2013- 2014 (1B)

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Amis tutorial 1st year Mathematics 1B.

1

VSAQ

Q.No .1 and 2

1. If area of triangle formed by lines x = 0: y = 0 and3x + 4y = a is 6 sq units. Find a

2. Transform the equation x+y+1=0 into normal form.3. Find the equation of line passing throughA(-1,3) and perpendicular to the straight line passing

through B(2,-5) and C(4,6)

4. Find the value of y if the line joining (3,y) and (2,7)is parallel to the line joining the points

(-1, 4) and (0, 6).

5. Find condition for points (a,0) (h,k) and (0,b) becollinear

6. find distance between parallel lines given by 5x-3y-4=0 and 10x-6y-9=0

7. If is angle between lines x/a +y/b=1&x/b+y/a=1 findthe value of sin

8. Find value of x if slope of the line joining points(2,5) and (x,3) is 2.

9. Find the equation of the straight line passing through (-4,5) and cutting off equal intercepts on the coordinate

axis.

10. Find the value of k if straight lines 6x-10y+3=0and kx-5y+8=0 are parallel.

11.Transform the following equation 3 4 0x y into i) slope intercept form ii) normal form

Q.No .3

1. If (3, 2, -1), (4, 1, 1), (6, 2, 5) are three vertices and (4,2, 2) is the centroid of a tetrahedron, find the fourth

vertex.

2. Find the fourth vertex of the parallelogram whoseconsecutive vertices are (2, 4, -1), (3, 6, -1), (4, 5, 1).

3. Find the ratio in which YZ-plane divides the line joining A(2, 4, 5) and B (3, 5, -4). Also find the point of

intersection.

4. Find the x if the distance between (5, -1, 7) and (x, 5, 1)is 9 units.

5. Show that the point A (-4, 9, 6), B (-1, 6, 6) and C (0, 7,10) form a right angled isosceles triangle.

6. Find the coordinates of the vertex C of triangle ABC if itscentroid is the origin and the vertices A, B are (1, 1, 1)

and (-2, 4, 1) respectively.

7. For what value of t the points (2, -1, 3), (3, 5, t), (-1, 11,9) are collinear.

Q.No .41. Find intercepts of plane 4x s+3y-2z+2=0 on the axes.2. Find the angle between planes

x+2y +2z-5=0, 3x+3y+2z-

8=0. (ii) 2x-y+z=6, x+y+2z=7.

3. Reduce the equation x+2y-3z-6=0 of the plane into thenormal form.

4. Transform the equation 4x-4y+2z-5=0 into interceptform.

5. Find the equation of the plane whose intercepts on X, Y,Z-axes are 1, 2, 4 respectively.

6. Find equation of plane passing through point (1,1,1)and parallel to the plane x+2y+3z-7=0.

7. Find equation of plane passing through point (-2, 1, 3)and having (3, -5, 4). As direction ratios of its normal.

Q.No .5 and 6

1. Compute

ax

xaax

ax

sinsinlim

2. Compute

11

13lim

0 x

x

x

3. Compute

xx

x

x 11lim

0

4. Compute20

1 cos 2lim

sinx

mx

nx

5. Computex

bxabxa

x

)sin()sin(lim

0

6. Compute 22

5 2lim

2 5 1x

x x

x x

7. Compute

11

1lim

0x

ex

x

8. Compute 8 3lim

3 2x

x x

x x

9. If the function f is defined by2 if x 1

( ) 2 1 x it is continuous on R

find k

k x k

f x

10. Show that 2

limx

x x x

=1/2

11. .12. Find .13. Evaluate .14. Evaluate .

Q.No .7 and 8

1. Differentiate esinx with respect to sinx.2. find derivative of

2xe logx w.r.t x

3. If y = ),( axxaxa

find dx

dy

4. If y= log(sin-1x) find dy/dx5. If f(x) = log ),tan(sec xx find f(x)6. if y=Cos(log(cotx)) find dy/dx7. find derivative of y= cos-1(4x3-3x)8. 1 1 costan

1 cos

xy

x

find dy/dx

9. if x=at2 y=2at find dy/dx10. if y=xx find dy/dx11. 1Find derivative of sin tan xe 12. If x= a(cost+tsint) y=a(sint-tcost) find dy/dx13. find dy/dx 3 3if x=3cost-2cos y= 3sint-2sint t14. If f(x) = 3 37x x then find )(xf .

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15. sinx dyif y=x finddx

16. if 3 3 3 dyx finddx

y a

Q.No .9

1. Find the approximate value of

,

, sin6001,

.

2. Find dy and if y=x2+3x+6, when x=10, 3. Find dy and if y= when x=8, 4. Find dy and if y=x2+x, when x=5, 5. If the increase in the side of a square is 4% find the % of

change in the area of the square.

6. The diameter of a sphere is measured to be 40cm, if anerror of 0.02cm. Occurs in this, find the error in volume

and surface area of the sphere?

Q.No .10

. Verify Rolles Theorem for the function f(x) =x2 +4 on [-3, 3].

. Verify Rolles Theorem for the function f(x) =x(x+3) e -x/2 on [-3, 0].

. Verify Rolles Theorem for the function f(x) =sinx-sin2x on[0,].. Verify the conditions of the Lagranges mean value theoremfor the following functions in each case find a point c in the

interval as stated by the theorem.(i)x2-1 on[2, 3], (ii) sinx-

sin2x on [0,].SAQ Q.No .11

1. A (2, 3), B (-3, 4) are two points. Find the equation oflocus of p so that the area of the triangle PAB is 8.5.

2. Find the equation of the locus of P, if the ratio of thedistances from P to A(5, -4), and B(7, 6) is 2:3.

3. The ends of the hypotenuse of a right angled triangle are(0, 6), and (6, 0). Find the equation of its third vertex.

4. A(1, 2), B(2, -3) and C(-2, 3)are three points. A point Pmoves such that PA2+PB2=2PC2.show that the equation of

the locus of P is 7x-7y +4=0.

5. Find the equation of locus of P, if A(4, 0),B(-4, 0) and |PA-PB|=4.

6. Find the equation of locus of P, so that the distance fromthe origin is twice the distance of P from A (1, 2).

Q.No .12

1. When the origin is shifted to the point (2, 3),thetransformed equation of the curve is Find the original equationof the curve.

2. When the origin is shifted to the point (-1, 2), by thetranslation of axes. Find the transformed equation

of .3. When the axes are rotated through an angle 300,find thetransformed equation of

4. When the axes are rotated through an angle 450,find thetransformed equation of

5. When the axes are rotated through an angle 450, thetransformed equation of the curve is find the original equation of the curve.

6. When the axes are rotated through an angle , find thetransformed equation of

7. Show that the axes are rotated through an angle of * + so as to remove the xy terms from theequation and through theangle

, if a=b.Q.No .13

1. If 3a+2b+4c=0, then show that ax+by+c=0represents a family of lines and find point ofconcurrency .

2. Find the equation line perpendicular to the line3x+4y+6=0 making an intercept -4 on the X-axis.

3. Find the equation of lines passing from point ofintersection of lines 3x + 2y + 4 = 0; 2x + 5y = 1 and

whose distance from (2, -1) is 2.

4. Find the equation of the straight line parallel to the line743 yx and passing through the point of intersection

of the lines 032 yx and 063 yx .

5. Find the value of k, if the angle between the straight lines4x y + 7 = 0 and kx 5y- 9 = 0 is 45.

6. If the straight lines 0,0 acybxcbyax and0 baycx are concurrent, then prove that a3 + b3 + c3 =

3abc.

7. Find the value of k, So that straight lines y 3kx + 4 = 0and (2k 1)x (8k-1) y 6 = 0 are perpendicular.

8. Find the equation lines passing from point (-3,2) andmaking an angle 45 with the line 3x-y+4=0

9. If the perpendicular distance of the straight line from theorigin is P, deduce that

222

111

baP

Q.No .14

1. S.T f(x) =

where a and b are real

constants, is continuous at 0.

2. Check the continuity of f given by { ()

3. Check the continuity of following function at 2

4. Find real constants a,b so that the function f given by

f(x)= is continuous on R5. Is f defined by

6. Is f defined by

7. , ()()

8. Compute xbxabxa

x

)sin()sin(lim

0

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9. sin sinShow that lim sina - acosax a

x a a x

x a

10. Compute1 1

8 8

0

(1 ) (1 )limx

x x

x

Q.No .15

1.

Find the derivative from First principle of function,, , . Logx2. If , then show that .3. If , then S.T .4. If f(x) = * +,g(x)= then differentiate f(x)

with respect to g(x).

5. If then P.T .6. If= .7. If .8. If y= * + *+ * + then show

that .Q.No .16

Find the equation of tangent and normal to the

(i) Curve y=5x4 at the point (1, 5).

(ii) Xy=10 at the point (2, 5).

(iii) Y= at the point (-1, 3).1. Show that length of subnormal at any point on the curve ( )2. Find the tangent and normal to the curve y=2 at the

point where the curve meets the Y-axes.

3. Show that the tangent at p(x1, y1) on the curve

.

4. Find the lengths of normal and subnormal at a point onthe curve y= .

5. At any point t on the curve x=a(cost+tsint)y=a(sint-tcost) find lengths of sub tangent ,subnormal.

Q.No .17

1. he distance time formula for the motion of particlealong straight line is s2=t3-9t2+24t-18 find when and

where velocity becomes zero.

2. A container in the shape of an inverted cone has height12cm and radius 6cm at the top. If it is filled with water at

the rate of 12cm3/sec, what is the rate of change in the

height of water level when the tank is filled 8cm?

3. A stone is dropped into a quiet lake and ripples move incircles at the speed of 5cm/sec. at the instant when the

radius of circular ripple is 8cm. how fast is the enclosed

area increases.

4. A balloon, which always remains spherical on inflation, isbeing inflated by pumping in 900c.c of gas per second.

Find the rate at which the radius of balloon increases

when the radius is 15cm.

5. The radius of a circle is increasing at the rate of0.7cm/sec. what is the rate of increases of it

circumference.

6. The radius of an air bubble is increasing at the rate of .at what rate is the volume of the bubbleincreasing when the radius is 1cm?

7. Suppose we have a rectangular aquarium with dimensionsof length 8m, width 4m and height 3m. suppose we are

filling the tank with water at the rate of 0.4m 3/sec. how

fast is the height of water changing when the water level

is 2.5m?

8. The volume of a cube is increasing a rate of 9 c.c persecond. How fast is the surface area increasing when the

length of the edge is 10c.ms?

9. Let a kind of bacteria grow in such a way that at time tsec. there are t3/2 bacteria. Find the rate of growth at time

t=4 hours.

Q18.Straght lines

1. Find the orthocenter of the triangle whose vertices are (-5, -7),

(13, 2), (-5, 6)

2. If the equations of the sides of the triangle are 7x+y-10=0, x-2y+5=0, x+y+2=0, find the orthocenter of the

triangle.

3. Find the circumcentre of the triangle whose vertices are(1, 3), (-3, 2), (5, -1)

4. Find the circumcentre of the triangle whose sides are3x-y-5=0, x+2y-4=0, 5x+3y+1=0.

5. If Q(h, k) is the foot of perpendicular from p(x1, y1) onthe line ax+by+c=0 then P.T

.findthe foot of perpendicular from p(4, 1) up on the line 3x-

4y+12=0.

6. If Q(h, k) is the image of p(x1, y1) w.r.to straight lineax+by+c=0 then P.T

.and find theimage of (1, 2)w.r.t 3x+4y-1=0.

7. If P and Q are lengths of perpendiculars from the originto the straight lines

X

.Q19.Pair of straight lines

1. Find the value of k, if the lines joining the origin to thepoints of intersection of the curve

2x2-2xy +3y2 +2x y -1=0 and the line x + 2y =k are

mutually perpendicular.

2. Show that the pair of straight lines6x2-5xy-6x2=0, 6x2-5xy-6x2+x+5y-1=0 form a square.

3. Show that the lines joining the origin to the points ofintersection of the curve

x2-xy +y2 +3x +3y -2=0 and the line

x -y-=0 are mutually perpendicular.4. Find the angle between the lines joining the origin to thepoints of intersection of the curve

x2+2xy +y2 +2x +2y -5=0 and

the line 3x -y =0.5. Find the centroid and area of the triangle formed by the

lines 12x2-20xy -7y2=0, and 2x-3y +4=0.

6. Find the condition for the the lines joining the origin tothe points of intersection of the circle

x2 +y2=r2 and the line lx+my=1 to coincide.(right angle

at the origin)

7. Find the value of k, if the equation

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2x2-kxy -6y2 +3x +y+1=0 represents a pair of straight

lines. Find the point of intersection of the lines and the

angle b/w the straight lines for this value of k.

Q20.Theorems:

(i) Cos=

|

(ii) h( )=(a-b)xy.(iii) Area= (iv) | |(v) =abc +2fgh a , (vi) , b 2 or 2 .(vii) Form an equilateral

triangle with .Q21direction cosine and direction ratios

1. If a ray makes angles with four diagonals of acube ,

S.T 2. Find the angle between two lines which are non-parallel

and whose direction ratios satisfy the equations

(i) 6mn+5lm-2ln=0, and 3l+m+5n=0.(ii) L +m + n=0, l2+m2 - n2=0(iii) L +m +n=0, 2lm +2nl-mn=0.(iv) L +m +n=0, 2mn +3nl-5lm=0.(v) Find the angle b/w two diagonals of a cube.

Q22Differentiation

1. If y=tan-1* + , find the value of2. If y = , find the value of3. If xy +yx = ab, S.T *+4. If + =a(x-y), find 5. If y = x +a2log[x + ], find 6. If f(x) =sin-1, g(x) =tan-1,

S.T f(x) =g(x).

7. F(x) =

,

S.T f(x) = (a+bcosx)-1.

9. If y= find.

Q23tangents and normal

1. If the tangent at any point on the curve Intersects the co ordinate axes in A, B. S .Tthe length AB is constant.

2. If the tangent at any point on the curve

Intersects the co ordinate axes in A, B, Showthat AP: PB is constant.

3. Show that the curves y2 = 4(x+1) and y2=36(9-x)intersect orthogonally.

4. at any point t on the curve x= a(t + sint),y=a (1-cost), find the length of tangent, normal, sub

tangent, subnormal.

1. At any point t on the curve x=a(cost + t)y=a( t cost) find lengths of sub tangent,subnormal.

5. Find the angle between (i)y2 = 4x, x2 + y2 =5, (ii)xy=2,x2+4y=0.

Q24.Maxima and minima

1. A wire of length l is cut into two parts which are bentrespectively in the form of a square and a circle. What are

the lengths of pieces of the wire so that the sum of the

areas is the least?

2. Find two positive numbers whose sum is 16 and sum ofwhose squares is minimum.

3. From a rectangular sheet of dimensions 30cmx80cm fourequal squares of side x cm are removed at the corners,

and the sides are then turned up so as to form an open

rectangular box. Find x so that the volume of the box is

the greatest.

4. If the curved surface of right circular a cylinder inscribedin a sphere of radius r is maximum, show that the height

of the cylinder is r.5. A window is in a shape of a rectangle surmounted by a

semi circle. If the perimeter of the window is 20 feet, find

the maximum area.

6. Prove that the radius of the right circular cylinder ofgreatest curved surface area which can be inscribed in a

given cone is half of that of the cone.

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