Maths IB Important
9 likes14,531 views
The document is a mathematics question bank focused on various topics including geometry, limits, continuity, and differentiation. It contains numerous problems and exercises categorized by difficulty, such as very short answer questions and more complex scenarios involving straight lines, planes, and applications of calculus. The content is intended for students studying mathematics, particularly those preparing for examinations.
![Mathematics –I (B) Questions Bank4
04. Compute )1b,0b,0a(
1b
1a
0x
Lim x
x
05. Compute
1x1
1e
0x
limL
x
06. Evaluate 2
x
bxcosaxcos
0x
Lim
07. Compute )xxx(
x
Lim 2
08. Find
2
x
xcos
2
x
Lim
09. Evaluate
7x5x13
4x3x11
x
Lim 3
3
10. Compute )Zn,m(,
nxsin
mx2cos1
0x
Lim 2
11. Evaluate
1x
)1xsin(
1x
Lim 2
12. Find
x
x1x1
0x
Lim
33
13. Find
222
2
)ax(
)ax(tan)axsin(
ax
Lim
14. Show that 1
2x
2x
2x
Lim
15. Show that 31x
x
x2
0x
Lim
16. Find
x2x3
x3x8
x
Lim
17. Evaluate
)2x)(1x2(
4x7x2
2x
Lim
2
18. Compute )x]x([
2x
Lim
and )x]x([
2x
Lim
](https://image.slidesharecdn.com/jrimp-140723114700-phpapp01/85/Maths-IB-Important-4-320.jpg)
Maths IB Important
- 1. Mathematics –I (B) Questions Bank1 Chapter 2M 4M 7M Total Locus - 1 - 04 Transformation Of Axes - 1 - 04 Straight Line 2 1 1 15 Pair Of Straight Lines - - 2 14 Three Dimensional Coordinates 1 - - 02 Direction Cosines And Direction Ratios - - 1 07 Plane 1 - - 02 Limits 2 - - 04 Continuity 1 - - 02 Differentiation 1 2 1 17 Application Of Differentiation Errors And Approximations 1 - - 02 Tangents And Normals 1 - 1 09 Rate measure - 1 - 04 Maxima and Minima - - 1 07 Partial Differentiation - 1 - 04 Total 10 7 7 97 Very Short Answer Questions (2M) Straight Line 01. Find the values of ‘t’ if the points (t, 2t), (2t, 6t) and (3, 8) are collinear. 02. Find condition for points (a, 0), (h, k) and (0, b), 0ab to be collinear. 03. Find the value of k, if the straight lines y -3kx +4 =0 and (2k -1)x –(8k -1)y -6 =0 are perpendicular. 04. Find distance between the parallel lines 5x -3y -4 =0 and 10x-6y -9 =0. 05. If the area of the triangle formed by the straight lines x =0, y =0 and 3x +4y =a (a>0) is 6. Find value of ‘a’. 06. Find the equation of straight line passing through the point (3, -4) and making X and Y –intercepts which are in the ratio 2 : 3. 07. Find the equation of the straight line passing through the points )at2,at( 1 2 1 and )at2,at( 2 2 2 . 08. Find the equation of the straight line passing through (-4, 5) and cutting off equal non -zero intercepts on the coordinate axes. 09. Find the equation of the straight line passing through the point (-2,4) and making intercepts whose sum is zero. 2000 - 2001
- 2. Mathematics –I (B) Questions Bank2 10. Find the length of the perpendicular drawn from the point (-2, -3) to the straight line 5x -2y +4 =0. 11. Find value of ‘k’ if the lines 6x -10y +3 =0 and kx -5y +8 =0 are parallel. 12. Find the value of ‘p’ if the straight lines 3x +7y -1 =0 and 7x –py +3 =0 are mutually perpendicular. 13. Find the angle between the straight lines 3x-y +5 = 0 and x+3y-2 = 0. 14. Find the area of the triangle formed by the straight line x -4y +2 =0 with the coordinate axes. 15. If a,b,c are in arithmetic progression. Then show that equation ax +by +c =0 represents a family of concurrent lines and find the point of concurrency . 16. Transform the equation (2+5k)x -3(1+2k)y +(2-k) =0 into the form L1 +λL2 =0 and find point of concurrency of the family of straight lines. 17. Find the equation of the straight line passing through (x1, y1) and parallel and perpendicular to ax +by +c =0. 18. The slope of a line through A (2, 3) is 3/4. Find the points on the line that are 5 units away from A. 19. Find the equation of the straight line passing through (1, 1) and at a distance of 3 units from (-2, 3). 20. Find area of the triangle formed by the straight line psinycosx and the coordinate axes. 21. Find the ratio in which the straight line 2x +3y -20 =0 divides the line segment joining the points (2, 3) and (2, 10). 22. A straight line meets the coordinate axes in A and B. Find the equation of the straight line, when (p, q) bisects AB . 23. Transform the equation x +y +1 =0 into normal form. 24. If 2x -3y -5 =0 is the perpendicular bisector of the line segment joining (3, -4) and ),( , find . 25. 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). 26. Find the equation of the straight line passing through the origin and making equal angles with the coordinate axes.
- 3. Mathematics –I (B) Questions Bank3 3D –Geometry 01. Find Centroid of the triangle whose vertices are (5, 4, 6), (1, -1, 3) and (4, 3, 2). 02. If (3, 2, -1), (4, 1, 1) and (6, 2, 5) are three vertices and (4, 2, 2) is the centroid of a tetrahedron, find the fourth vertex. 03. Find fourth vertex of the parallelogram whose consecutive vertices are (2, 4, -1), (3, 6, -1) and (4, 5, 1). 04. Find the ratio in which the point C(6, -17, -4) divides the line segment joining the points A(2, 3, 4) and B(3, -2, 2). 05. 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. 06. Find ‘x’ if the distance between (5, -1, 7) and (x, 5, 1) is 9 units. 07. Find the coordinates of the vertex ‘C’ of triangle ABC if its centroid is the origin and the vertices A, B are (1, 1, 1) and (-2, 4, 1) respectively. 08. For what value of ‘t’ the points (2, -1, 3), (3, -5, t) and (-1, 11, 9) are collinear. The Plane 01. Find the angle between the planes x +2y + 2z =0 and 3x +3y +2z -8 =0. 02. Reduce the equation x +2y -3z -6 =0 of the plane into the normal form. 03. Find intercepts of the plane 4x +3y -2z +2 =0 on the coordinate axes. 04. Find the direction cosines of the normal to the plane x +2y +2z -4 =0. 05. Find the equation of the plane passing through the point (1, 1, 1) and parallel to the plane x +2y +3z -7 =0. 06. Find the equation of the plane passing through the point (-2, 1, 3) and having (3, -5, 4) as direction ratios of its normal. 07. Find equation to the plane parallel to ZX –plane and passing through (0, 4, 4). 08. Find constant ‘k’ so that planes x -2y +kz =0 and 2x +5y –z =0 are at right angles. 09. Find equation of Plane whose intercepts on X, Y, Z –axes are 1, 2, 4 respectively. Limits 01. Compute x 1x1 0x Lim 02. Compute 1x1 13 0x limL x 03. Evaluate x )bxasin()bxasin( 0x Lim
- 4. Mathematics –I (B) Questions Bank4 04. Compute )1b,0b,0a( 1b 1a 0x Lim x x 05. Compute 1x1 1e 0x limL x 06. Evaluate 2 x bxcosaxcos 0x Lim 07. Compute )xxx( x Lim 2 08. Find 2 x xcos 2 x Lim 09. Evaluate 7x5x13 4x3x11 x Lim 3 3 10. Compute )Zn,m(, nxsin mx2cos1 0x Lim 2 11. Evaluate 1x )1xsin( 1x Lim 2 12. Find x x1x1 0x Lim 33 13. Find 222 2 )ax( )ax(tan)axsin( ax Lim 14. Show that 1 2x 2x 2x Lim 15. Show that 31x x x2 0x Lim 16. Find x2x3 x3x8 x Lim 17. Evaluate )2x)(1x2( 4x7x2 2x Lim 2 18. Compute )x]x([ 2x Lim and )x]x([ 2x Lim
- 5. Mathematics –I (B) Questions Bank5 19. Find 1x 3x2 x Lim 2 20. Evaluate 8x 2x 2x Lim 3 21. Evaluate 4x 4 2x 1 2x Lim 2 22. Evaluate )x1x( x Lim 23. Find 9x 3x 3x Lim 2 24. Find x1x1 x 0x Lim Continuity 01. Show that 0xif 2x bxcosaxcos 0xif)2a2b( 2 1 )x(f where a and b are constants, is continuous at x =0. 02. Is ‘f ‘defined by 0xif x sin2x 0xif0 )x(f is continuous at x=0? 03. Check the continuity of ‘f’ given by 3xand5x0if3)x22(x9)(x 3xif1.5 )x(f 2 is continuous at 3. 04. Find ‘a’ so that 3xif3ax 3xifx2x3 )x(f 2 is continuous at x= 3. 05. If ‘f’ is given by 1xif)kkx( 1xif2 )x(f 2 is continuous function on R, then find value of k. Differentiation 01. Find the derivative of the function xlogexsin5y x 02. If xsinxcos xcos y find dx dy .
- 6. Mathematics –I (B) Questions Bank6 03. If n1n bxaxy then prove that y)1n(nyx2 04. If 0axy3yx 33 find dx dy . 05. Find the derivatives of the following functions w.r.t x. i) )x3x4(cos 3l ii) cox1 cox1 Tan l 06. Differentiate f(x) w.r.t. g(x) if x ef(x) , xg(x) . 07. Find the derivatives of the following functions w.r.t. x. i) x 1x1 Tan 2 l ii) xtan)x(log 08. If )0x(7)x(f x33x , then find )x(f . 09. If )xlog(cotcosy , then find dx dy . 10. If ))e(log(siny xl , then find dx dy . 11. If )xtansec(y , then find dx dy . 12. Find the derivative of the function 2x2 ea)x(f . 13. If )etan(x y , then show that 2 y x1 e dx dy . 14. If xsinxe)x(f x , then find )x(f . 15. If tsinay,tcosax 33 , find dx dy . 16. If nxnx beaey then prove that yny 2 . 17. If ),0x(xy x then find dx dy . 18. If ),xsin(logy find dx dy . 19. If 1002 x..........xx1y then find )1(f . 20. If ),xtanxlog(sec)x(f find )x(f .
- 7. Mathematics –I (B) Questions Bank7 21. If 231 )x(coty , find dx dy . Errors and Approximations 01. Find dy and y if 6x3xy 2 , when x =10, x =0.01. 02. Find dy and y if x 1 y , when x =2, x =0.002. 03. Find dy and y if x2xy 2 , when x =5, x = -0.1. 04. Find dy and y if x ey , when x =0, x = 0.1. 05. Find dy and y if 2 xy , when x =20, x = 0.1. 06. If the increase in the side is 1%, find percentage of change in area of the square. 07. If an error of 3% occurs in measuring the side of a cube, find the percentage error in its volume. 08. Find the approximate value of 160sin 0 , 0175.0 180 09. Find the approximate value of 3 123 10. The radius of a circular plate is increasing in length at 0.01 cm/sec. what is the rate at which the area is increasing, when radius is 12 cm. 11. The diameter of a sphere is measured to be 20 cm. If an error of 0.02cm. occurs in this, find the errors in volume and surface area of the sphere. Tangents and Normals 01. Show that length of sub –normal at any point on the curve ax4y2 is constant. 02. Find equations of normal to the curve x2x4xy 2 at (4, 2). 03. Find equations of tangent and normal to the curve 23 x4xy at (-1, 3). 04. Find equations of tangent and normal to the curve 33 sinay,cosax at 4 . 05. Find the slope of the tangent to the curve 1x 1 y at 2 1 ,3 . 06. Find the slope of the tangent to the curve 2x3xy 3 at the point whose x –coordinate is 3. 07. Find the points at which the tangent to the curve 7x9x3xy 23 is parallel
- 8. Mathematics –I (B) Questions Bank8 to the x –axis. 08. Show that at any point on the curve a x bey , length of sub –tangent is constant. 09. Show that at any point on the curve ax4y2 , length of sub –normal is constant. Short Answer Questions (4M) Locus 01. Find the equation of locus of a point, the difference of whose distances from (-5, 0) and (5, 0) is 8. 02. A (-9, 0) and B (-1,0) are two points. P(x, y) is a point such that PB3PA . Show that the locus of P is .3yx 22 03. Find the equation of locus of a point, the sum of whose distances from (0, 2) and (0, -2) is 6 units. 04. Find equation of locus of P, if A (4, 0), B (-4, 0) and 4PB-PA 05. Find equation of locus of P, if A (2, 3) and B (2,-3) such that PA +PB =8. 06. Find the equation of locus of P, if the line segment joining (2, 3) and (-1, 5) subtends a right angle at P. 07. Find the equation of locus of P, if the ratio of the distance from P to (5, -4) and (7, 6) is 2 : 3. 08. A (5, 3) and B (3, -2) are two fixed points. Find the equation of locus of P, so that the area of triangle PAB is 9 sq. units. 09. A (1, 2), B (2, -3) and C (-2, 3) are three points. A point P moves such that 222 PC2PBPA . Show that equation to the locus of P is 7x -7y +4. 10. Find the equation of locus of a point P, if the distance of P from A (3, 0) is twice the distance of P from B (-3, 0). 11. Find equation of locus of a point P, if A (4, 0), B (-4, 0) and 4PBPA 12. An iron rod of length l2 is sliding on two mutually perpendicular lines. Find the equation of locus of the mid point of the rod. Transformation of axes 01. when the axes are rotated through an angle 4 , find the transformed equation of .9y3xy10x3 22 02. when the axes are rotated through an angle 6 , find the transformed equation of
- 9. Mathematics –I (B) Questions Bank9 .a2yxy32x 222 03. When origin is shifted to the point (2, 3), the transformed equation of a curve is .011y7x17y2xy3x 22 Find the original equation. 04. When the axes are rotated through an angle 45°, transformed equation of a curve is .225y17xy16x17 22 Find the original equation. 05. Show that the axes are to be rotated through an angle of ba h2 Tan 2 1 l , so as to remove the ‘xy’ term from the equation 0byhxy2ax 22 , if ba and through the angle 4 if ba . 06. When the axes are rotated through an angle α, find the transformed equation of psinycosx . 07. When the origin is shifted to (-1, 2) by the translation of axes, find the transformed equation of .01y4x2yx 22 08. Find the point to which the origin is to be shifted by translation of axes so as to remove first degree terms from the equation 0cfy2gx2byhxy2ax 22 where .abh2 09. Find the transformed equation of .0y12x12y8xy4x5 22 When the origin is shifted to . 2 1 ,1 by translation of axes. 10. Find the transformed equation of .0y5xy4x3 22 When the origin is shifted to (3, 4) by translation of axes. Straight Line 01. Transform the equation 1 b y a x into the normal form when 0a and ,0b if the perpendicular distance of straight line from the origin is p, deduce that . b 1 a 1 p 1 222 02. Transform the equation 4yx3 into (a) slope -intercept form (b) intercept
- 10. Mathematics –I (B) Questions Bank10 form and (c) normal form. 03. Find the value of k, if the limes 2x -3y +k =0, 3x -4y +8 =0 and 8x -11y -13 =0 are concurrent. 04. A straight line through Q )2,3( makes an angle 6 with the positive direction of X –axis. If the line intersects the line at 08y4x3 at p, find distance PQ. 05. Find equation of straight line making non –zero equal intercepts on coordinate axes passing through the point of intersection of lines 2x -5y +1 =0 & x -3y -4 =0 06. Find the equations if the straight lines passing through the point (-3, 2) and making an angle of 45° with the straight line 3x –y +4 =0. 07. Find the equation of the straight line parallel to the line 3x +4y =7 and passing through the point of intersection of the lines x -2y -3 =0 and x+ 3y -6 =0. 08. Find the equation of the straight line perpendicular to the line 2x +3y =0 and passing through the point of intersection of the lines x +3y -1 =0 and x -2y +4 =0. 09. Find equations of the straight lines passing through point of intersection of the lines x +y +1 =0, 2x -y =1 and whose perpendicular distance from (1, 1) is 2 units. 10. Find the points on the line 4x -3y -10 =0 which are at a distance of 5 units from the point (1, -2). 11. Find the length of the perpendicular drawn from the point of intersection of the lines 3x +2y +4 =0 and 2x +5y -1 =0 to the line 7x +24y -15 =0. 12. Find value of k, if the angle between the lines 4x –y +7 =0 and kx -5y -9 =0 is 45°. 13. A straight line forms a triangle of area 24 sq. units with the coordinate axes. Find the equation of that straight line if it pass through (3, 4). 14. The base of equilateral triangle is x +y -2 =0 and the opposite vertex is (2, -1). Find the equations of the remaining sides. 15. Find the equation of the line perpendicular to the line 3x +4y +6 =0 and making an intercept -4 on the X –axis. 16. If (-2, 6) is the image of the point (4, 2) w.r.t. the line L. Find the equation of L. And also find the image of (1,2) w.r.t. the line 3x +4y -1 =0. 17. Find the angles of triangle whose sides are x +y -4 =0, 2x +y -6 and 5x +3y -15 =0. 18. A straight line is parallel to the line ,x3y passes through Q(2, 3) and cuts the line 2x +4y -27 =0 at P. Find the length PQ.
- 11. Mathematics –I (B) Questions Bank11 19. A variable straight line drawn through the point of intersection of lines 1 b y a x . and 1 a y b x meets the coordinate axes at A and B. Show that the locus of the mid point of AB is )yx(abxy)ba(2 . 20. If the straight lines ax +by +c =0, bx +cy +a =0 and cx +ay +b=0 are concurrent, then prove that .abc3cba 333 Differentiation 01. Find the derivative of the function sinxf(x) from first principle. 02. Find the derivative of the function sin2xf(x) from first principle. 03. Find the derivative of the function xcosf(x) 2 from first principle. 04. Find the derivative of the function tan2xf(x) from first principle. 05. Find the derivative of the function sec3xf(x) from first principle. 06. Find the derivative of the function 1xf(x) from first principle. 07. Find the derivative of the function logxf(x) from first principle. 08. Find the derivative of the function cosaxf(x) from first principle. 09. Find the derivative of the function b)tan(axf(x) from first principle. 10. If ,ex yxy then show that . )xlog1( xlog dx dy 2 11. If ,xy y then show that . )ylogy1(x y )ylog1(x y dx dy 22 12. If ),tsintt(cosax ),tcostt(sinay then find dx dy . 13. 14. Differentiate xsin e w.r.t. sinx. 15. If ),yasin(xysin then show that . asin )ya(sin dx dy 2 (a is not a multiple of .) 16. If )1x( x1 x1 Tany l , then find dx dy . 17. If 1x0for x1x1 x1x1 Tany 22 22 l , find dx dy . 18. If , x1 xsinx y 2 l then find dx dy . 19. If .tsin2tsin3y,tcos2tcos3x 33 Then find dx dy . 20. If .tsinay,tcosax 33 Then find dx dy .
- 12. Mathematics –I (B) Questions Bank12 21. If ,)x(sinxy xcosxtan find dx dy . 22. If ),x4x3(siny 3l then find dx dy . 23. If , xcos1 xcos1 Tany l then find dx dy . 24. If )0b,0a(, xsinba xsinab siny l . Then find dx dy . 25. If ),tsint(ax )tcos1(ay then find 2 2 dx yd . 26. If 1byhxy2ax 22 , then prove that . )byhx( abh dx yd 3 2 2 2 27. If ,xsin)x2b(xcosay then show that .xcos4yy Rate measure 01. A man 6 ft. high walks at a uniform rate of 4 miles per hour away from a lamp 20 ft. high. Find the rate at which length of his shadow increases. (1 mile=5280 ft.). 02. A man 180 cm. high walks at a uniform rate of 12 km. per hour away from a lamp post of 450 cm. Find the rate at which the length of his shadow increases. 03. Sand is poured from a pipe at the rate of 12 Cc/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always One –sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm. 04. Water is dripping out from conical funnel, at a uniform rate of 2 cm3/sec. through a tiny hole at the vertex at the bottom. When the slant height of the water is 4 cm. find the rate of decrease of the slant height if the water given that the vertical angle of the funnel is 120° . 05. The displacement ‘s’ of a particle traveling in a straight line in ‘t’ seconds is given by .tt11t45s 32 Find the time when the particle comes to rest. 06. A balloon is in the shape of an inverted cone surmounted by hemisphere. Diameter of the sphere is equal to the height of the cone. If h is the total height of the balloon, then how does the volume of the balloon changes with h? What is the rate of change in volume when h=9 units. 07. A point P is moving on the curve .x2y 2 The x -coordinate of P is increasing at
- 13. Mathematics –I (B) Questions Bank13 the rate of 4 units per second. Find the rate at which the y -coordinate is increasing when the point is at (2, 8). 08. The distance –time formula for the motion of a article along the straight line is 18t24t9ts 23 . Find when and where the velocity is zero. 09. A ship starts moving from a point at 10 a.m. and heading due east at 10 km/hr. another ship starts moving from the same point at 11 a.m. and heading due south at 12 km/hr. How fast is the distance between the ships increasing at 12 noon. 10. Show that curves 0y2x5x6 2 and 3y8x4 22 touch each other at 2 1 , 2 1 . Partial Differentiation 01. Using Euler’s theorem show Tanu 2 1 yuxu yx for the function yx yx sinu l . 02. If yx yx Tanu 33 l , show that u2sinyuxu yx . 03. If 33 33 l yx yx Tanu , show that 0yuxu yx . 04. If )yx(sinu l , then show that Tanu 2 1 yuxu yx . 05. If )ytanxlog(tanz , show that .2z)y2(sinz)x2(sin yx 06. If the function x y Tanf l , show that 0ff yyxx . 07. Show that for the function )yxlog(f 22 , 0ff yyxx . 08. If )y(g)x(fez yx , show that yx xy ez . 09. If 222 )by()ax(r , find yyxx rr . 10. State and prove Euler’s theorem. 11. Verify Euler’s theorem for the function 33 2 yx yx f . 12. If f(x, y) is a homogeneous function of degree ‘n', whose first and second order
- 14. Mathematics –I (B) Questions Bank14 derivatives exist, then show that .f)1n(n yx f xy2 y f y x f x 2 2 2 2 2 Long Answer Questions (7M) Straight Line 01. Find the Orthocentre of the triangle with the vertices (-2, -1), (6, -1) and (2, 5). 02. Find the Orthocentre of the triangle formed by the lines x +2y =0, 4x +3y -5 =0 and 3x +y =0. 03. If the equations of sides of a triangle are 7x +y -10 =0, x -2y +5 =0 and x + y +2 =0. Find the Orthocentre of the triangle. 04. Find the Circumcentre of the triangle whose sides are 3x –y -5 =0, x +2y -4 =0 and 5x +3y +1 =0. 05. Find the Circumcentre of the triangle whose vertices are (1, 3), (-3, 5) and (5, -1). 06. Find the In –centre of the triangle formed by the lines x3y,x3y and y=3. 07. If p and q are lengths of the perpendiculars from the origin to the straight lines aeccosysecx and 2cosasinycosx , prove that .aqp4 222 08. Find the equation of the straight line passing through (1, 2) and making an angle 60° with the line .02yx3 09. Find the equations of the straight lines passing through the point of intersection of the lines 3x +2y +4 =0, 2x +5y =1 and whose distance from (2, -1) is 2 units. 10. Find the equations of the straight lines passing through the point (5, -3) and having intercepts whose sum is 6 5 . 11. If Q (h, k) is the foot of the perpendicular from P(x1, y1) on the line ax +by +c = 0 then prove that 22 1111 ba:)cbyax(b:)yk(a:)xh( , also find the foot of the perpendicular from (-1, 3) on the straight line 5x -y -18 =0. 12. If Q (h, k) is the image of the point P(x1, y1) w.r.t. the straight line ax +by +c = 0 then prove that 22 1111 ba:)cbyax(2b:)yk(a:)xh( , also find the image of (1, 2) w.r.t. the straight line 3x +4y -1 =0.
- 15. Mathematics –I (B) Questions Bank15 Pair of Straight Lines 01. Show that the product of the perpendicular distances from a point ),( to the pair of lines 0byhxy2ax 22 is 22 22 h4)ba( bh2a . 02. Show that the area of the triangle formed by the lines 0byhxy2ax 22 and Lx +my +n=0 is 22 22 blhlm2am abhn sq. units. 03. If 0cfy2gx2byhxy2axS 22 represents a pair of parallel lines, then show that (i) abh2 (ii) 22 afbg and (iii) the distance between the two parallel lines is )ba(a acg 2 2 or )ba(b bcf 2 2 . 04. If the equation 0cfy2gx2byhxy2axS 22 represents a pair of straight lines then show that (i) 2 2 2 abc 2fgh-af bg ch 0 and (ii) 2 2 2 h ab, g ac and f bc . 05. Show that the equation of the pair of bisectors of the angles between the pair of lines 0byhxy2ax 22 is 0)ba(xy)yx(h 22 . 06. Show that the area of the equilateral triangle formed by pair of lines 0)lymx(3)mylx( 22 and the line lx +my +n=0 is )ml(3 n 22 2 sq. units. 07. An angle between the pair of lines 0byhxy2ax 22 is then show that 2 2 a b cos (a b) 4h 08. If ),( is the Centroid of the triangle formed by the lines 0byhxy2ax 22 and lx +my =1, prove that )amhlm2bl(3 2 hlamhmbl 22 09. Find the value of k, if the lines joining the origin to the points of intersection of the curve 01yx2y3xy22xS 22 and the line x +2y =k are mutually perpendicular.
- 16. Mathematics –I (B) Questions Bank16 10. If the straight lines joining the origin to the points of intersection of the curve 04y3x2y3xy3x 22 and the line 2x +3y =k are perpendicular, prove that 052k5k6 2 . 11. Find the angle between the lines joining the origin to the points of intersection of the curve 05y2x2yxy2xS 22 and the line 3x –y +1 =0. 12. Show that the lines joining the origin to the points of intersection of the curve 02y3x3yxyx 22 and the straight line 02yx are mutually perpendicular. 13. Find Centroid and area of triangle formed by the lines 0y7xy2012xS 22 and 2x -3y +4 =0. 14. Find the condition for the lines joining the origin to the points of intersection of the circle 222 ayx and the line lx +my =1 to coincide. 15. Show that the pairs of straight lines 0y6xy56x 22 and 01y5xy6xy56x 22 form a square. 16. Show that the straight lines represented by 0y23xy483x 22 and 3x -2y +13 =0 form an equilateral triangle of area 3 13 sq. units. 17. Show that the equation 06y23xy7xy13x2 22 represents a pair of straight lines. Also find the angle between them and the coordinates of the point of intersection of the lines. Direction Cosines and Direction Ratios 01. Find the angle between the lines whose direction cosines are given by the equations 3l +m +5n =0 and 6mn -2nl +5lm =0. 02. If a ray makes the angles and,, with four diagonals of a cube then find 2222 coscoscoscos . 03. Find the angle between the lines whose direction cosines satisfy the equations l +m +n =0, .0nml 222 04. Find the direction cosines of two lines which are connected by the relations 2l -m +2n =0 and lm +mn +nl =0.
- 17. Mathematics –I (B) Questions Bank17 05. Find the direction cosines of two lines which are connected by the relations l +m +n =0 and mn -2nl -2lm =0. 06. Find the angle between two diagonals if a cube. Differentiation 01. If )yx(ay1x1 22 , then show that 2 2 x1 y1 dx dy . 02. If 22222 xaxlogaxaxy , then prove that 22 xa2 dx dy . 03. If 22 22 l x1x1 x1x1 Tany , for 1x0 , find . dx dy 04. If xcosxtan )x(sinxy , then find . dx dy 05. Differentiate 1x1Tan 2l w.r.t. xTan l . 06. Find the derivative of 2 l x1 x2 Tan)x(f w.r.t. 2 l x1 x2 sin)x(g . 07. If 42 3 l 2 3 l 2 l xx61 x4x4 Tan x31 xx3 Tan x1 x2 Tany , then show that . x1 1 dx dy 2 08. If bxy ayx , then show that 1xy x1y xyxlogx ylogyyx dx dy . 09. If xy yx then show that . )xxlogy(x )yylogx(y dx dy 10. If )cos1(siny),cos1(cosx , then find . 2 at dx dy 11. If )cos(sinay),sin(cosax , then find . dx dy 12. If ysinx xy , then find . dx dy 13. If yxy ex , then find 2 )xlog1( xlog dx dy . 14. If xlogx ylog , then prove that . )x(logx )ylog.xlog1(y dx dy 2 15. Find the derivative of logx sin x(sin x ) x w.r.t.’ x’. 16. If x x Tan)x(gand x sin)x(f ll then show that )x()x(g)x(f .
- 18. Mathematics –I (B) Questions Bank18 Tangents and Normals 01. Show that the curves )1x(4y2 and )x9(36y2 intersect orthogonally. 02. If the tangent at any point P on the curve )0mn(ayx nmnm meets the coordinate axes in A and B, then show that AP : BP is a constant. 03. If the tangent at any point on the curve 3 2 3 2 3 2 ayx intersects the coordinate axes in A and B, then show that the length AB is a constant. 04. Show that the equation of tangent to the curve ayx at the point (x1, y1) is 2 1 2 1 1 2 1 1 ayyxx . 05. Find the equations of (i) tangent (ii) normal and (iii) lengths of sub –tangent and sub –normal to the curve 6x4x3y 2 at (1, 1). 06. At any point ‘t’ on the curve x=a(t +sint), y=a(1 -cost), find the lengths of tangent, normal, sub –tangent and sub –normal. 07. show tat the condition for the orthogonality of the curves 1byax 22 and 1ybxa 2 1 2 1 is . b 1 a 1 b 1 a 1 11 08. The tangent to the curve )0a( a x sinaxa4y2 at a point P on it is parallel to X –axis. Prove that all such points P lie on the curve .ax4y2 09. If p, q denote the lengths of perpendiculars drawn from the origin in the tangent and normal at a point respectively on the curve 3 2 3 2 3 2 ayx , then show that 222 aqp4 . 10. Find the angle between the curves x4y2 and 5yx 22 . 11. Find the angle between the curves 3y3x 2 and 025yx 22 . 12. show that the curves ax4y2 and 2 cxy cut orthogonally if 44 a32c .
- 19. Mathematics –I (B) Questions Bank19 Maxima and Minima 01. Find the rectangle of maximum perimeter that can be inscribed in a circle. 02. Show that the semi –vertical angle of the right circular cone of maximum volume and of given slant height is 2Tan l . 03. Show that when the curved surface of right circular cylinder inscribed in a sphere of radius ‘R’ is maximum, then the height of the cylinder is R2 . 04. Show that the area of a rectangle inscribed in a circle is maximum when it is a square. 05. From a rectangular sheet of dimensions 30 cm. X 80 cm. four equal 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 the value of ‘x’, so that the volume of the box is the greatest. 06. A window is in the shape of a rectangle surmounted by a semicircle. If the perimeter of the window be 20 ft., find the maximum area. 07. A wire of length l is cut into two parts which are bent respectively in the form of a square and a circle. What are the lengths of pieces of so that the sum of the areas is least. 08. The sum of the height and diameter of the base of a right circular cylinder is given as 3 units. Find the radius of the base and height of the cylinder so that the volume is maximum. 09. Find the maximum and minimum values of x2sinxsin2 over 2,0 . 10. Find the absolute maximum value and the absolute minimum value of the function 2,0inx2sinx)x(f . "If you think you can win, you can win. Faith is necessary to victory."