Super Sixer,
SECTION
V1, ) Find the value of x, ifthe slope of the line passing through (2, 5) (x, 3) is 2.
(1) Find the equation of thestraight line passing through (- 4,5) and cutting of equal
intercepts on co-ordinate axes.
() Find the equation of the straight linepassing through the points (ar, 2at,). (at, 2at
(iv) Transform the equation 3x +y=4 into
a) Slope - intercept form b) Intercept form c) Normal form.
(v) Transformthe equation x +y+l=0 into normal form.
(vi) Findthe area of the triangle formed by the linex cos a. + y sin a =p with co-ordinate axes.
/2. () Find the value of 'p' if the lines 3x + 4y =5, 2x +3y =4, px + 4y =( 6 are concurrent.
al 2 (ü) Findthe point of concurrenceof (2+ 5k) x-3(1 +2k) y+(2- k) =0 1)20
(Gi) Find the valueof k, if the angle between the lines kx +y+9 =0, 3x -y +4=0is 4
ashts (iv) Find the distance betwen the parallel lines Zx 3y -4=0 and 10x 6y -9 =0.
Find thevalue of'p'ifthe straight lines 3x +7y - l=0,7x -py +3 =0 are mutually
perpendicular.
(vi) Find the equation of the straight line perpendicular to the line 5x 3y +1=0 and
passing through (4, - 3) 4|2
23/3. (i) Find x, if the distance between (5, - 1, 7) and (x, 5, 1) is 9 units.
(ü) Show that the points (1, 2, 3) , (2, 3, 1), (3, 1, 2) form an cquilateral triangle.
() For what values oft,if the points (2, 1, 3) (3, 5, t)(-1, 11, 9) are collinear ?
(iv) In AABC, the centroid is the origin and the vertices of A, Bare (1, 1, I) and (-2, 4, 1)
respectively then find C.
(v) If(3,2, - 1) (4,1, 1) and (6, 2, 5) are three vertices and (4, 2, 2) is the centroid of a
tetrahedron then find the fourth vertex.
(vi) Find the fourth vertex of the parallelogram whose consecutive vertices are
(2,4,- 1), (3,6,- 1I) and (4,5, 1)
/4. ) Reduce the equation x+2y- 3z-6 =0 of the plane tonormal form.
(ü) Find the d.c's of the normal to theplane x + 2y + 2z- 4 =0.
(üi) Write the cquation of the plane 4x 4y +2z +5 =0into intercept form.
(iv) Findthe angle between the planes x+ 2y +2z -5 =0and 3x+ 3y +2z- 8=0.
(v) Find the angle between the planes 2x-ytz= 6 and x + y + 2z=7
(vi) Find the cquation of the plane passing thr ough (1, 1, 1) and parallel to the plane
x+ 2y +3z-7=0,
211
Jr Mathematics IB Super Sixer
Eraluste the following :
b) 3- 1
X I
X01+x- 6o) c)
Xsin a -a sin x
b) It
X-a
tan(x-a) S 19
() jp) b) It sin(x -a)tan (*-a)
Xa
sin (a+bx)-
-sin(a -bx) h
X
sinax
X’0 XCosx
e'-sin x-1 e'-e'
()a) X b) It
I’3 x-3
(v3)a) t
COS ax-cos bx L3|1-3
2 1-cos mx
b) -0
It 1-cos nx
n0
Evaluatethefollowing:
11x-3x +4
76. 0) a) t 13x-5x-7 b) lim o4
2x'-1
() a)
It
5x+4
() a) 101 b) It +x*1 3y
X0+ X
,al ot
(iv) a) It (1+x)--x/8
X
b) It
X0
() Is f, defined by f(x) = x if x>1' is continuous on R.
sin2x
, if x *0
(vi) If fis dfined by f(x) = continuous at '0'?
1 ifx=0
/7 0) a)If f() e log (3x +4), then find f'()
sinx, then find f's), )w
b) Iff()=xe'
ro)
) Ir O)-1tx+x'tont then find
)If fo)ya'sx then find r'()
(yiro)-2logs, find r'() 10
lsin(ogx), find/3
(0Iy-log dx
dy 21/13
() Ify=log (scox +tanx), find d
23)M
V8. @ y =se(Vanx) find
fewa}, 32
@ Fdthedrivative of
3)
(OFindthe derivative ofsin (x-) wrt
(v) Find the derivative of sinh
(Ifx=a cost,y=a sin't find dy
dx
(6) If y = a+be, then prove that y =n'y
J9. 0 f y=x+3x +6, then find Ay and dy when x= 10, Ax =0.01l
Find Ay and dy of y= f)=+x at x=10 when Ax = 0.1
(n Find Ay and dy for y= 5x+6x +6 ,x=2, Ar =0.001
() Find theapproximate value of 82
()If the increase in the side of a square is 4% find the percentage of increase in the area of the
Square.
() lf the increase in the side of a square is 2% then find the approximate percentage of increase in
Its area
S10.0 Verify Rolle's theorem for the function x?-lon [- 1, ]
O Verify Rolle's theorenm for thefunction flx) = - 5x+6 in the interval [E
( Veify the condition of the Lagrange's mean value theorem for the function
fx) =x-lon [2, 3]
(v) Find 'c for the function f(x) = N on [2, 4] by Legrange's theorem.
(v) Verify the condition of theLegrange's theorem for the function -I on 2, 3]
(vi) Show that there is no real number kfor which the equation - 3x +k=0 has two
distinct roots in (0, 1)
SECTION
/1.0 Find the equation of locus of apoint P such that the distance of P from the origin is twice the
distance of P from A(1, 2)
(0) A(1,2) B (2,-3) C(-2,3) are three points. Apoint P moves such that PA + PB=2PC
Findthe Locus of P. 15
(m) Find the locus ofP such that the line segment joining (2,3) &(1,5)
subtends a rightangle
at P.
(iv) The ends of the hypotenuse of aright angled triangle are (0,6) and (6, 0). Find the
cquation of the locus of its third vertex.
(v) Find the equation of Locus of P,ifA(5 ,0) ,B(-5,0) and PA -
PB| = 8.
(vi) Find the equation of locus ofP, ifA (4, 0), B(- 4, 0) and|
PA- PB|=4.
Jr Mathematics - IB 213)
Super Sixer
/12.) Whenthe origin is shifted to (- 1, 2) by the translation of axes, find the transformed
equation of x +y+ 2x- 4y + |
=0,
When the origin is shifted to the point (2.3),thetransformed cquation of a curve is
2+ 3xy- 2y" + 17x -7y- 11=0. Findthe
(i) When the axes are rotated through an angle TAoriginal
then equation of the curve. equation of
find the transformed
3x+ 10xy +3y' =9
(v) When the: axes are rotated 5)
through an angle 6 Find the transformedequation of
x'+ 2N3xy-y' =2a'
(v) When the axes are rotated through an angle 45° the
17-16xy + 17y =225. Findthe original equation.transformed equation of a curve s
(vi).Show that the axes are to rotate 2h
through an angle 1 tan So as to remove the xy
2 a-b
term from the equation ax+ 2hxy + by² =0. if a +b If a=b.
and through the angle
4
X
/13.0) Transform the equation +=1 into normal form, If the perpendicular distance of the
b
straight line from origin is 'p' deduce that 1 1 a+ 1
p a'
() Find the value of k,if the lines 2x -3y +
are concurrent.
k=0,3x-4y -13 =0,8x -lly -33 =0.
(m) Find theequation of the line perpendicular to the line 3x + 4y
+6=0 and making an intercept-4 on
the X- axis.
(iv) Find the equation of the straight line passing through the point of
intersection of the lines
x+y+1=0and 2x-y +5=0 and containing the point (5, - 2)
(v) Find the value of kif theangle between the straight lines 4x-y
+7=0,kx- 5y-9 = 0is 45°
(vi) Findthe image of (1, - 2) w.rtthe straight line 2x- 3y + 5 =0.
1
/14.() Check the continuity of fgiven by f(x) =2 -4) , ifx<2 at the point 2.
|2-8x3 ,if x>2
(ü) Check the continuity of the following function given by
-9 , if 0<x<3and x#3
f(x) = x²-2x-3 at the point 3.
1.5 if x=3 |sin x ,if x s0
(u) Find the real constants a, b so that the function f given by f(x) = x+a ,if 0<x<
bx +3,if I<x<3
is continuous on R.
cosax-cosbx
,ifx #0 e
-3if x>3
(iv) Show that f(x) = where a,b are real ,is
,if x = 0 continuous at'0'.
Jr Mathematics - IB 214
Super Sixer
Kx-k, if x>1;is a continuous function on R. then find'k'.
(v) Iff,given by f(x) = 2 if x<1
bri
x+2 if -f<x<3
(v) Check whether the function has lmit exists or not, f(x) = |x' if 3<x<sas3
V15.0) Find the derivatives ofthefollowing using first principle.
a) sin 2x b) cos ax c) tan 2x d) sec 3x e) x sin x bri
dy
() If x=a (cost +tsint), y=a(sint - tcost) find dx
2x dy
(m) If y = tan find
dx (iv) If x= X-y. find dy + z
d b
(v) If y=x, then show that dy y'
dx
x(1-y logx)
(vi) Ify= axt+ bx then prove that xy" = n(n + )) y
V16.(1) The distance -time formula for the motion of a particle along a straight line is
S=p-9t+ 24t 18. Find when and where the velocity is zero.
(ü) The displacement 's' of a particle travelling in a straight line in,t seçonds is given by
s=45t+ llt' -t. Find time the particle comes to rest.
4/16
(iü) The volume of acube is increasing at the rate of 9 cubic centimeters per second. How
fast
is the surface area increasing when the length of the edge is 10 centimeters? si!
(iv) The volume of a cube i_ increasing at the, rate of 8cm/sec. How fast is the
surface arca
increasing when the length of an edge is l2 cm?
(v) A container is in the shape of an inverted cone has height 8m and radius
6m at the top. If it
is filled with water at th¹ rate of 2m³/minute, how fast is the
height of water changing when
the level is 4m?
(vi) A point P is moving on the curve'y=2x. The
x-coofdinate of Pis increasing at the rate
of 4 units per second. Find,the rate at which the y -
is at (2, 8).
coordinate is increasing when the point
V17.) Find the equation of tangent ¬mal to the curve y=5x* at (1, 5)
(i) Find the equation of the tangent and the normal to the
curve y*= ax at (a, a)
(1i) Show that the tangent at any point on the
curve x =C sec , y=ctan ) is
y sin 9 =X-c cos .
(iv) Findthe lengths of subtangent and subnormal at a point on the
curvey bs i n 2
() Show that at any point (x, y) on the curve y =
bea the length of subtangent is a constant
and the lenght of the subnormal is y/a.
(vi) Show that the curves 6x- 5x + 2y =0 and 4x +
8y =3touch cach other at 1
Jr Mathematics -IB 215
SECTION C
/18.() Find the circumcenter of the
triangle whose vertices are
i)e2,3),(2,-1),(4,
() Find the 0)222 i) (1,3),(-3, 5), (5,-1)
orthocenter the triangle whose
of
vertices are
i)-2, -1) (6,- 1) (2, 5)S24 i) (6,-2),(- 1,
in) Find the
circumcenter of the triangle whose sides are 2), (1, 4)
i) 3x- y-5=0, x+ 2y- 4
=0, 5x+ 3y +1=0
)xty=0,2x +y+5=0, x-y=2
(iv) Find the orthocenter of the
triangle whose sides are given by
i) x+ y+ 10 = 0,x - y - 2 =
ii) 7x +y - 10=0, x-2y +5=0, x
0,2x + y -7 =0 1239
+y+2=0.
(v) IfQ (h, k) be the foot ofthe perpendicular from P (x, , y,)on the
straight line
ax + by +tc=0,then show that h - X1 k- y1 (ax1+ byj+ c)
b +b
(vi) If Q (h, k) is the image of P (x,.v.) with respect to the line ax + by +c =0
then prove that h-x1k-y| -2 (axj+byj+c)
a
a+b
/19.0) Show that the equation to the pair of bisectors of angles between the pair of
lines ax + 2hxy +by=0 is h(x- ) =(a -b) xy
() Show that the product of perpendiculars from (a,B) to the pair of lines
+2haß + bB
ax +2hxy+ by' =0 is
(n) If e is the angle between the pair of lines represented by
ax'+ 2hxy + by² = 0then prove that cos 8 = ya-b)+4h2
la +bl
23/
lines
(iv) Show that the area of the triangle formed by the
-ab
ax + 2hxy + by = 0 and Ix + mytn=0 is am-2hlm+bl²
represents a pair of
(v) Ifthe equation ax + 2h xy + by" + 2gx + 2fy + c=0
(i) af =bg' and
parallel lines then prove that ) h=ab
(11i) the distance between the
parallel lines =
g-ac 2 2-be
´a(a+b)b(ab)
2x-13xy-7y +x+ 23y - 6=0 represents a pair of straight lines.
(vi) Show that the equation
between them and the coordinates of the point of intersection ofthe lines.
Also find the angle
216
SUper Sixer
/20.0) Find theangle betwecen the lines joning theorigin to the points of intersection of the curve
'+ 2xy+y +2x + 2y - 5 = 0 and the line 3x-y +1 = 0 j2
(n) Show that the lines joining the origin to the points of intersection of the curve
x-xy t y+ 3x +3y -2 = 0 and the line x-y- J =0 are mutualy perpetidiular.3|6 )
(m) Find the value of k, if thelines joining the origin to the points of intersection of the cnrve
2x- 2xy +3y +2x -y-l = 0and the line x+ 2y = kare mutually perpendicalar.}" |o
(iv) Show that the lines joining the origin to the points of intersection of the curve
7x-4xy + 8y´ + 2x- 4y - 8 = 0 with the line 3x - y = 2are mutually perpenidicular.
(v) Write down theequation of the pair of straight lines joiningthe origin to the points of69
intersection of the line 6x - y +8=0with the pair of straight lines
3x*+ 4xy - 4y- 1lx + 2y + 6 = 0. Show that the lines so obtained make equal angles with the
coordinate axes.
(vi) Findthe condition for the chord Ix +my =I ofthe circle x +y'=a' (whose cefter is the
origin) tosubtend aright angle at the origin.|t
/21.()) Find the angle between the lines whose direction cosines satisfy the equations
m+n=0,P+m² -n' =0|g bu f
(ü) Find the angle between the lines whose direction cosines are given by the relations
91 3l +m +5n =0, 6mn - 2n/ +5lm =0.
(11) Find the direction cosines of two lines which are connected by the relations
2)l+m+n=0 and mn - 2n/ 2/m =0
(iv) Find the direction cosines of two lines which are connected by the relations
u40 - Sm + 3n = 0, and 7 + Sm'-3n=0.
(v) Findthe angle between the two diagonals of acube.
(vi) If a ray makes angles a, B, 1, 8with the four diagonals of a cube then find
a cos a +cos?B+cos?y+ cos? 8
/22.0-1f i-' +-y' =a(x-y) then prove that dydx Vi-x?
(Af y = xa+x'+a'log x+a? +x then show that ydx -2a+x?
().If x' + y' = a then prove that dy yx-y'logy
dx logx +xy*-l
(iv) Find the derivative of a) x lanx + sin x c0sX b) sinx lops + x six
X-B gx) = tan -1X-B
(v) If f(x) = sin-l Va-ß then prove that f'(x)=g(x)
Va-x
/= tan,-1 then find dy
dx
217
Super Sixer
as0 thetangent at any point on the curve g2 + y2 a interscçts the
coordinate axesin
Aand B, then show that
the lenoth AB is a 7/148 19
constant. mccts the coordinate axes in
)Ithe tangent at any point Pon the
curve x"y =atn
y
2/145
A, B then show that AP: BP IS a constant.
At any pont t on the curve x = a(t +sin t) y = a(l|- cost). Findthe lengths of tangent,
nomal, subtangent and subnormal. 3)53
(v) Find the lengths of subtangent , subnormal at a point 'e on the curve
X= a(cos t + tsint) , y= a (sin t t cos t)
atv) a), 0) Define angle between the curves.
Q(ii) Find the angle between the curves xy =2,x'+4y=0
b) Find the angle between the curves y² =8x and 4x2 +y² =32
10()Show that the curves y = 4(x +1) and y 36 (9 - x) intersects orthogonally.
D4.0 a) The sum of two numbers is 16. Find the numbers so that the sum of squares is
minimum.
b)Fina tWopositive integers x and y such that x + y= 60and xy is maximum.
() Awindow is in the shape ofa rectangle surmounted by a semicircle. If the perimeter of the
window be 20 ft, find the maximum area.
of radius
(m) Show that when the curved surface of a right circular cylinder, inscribed in a sphere
Ris maximum, then the height of the cylinder is 2R 39
form of a square
(iv) A wireof length l cut into two parts which are bent respectively in the
of areas is least?
and a circle. What are the lengths of pieces of wire so that the sum
(v) The profit function P(x) of a company sclling x items per day is given by
the company should manufacture
P(x) =(150 - x) x - 1000. Find the number of items that
profit.21
to get maximumprofit. Also find the maximum square of sidexcm, are
cm, four equal
(vi) Fromarectangle sheet of dimension 30 cm x 80
up so as to form an open rectangular
removed at the coners , and thesides are then turned
the box is grcatest.
box. Find the value of x, so that the volume of
218
