North Maharashtra University 2007 B.Sc Mathematics FY 3 - Question Paper

dt dt dt

11.    If u is differentiable vector function of scalar variable t & 0 is differentiable scalar

function of scalar variable t then prove that (d> u) = u + 0

^F dt ^y dt dt

12.    If u is differentiable vector function of scalar variable s and s is differentiable scalar

-t    *11,,1    tit du du ds ds du

function of scalar variable t then prove that = . = . .

dt ds dt dt dt

13.    If f(t) = /xCt)! + /₂(t)J + fr$(t)k is a differentiable vector function of the scalar variable

, ,i    a 4. &    ^dfi(t) - , df2(t) - , df₃(t) y

t, then prove that fit) =-1 H--] H--k

^F dt dt dt ^J dt

14.    Prove a non-constant vector function u(t)is of constant direction iff ux = 0.

Q.3 Examples    ( 4- marks each )

1.    Find the value of ax(b xc') ,if a = i 2j + k, b = 2T + j + k , c = i + 2j k

2.    Find the value of a x(b Xc)

if a = 2T 10/ +2k , b = 3T + J+ 2k , c = 2T + J+3k.

3.    If a = 2T J+ 3k , b = T + J 3k , c = 3! + 3J+ 2k.

Verify that ax(6xc) = (a.c)b- (a. b)"c

4.    If a = 2T J+ 3k , b = T + J 3k , c = 3! + 3J+ 2k.

Verify that b X (a Xc) = (b. c)"a (b.a)"c

5.    If a = 2T+3y + 4/c, b = i+}k, c = i}+ k. Find ax(b Xc) and verify that a x(b xc) is perpendicular to both a and (i x c).

6.    If a = 3! + 2} k .Find Ix (a XT) + }x (a xj) + k x(a x fc).

7.    Show that !x(axl)+ J x (a xj) + k x(a x/c) = 2a.

8.    Verify that ax(ixc) = (a.c)b- (a. b) c given that a = 1+ 2J + 3/c, b = 2i}+k, c = 3! + 2/ 5/e.

9.    If A = I+2//c, B = 2i+}+3k, C =! } + k. and D = 3i+} + 2k

Evaluate (A xB) .(Cx D)

10.    If A = i + 2} k, B = 2T + }+3k, C =! } + k. and D = 3T + J+ 2k Evaluate (A xB) x(Cx D)

11.    If A = T2J 3k , B = 2T + }k, C = T+3} 2k. find that |(A xB)xC|

12.    If A = i 2J 3k , B = 2T + }k, C =T+3J 2k. find that IA x(B xC)|

13.    If A = i 2J 3k , B = 2T + }k, C =T+3J 2k. find that (AxB)x(BxCj

14.    If A = T2J 3k , B = 2T + }k, C =T+3J 2k. find that (AxB).(BxQ

15.    If a = 2i+}k, b = i + 2j 4k, c = i+} + k. find axb.(axc)

16.    If a = 1+2} k, b = 3T 4k , c = i+} and d = 2T J 3k, find (a x &). (c x d).

17.    If a = 1+2} k, b = 31 4/c, c = i+} and d = 2i} 3k, find

(axii)x(cx d).

18.    If a = T + }k, b = 2T + }3k, c = I J+3/c & d = 31 + 4} 2k, then find (a x b). (c xd) + (cx a).(b xd) + (dx a). (c x b)

19.    Prove that

(AxB)x(CxD) + (AxC)x(DxB) + (AxD)x(BxC) = ~2[B C D]A .

20.    Prove that (B x C). {A x D) + (C x A). (B x D) + (Ax B). (C x D) = 0

21.    Prove that

[axb b Xq cxf] + [axq b Xr c xb] + \axr b Xp cxq] = 0

22.    Find a the set of vector reciprocal to the set of vectors 2T+3}k, T }2k, T+2j + 2k.

23.    Find a the set of vector reciprocal to the set of vectors i + j + k, i + j + k, i+y k.

24.    Find a the set of vector reciprocal to the vectors a. b and axb.

25.    If a,b ,c is a set non coplaner vectors and a' = _r ^b-^C _n b' = _r ^c_^a _n &

^r    [a b c]    [a b c]

.    axb ,    , _    b'xc' y    c'xal _    alxb'

c = - then prove that a = _r_ _ __n , b = _r_ _ __n & c = _r_ _ __n

[a b c\    [a' br c']    [a' br c']    [a' br c']

26.    If a,b,c & a', b', c' are reciprocal system of vectors then prove that

ax a' = b x b' = cx c' = 0

27.    // a,b, c & a', b', c' are reciprocal system of vectors then prove that

a+b+c

a' xb' + b' x c' + c' x a' =

[a b c]

28.    .If a,b, c & a', b', c' are reciprocal system of vectors then prove that

a. a' + b.b' + c.c' = 3

29.    Evaluate lim_t_₀ [(t² + 1)1 +    + (1 + 2t)¹/t

30.    If fit) = ^sin2t! + cos tj , t 0 and /(0) = xi + J is continuos at

t = 0 , find x.

sin 3t_ log(l+t)

31.    If /(t)=pi+ _t     t = 0 , then find f (0).

32.    f(t) = cos t! + sin tj + tan tk , find f'(t) and

33.    If r = (t² + 1)1 + (4t 3)J + (2t² 6t)k, find & |~ t² + 1)1+ (4t - 3)J+ (2t²

35.    If r = (t² + 1)1 + (4t 3)7 + (2t² 6t)k, find

36.    If F = e^_tT + log(t² + 1)7 tan t k find

37.    If r = e^_tT + log(t² + 1)7 tan t k find

7 +

38.    If a = t²i + ty + (2t + 1)fc and b = (2t 3)i +j tk find ~~(a. b) at t = 1

39.    If a = t²! +t7+ (2t + 1)fc and b = (2t 3)1 + }tk find |a x l)| at t = 1

40.    If a = t²i + ty + (2t + 1)fc and b = (2t 3)i +j tk find ~(^aX    t

41.    If h = 3t²! (t + 4)7 + (t² 2t)k and v sin ti + 3e~^tJ 3 cos tk find

d^z _ _

- (uxv) at t = 0 dt^z

42.    If r = 4asin³9 I + 4acos³07 + 3bcos29 k,find I x I

^{y 7} '^J\d8d8²\

43.    If f = 4asin³d l + 4acos³9 J+3bcos29 k,find \ -1

^{1 J} \_dd dd² dd³i

44.    If r = ae^2t + be^3t,prove that -7 5 + 6r = 0.

^J r dt² dt

45.    Show that r = e~^t(acos2t + bsin2t), where a & & are constant vectors is solution of the differential equation + 2 + 5f = 0.

46. // r = a cost 1 + a smt j + at tana k, find

47.    If r = a cost i + a sintj + at tana k, find    

48.    If r = cosnt i + sinntj, where n is constant show that r. ~ = 0

49.    If r = cosnt i + sinntj, where n is constant show that r x = nk.

50.    If a = sind i + cosd j+6k, b = cosd i sind J 3k, c = i + 2j 3k

find j--j[flx(i)xc)] at 9 =

_    ,, , d (_ dr d²r\ _ dr d³r

51. Prove that (r. x -) = r. x -

dt V dt dt²J dt dt³

52.    Show that r = ae + be is a solution of the differential equation qr +

P~~: + ⁼ 0 , where k, I are roote of the equation m² + pm + q = 0, a,b & p, q being constant vectors & scalars respectively.

   dr d²r ²

53.    If r = acosti + asintJ+btk,show that x- =a²(a²+b²)

^{J J} ' dt dt²

I²1

55. If r = a cosuit + b sinwt, where a, b are constant vectors & is constant

scalar, prove that f x = w(axii)

56.    If r = a cosuit + b sinu>t, where a, b are constant vectors & is constant

scalar, prove that    u>²?.

57.    Find the unit tangent vector & curvature at point P (x, y, z ) on the curve F(t) = 3 costT+ 3 sintj + 41 k.

58.    Find the unit tangent vector & the curvature at point P (x, y, z)on the curve x = a cosd ,y = a sind , z = ad tana, where a & a are constants

59.    A particle moves along the curve r = e~^tT + 2 cos3tJ + 2 sin3t k, find the velocity & acceleration at any time t & also their magnitude at t = 0.

60.    Find the velocity & acceleration vector of a particle moving along the curve x=2sin3t, y = 2 cos3t, z = 8t & also their magnitude.

61.    Find the acute angle between the tangents to the curve r = t²i 2tj + t³k, at the points t = 1 & t = 2.

62.    Show that the acute angle between the tangents to the curve x = t, y = t², z = t³ at t = 1 & t = -1 is cos^-1-.

7

63.    Find the cosine of acute angle between the tangents to the curve r = t²i +2tJ -t²k at t = 1 & t = -3.

64.    Find the curvature of the curve r = a(3t t³)i + 3at²J + a(3t + t³)k.

Find the component of its velocity in the direction i + j + 3k

66.    A particle moves along the curve r = cost i + sintj + t tana k, where a is constant & t is time variable. Find tangential & normal components of acceleration .

67.    A particle moves along the curve r = (t³ 4t)i + (t² + 4t)j + (St² 3t³)k, where t is time variable. Find the tangential & normal components of acceleration at t = 2.

68.    A particle moves along the curve r = 2t²i + (t² 4t)j + (3t 5)k, obtain the components of velocity & acceleration at t = 1 , along the direction I 3J+ 2 k.

69.    A particle moves along the curve r = + e~^tJ + V2 tk.Find i)v & a ii)T & N

iii) the tangential & normal component of a.

70.    If r = xcosyT + xsinyj + ae^myk,findi) ii) iii) iv)

^{J    y    y    y} dx ^y dy ^y dx² ' dxdy

71.    If r = cosxy 1+ (3xy 2x²)J (3x + 2y)k,find i) ii) iii) iv) --p

72.    If r = x cosy T + x siny J + c log(x + Vx² c²)fc, find i) ii)

..._A d²r >. d²r

0 T-7 iv)

dy²    dxdy

⁷³.    f = (.x + y)l + l(x-y)j-^xfk , find i) ii) g iii) 0 iv)

dr dr

74.    If r = x cosy i + y sinyj + ae^myk,find .g* %..

Id* ^X3y I

75.    If u = x²yz i 2xz³ J + xz²k, and v = 2zi + yJ x²k, find

d² dx dy

⁷⁶.    If r = (x + y)i + ^b-(x-y)j-xyk find i) [ g ]

d²r

11

dy dxdy J

77.    If u = z³ i x²k, and v = 2xyzj,and w = 5xyi + 3zk

find

dxdydz

78.    If F = a cosu sinv i + a sinu sinv J + acosvk show that

j is a unit vector

,2 o. ;n _ _vj7t_ 2_T

79. If 0(x,y, z) = xy z & u = xzi xy²} + yz²k find

d³

(0 u)at the point (2, 1,1).

d^zxd^zz

80. If r = e~^Ax(asinAy + bcosAy) where a & b are constant vector & A is constant scalar

d²v d²v

.show that + = 0.

ox* oy^z

(Differential Operators, Vector Integration)

Q.1 Objective Questions    ( 2 marks each )

A)    Fill in the blanks

i)    The gradient of a scalar point function is a------function.

ii)    The angle between two surfaces is defined as the angle between their-----at the point of

Intersection.

iii)    The directional derivative of scalar point function (p at point P along a is equal to-----.

iv)    The divergence of a vector point function is a-------function.

v)    A scalar point function which satisfies Laplaces equation is called-------

vi)If    u & v are irrotational then ux vis--------

vii)    The curl of a vector point function is--------

viii)    If C is a simple closed curve then the line integral of f over C is called----------

ix)    Line integral may or may not depend upon------

x)    A vector field f about any closed curve in the region is------

B)    Multiple choice Questions

i)    The dot product of two vectors is ------

a) Vector b) Scalar c) Vector field d) None of these

ii)    The cross product of any vector with itself is-----

a) 0 b) 1 c) 2 d) None of these

iii)    The angle between the vectors (2i + 3J+fc) and (2i J k) is-----

^{a) b)} _T ^c) - ^d) -

iv)    If \a.b\ = |a xb \ then the angle between a & b is-----------

v)    The directional derivative of 0 = xyz at (1, 1, 1) in the direction of j is equal to------

a) 1 b) -1 c) 0 d) none of these

vi)    If 0(x,y,z) = 2x²y³ 3y²z³ then V0 at point (1, -1, 1) is equal to---------------

a) 4i + 12J + 9k b) 4! 12/ + 9fc c) 4i + 12J 9k d) none of these

vii)    If / = x²i + y²J + z²k then div.(curl/) is equal to--------------

a) 1    b) 2    c) 0 d) none of these

viii)    If Vxv = 0 then vector point function v is..............

a) Solenoidal b) Irrotational c) Harmonic function d) none of these

ix)    If u = ti t²J +(t V)k and v = 2t²! + 6t/c then Ju.vdt is equal to-----------

a)12    b) 16    c) 14    d) none of these

x)    The Value of sin²x dx is equal to------------

a) ^{0    b}) f    c) f ^d) f

C)    Numerical Examples

1)    Find Vr ,Where r = xi + yj + zk and |f| = r

2)    Find the gradient of x² +y² z = 1 at the point (1, 1, 1).

3)    Find divT, Where r = xi + yj + zk

4)    Find Curl f, Where r = xi + yj + zk

5)    Find the directional derivative of f = xy + yz + zx in the direction of vector I + 2/ + 2fc at the point (1, 2, 0 )

6)    Find Vrⁿ, Where r = xi + yj + zk

7)    If f = yzi + zxj + xyk then find divf

8)    If u = y² z² then find V²u

9)    If a = 2yz! x²yj + xz²k and = 2 x²yz³ then find a. V<p

10)    Evaluate f_c f .dr, where f = 2xy! + x²J from (0 ,0) to (1, 1) along the straight line (0, 0) to (1, 1)

D)    Define the following

1) Scalar point function

2)    Vector point function

3)    The Vector differential Operator (V)

4)    Gradient of a Scalar point function

5)    Divergence of Vector point function

6)    Solenoidal Vector function.

7)    Curl of a Vector point function

8)    Irrotational Vector function.

9)    Line integral.

10)    Conservative vector field.

Q.2 Theory Questions    ( 4 marks each )

1)    If 0 & 0 are scalar point functions & if    exit in a given region R then prove that V (0 0) = V$ i. e. grad (0 0) = grad + grad 0

2)    Prove that a necessary & sufficient condition for a scalar point function to be constant is that = 0.

3)    If are scalar point functions if    exist in a given region R then prove that V (00) = 0 W + V$ i. e. grad (O) = O gradW + W gradO

4)    If are scalar point functions if    exist in a given region R then prove

that _{v    =}    . _e. grad () ₌    Provided * #0

5)    If (x, y, z) be a scalar point function defined in a region R. Let P(x, y, z) be a point in R & let a be a unit vector then prove that the directional derivative of at P along

a is given by = V0.a

6)    If n be a unit vector normal to the level surface (x, y, z) = c at a point P of (x, y, z) in the direction of increment increasing & n be a distance along the normal , then

prove that grad .n

7)    If u & v be the vector point functions then prove that div (u v) = div u + div v i. e. V. (u v) = V. u V.v

8)    If u & v be the vector point functions then prove that curl (u v) = curl u + curl v

i. e. Vx(u v)=Vxu + Vxp

9)    If u be a vector point function &  be any scalar point function then prove that Div (u ) = (grad ). u + div u    i. e. V. ) = (V). u + 4>(V. u)

10)    If u be a vector point function &  be any scalar point function then prove that

Curl (tyu) = (grad)x u + (curl u) , i. e. V x(u) = (V) xu + 4>(V x it)

11)    If u & v be the vector point functions then prove that div(u xv) = v.curlu u. curlv i.e. V.(uXf)=p.(Vxu ) u .(Vxv)

12)    If u & v be the vector point functions then prove that curl(u Xv) = (v.V)u v. divu - (u. V)v + u divv i.e. V. (uxv) = (v. V)u v(V. u) + (u. V) v +u(V. v)

13)    If u and v are two vector point functions then prove that grad(u.v) = (v.V)u + (u.V)v + vx (crulu) + ux (crulv) i.e. V(u. v) = (v.V)u + (u. V)v + vx (Vxu) + u x(Vxv)

14)    If be a scalar point function then prove that curl (grad ) = 0

i.e. V x (V) = 0

15)    If u be a vector point function then prove that

Div .(curlu) = 0

i. e.V.(V xu) = 0

16)    If u be a vector point function then prove that

Curl (curl u) = grad(divu) V²u

i.e. Vx(Vx u) = V(V. u) V²u

17)    If be a Scalar point function then prove that

(V. V) = V(V. ) = V² where V² be a Laplacian Operator

18)    prove that a vector field / is conservative if and only if the circulation of f about any closed curve in the region is zero

19)    Prove that if f be a continuously differentiable field on a region R then f is conservative if and only if it is the gradient of scalar point function defined on R(i.e.

V= f )

20)    Prove that if f be a continuously differentiable field on a region R then f is conservative if and only if it is irrotational (i.e. V X f = 0 )

Q.3 Examples    (4 marks each)

1.    If a = xyzi + xz²J y³k and b = x³! xyzj + x²zk then find

d²a d²b ,    i

^{at the} p^{int (1, 1, 0)}.

2.    If f = x! + y/ + zk and |f| = r then Prove that

i) V(r) = '(r)Vr ii) Vr = = f

3.    Prove that Vrⁿ = nrⁿ~²r , where r = xi +yj + zk

4.    Find grad (gradu.gardv) ,where u = 3x²y and v = xz² 2y

5.    Find (x,y, z)if = 2xyz³! + x²z³J + 3x²yz²k and (1,-2,2) = 4

2 2

6.    Find the gradient and unit normal to the surface x + y - z = 1 at (1, 1 ,1)

7.    Find the equation of tangent plane and the normal to the surface xy + yz + zx = 7 at the point (1, 1 ,3)

8.    Find the acute angle between the tangents to surfaces xy²z = 3x + z² and 3x² - y² + 2z = 1 at the point (1, -2, 1)

2 2 2 2 2

9.    Find the cosine of the acute angle between the surfaces x + y + z =9 and z = x + y - 3

at the point (2, -1, 2)

10.    Find the directional derivative of 2xy + z² at the point (1, -1, 3) in the direction of the vector i + 2J+ 2fc

11.    Find the directional derivative of (x,y, z) = x²y + xz² -2 at the point A(1, 1, -1) along AB where B is the point (2, -1, 3)

12.    Find the value of a & b if the surfaces ax² - byz = (a + 2)x & 4x²y + z³ = 4 are orthogonal at the point (1, -1, 2)

13.    Find the value of the constant a, b, c so that the directional derivative of of

(x, y, z) = axy² + byz + cz²x³ at the point (1, 2, -1) as a maximum magnitude 64 in the direction parallel to Z-axis.

14.    What is the greatest rate of increase of u = xyz² at (1, 0 ,3) ?

15.    If r = xi +yj + zk and |f| = r then find (i) div (rⁿ f) , (ii) curl (rⁿ f)

16.    If r = xi + yj + zk and |f| = r then find Laplacian of rⁿ i.e. V²rⁿ

17.    Prove that curl ( grad) = 0 where be any scalar point function.

18.    Given that = 2x³y²z⁴ , find div ( grad

19.    Determine the constant a so that the vector function

v = (x + 3y)T + (y 2z)J + (x + az)kis solenoidal

20.    If the f = (axy z³)i + (a 2)x²/ +(1 a)xz²k is irrotational then find the value of a

21.    Find the constant a, b, c so that the vector function

/ = (x + 2y + az)T + (bx 3y z)J + (4x + cy + 2z)k is irrotational

22.    If is a constant vector & v = x r,prove that div(v) = 0

23.    If u & v are irrotational the prove that u x vis solenoidal

24.    Prove that f = /i(y, z)T + f₂(z, x)J + /₃(x,y)fc is solenoidal

25.    Prove that    

26.    Prove that the vector function f(r) f is irrotational

27.    If f = ti 3j + 2tk,g = i 2j + 2k & h = 3i + tjk then evaluate

28.    If f = yi + 2xJ + 2yk,evaluate f_c f.dr , where C is the curve given by f = t!+ t²J +t³k from t = 0 to t = 1.

29.    Find the total work done in moving a particle in a force field f = 3xy! 5zj + 10xk Along the curve x = t² + 1, y = 2t², z = t³ from t = 1 to t = 2

30.    Evaluate f_c [(x² y²)i + 2xyj].dr around a rectangle with vertices at (0, 0), (a, 0),

(a, b) & (0, b) transverse in the counter-clockwise direction.

31.    Evaluate f_c f.dr, where f = x²i + y³J & C is the arc of the parabola y = x² in the XY plane from (0, 0) to (1, 1)

32.    If f = 3xyi y²J,evaluate f_c f.dr, where C is the curve in the XY plane y = 2x² from (0, 0) to (1, 2).

33.    Evaluate f_c f.dr, where f = yzi + (zx + 1)/ + xyk & C is any path from (1, 0, 0) to (2, 1, 4).

34.    If f = (3x² + 6y)I 14yzJ + 20xz²k, evaluate f_c f.dr from (0, 0, 0) to

(1, 1, 1) along the paths (i) x = t, y = t², z = t³ , (ii) the straight line joining (0, 0, 0) to (1, 1, 1).

35.    Evaluate J(x dy y dx) around the circle x² + y² = 1.

36.    Find the circulation of f round the curve C, where f = yi+ zj + xk & C is the circle

x² + y² = 1, z = 0

37.    If u(t) = ti t²J + (t 1)k & v(t) = 2t²i + 6tk,then evaluate f₍(u x v)dt

38.    The acceleration of a particle at any time t is given by

a = e~^tT 6(t + 1)J + 3sintk. If the velocity v & displacement r are zero at t = 0 find v & r at any time t .

39.    Show that f = (2xy + z³)! + x²J + 3xz²k is a conservative vector field. Find the scalar point function such that / = V

40.Determine    whether the force field , f = 2xz! + (x² y)J + (2z x²)k is conservative or non- conservative .

(Change of Axes, General Equation of Second Degree) Q.1 Objective Questions    ( 2 marks each ) (A)    Fill in the blanks

i.    The two types of change of co-ordinate axes are------&------.

ii.    The equations of translation are-----&------.

iii.    If by rotation of axes, without change of origin, the expression ax2 + 2hxy + by2 Becomes AX2 + 2HXY + BY2 then the invariants are ------- .

iv.    Parabola is a-----conic (central/ non-central)

v.    The general equation of a conic is------.

vi.    The centre of a conic given by ax² + 2hxy + by² + 2gx + 2fy + c = 0 is -------

vii.    The conic given by ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents a parabola if-----.

viii.    In order to remove the xy terms from the equation ax² + 2hxy + by² + 2gx + 2fy + c = 0 the axes should be rotated through an angle-----

ix.    The equation ax2 + by2 + 2gx + 2fy + c = 0 represents a circle if------.

x.    The equation ax² + 2hxy + by² + 2gx + 2fy + c = 0 represents------when ab - h² = 0 &

hg - bg = 0

(B)    Define the following:

i)    If the origin is shifted to the point (2, 1), the directions of the axes remains the same then the equations of translation are

a)x = 2 - X, y = 1 - Y    b) x = 1 - X , y = 2 - Y

c) x = 2 + X, y = 1 + Y d) x = 1 + X, y = 2 + Y.

ii)    If by rotating the axes through an angle 45⁰ the equation

x² + 2xy + 5y² + 3x - 6y + 7 = 0 becomes px² + 2rxy + qy² + sx + ty + u = 0 then the value of pq - r² is a) 4 b) 6 c) 0 d) 5

iii)    If the axes are rotated through an angle 45⁰ then the equations of rotation are

.    X+Y    X-Y    . .    X-Y    X+Y

^a)^{x =} nr-y= nr ^b'> ^x= nr -y= nr

_s    X+Y    X-Y    X-Y    X+Y

c)x = , y p    d) * = , y =

iv)    A conic is a parabola if a) e < 1 , b) e > 1, c) e = 0 , d) e = 1

v)    The equation ax² + 2hxy + by² + 2gx + 2fy + c = 0 represents an ellipse if

a) h² - ab > 0 b) h² - ab < 0 c) h² - ab = 0 d) h² + ab > 0 (Provided that abc + 2fgh - af² - bg² - ch² 0 )

vi)    The equation 16x² - 24xy + 9y² - 6x - 8y - 1 = 0 represents

a) an ellipse b) a parabola c) a circle d) a hyperbola

vii)    To remove the xy terms from the equation 7x² + 8xy + y² - 52x - 22y + 76 = 0, tan20 must

be a) b) c) 1 d) 0

viii)If    a plane is perpendicular to an axis of the cone & cuts it , the section is

a) A parabola b) an ellipse c) a hyperbola d) a circle

ix)    The centre of the conic given by x² - 4xy - 2y² + 10x + 4y = 0 is at

a) (1, 2) b) (1, -2) c) (-1, 2) d) ( -1, -2)

x)    Length of the latus rectum for a conic y2 = is a) | b) c) d)

i.    Find the transformed form of the equation xy - x - 2y + 2 = 0 if the origin is shifted to the point (2, 1)

-1 3

ii.    If the axes are rotated through an angle 6 = sin ¹ - keeping the origin fixed then find the equations of rotations.

iii.    Find 6 through which the axes should be rotated in order to remove the xy term from the equation 7x² + 12xy - 5y² + 4x + 3y - 2 = 0.

iv.    Find the centre of the conic given by the equation 5x² + 6xy + 5y² - 10x - 6y - 3 = 0.

v.    Identify the conic given by 16x² - 24xy + 9y² - 6x - 8y - 1 = 0.

vi.    Identify the conic given by 5x² - 6xy + 5y² + 18x - 14y + 9 = 0.

vii.    Through which angle the axes should be rotated to remove the xy term from x² + 2xy + y² - 2x - 1 = 0.

viii.    Find the length of axes of the hyperbola 2y² - 3x² = 1.

ix.    Find the length of the latus rectum of the ellipse 2x² + 3y² = 6.

x.    If by rotation of axes keeping the origin fixed, the equation

x² + 2xy + 5y² + 3x - 6y + 7 = 0 transform to px² + 2rxy + qy² + sx + ty + u = 0., then find the values of p + q and pq - r² .

Q.2 Theory Questions    ( 6 marks each)

i.    Obtain the equations of translation when the origin is shifted to the point (h, k), directions of the axes remaining the same.

ii.    Obtain the equations of rotations when the axes are rotated through an angle 6 keeping the origin fixed.

iii.    If by change of axes, without change of origin, the expression ax² + 2hxy + by² becomes a'x'² + Ih'x'y' + b'y'² then prove that a + b = a + b & ab - h² = ab - h².

iv.    Show that equation of a conic is a second degree equation in x & y ( Hint: Use focus-directrix property).

v.    If (x, y) & (x, y) are the co-ordinates of the same point referred to two sets of rectangular axes with the same origin & if ux + vy becomes ux + vy , where u & v are independent of x & y then show that u² + v² = u² + v².

vi.    Show that if the set of rectangular axes is turned through some angle keeping the origin fixed then g² + f² in the equation ax² + 2hxy + by² + 2gx + 2fy + c = 0 is invariant.

vii.    Show that if the equations ax² + 2hxy + by² = 1 & ax² + 2hxy + by² = 1 represent the same conic & if the axes are rectangular then prove that

(a - b) ² + 4h² = (a - b)² + 4h². ( Hint: use a+ b = a + b & ab - h² = ab - h²).

viii.    If by rotation of the axes the equation ax² + 2hxy + by² + 2gx + 2fy + c = 0 becomes a'x'² + 2h'xy + b'y'² + 2gx + 2fy + c = 0 then prove that

a' = [(a + b) + 4h² + (a b)²] & b' =[(a + b) 4h² + (a b)²]

(Hint: Use a' = [(a + b) + (a b) cos 2d + 2h sin 20],

1 ?h b' = - [(a + b) (a b) cos 26 2h sin 20] & then use tan 20 = )

ix. Prove that every general equation of second degree represents a conic.

Q.3 Examples    (4 marks each)

1)    If the origin is shifted to the point (h, 2), find the value of h so that the new equation of the locus given by the equation x² + 4x + 3y - 5 = 0 will not contain a first degree terms in x.

2)    The origin is shifted to the point (-2, k), find the value of k so that the new equation of the locus given by 2y² + 3x + 4y - 7 = 0 will not contain the first degree term in y.

3)    Obtain the new equation of the locus given by 3x² + y² + 18x - 8y - 16 = 0 when the origin is shifted to the point (-3, 4).

4)    Obtain the transformed equation of the locus given by 3x² + 2V3 xy + 5y² = 1 when the axes are rotated through an angle of 60⁰.

5)    Transform the equation 3x² + 2xy + 3y² + 8x + 3y + 4 = 0 by rotating the axes through an

angle 0, where 0 = sin^-1 , 0 < , keeping the origin fixed.

6)    Find the new equation of the locus given by x² + 3y² + 4x + 18y + 30 = 0 when the origin is shifted to the point (-2, -3) directions of the axes remaining the same.

7)    If the axes are turned through an angle of 45⁰ keeping the origin fixed, show that the equation of the locus given by x² - 4xy + y² = 0 changes to 3y² - x² = 0.

8)    The equation of the curve referred to the axes through (-1, 2) as origin & parallel to the original axes is 2X² + 3Y² = 6. Find the equation of the curve referred to original set of axes.

9)    What does the equation 3x² - 4xy + 25y² = 0 become when the axes are rotated through an angle tan^-1 2 ?

10)    Find the co-ordinates of a point to which the origin should be shifted so that the new equation of the locus given by x² - 2xy + 3y² - 10x + 22y + 30 = 0 will not contain the first degree terms in the new co-ordinates.

11)    The axes are changed by changing the origin to (a, 2) .By this transformation the line given by x + 2y + 3 = 0 passes through the origin .Find the value of a.

12)    Transforms the equation of a circle x² + y² + 2x + 2y + 1 = 0 to standard form.

13)    Transform the equation x² + 4xy + y² - 2x + 2y - 6 = 0 when the origin is shifted to the point (-1, 1) and then the axes are turned through an angle of 45⁰.

14)    What does the equation x² - 5xy + 13y² - 3x + 21y = 0 when the origin is changed to

(-1, -1) and then the axes turned through an angle tan^-1 Q).

15)    Transform the equation 7x² - 8xy + y² + 14x - 8y - 2 = 0 when the origin is shifted to the point (-1, 0) and then the axes are turned through an angle of tan^-1 ()

16)    The equation 3x² + 2xy + 3y² - 18x - 22y + 50 = 0 transforms to 4x²+2y²=1 when the origin is shifted to the point (2, 3) and then the axes are rotated through an angle S.Find the measure of an angle 6

17)    Change the origin to (1, 2) and transform 3x² - 10xy + 3y² + 14x - 2y + 3 = 0.Further rotate the axes through d = - and find the final transform of the equation.

4

18)    Transform the equation 11x² + 24xy + 4y² - 20x - 40y - 5 = 0 to rectangle axes through the point (2, -1) inclined at an angle tan^-1 to the original axes.

19)    Transform the equation 5x² + 6xy + 5y² - 10x - 6y - 3 = 0 when the origin is changed to (1, 0) and then the axes are rotated through an angle ().

20)    Obtain the equation of rotation in order to remove the xy term form x² + 6xy + 8y = 7y² + 8x + 20.

21)    Find the centre of a curve and identify it .3x² + 8xy - 7y² - x + 7y - 2 = 0

22)    Find the centre of the following conics and identify each of them 14x² - 4xy + 11y² - 44x - 58y + 71 = 0

23)    Find the centre of the following conics and identify each of them 3x² - 10xy + 3y² + 14x - 2y + 3 = 0

24)    Find the centre of the following conics and identify each of them 5x² + 6xy + 5y² - 10x - 6y - 3 = 0

25)    Find the centre of the following conics and identify each of them 55x² - 30xy + 39y² - 40x - 24y - 464 = 0

26)    Find the centre of the following conics and identify each of them 8x² - 24xy + 15y² + 48x - 48y + 7 = 0

27)    Reduce the equation of a parabola 16x² - 24xy + 9y² - 6x - 8y - 1 = 0 in the standard form

28)    Find the centre of a conic 7x² + 8xy + y² - 52x - 22y + 76 = 0 and reduce it to its standard form.

29)    Transform the equation of a conic x² - 4xy - 2y² + 10x + 4y = 0 to its standard form.

30)    Find the co-ordinate of the centre of the conic 5x² + 6xy + 5y² - 4x - 4y - 4 = 0 and reduce the equation of the conic to its standard form.

31)    Transform the equation 3(x² + y² + 1) = 2y (12x + 1) - 14x (y + 1) to the form ax² + fiy² = 1

32)    Show that the equation x² + 2xy + y² - 2xy - 1 = 0 represented a parabola .Reduce the equation to its standard form .Also find the length of the latus rectum.

33)    Show that the equation x² - 4xy - 2y² + 10x + 4y = 0 represent a hyperbola .Find its centre .Also find the equation of the asymptotes.

34)    Find the centre of the conic 5x² - 6xy + 5y² + 18x - 14y + 9 = 0 and reduce it to the standard form .Also find the eccentricity of the conic

35)    Determine the nature of the following conics. Also find the centre and length of axes in each case,5x² + 6xy + 5y² - 10x - 6y - 3 = 0

36)    Determine the nature of the following conics. Also find the centre and length of axes in each case 36x² + 24xy + 29y² - 72x + 126y + 81 = 0

37)    Determine the nature of the following conics. Also find the centre and length of axes in each case x² + 4xy + y² - 2x + 2y - 6 = 0

38)    Determine the nature of the following conics. Also find the centre and length of axes in each case 9x² + 24xy + 16y² - 44x + 108y - 124 = 0

39)    Determine the nature of the following conics. Also find the centre and length of axes in each case 32x² + 52xy - 7y² - 64x - 52y - 148 = 0

40)    Determine the nature of the following conics. Also find the centre and length of axes in each case 16x² - 24xy + 9y² - 104x - 172y + 44 = 0.

(Sphere)

Q.1 Objective questions    ( 2 marks each )

A)    Fill in the blanks:

i.    A......is the locus of a point which moves in a space so that it is always at a

constant distance from a fixed point.

ii.    Equation of a sphere is.......degree equation in x, y, z.

iii.    The intersection of the sphere & the plane is -------- .

iv.    A line which meets a sphere in two coincident points is called the ------line to the

sphere.

v.    The locus of the tangent lines to a sphere at a point on it is called the ------plane at

that point.

vi.    The plane is tangent plane to the sphere iff length of the perpendicular from the centre is equal to........

vii.    Two spheres are said to cut orthogonally if they intersect each other at.......

angles.

viii.    Two spheres are non-intersecting if distance between the centres is greater than the sum of.......of the spheres.

ix.    Two spheres touch each other externally if distance between the centres is equal to the sum of the ------of the spheres.

x.    The plane of the great circle passes through the......of the sphere.

B)    Define the following:

i.    Sphere

ii.    Tangent to the sphere.

iii.    Tangent plane to the sphere.

iv.    Normal to the sphere at a point.

v.    Great circle.

vi.    State condition of tangency.

vii.    Orthogonal sphere.

viii.    State condition of orthogonality.

ix.    State general equation of the sphere.

x.    State equation of sphere having centre at origin & radius is a.

C)    Numerical problems:

i.    Find the centre of the sphere 2x² + 2y² + 2z² + 3x + 4y - 6z - 4 = 0.

ii.    Find the radius of the sphere 2x² + 2y² + 2z² + 3x + 4y - 6z - 4 = 0.

iii.    Find the centre & radius of the sphere x² + y² + z² + 4x - 6y - 8z = 2.

iv.    Find the radius of the sphere passing through the point (2, 1, 3) & having the centre at (1, -3, 4).

i.    The general equation of the sphere is

a) Linear b) second degree c) third degree d) none of these

ii.    The section of the sphere taken by the plane is

a) Sphere b) circle    c) plane    d) none of these

iii.    The centre of the great circle & the corresponding sphere are

a) Different b) same c) not repeated to each other d) none of these

iv.    The number of tangent lines at a point on the sphere are

a) Two b) three    c) infinite    d) none of these.

v.    The tangent plane to the sphere touches the sphere at

a) One point b) two point c) three point d) none of these.

vi.    The normal line to the sphere at a point passes through

a)    Centre of the sphere    c) tangent plane

b)    Great circle    d) none of these

vii.    If equation S + AS = 0 represents a radical plane then

a) A = 1 b) A = -1 c) A = 0 d) none of these.

viii.    The angle between the tangent planes at a common point of the orthogonal spheres is a)    b)    c) 0    d) none of these.

ix.    The radius of the spheres x² + y² + z² + 4x - 6y - 8z = 2 is

a) V32    b) V3T    c) 31    d) none of these.

x.    If the sphere x² + y² + z² + 2ux + 2vy + 2wz + d = 0 passes through origin then

a) d= -1    b) d= 0    c) d= 1    d) none of these

Q.2 Theory Questions    (6- marks each)

1.    Show that the equation x² + y² + z² + 2ux + 2vy + 2wz + d = 0 represents a sphere .Find the centre and radius.

2.    Obtains the equation of the sphere which passes through the origin and makes intercepts a, b, c on co-ordinate axes .Hence find the equation of the sphere passing through the origin and making intercept 2, 3, 4 on the axes .

3.    Find the equation of the sphere with line joining the points A (x₁ , y₁, z₁) and B(x2, y2, z2) as one the diameters. Hence obtain the equation of the sphere described on (2, -3, 1) and (3, -1, 2) as extremities of a diameter.

4.    Find the condition that the plane lx + my+ nz = p may touches the sphere x² + y² + z² = a² and find the point of contact.

5.    Find the condition that the plane lx + my+ nz = p may touches the sphere x² + y² + z² + 2ux + 2vy + 2wz + d = 0

6.    Find the equation of the sphere passing through the four points P(x₁ ,y₁ ,z₁),

Q( x2, y2, z2), R (x3, y3, z3) and S(x4, y4 ,z4).

7.    Find the equation of the tangent plane to the sphere

x² + y² + z² + 2ux + 2vy + 2wz+ d = 0 at the point A (a, fi, y).

8.    Obtain the equation of the normal to the sphere

x² + y² + z² + 2ux + 2vy + 2wz+ d = 0 at the point A (a, fi, y).

9.    Let the equation of a circle be S = x² + y² + z² + 2ux + 2vy + 2wz+ d = 0 and

U = lx + my + nz - p = 0. Show that S + XU = 0, where A is a parameter represents a family of sphere through the given circle.

10.    Define orthogonal spheres. Obtain the contain that the spheres x² + y² + z² + 2uix + 2viy + 2wiz+ di = 0 and

x² + y² + z² + 2u₂x + 2v₂y + 2w₂z+ d₂ = ⁰ are orthogonal to each other.

11.    Define a sphere. Obtain the equation of the sphere whose centre is (a, b, c) and radius is r and states the characteristics of the equation of the sphere.

Q.3 Examples    (4- marks each)

1.    Find the equation of the sphere passing through O (0, 0, 0), A (a, 0, 0), B (0, b, 0),

C(0, 0, c)

2.    Find an equation of the sphere with the centre at (1,-3, 4) and passing through the point (2, 1, 3).

3.    Find the equation of the sphere which passes through the points (2, 4, -1), (0, -4, 3) , (-2, 0, 1) and (6, 0, 9 ).

4.    Find the equation of the sphere which passes through the points (1, 2, 3), (0, -2, 4),

(4, -4, 2) and (3, 1, 4).

5.    Find the equation of the sphere which passes through the points A(1, 0, 0), B(0, 1, 0), C(0, 0, 1) and has its radius as small as possible .

6.    Find the equation of the sphere described on (2,-3, 1) and (3, -1, 2) as extrimities of a diameter.

7.    Find the equation of the sphere with centre at (-1, 2, 3) , and passing through the point (1, -1, 2).

8.    A plane passes through a fixed point (a, b, c) .Show that the locus of the foot of the perpendicular to it from the origin is the sphere.x²+ y² + z² - ax - by - cz = 0.

9.    Find the equation of the sphere passing through the points (1, 2, 3), (0, -2, 4), (4, -4, 2) and having its centre on the plane 2x - 5y - 2z - 5 = 0 .

10.    Find the equation of the sphere passing through the points A (3, 0, 2), B(-1, 1, 1),

C (2, -5, 4) and having its centre on the plane 2x + 3y + 4z - 6 = 0

11.    Find the equation of the sphere passing through the points (0, 0, 0), (-1, 2, 0), (0, 1, -1) and (1, 2, 3).

12.    A sphere of radius k passes through the origin and meets the axes in A, B, C .Prove that the locus of the centroid of the triangle ABC is the sphere g( x² + y² + z² ) = 4k².

13.    Find the co-ordinate of the centre and radius of the circle x² + y² + z² - 2y - 4z = 11,

x + 2y + 2z = 15.

14.    Find the centre and radius of the circle x² + y² + z² - 2x - 4y - 6z - 2 = 0,

x + 2y + 2z = 20.

15.    Obtain the equation of the sphere through the three points (1, -1, 1), (3, 3, 1),

(-2, 0 ,5) and having its centre on the plane 2x - 3y + 4z - 5 = 0.

16.    Find the co-ordinates of the points of intersection of the line and the sphere

, x² + y² + z² = 49

4    3    -5    ^J

17.    Show that the plane 2x - 2y + z + 16 = 0 touches the sphere

x² + y² + z² + 2x - 4y + 2z - z = 0 and find the co-ordinate of the points of contact.

18.    Show that the line    touches the sphere x² + y² + z² - 2x - 4z - 4 = 0.

Also find the co-ordinates of the point of contact.

19.    Find the equation of the tangent plane to the sphere

3(x² + y² + z²) - 2x - 3y - 4z - 22 = 0 at the point (1, 2, 3).

20.    Find the equation of the tangent plane to the sphere x² + y² + z² - 2x - 10z - 9 = 0 at the point (4, 5, 6).

21.    Find the equation of the normal plane to the sphere x² + y² + z² - 2x - 10z - 9 = 0 at the point (4, 5, 6).

22.    Show that the plane 2x - 2y + z + 12 = 0 touches the sphere x² + y² + z² - 2x - 4y + 2z = 3 and the point of contact.

23.    Find the equation of the tangent plane to the sphere x² + y² + z² - 2x - y - z -5 = 0 at the point (1, 1, -2).

24.    Find the equation of the sphere passing through the circle x² + y² + z² - 4 = 0,

2x + 4y + 6z - 1 = 0 and having its centre on the plane x + y + z = 0

25.    Find the equation of the sphere passing through the circle

x² + y² + z² + 10y - 4z - 8 = 0, x + y + z = 3 as a great circle.

26.    Find the equation of the sphere passing through the circle

x² + y² + z² - 3x + 4y - 2z - 5 = 0, 5x - 2y + 4z + 7 = 0 as a great circle.

27.    Find the equation of the tangent plane to the sphere x² + y² + z² - 6x - 4y + 10z = 0 at the origin

28.    Find the equation of the sphere passing through the circle

x² + y² + z² + 7y - 2z + 2 = 0, 2x + 3y + 4z = 8 as a great circle .

29.    Find the equation of the sphere passing through the circle x² + y² + z² = 5 x + 2y + 3z = 3 and touch the plane 4x + 3y = 15 .

30.    Find the equation of the sphere passing through the circle x² + y² + z² - 2x - 4y = 0, x + 2y + 3z = 8 and touch the plane 4x + 3y = 25.

31.    Show that the sphere x² + y² + z² = 25 and x² + y² + z² - 24x - 40y - 18z + 225 = 0 touch each other externally .

32.    Show that the sphere x² + y² + z² + 4y - 5 = 0 and x² + y² + z² - 6y + 5 = 0 touch each other externally and find the co-ordinate of the point of contact.

33.    Show that the sphere x² + y² + z² = 64 and x² + y² + z² - 12x + 4y - 6z + 48 = 0 touch each other externally and find the co-ordinate of the point of contact.

34.    Find the equation of the sphere passing through the circle

x² + y² + z² + 4x -2y + 4z - 16 = 0, 2x +2y + 2z + 9 = 0 and a given point (-3, 4, 0 ).

35.    Find the equation of the sphere passing through the circle x² + y² + z² = 9,

2x + 3y + 4z = 5 and a given point (1, 2, 3).

36.    Show that the sphere x² + y² + z² + 6y + 2z + 8 = 0 and x² + y² + z² + 8y + 4z + 20 = 0 cut orthogonally.

37.    Show that the sphere x² + y² + z² + 7x + 10y - 5z + 12 = 0 and x² + y² + z² - 4x + 6y + 4 = 0 intersect orthogonally

38.    Find the equation of the sphere passing through the circle

x² + y² + z² + x - 3y + 2z - 1 = 0 , 2x + 5y - z + 7 = 0 and cuts orthogonally the sphere x² + y² + z² - 3x + 5y - 7z - 6 = 0

39.    Find the equation of the sphere passing through the circle

x² + y² + z² - 2x + 3y - 4z + 6 = 0 , 3x - 4y + 5z - 15 = 0 and cuts orthogonally the sphere x² + y² + z² + 2x + 4y - 6z + 11 = 0

40.    Show that the sphere x² + y² + z² - 14x + 45 = 0 and x² + y² + z² + 4x - 117 = 0 touch each other externally and find the co-ordinate of the point of contact.

(Cone, Cylinder and Conicoids)

A)    State true or false and justify your answer.

1.    Equation of cone with vertex at origin is non homogenous.

2.    Every homogenous equation of degree in x, y, z represents a cone with vertex at the origin.

3.    If a, b, c are the direction ratio of any generator of the cone f(x, y, z) = 0 with vertex origin then f (a, b, c) = 0.

4.    If the number a, b, c satisfy equation of the cone with vertex origin then a, b, c are the direction ratios of some generator of that cone.

5.    A section of a right circular cone by a plane perpendicular to its axis and not passing through the vertex is a circular.

6.    The point (1, 1, 1) the vertex of the cone

5x² + 3y² + z² - 2xy - 6yz - 4zx + 6x + 8y +10z - 26 = 0.

B)    Fill in the blanks

1.    The equation of the right circular cylinder of radius 2 whose axis is the line

x-1 _ y-2 z-3 .

2 1 ^_ 2 ^lS

5x² + 8 y² + 5 z² 4 xy + (--) 8 zx + (--) 16y 14z + (--) = 0

2.    The equation of the tangent plane at the point P (-2, 2, 3) to the ellipsoid 4x² + y² + 5z² = 65 is 8x + () - 15z+() = 0

3.    The equation of the tangent plane at the point P (-2, 2, 3)to the ellipsoid 4x² + y² + 5z² = 65 is 8x - 2y - 15z - 65 = 0 then the normal at point is

x+2 _ y-2 ---

^{_ _} -15

4.    The equation of a right circular cone with vertex (2, 1, -3) whose axis parallel to Y-axis & semi vertical angle 45⁰ is x² + (---) + z² - 4x + () + () + () = 0 .

C)    Define the following terms.

1.    Cone & Guiding curve

2.    Quadric cone

3.    Right circular cone

4.    Enveloping cone of the sphere

5.    Cylinder

6.    Right circular cylinder

7.    Enveloping cylinder

8.    Tangent line & tangent plane

9.    Director sphere

10.    Normal at point & foot of the normal

D)    Multiple choice Questions:

1.    (A) Every homogeneous equation of degree in x, y, z represents a cone with vertex at the origin (B) If the numbers a, b, c satisfy equation of the cone with vertex at origin then a, b, c are the direction ratios of some generator of that cone.

a) A is false, B is false    b) A is true , B is true,

c) A is false , B is true,    d) A is true, B is false

2.    ax² + by² + cz² + 2fyz + 2gzx + 2hxy = 0 this is equation of

a)    Quadratic cone with vertex at origin    c) right circular cone

b)    cylinder when guiding curve is on XY plane d) cone with vertex at (a, fi, y)

2 2 2

3.    - - -=1, this equation represents

a) Ellipsoid b) hyperboloid of one sheet c) hyperboloid of two sheet d) none of these

4.    The equation 5x² + 3y² + z² - 2xy - 6yz - 4xz + 6x + 8y + 10z - 26 = 0 represent cone then vertex of this cone is a) (3, 1, 2) b) (1, 2, 3) c) (2, 3, 1) d) (2, 1, 3)

5.    The line - = - = touches ellipsoid + + = 1 then the point of

23 -4    8 9 4    

contact is a) ( , ,2 ) b) (2,    c) (2, , 1) d) none of these

6.    The vertex of the cone 4x² + 3y² - 5z² - 6yz - 8x + 16z - 4 = 0 is

a) (1, 1, 1) b) (2, 2, 2)    c) (1, 2, 3)    d) none of these

7.    (A) I---I--= p is condition that the plane lx + my + nz = p is tangent plane to

ABC

000    x v z

the conicoid Ax + By + Cz = 1 & (B) -2 - = 1, this equation represents hyperboloid of one sheet

a)    A is true, B is true    c) A is true, B is false

b)    A is false, B is true    d) A is false, B is true

8.    The plane 6x - 5y - 6z = 20 touches the hyperboloid 4x² - 5y² + 6z² = 40 then the point of contact is

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

9.    The plane 7x + 5y + 3z = 30 touches the ellipsoid 7x² + 5y² + 3z² = 60 then the point of contact is

a) (1, 1, 1) b) (2, 2, 2) c) (3, 3, 3)    d) ( 4, 4, 4)

10.    The plane 3x + 12y - 6z = 17 touches the conicoid 3x² - 6y² + 9z² + 17 = 0 then the point of contact is

a) (1, 2, 3) b) (-1, 2, 3) c) (-1, 2, 2/3) d) (1, 2, 2/3)

E)    Numerical problems:

1.    Define quadric cone , write down the general equation of quadric cone with vertex at the origin

2.    State the general equation of a cone with vertex at the point V(a, fi, y)

3.    Write down the condition that the general equation of the second degree should represent a cone.

4.    Write down the equation of right circular cone whose vertex at origin, semi vertical angle 9 & direction ratios of the axis a, b, c.

5.    Write the equation of the right circular cone whose vertex at (a, fi, y), semi vertical

angle & direction cosine of axis are l, m, n.

6.    State the equation of right circular cone whose vertex at origin, Z axis as the axis, semi vertical angle.

7.    Write down the condition that the plane is tangent plane to the conicoid & also write the co-ordinate of point of contact.

8.    Write down the length of the perpendicular P on tangent plane

Axx₁ + Byy₁ + Czz₁ = 1 from the origin also writes the direction cosine of the normal to the above plane.

y² z²

9.    If the conicoid is ellipsoid + - + = 1 write down the equation of the normal at

(x₁, y₁, z₁) in terms of direction cosines.

10.    Write down the equation of right circular cylinder whose axis is the line

X Xi V-Vi Z-Z-x n 1    1-=-=- & whose radius is r.

I    m    n

Q.2 Theory Questions:    ( 4 marks each )

1.    Show that the equation of the cone with vertex at the origin is homogeneous.

2.    Show that every homogeneous equation in x, y, z represents a cone with vertex at the origin.

3.    Find the equation of a cone with vertex at V(a, fi, y)

4.    Find the condition that the general equation of the second degree should represent a cone.

5.    Find the equation of the right circular cone vertex at (a, fi, y), semi vertical angle 9, direction ratios of the axis are a, b, c.

6.    Find the equation of the right circular cone satisfying the following (i) Vertex at origin, semi vertical angle 9, direction ratios of the axis a, b, c. (ii) Vertex at

(a, fi, y), semi vertical angle 9, direction cosine of the axis l, m, n. (iii) Vertex at origin, semi vertical angle 9, direction cosine of the axis l, m, n.

7.    Find the equation of a right circular cone satisfying the following (i) vertex at the origin, z-axis as the axis, semi vertical angle 9 (ii) vertex at the origin, X-axis as the axis, semi vertical angle 9 (iii) vertex at the origin, Y-axis as the axis, semi vertical angle 9

8.    Find the equation of the tangent plane to the cone at P (x₁, y₁, z₁).

9.    Find the equation of cylinder when guiding curve is on XY plane whose generators are parallel to the line - = = -

l m n

10.    Find the equation of cylinder whose generators intersect the guiding curve & parallel to the line - = = -

l m n

11.    Find the equation of right circular cylinder whose axis is the line & whose radius is r.

12.    Find the point of intersection of the line    ₌ _Ei & the conicoid

l    m    n

Ax² + By² + Cz² = 1

13.    Find the condition that the line    may be tangent to the conicoid

l    m    n

Ax² + By² + Cz² = 1

14. Find the condition that the line lx + my + nz = p is tangent to the conicoid Ax² + By² + Cz² =

Find the equation P (x_1; y_1; z₁) on it.

16. Find the equation of normal to the conicoid Ax² + By² + Cz² = 1 at the point P (x_1; y_1; z₁) on it.

Q.3 Examples    ( 4 marks each )

1.    Find the equation of cone whose vertex is at the origin & which passes through the curve given by the equation ax² + by² + cz² = 1 & lx + my + nz = p.

2.    Prove that the equation of the cone whose vertex is the origin & base the curve z = k, f(x, y) = 0 is/(f, Zt) = 0.

3.    The plane + ⁼ 1 meets the co-ordinate axis in points A, B, & C. Prove that the equation of the cone generated by the lines drawn from the origin to meet the circle

^{ABC is} yzg + ) + zx(f + f) + xy(f + i) =

4.    If O is the origin, find the equation of the cone generated by the line OP as the point P describes the curve whose equation are x² + y² + z² + x - 2y + 3z - 4 = 0 ,

x² + y² + z² + 2x - 3y + 4z - 5 = 0.

5.    Obtain the general equation of the cone which passes through the three axes.

6.    Obtain the equation of the cone which passes through the axes & the lines = |

& ₌ Z ₌ .

-3 1 -2

7.    Obtain the equation of the cone which passes through the axes & the lines - = - = -& * = = . ¹ 2 3

3 1-4

X V z

8.    Obtain the equation of the cone which passes through the axes & the lines - = = -& * = = *.

1-13

9.    Find the equation of the cone with vertex at the origin & containing the curve x² + y² = 4 & z = 5.

10.    Examine whether the following equation represents a cone

5x² + 3y² + z² - 2xy - 6yz - 4xz + 6y + 8y + 10z - 26 = 0, if it represents a cone, find its vertex.

11.    Find the equation of right circular cone with its vertex at (1, -2, -1) semi vertical

i    _c x-1 y+2 z+1

angle 60 & axis    = .

12.    Find the equation of right circular cone passes through the point (1, 1, 2) has its axis the line 6x = 3y = 4z & vertex at the origin.

13.    Find the equation of the cone with its vertex at the origin & passing through the curve

x² + y² + z² - 2x + 2y + 4z - 3 = 0 & x² + y² + z² + 2x + 4y + 6z - 11 = 0.

14.    Find the enveloping cone of the sphere x² + y² + z² - 2x + 4z - 1 = 0 with its vertex at (1, 1, 1).

X V Z    2 2 2

15.    Verify that the line - = = - is the generator of the cone x + y + z + 4xy - xz = 0

16.    The line 3x + 2y - z = 0 , x + 3y + 2z = 0 is a generator of the cone 2x² + y² - z² + 3yz

-    2xz + axy = 0, find the value of a.

17.    Prove that the equation 4x² - y² + 2z² + 2xy - 3yz + 12x - 11y + 6z + 4 = 0 represents a cone whose vertex is (-1, -2, -3).

18.    Prove that the equation x² - 2y² + 3z² - 4xy + 5yz - 6xz + 8x - 19y - 2z - 20 = 0 represents a cone & find the vertex.

19.    Show that the equation x² - 2y² + 4z² + 4xy + 6yz - 2zx + 6x - 30y + 14z = 0 represents a quadratic cone & find its vertex.

20.    Examine whether the following equation represents a cone

4x²+ 3y² - 5z² - 6yz - 8x + 16z - 4 = 0 if it represents a cone find its vertex.

21.    Prove that the equation ax² + by² + cz² + 2ux + 2vy + 2wz + d = 0 represents a cone if

7    7    7