e

e

1

1= a) cos + isin b) cos - isin c) sin + icos d)sin + icos

13. If zn = cos3

n+ isin

3

nthen z1, z1…z6 is a) 1 b) -1 c) I d) -i

14. The complex conjugate of –4 – i9 is

a) –4 + i9 b) 4 + i9 c) 4 – i9 d) –4 – i9

15. The axis of the parabola y2 – 2y + 8x – 23 = 0 is

a) y = -1 b) x = -3 c) x = 3 d) y = 1

16. The angle between the two tangents drawn from the point (-4,4) to

y2 =16x is a) 45 b) 30 c) 60 d) 90

17. The line 5x – 2y + 4k = 0 is a tangent to 4x2 – y

2 = 36 then k is

a) 9

4 b)

3

2 c)

4

9 d)

16

81

18. The equation of the directrix of the parabola y2 = –8x is

a) y + 2 = 0 b) x – 2 = 0 c) y – 2 = 0 d) x + 2 = 0

19. The gradient of the curve y= -2x3 + 3x + 5 at x= 2 is

a) -20 b) 27 c) - 16 d) - 21

20. A spherical snowball is melting in such a way that its volume is decreasing

at a rate of 1cm3/ min. The rate at which the diameter is decreasing when

the diameter is 10 cms is

a) 50

1cm/min b)

50

1 cm/min c) min/

75

11cm

d) min/

75

2cm

21. The equation of the tangent to the curve y= 5

3x at the point (-1,

5

1) is

a) 5y + 3x = 2 b) 5y - 3x = 2 c) 3x – 5y = 2 d) 3x + 3y =2

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22. The angular displacement of a fly wheel in radians is given by = 9t2 – 2t

3.

The time when the angular acceleration zero is

a) 2.5s b) 3.5s c) 1.5s d) 4.5s

23. If x = r cos , y = r sin then x

r

is equal to

a) sec b) sin c) cos d) cosec

24. The differential of y if y = x5 is, a) 5x

4 b) 5x

4dx c) 5x

5dx d) 5x

5

25. The value of

0

32 cossin dxxx is a) b) 2

c)

4

d) 0

26. The area bounded by the line y = x, the x – axis, the ordinates x = 1,

x = 2 is a) 2

3 b)

2

5 c)

2

1 d)

2

7

27. The volume of the solid obtained by revolving 16

y

9

22

x

= 1 about the

minor axis is a) 48 b) 64 c) 32 d) 128

28. b

adxxf )(

a) – b

adxxf )( b) –

a

bdxxf )( c) –

a

dxxf0

)( d) –2 b

dxxf0

)(

29. The differential equation of all circles with centre at the origin is

a) x dy + y dx = 0 b) x dy – y dx = 0

c) x dx + y dy = 0 d) x dx – y dy = 0

30. The complementary function of (D2 + 1) y = e

2x is

a) (Ax + B)ex

b) A cos x + B sin x c) (Ax + B)e2x

d) (Ax + B)e-x

31. A particular integral of (D2 – 4D + 4) = e

2x is

a) xe

x 22

2 b) xe

2x c) xe

-2x d)

xex 2

2

32. The order and degree of the differential equation are 3

2

3 )y(y y

a) 2, 3 b) 3, 3 c) 3, 2 d) 2, 2

33. The number of rows in the truth table of ~[p (~q)] is

a) 2 b) 4 c) 6 d) 8

34. The conditional statement p q is equivalent to

a) p q b) p ~ q c) ~ p q d) p q

35. Which of the following is a tautology?

a) p q b) p q c) p ~ p d) p ~ p

36. In the group (Z5 – {[0]},.5), 0([2]) is a) 5 b) 3 c) 4 d) 2

37. Given E (X +c) = 8 and E (X – c) = 12 then the value of c is

a) -2 b) 4 c) -4 d) 2

38. Var (4X + 3) is a) 7 b) 16 Var (X) c) 19 d) 0

39. In 5 throws of a die, getting 1 or 2 is a success. The mean number of

successes is a) 3

5 b)

5

3 c)

9

5 d)

5

9

40. A discrete random variable X has probability mass function p(x), then

a) 0 p(x) 1 b) p(x) 1 c) p(x) 1 d) 0< p(x)< 1

II. Answer any ten questions: - 10×6=60

41. 2x – 3y + 7z = 5, 3x + y – 3z = 13, 2x + 19y – 47z = 32 check consistency.

42. State and prove reversal law for inverses of matrices.

43. If , , are the position vectors of the vertices A, B, C of a triangle ABC,

then prove that the area of triangle ABC is

| × × × × × | Deduce

the condition for points , to be collinear.

44. (a) Find the area of parallelogram ABCD whose vertices are A(-5, 2, 5),

B(-3, 6, 7), C(4, -1, 5) and D(2, -5, 3)

(b) If the edges = = =

meet at a vertex, find the volume of the parallelepiped.

45. Prove that the complex numbers 3+3i, -3-3i, -3 + i are the

vertices of an equilateral triangle in the complex plane.

46. Prove that the product of perpendiculars from any point on the hyperbola

-

=1to its asymptotes is constant and the value is

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47. At 2.00 p. m a car’s speedometer reads 30 miles / hr., at 2.10 pm it reads

50 miles / hr. show that sometime between 2.00 and 2.10 the acceleration

is exactly 120 miles / hr2.

48. (a) Prove that identity element of a group is unique.

(b) show that (a-1

)-1

= a for all a , a group.

49. Locate the extreme point on the curve y = 3x2 – 6x and determine its nature

by examining the sign of the gradient on either side.

50. Find the volume of the solid that results when the ellipse

= 1

(a>b>0) is revolved about the minor axis.

51. Find the differential equation that will represent the family of all circles

having centres on the x – axis and the radius is unity.

52. Show that p q ((~ p) q)

53. A pair of dice is thrown 10 times. If getting a doublet is considered a

success find the probability of (i) 4 success (ii) no success.

54. Suppose that the amount of cosmic radiation to which a person is exposed

when flying by jet across the United States is a random variable having a

normal distribution with a mean of 4.35 m rem and a standard deviation of

0.59 m ram. What is the probability that a person will be exposed to more

than 5.20 m rem of cosmic radiation of such a flight.

55. (a) Show that for any polynomial equation p(x) = 0 with real coefficients,

imaginary roots occur in conjugate pairs.

(b) Show that the percentage error in the nth

root of a number

approximately

times the percentage error in the number

III. Answer any ten questions: - 10×10=100

56. x + y - z =1, 2x + 2y – 2z = 2, -3x – 3y + 3c = -3 examine the consistency

of the system of equations. If it is consistent then solve the same.

57. (a) With usual notations prove (i) Cos A =

(b) Find the equation of the plane passing through the intersection of the

planes 2x-8y+4z=3 and 3x-5y+4z+10=0 and perpendicular to the plane

3x-y-2z-4=0

58. Find the vector and Cartesian equation of the plane

=

=

and

perpendicular to the plane

59. Find the equation of the director circle of the hyperbola if:

i) The centre of the hyperbola is same as the centre of the ellipse

ii) The length of the latus rectum is

and the eccentricity is

iii) the equation of the conjugate axis is x = 1

60. Find the eccentricity, centre, foci and vertices of the hyperbola

9x2-16y

2-18x-64y-199=0 and also trace the curve.

61. Find the equation of the rectangular hyperbola which has for one of its

asymptotes the line x+2y-5=0 and passes through the points (6,0) and

(-3,0)

62. Find the equations of those tangents to the circle x2+y

2=52, which are

parallel to the straight line 2x+3y=6.

63. Show that of all the rectangles with a given perimeter the one with the

greatest area is a square.

64. Find the area of the region enclosed by y2 = x and y = x – 2

65. Find the surface area of the solid generated by revolving they arc of the

parabola y2=4ax, bounded by its lattes rectum about x-axis.

66. Find the cubic polynomial in x which attains its maximum value 4 and

minimum value 0 at x = -1 and 1 respectively.

67. Solve : (1+ex/y

)dx + ex/y

(1-x/y)dy =0 given that y = 1, where x= 0

68. If the number of incoming buses per minute at a bus terminus is a random

variable having a passion distribution with λ = 0.9, find the probability that

there will be i) Exactly 9 incoming buses during a period of 5 minutes

ii) Fewer than 10 incoming buses during a period of 8 minutes.

iii) Atleast 14 incoming buses during a period of 11 minutes.

69. Show that the nth roots of unity form an Abelian group of finite order with

usual multiplication.

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70. (a) If w = u2e

v where u =

and v = y log x, find

and

(b) If a and b are the roots of x2+2 +4=0 then find the value of a

n + b

n

also deduce the value of a12

+b12

(n-is an integer)

Prepared by Mr.R.RAJASEKARAN, M.SC.,B.ED.,M.PHIL.

DAYANANDA VIDYALAYA MAT HR SEC SCHOOL

KURUSILAPATTU

VELLORE DT

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jahde;j tpj;ahyah nkl;upf; Nky;epiyg; gs;sp FUrpyhg;gl;L – NtY}H khtl;lk; – 635 702

tFg;G : XII CENTUM SPECIAL TEST Neuk; : 3 kzp ghlk; : fzpjk; kjpg;ngz;fs;: 200

I. Choose the correct answer: - 40×1=40

1. + - vd;w ntf;lu; xU %iytpl;lkhfTk;> -3 +4 vd;w ntf;liu xU gf;fkhfTk; nfhz;l ,izfuj;jpd; gug;G

a. 10 b. 6 c.

d. 3

2. vd;gd a, b, c Mfpatw;iw kl;Lf;fshff; nfhz;l tyf;if mikg;gpy; xd;Wf;nfhd;W nrq;Fj;jhd ntf;lu;fs;

vdpy; [ ] d; kjpg;G

a. a2 b

2 c

2 b. 0 c.

d. abc

3. = + vd;w rkd;ghL Fwpg;gJ

a. kw;Wk; Gs;spfis ,izf;Fk; Neu;NfhL b. xoy jsk; c. yoz jsk; d. zox jsk;

4. - kw;Wk; - vd;w ntf;lu;fSf;F ,ilg;gl;l Nfhzk;

a.

b.-

c.-

d.

5. MjpapypUe;J ( +4 +12 )= 26 vd;w jsj;jpw;F tiuag;gl;l nrq;Fj;jpd; ePsk;

a. 2 b.

c. 26 d.

6. (-4> 4) vd;w Gs;spapypUe;J y2=16xf;F tiuag;gLk; ,U

njhLNfhLfSf;F ,ilNaAs;s Nfhzk; a. 45

o b. 30

o c.60

o d.90

o

7. 9x2+16y

2=144 vd;w $k;G tistpd; ,af;F tl;lj;jpd; Muk;

a. b. 4 c. 3 d. 5

8. xy = c2vd;w nrt;tf mjpgutisaj;jpy; ‘t1’ vd;w

Gs;spaplj;Jr; nrq;NfhL kPz;Lk; mt;tistiuia ‘t2’–y; re;jpf;fpwJ vdpy; t1

3 t2 a. 1 b. 0 c. -1 d. -2

9. y = 3x2+3sin x vd;w tistiuf;F x = 0 tpy; njhLNfhl;bd;

rha;T a. 3 b. 2 c. 1 d. -1

10. x3-2x

2+3x+8 vd;w mjpfupf;Fk; tPjkhdJ x mjpfupf;Fk;

tPjj;ijg; Nghy; ,Uklq;F vdpy; x-d; kjpg;Gfs;

a. (-

, -3) b. (

, -3) c. (-

, 3) d. (

, 1)

11. NfhLfs; y = x, y = 1 kw;Wk; x = 0Mfpait Vw;gLj;Jk; gug;ig mr;irg; nghWj;Jr; Row;wg;gLk; NghJ cUthFk; jplg;nghUspd; fd msT

a.

b.

c.

d.

12.

+

vd;w tistiuapd; tpy;ypd; ePsk; a. 48 b. 24 c. 12 d. 96

13.

dx

a.

b.

c.

d.

14.

+5y1/3

= x vd;gjpd; tupir> gb KiwNa

a. 2, 1 b. 1, 2 c. 1, 6 d. 1, 3

15.

– y tan x =cos x d; njhiff; fhuzp

a. sec x b. cos x c. etan x

d. cot x

16. (G, .) vd;w Fyj;jpy; G = {1, -1, i, -i} vdpy; -1 ,d; tupir

a. -1 b. 1 c. 2 d. 0

17. f(x) =

kw;nwq;fpYk;

vd;gJ epfo;jfT mlu;j;jp rhu;G vdpy; k d; kjpg;G

a.

b.

c.

d.

18. var (4X+3) vd;gJ

a. 7 b. 16 var (X) c. 19 d. 0

19. xU rktha;g;G khwp X gha;rhd; gutiyg; gpd;gw;WfpwJ> NkYk; E (X

2) = 30

a. 6 b. 5 c. 30 d. 25

20. ,ay;epiyg; gutiyg; nghWj;J fPo;f;fz;ltw;wpy; vJ / vit rup? I. X = (ruhrup) vd;w Nfhl;bw;F rkr;rPuhdJ

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II. xU Kfl;Lg; guty; III. ruhrup = ,ilepiy msT = KfL IV. X = ruhrup a. I, II kl;Lk; b. II, IV kl;Lk; c. I, II, III kl;Lk; d. ,it midj;Jk;;

21. A vd;w xU jpirapyp mzpapd; tupir 3> jpirapyp k 0 vdpy; A

-1vd;gJ

a.

I b.

I c.

I d. kI

22. aex + be

y = c, pe

x + qe

y = d kw;Wk; =

, =

, =

vdpy; (x, y) ,d; kjpg;G

a.

b.

c.

d.

23.

vd;w mzpapd; juk; 2 vdpy; tpd; kjpg;G

a. 1 b. 2 c. 3 d. VNjDk; xU nka;naz; 24. rkgbj;jhd Neupar; rkd;ghl;Lj; njhFg;ghdJ

a. vg;nghOJk; xUq;fikT cilajhFk; b. ntspg;gilj; jPu;T kl;LNk nfhz;Ls;sJ c. vz;zpf;ifaw;w jPu;Tfs; nfhz;Ls;sJ d. xUq;fikT cilajhf ,Uf;fj; Njitapy;iy

25. ; ( ) + ) + ( ), vdpy;

a. xU xuyF ntf;lu; b. = c. d. = + +

26.

vd;w fyg;ngz;zpd; kl;L> tPr;R KiwNa

a. e9,

b. e

9,-

c. e

6,-

d. e

9,-

27. a = 3 +i kw;wk; z = 2 – 3i vdpy; az, 3az kw;Wk; -az vd;gd Xu; Mu;fd; jsj;jpy; a. nrq;Nfhz Kf;Nfhzj;jpd; Kidg;Gs;spfs;

b. rkgf;f Kf;Nfhzj;jpd; Kidg;Gs;spfs; c. ,Urkgf;f Kf;Nfhzj;jpd; Kidg;Gs;spfs; d. xNu Nfhliktd

28. Zn = cos

+ I sin

vdpy; Z1. Z2 …… Z6 vd;gJ

a. 1 b. -1 c. I d. -i

29. | Z – Z1| = |Z – Z2| vdpy; fyg;ngz; Z ,d; epakg;ghij

a. Mjpia ikakhff; nfhz;l xU tl;lk; b. Z1 I ikakhff; nfhz;l xU tl;lk; c. Mjp topr;nry;Yk; Neu;NfhL d. Z1kw;Wk; Z2 f;fis ,izf;Fk; Nfhl;bd; nrq;Fj;J ,Urkntl;b

30. 16x2-3y

2-32x-12y-44=0 vd;gJ

a. xU ePs;tl;lk; b. xU tl;lk; c. xU gutisak; d. xu; mjpgutisak;

31. f(a) = 2, f1(a) = 1, g (a) = -J, g

1 (a) = 2 vdpy;

d; kjpg;G

a. 5 b. -5 c. 3 d. -3

32. f1(x) =0 vd;w rkd;ghl;bw;F x = x0 vd;w %ykhdJ ,ul;il tupir nfhz;Ls;sJ vdpy; x = x0 MdJ

a. ngUk Gs;sp b. rpWk Gs;sp c. tisT khw;Wg; Gs;sp d. khWepiyg; Gs;sp

33. u = y sin x vdpy;

=

a. cos x b. cos y c. sin x d. 0

34. y2(a+2x) = x

2 ( 3a-x) vd;w tistiuapd; njhiyj; njhLNfhL

a. x = 3a b. x = -

c. x =

d. x = 0

35.

x x dx ,d; kjpg;G

a. b.

c.

d.0

36. (D2-4D+4) y = e

2x d; rpwg;G jPu;T (P.I)

a.

e

2x b. xe

2x c. xe

-2x d.

e

-2x

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37. m vd;w khwj;jf;f khwpypiaf; nfhz;l rkd;ghL y = emx ,d;

tiff;nfOr; rkd;ghl;by; m vd;gJ

a.

b.

c. y

1 d. y

38. ~ [p (~q)] ,d; nka; ml;ltizapy; epiufspd; vz;zpf;if

a. 2 b. 4 c. 6 d. 8

39. gpd;tUtdtw;wpy; vJ R ,y; <UWg;Gr; nrayp my;y? a. a * b = ab b. a * b = a – b c. a * b = d. a * b =

40. (Z9, +9) ,y;[7] ,d; tupir

a. 9 b. 6 c. 3 d. 1 II. Answer any ten questions: - 10×6=60

41. 2x – 3y + 7z = 5, 3x + y – 3z = 13, 2x + 19y – 47z = 32 xUq;fikTj; jd;ikia Muha;f.

42. Neu;khWfSf;Fupa tupir khw;W tpjpia vOjp epUgp. 43. Kf;Nfhzk; ABC ,d; cr;rpfshd A, B, C apd; epiyntf;lu;fs; ,

, vdpy; Kf;Nfhzk; ABC ,d; gug;ghdJ is

| × + × + × │

Vd epWTf. ,jdpd;Wk; , , xNu Neu;f;Nfhl;lik ntf;lu;fshapUf;f epge;jidiaf; fhz;f.

44. (a) A(-5, 2, 5), B(-3, 6, 7), C(4, -1, 5) kw;Wk; D(2, -5, 3) Mfpaw;iw cr;rpfsha;f; nfhz;l ,izfuj;jpd; gug;Gf; fhz;f.

(b) = = = vd;w ntf;lu;fs; xU Kidapy; re;jpf;Fk; tpspk;Gfshf; nfhz;l ,izfuj;jpz;kj;jpd; fd msT fhz;f.

45. 3+3i, -3-3i, -3 + i vDk; fyg;ngz;fs; xU rkgf;f Kf;Nfhzj;ij Mu;fd; jsj;jpy; cUthf;Fk; vd;W fhl;Lf.

46.

-

=1 mjpgutisaj;jpd; VNjDk; xU Gs;spapypUe;J mjd;

njhiynjhLNfhLfspd; nrq;Fj;Jj; J}uq;fspd; ngUf;Fj; njhif

xU khwpyp vd;Wk; mjd; kjpg;G

vdTk; fhl;Lf.

47. kjpak; 2.00 kzpf;F xU rpw;We;jpd; Ntfkhdp 30 iky;fs; / kzp vdTk; 2.10 kzpf;F Ntfkhdp 50 iky;fs; / kzp vdTk; fhl;LfpwJ. 2.00 kzpf;Fk; 2.10 kzpf;Fk; ,ilg;gl;l VNjh xU

rkaj;jpy; KLf;fk; rupahf 120 iky;fs; / kzp2 Mf ,Ue;jpUf;Fk; vdf; fhl;Lf.

48. (a) xU Fyj;jpd; rkdp cWg;G xUikj; jd;ik tha;e;jJ vd epUgp. (b) xU Fyj;jpy; cs;s a f;F (a

-1)

-1 = a vd epUgp.

49. y = 3x2 – 6x vd;w tistiuapd; Kfl;Lg; Gs;spfisf; fhz;f kw;Wk;

,U gf;fKk; cs;s rha;tpd; Fwpiaf; nfhz;L mjd; jd;ikia Muha;e;J mwpf.

50.

= 1 (a>b>0) vd;w ePs;tl;lk; Vw;gLj;Jk; gug;gpid Fw;wr;irg;

nghWj;Jr; Row;wpdhy; Vw;gLk; jplg;nghUspd; fd msTf; fhz;f. 51. x – mr;rpd; kPJ ikak; kw;Wk; XuyF Muk; nfhz;l tl;lj;

njhFg;gpd; tiff;nrOr; rkd;ghl;il mikf;f. 52. epUTf: p q ((~ p) q)

53. xU N[hbg; gfilfs; 10 Kiw Rz;lg;gLfpd;wd. ,U gfilfSk; xNu vz; fhl;Ltij ntw;wp vdf; nfhz;lhy; m) 4 ntw;wpfs; M) G+r;rpa ntw;wp Mfpatw;wpd; epfo;jfTf; fhz;f.

54. mnkupf;f fz;lj;jpy; n[l; tpkhdj;jpy; gazk; nra;Ak; xU egu; fh];kpf; fjpupaf;fj;jpdhy; ghjpf;fg;gLtJ xU ,ay;epiy gutyhFk;. ,jd; ruhrup 4.35 m rem MfTk;> jpl;l tpyf;fk; 0.59 m

rem MfTk; mike;Js;sJ. xU egu; 5.20 m rem f;F Nky; fh];kpf; fjpupaf;fj;jpdhy; ghjpf;fg;gLthu; vd;gjw;F epfo;jfT fhz;f.

55. (a) nka;naz; Fzfq;fisf; nfhz;l p(x) = 0 vd;w gy;YWg;Gf; Nfhitr; rkd;ghl;bd; fyg;ngz; %yq;fs; ,iznaz; ,ul;ilahfj;jhd; ,lk;ngWk; vd ep&gpf;f. (b) xU vz;zpd; n

Mk; gb %yk; fzf;fplg;gLk; NghJ Vw;gLk;

rjtPjg; gpio Njhuhakhf> me;j vz;zpd; rjtPjg; gpioapd;

klq;F MFk; vdf; fhl;Lf. III. Answer any ten questions: - 10×10=100

56. gpd;tUk; rkd;ghLfspd; njhFg;G xUq;fikT cilajh vd;gij Muha;f. xUq;fikT cilajhapd; mtw;iwj; jPu;f;f. x + y - z =1,

2x + 2y – 2z = 2, -3x – 3y + 3z = -3

57. (a) tof;fkhd FwpaPLfSld; Cos A =

epUgp.

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(b) 2x-8y+4z=3 kw;Wk; 3x-5y+4z+10=0 vd;w jsq;fspd; ntl;Lf;NfhL topNar; nry;yf; $baJk; 3x-y-2z-4=0 vd;w jsj;jpw;F nrq;Fj;jkhdJkhd jsj;jpd; rkd;ghL fhz;f.

58.

=

=

vd;w Nfhl;il cs;slf;fpaJk;

vd;w jsj;jpw;Fr; nrq;Fj;jhfTk; mike;j jsj;jpd; ntf;lu; kw;Wk; fhu;Brpad; rkd;ghLfisf; fhz;f.

59. fPo;f;fhZk; Gs;sp tptuq;fspd; gb fpilf;fg;ngWk; mjpgutisaj;jpd; ,af;F tl;lj;jpd; rkd;ghL fhz;f.

i)

vd;w ePs;tl;lj;jpd; ikak;> mjpgutisaj;jpd;

ikakhf mikfpwJ.

ii) mjpgutisaj;jpd; nrt;tfyj;jpd; ePsk;

kw;Wk; ikaj;

njhiyj;jfT

iii) Jizar;rpd; rkd;ghL x = 1

60. 9x2-16y

2-18x-64y-199=0 vd;w mjpgutisaj;jpd; ikaj; njhiyj;

jfT> ikak;> Ftpaq;fs;> cr;rpfs; Mfpatw;iwf; fhz;f. NkYk; mjd; tistiuia tiuf.

61. x+2y-5=0 I xU njhiyj;njhLNfhlhfTk; (6,0) kw;Wk; (-3,0) Mfpa Gs;spfs; topNa nry;yf; $baJkhd nrt;tf mjpgutisaj;jpd; rkd;ghL fhz;f.

62. x2+y

2=52, vd;w tl;lj;jpw;F 2x+3y=6 vd;w Neu;f;Nfhl;bw;F ,izahf

tiuag;gLk; njhLNfhLfspd; rkd;ghLfisf; fhz;f. 63. nfhLf;fg;gl;l xU Rw;wstpidf; nfhz;l nrt;tfq;fSs; rJuk;

kl;LNk ngUk gug;gsitf; nfhz;bUf;Fk; vdf; fhl;Lf. 64. tistiu y

2 = x kw;Wk; y = x – 2 vd;w Nfhl;bdhy; milagLk;

gug;gpidf; fhz;f. 65. y2

=4ax, vd;w gutisaj;jpd;> mjd; nrt;tfyk; tiuapyhd gug;gpid x mr;rpd; kPJ Row;Wk; NghJ fpilf;Fk; jplg;nghUspd; tisgug;igf; fhz;f.

66. xU Kg;gbg; gy;YWg;Gf; Nfhit x = -1 vDk; NghJ ngUk kjpg;G 4 MfTk;> x = 1 vDk; NghJ rpWk kjpg;G 0 MfTk; ,Ug;gpd; mf;Nfhitiaf; fhz;f.

67. x= 0 Mf ,Uf;Fk; NghJ y = 1 vd ,Uf;Fkhdhy;

(1+ex/y

)dx + ex/y

(1-x/y)dy =0 vd;w rkd;ghl;bd; jPu;T fhz;f. 68. xU NgUe;J epiyaj;jpy; xU epkplj;jpw;F cs;Ns tUk;

NgUe;Jfspd; vz;zpf;if gha;]hd; gutiyg; ngw;wpUf;fpwJ. ,q;F λ = 0.9 vdf;nfhz;L m) 5 epkpl fhy ,ilntspapy; rupahf 9 NgUe;Jfs; cs;Ns tu M) 8 epkpl ,ilntspapy; 10f;Fk; Fiwthf NgUe;Jfs; cs;Ns tu ,) 11 epkpl ,ilntspapy; Fiwe;jgl;rk; 14 NgUe;Jfs; cs;Ns tu epfo;jfT fhz;f.

69. tof;fkhd ngUf;fypd; fPo; 1 ,d; n Mk; gb %yq;fs; Kbthd Fyj;ij mikf;Fk; vdf;fhl;Lf.

70. (a) w = u2e

v vd;w rhu;gpy; u =

kw;Wk; v = y log x, vDkhW ,Ug;gpd;

kw;Wk;

fhz;f.

b) a kw;Wk; b vd;git x2+2 +4=0 vd;w rkd;ghl;bd; %yq;fshf

,Ug;gpd; an + b

n d; kjpg;gpidf; fhz;f. ,jpypUe;J a

12+b

12 d;

kjpg;gpidj; jUtpf;f. (n vd;gJ xU KO vz;) R.RAJASEKARAN, M.SC.,B.ED.,M.PHIL.

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