248461097 MATHS IB QUESTION BANK Chapter Wise Important Questions for IPE PDF

March 31, 2018 | Author: Angela Jones | Category: Triangle, Line (Geometry), Elementary Geometry, Geometry, Elementary Mathematics


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JUNIOR MATHS IB - Chapter wise important questions for IPE by MN RaoLOCUS SHORT ANSWER TYPE QUESTIONS 1. 2. 3. 4. 5. 6. 7. 8. 9 10. Find the equation of locus of a point P such that PA2 + PB 2 = 2c 2 , where A = (a, 0), B = (-a, 0)and 0< a < c . Find the equation of locus of P, if the line segment joining (2, 3) and (-1, 5) subtends a right angle at P. The ends of the hypotenuse of a right angled triangle are (0, 6) and (6, 0). Find the equation of locus of its third vertex. Find the equation of locus of P, if A = (4, 0), B = (-4, 0) and PA − PB = 4. Find the equation of locus of P, if A = (2, 3), B = (2, -3) and PA + PB = 8, 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. If the distance from P to the points (2, 3) and (2, -3) are in the ratio 2 : 3, then find the equation of locus of P. A (1, 2), B (2, -3) and C (-2, 3) are three points. A point P moves such that PA2 + PB 2 = 2PC 2 . Show that the equation to the locus of P is 7x – 7y + 4 = 0. 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) A (2, 3) and B (-3, 4) be two given points. Find the equation of locus of P so that the area of the triangle PAB is 8.5 2. TRANSFORMATION OF AXES SHORT ANSWER TYPE QUESTIONS : 1. When the origin is shifted to (-1, 2) by the translation of axes, find the transformed equations of the following 2 x2 + y 2 − 4x + 4 y = 0 2. The point to which the origin is shifted and the transformed equation are given below. Find the original equation. ( −1, 2) ; x 2 + 2 y 2 + 16 = 0 3. Find the point to which the origin is to be shifted so as to remove the first degree terms from the equation 4 x 2 + 9 y 2 − 8 x + 36 y + 4 = 0 4. Find the angle through which the axes are to be rotated so as to remove the xy term in the equation x 2 + 4 xy + y 2 − 2 x + 2 y − 6 = 0 5. When the origin is shifted to the point (2,3), the transformed equation of a curve is x 2 + 3xy − 2 y 2 + 17 x − 7 y − 11 = 0 . Find the original equation of the curve. 6. When the axes are rotated through an angle 45°, the transformed equation of a curve is 17 x 2 − 16 xy + 17 y 2 = 225. Find the original equation of the curve. π 7. When the axes are rotated through and angle , find the transformed equation of x 2 + 2 3xy − y 2 = 2a 2 6 π 8. When the axes are rotated through an angle , find the transformed equation of 3x 2 + 10 xy + 3 y 2 = 9 4 1  2h  9. Show that the axes are to be rotated through an angle of Tan−1   so as to remove the xy term 2  a −b  π from the equation ax 2 + 2hxy + by 2 = 0 , if a ≠ b and through the angle , if a = b 4 Vanithatv – GUIDE education and Career Live show daily at 8.30pm 1 8. 10 In what follows. Find the equations of the straight lines which make the following angles with the positive X-axis in the positive direction and which pass through the points given below. -3) and (0. Find the ratios in which the following straight lines divide the line segments joining the given points. Find the equation of the straight line passing through (-2. -3) 20. 3x – 4y = 7 13. 2. Find the value of x. if the straight lines 3x + 7y – 1 = 0 and 7x – py + 3 = 0are mutually perpendicular Vanithatv – GUIDE education and Career Live show daily at 8. (k + 1)x + (k + 2)y + 5 = 0 14. 22. -2) 4. -3) 21. 5) and (x. x – 4y + 2 = 0 1 2 15. Find the equation of the straight line parallel to the line 2x + 3y + 7 = 0 and passing through the point (4. A straight line passing through A(-2. (-2. Transform the following equations into the form L1 + λ L2 = 0 and find the point of concurrency of the family of straight lines represented by the equation. STRAIGHT LINE VERY SHORT ANSWER TYPE QUESTION : 1. 150° and (-2. Find the length of the perpendicular drawn from the point given against the following straight line 5x – 2y + 4 = 0. Find the equation of the straight line. Find the points on the straight line whose distance from A is 4 units. 3 3 7. p denotes the distance of the straight line from the origin and α denotes the angle suur made by the normal ray drawn from the origin to the straight line with OX measured in the anticlockwise sense. Find the equation of the straight line passing through the point (3. Find the area of the triangle formed by the following straight lines and the coordinate axes. Find the distance between the following parallel lines 5x – 3y – 4 = 0. if the slope of the line passing through (2. a(x – y) + b(x + y) = 2b 17. ax + by = a + b. α = i) p = 5. -3) 18. -3). 1) makes an angle of 30° with OX in the positive direction. The angle made by a straight line with the positive X-axis in the positive direction and the Y-intercept cut off by it are given below. Find the equations of the straight lines passing through the origin and making equal angles with the coordinate axes. Find the value of k. 3) is2. y = x− 3 3 16. Find the equation of the straight line passing through (-4. 2 Tan −1   . Find the equation of the straight line perpendicular to the line 5x – 3y + 1 = 0 and passing through the point (4. State whether the points lie on the same side or on either side of the straight line. 4) and making non-zero intercepts whose sum is zero 9. 5) and cutting off equal nonzero intercepts on the coordinate axes. -1) 5. Find the equation of the line containing the points (2. 12.30pm 2 . Find the equations of the straight line with the following values of p and α 7π ii) p = 6. α = 150° iii) p = 1.3. 6. 135° and (3. if the straight lines 6x – 10y + 3 = 0 and kx – 5y + 8 = 0 are parallel. Find the value of p. 10x – 6y – 9 = 0 19. α = 60° 4 suur 11. Find the equations of the straight lines which make the following angles with the positive X-axis in the positive direction and which pass through the points given below. 3. -4) and making X and Y-intercepts which are in the ratio 2 : 3. y = − 3x + 5. 4. px + 4y = 6 9. 5). (-2. (5. b+c). 8. If the 6 straight line intersects the line 3 x − 4 y + 8 = 0 at P. A triangle of area 24 sq. find the coordinates of B. 2at1 ) and ( at22 . y = 0 and 3x + 4y = a (a > 0) is 6. (c. A straight line L is drawn through the point A(2. 3x + 2y – 2 = 0 and 2x – 3y – 23 = 0 are concurrent and find the point of concurrency. write the equation of the straight line. If the portion of a straight line intercepted between the axes of coordinates is bisected at (2p. 6) and (3. 4). 14. (b – c)x + (c – a)y = a – b and (c – a)x + (a – b)y = b – c are concurrent. SHORT ANSWER QUESTION 1. 27. Find the equations of the straight lines passing through (1. Find the point on the straight line 3x + y + 4 = 0 which is equidistant from the points (-5. 7) ii) (1. 9) and C (-2. 3). 26. 18. 13. 2) in the straight line 3x + 4y – 1 = 0 Find the locus of the foot of the perpendicular from the origin to a variable straight line which always passes through a fixed point (a. Find the equation of the straight line. (10. A straight line through P(3. 2) 11. 3x + 4y = 5. Find the value of p. 16. -3). 2) and (-4. 2q). if it passes through (3. 2 makes an angle π with the positive direction of the X-axis. 2at2 ) 24. Find the equations of the straight lines passing through the point (-3. 15. A (10.3) and (i) parallel to (ii) perpendicular to the line passing through the points (3. -3) are on the same side or on opposite sides of the straight line 2x – 3y + 4 =0 Find the ratios in which (i) the X – axis and (ii) the Y-axis divide the line segment AB joining A (2. -5) and (-6. 1) upon the straight line 3x – 4y + 12 = 0 x – 3y – 5 = 0 is the perpendicular bisector of the line segment joining the point A. B (-4. Find the angle which the line L makes with the positive direction of the X-axis Show that the straight line (a – b)x + (b – c)y = c – a. if the following lines are concurrent. 4) makes an angle of 60° with the positive direction of the X-axis. If the area of the triangle formed by the straight lines x = 0. B If A = (-1. ( ) 3. State whether (3. If 3a + 2b + 4c = 0. Find the value of a. Transform the following equations into (a) slope-intercept form (b) intercept form and (c) normal form iii) x + y + 2 = 0 iv) 3 x + y + 10 = 0 1) ex + 4y = 5 ii) 3 x + y = 4 5. 3x + 4y + 9 = 0 and 7x + y – 54 = 0 form a right angled isosceles triangle. Show that following sets of points are collinear and find the equation of the line L containing them. -6). then show that the equation ax + by + c = 0 represents a family of concurrent straight lines and find the point of concurrency. Find the equations of suur i) ABsuurii) the median through A iii) the altitude through B iv) the perpendicular bisector of the side AB 3. 1) such that its point of intersection with the straight line x + y = 9 is at a distance of 3 2 from A. 2) and making an angle of 45° with the straight line 3x – y + 4 = 0 A straight line through Q Vanithatv – GUIDE education and Career Live show daily at 8. (2. -6) Find the distance between the parallel straight line 3x + 4y – 3 = 0 and 6x + 8y – 1 = 0 25. 4) 7. find the distance PQ. -1) are the vertices of a triangle. (b. Find the equation of the straight line passing through the point ( at12 . units is formed by a straight line and the coordinate axes in the first quadrant.23. 10. c + a ). 17. Find the image of the point (1. 6) iii) (a. 2x + 3y = 4. b) Show that the lines x – 7y – 22 = 0. i) (-5. 1) Find the foot of the perpendicular drawn from (4. Find the coordinates of the points on the line which are 5 units away from P 12. Show that the lines 2x + y – 3 = 0. 1). 19. a + b) 2.30pm 3 . 6.-3) and B (3. 2.6 = 0 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 A straight line parallel to the line y = 3 x passes through Q (2. 3). x + y + 4 = 0 ii) 3 x 2 − 4 xy + y 2 = 0. (6. bx + cy + a = 0 and cx + ay + b = 0 are concurrent. 22. 2x + 5y = 1 and whose distance from (2. 9. 5. y1 ) and B ( x2 . 3. PAIR OF STRAIGHT LINES LONG ANSWER QUESTIONS 2 1.30pm 4 . then prove that a3 + b3 + c3 = 3abc x y A variable straight line drawn through the point of intersection of the straight lines + = 1 and a b x y + = 1 meets the coordinate axes at A and B. -1) is 2 Find the area of the parallelogram whose sides are 3x + 4y + 5 = 0. 23. The ratio in which the straight line L ≡ ax + by + c = 0 divides the line segment joining the points A ( x1 . 2 x − y = 6 3. x – y – 2 = 0 and 2x + y – 7 =0 Find the circum center of the triangle whose vertices are (1.2y – 3 = 0 and x + 3y . -1) and (2. 8. 5) Find the circumcenter of the triangle whose vertices are given below (-2. 6. 2) and (3. 25. 21. 2x + 3y + 1 = 0 and 2x + 3y – 7 = 0 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 . 28. 5x – y – 2 = 0 and x – 2y + 5 = 0 If p and q are the lengths of the perpendiculars from the origin t the straight lines x sec α + y cosec α = a and x cos α -y sin α = a cos2 α . 24. 0). 26. 4x + 3y – 5 = 0 and 3x + y = 0 Find the circumcenter of the triangle whose sides are given by x + y + 2 = 0. Show that the straight lines represented by ( x + 2a ) − 3 y 2 = 0 and x = a form an equilateral triangle. 2.20. -1) and (4. Find the equation of the pair of lines intersecting at (2. Find the equations of the straight lines passing through the point of intersection of the lines 3x + 2y + 4 = 0. Find the centroid and the area of the triangle formed by the following lines i) 2 y 2 − xy − 6 x 2 = 0. (2. 2) Find the circumcenter of the triangle formed by the straight lines x + y = 0. 0) Find the orthocenter of the triangle formed by the lines x + 2y = 0. 27. 10. Find the orthocenter of the triangle whose sides are given by x + y + 10 = 0. y0 ) to the straight line ax + by + c = 0 is ax0 + by0 + c a 2 + b2 LONG ANSWER QUESTIONS 1. 4. -1). prove that 4 p 2 + q 2 = a 2 Find the in centre of the triangle formed by the following straight lines X + 1 = 0. (-1. -1) and Vanithatv – GUIDE education and Career Live show daily at 8. Find the length of PQ. 2x + y + 5 = 0 and x – y = 2 Find the equations of the straight lines passing through the point (1. 2) and making an angle of 60° with the line 3 x + y + 2 = 0 Find the orthocenter of the triangle with the following vertices (-2. 3x – 4y = 5 and 5x + 12y = 27 4. 3) and cuts the line 2x + 4y – 27 = 0 at P. y2 ) is − L11 : L22 If the straight lines ax + by + c= 0. Show the locus of the mid point of AB is 2 (a + b) xy b a = ab (x + y) The length of the perpendicular from the point P ( x0 . 3x + 4y – 2 = 0. 7. 4. 10. k) is the foot of the perpendicular from P ( x1 . ( h − x1 ) : a = ( k − y1 ) : b = − ( ax1 + by1 + c ) : ( a 2 + b2 ) Vanithatv – GUIDE education and Career Live show daily at 8. 5. If the equation ax 2 + 2hxy + by 2 + 2 gx + 2 fy + c = 0 represents a pair of intersecting lines. 16. Write down the equation of the pair of straight lines joining the origin to the points of intersection of the line 6x – y + 8 = 0 with the pair of straight lines 3 x 2 + 4 xy − 4 y 2 − 11x + 2 y + 6 = 0 Show that the lines so obtained make equal angles with the coordinate axes. 8. Also show f 2 + g2 if the given lines are perpendicular. Prove 2 that ( a + b ) = 4 h 2 . prove that 9. y1 ) on the straight line ax = by + c = 0. If Q (h. 20. 11. Show that the two points of lines 3 x 2 = 8 xy − 3 y 2 = 0 and 3 x 2 + 8 xy − 3 y 2 = 2 x − 4 y − 1 = 0 form square.unit 3 If one line of the pair of lines ax 2 + 2hxy + by 2 = 0 bisects the angle between the coordinate axes.30pm 5 . then show that 12. and ii) parallel to the pair 6 x 2 − 13 xy − 5 y 2 = 0 Find the equation of the bisector of the acute angle between the lines 3x – 4y + 7 = 0 and 12x + 5y – 2 = 0 2 2 Show that the lines represented by ( lx + my ) − 3 ( mx − ly ) = 0 and lx + my + n = 0 form an equilateral n2 3 (l 2 + m2 ) triangle with area 6. 13. 19. Show that the straight lines y 2 − 4 y + 3 = 0 and x 2 + 4 xy + 4 y 2 + 5 x + 10 y + 4 = 0 form a parallelogram and find the lengths of its sides Show that the product of the perpendicular distances from the origin to the pair of straight lines c represented by ax 2 + 2 hxyy + by 2 + 2 gx + 2 fy + c = 0 is 2 ( a − b ) + 4h 2 14. α bl − hm = β am − hl = 2 3 ( bl − 2hlm + am2 ) 2 Find the value of k. i) perpendicular to the pair 6 x 2 − 13 xy − 5 y 2 = 0 . If (α . Show that the lines x 2 + 2 xy − 35 y 2 − 4 x + 44 y − 12 = 0 and 5x + 2y – 8 = 0 are concurrent. β ) is the centroid of the triangle formed by the lines ax 2 + 2hxy + by 2 = 0 and lx + my = 1. Show that the straight lines represented by 3 x 2 + 48 xy + 23 y 2 = 0 and 3x – 2y + 13 = 0 form an 13 equilateral triangle of area sq. 7. 17. then that the square of this distance is 15. 21. the square of the distance of their point of intersection from the origin is c (a + b) − f 2 − g2 ab − h 2 . 18. Find the point of intersection of the lines and the angle between the straight lines for this value of k. h2 + b 2 Show that the lines joining the origin to the points of intersection of the curve x 2 − xy + y 2 + 3 x + 3 y − 2 = 0 and the straight line x − y − 2 = 0 are mutually perpendicular Find the values of k. if the lines joining the origin to the points of intersection of the curve 2 x 2 − 2 xy + 3 y 2 + 2 x − y − 1 = 0 and the line x + 2y = k are mutually perpendicular Find the angle between the lines joining the origin to the points of intersection of the curve x 2 + 2 xy + y 2 + 2 x + 2 y − 5 = 0 and the line 3x – y + 1 = 0 Find the condition for the chord lx + my = 1 of the circle x 2 + y 2 = a 2 (whose centre is the origin) to subtend a right angle at the origin Find the condition for the lines joining the origin to the points of intersection of the circle x 2 + y 2 = a 2 and the line lx + my = 1 to coincide. if the equation 2 x 2 + kxy − 6 y 2 + 3 x + y + 1 = 0 represents a pair of straight line. -1. 11. B(5. 4. -2) and (1. 5. -4). -4) is divided by the plane 2x – 3y + z + 6 = 0 Vanithatv – GUIDE education and Career Live show daily at 8. 0. 23. Also find the point of intersection. Show that the product of the perpendicular distances from a point (α . -4). 4) are collinear 4.e. If (3. (3. (2. -3. 6) If the equations of the sides of a triangle are 7x + y – 10 = 0. If A = (-2.t the straight line ax + by + c = 0 Find the orthocenter of the triangle whose vertices are (-5. Find the point equidistant from the four points (-1. Find the distance between the points (3. (0. 2. 3). 5). 32. 0. Show that the points (2. 3. 3). 1) and (-2. 9) are collinear ? SHORT ANSWER QUESTION 1. 2) are three points. Find the incenter of the triangle formed by the straight lines y = 3 x . 2) is the centroid of a tetrahedron. Find the ratio in which YZ – plane divides the line joining A (2. (-1. -1). 3.30pm 6 . 4. 3. 2. 26. 25.k) is the image of the point P ( x1 . B(1. 4. 4. -1) satisfies the equation 8 x 2 + 9 y 2 + 8 z 2 − 18 x − 36 y + 18 z + 54 = 0 4. Find the ratio in which the line joining (2. (13. 31. 1. 3). 7) 2. Find the centroid of the triangle formed by the lines 12 x 2 − 20 xy + 7 y 2 = 0 and 2x – 3y + 4 = 0 Prove that the lines represented by the equations x 2 − 4 xy + y 2 = 0 x + y = 3 form an equilateral triangle. Show that the points (1. C(4. 3). 1. (7. 6. -1) and (4. 2. y1 ) w. If the equation ax 2 + 2hxy + by 2 = 0 represents a pair of distinct (i. intersecting) lines. 8. 6) and (3.r. 2). find the orthocenter of the triangle. y = − 3x and y = 3. the points (2. β ) to the pair of straight lines 2 2 ax + 2hxy + by = 0 is aα 2 +2hαβ+bβ 2 ( a-b ) 2 +4h 2 5. 5. 1) and (6. 5. Let the equation ax 2 = 2hxy + by 2 = 0 represent a pair of straight lines. 2. 13) prove that P satisfies the equation x 2 + y 2 + z 2 + 28 x − 12 y + 10 z − 247 = 0 3. (-3. 5. 5. 28.. 16) and (3. THREE DIMENSIONAL COORDINATES VERY SHORT ANSWER QUESTIONS 1. 24. If Q (h. 4. P is a variable point which moves such that 3PA = 2PB. 27. 3) and B = (13. -1) Find the circumcenter of the triangle whose sides are 3x – y – 5 = 0. 2. 6). 2) form a right angled isosceles triangle. Show that the point whose distance from Y-axis is thrice its distance from (1. 2. 3). -7). 5) are three vertices and (4. 2. 5. A(5. x – 2y + 5 = 0 and x + y + 2 = 0. 2. x + 2y – 4 = 0 and 5x + 3y + 1 = 0. -3. Find the circumcenter of the triangle whose vertices are (1. t) and (-1. For what value of t. Find the coordinates of the point in which the bisector of BAC meets the side BC . -1. Show that the points A (3. (4. 1. -5. Then the angle θ between the a+b lines is given by cos θ = 2 ( a − b ) + 4h 2 29. -10) are collinear and find the ratio in which B divides AC . 2) 7. 2) and (-5. find the fourth vertex 3. 5) and (5.22. -6) and C(9. 2. 5) and B(3. then the combined equation of the pair of bisectors of the angle between these lines is h ( x 2 − y 2 ) = ( a − b ) xy 30. 2. γ with the positive directions of X. 3). Find the point of intersection of the lines AB and CD where A = (7. B = (4. 2. DIRECTION COSINES AND DIRECTIONS RATIOS VERY SHORT ANSWER QUESTIONS 1. 7. 2. 2. 5) are vertices of a parallelogram A (1. 5) Show that the points (4. 5. 4. (3. ( 3. Find the coordinates of the foot of the perpendicular drawn from A to BC Show that the four points (5. 1). 1) are three points and D is the foot of the perpendicular from A to BC. -1). -2. -3). 5. 5). A(6. (1. 3). 4) and D (1. 8. -2. β . 4. 0 )  12 −3 −4   4 12 3  Show that the lines with direction cosines  . − 3. -6. β . 8). 2. (1. 4). Z – axes. If a line makes angle α . -18. -4. 7) taken in order form a rhombus Vanithatv – GUIDE education and Career Live show daily at 8. -3. (-1. 1) and (1. R. suur suur Lines OA. (-1. -1. . 8. (-1. -2. 11) 6. 3. . β . 1). OB are drawn from o with direction cosines proportional to (1. -16. S are the points (2. 3. Find the direction cosines of the normal to the plane AOB. 4). -1. 4.LONG ANSWER QUESTIONS suur suur 1.1. 6. 7. B(0.  and  . -11. Show that the lines whose d. 4). (4. 1). B = (17. -3). 4. C = (1. OZ find the value of cos 2α + cos 2β + cos 2γ . A straight line is inclined to the axes of X and Y at angles of 60° and 45° respectively find its inclination to the Z-axis uuur uuur uuur If aline makes angles α . 3.30pm 7 . -2. -6. 8). 5. 3. δ with the four diagonals of a cube find cos 2 α + cos 2 β + cos 2 γ + cos 2 δ Find the angle between the lines whose direction cosines are given by the equations 3l + m + 5n = 0 and 6mn – 2nl + 5lm = 0 Find the angle between two diagonals of a cube. Find the direction cosines of two lines which are connected by the relations l − 5m + 3n = 0 and 7l 2 + 5m 2 − 3n 2 = 0 . Find the coordinates of D. Find the angle between the lines whose direction ratios are (1. what is the value of sin 2 α + sin 2 β + sin 2 γ ? 2. 6. l 2 + m2 − n 2 = 0 If a ray makes angle α . B(3. 4). 5) respectively Find the angle between DC and AB where A = (3. 2mn + 3nl – 5lm = 0 are perpendicular to each other Find the angle between the lines whose direction cosines satisfy the equations l + m + n = 0. 3.c’s are given by l + m + n = 0. 0) are vertices of a triangle. γ with the coordinate axes OX . SHORT ANSWER QUESTIONS 1.  are perpendicular to each  13 13 13   13 13 13  other. Q. C = (-1. C(2. 2 ) . -3. OY . LONG ANSWER QUESTIONS 1. C(-2. 4). Y. 7. 5. 4). 4. 10) and (7. suur suur Show that the lines PQ and RS are parallel where P. γ . (2. 0. -4. -5) and D = (3. LIMITS AND CONTINULTY VERY SHORT ANSWER TYPE QUESTIONS 2M. Compute the following limits. if 3 < x < 5 a=2 lim x − 2 = −1 x → 2 x−2 lim  2 x  show that + x + 1 = 3  x→2 x  4. Show that the points (0.c) show that the equation x y z to the plane is + + = 3 . 1.1)(1. 2. 2. 6) 2. -1. 6. 2). if x ≤ 1  1  f ( x ) =  2x + 1 if 1 < x ≤ 2. 3. (3. x2 − a2 x−a if -1< x ≤ 3 . -3. Show that the plane through (1. -1. 3) and having (3. THE PLANE VERY SHORT ANSWER QUESTIONS 1. Find the equation of the plane through (4. -5. 0. 4. Find the angle between the planes x + 2y + 2z – 5 = 0 and 3x + 3y + 2z – 8 = 0 SHORT ANSWER QUESTIONS 1. 4. 1. 1) and (-7. 4. Find the equation of the plane passing through (2. Find the intercepts of the plane 4x + 3y – 2z + 2 = 0 on the coordinate axes. 3) and (-2. B. 0) are coplanar 3. -5) is parallel to Y-axis 8. Find the equation of the plane passing through the point (-2.1) and parallel to the plane x + 2y + 3z – 7=0 2. Show that 3.. C. 2) and perpendicular to the join of (1. lim x→a  x+2 f (x) =  2 x a=3 1. Compute 1. If the centroid of ∆ABC is (a. 1.b. 1. -3. 4) 3. (7.1.30pm 8 . Reduce the equation x + 2y – 3 – 6 = 0 of the plane to the normal form 2. FUNCTIONS. 2. 0. A plane meets the coordinate axes in A. (3. -1). 4) and perpendicular to x – 2y + z = 6 LONG ANSWER QUESTIONS 1. 3.1. 3. 1. 3. 0) and perpendicular to the planes 2x + y + 2z + 3 = 0 and 3x + 3y + 2z – 8 = 0 4. (1.  3x if x >2  2.7. 1). -1). Find the equation of the plane through the points (2. 1) and (3. 0). (2. Find the equations of the plane passing through the point (1. . 4) as d..r’s of its normal 4. Compute lim x→2+ lim x→2− ([ x ] + x ) and lim x→2− 2 − x ( x < 2 ) What is ([ x ] + x ) lim x→2 2− x ? Vanithatv – GUIDE education and Career Live show daily at 8. a b c 4. Find the equation of the plane through (-1.  6 if x < −2   x − 10  k 2 x − k if x ≥1 If f. 1. 1. 2. 9. 1. Vanithatv – GUIDE education and Career Live show daily at 8. given by f ( x ) =  .5 if x = 3   x −1  x − 1 if x >1  If f is a function defined by f ( x ) =  5 − 3 x if −2 ≤ x ≤1. 10. 11. 2.lim 5. 1. cos x π 1. 2. 3. π  2 x−  2  lim sin ( a + bx ) − sin ( a − bx ) x→0 x lim tan ( x − a ) ( a ≠ 0) x → a x2 − a2 lim  x sin a − a sin x   x → a  x−a  lim  cos ax − cos bx   x → a  x2  lim  3x − 1    x → 0  1 + x −1  x→ 2. if x < 1  2 is a continuous function on R. then find the values of k.. lim 3 1 + x − 3 1 − x x→0 x 3. then discuss the continuity of f. 1. 6. 2. 7. lim sin ( x − a ) tan 2 ( x − a ) 2 x→a ( x2 − a2 ) 1. 3. lim 8 x + 3x x → ∞ 3 x − 2x 2. 1.30pm 9 . lim 11x 3 − 3 x + 4 x → ∞ 13 x 3 − 5 x 2 − 7 lim x2 + x − x x→∞ lim  2 x + 3    x → ∞  x2 −1  ( ) lim 2 + cos 2 x x → ∞ x + 2007  sin 2 x if x ≠ 0 Is f defined by f ( x ) =  continuous at 0 ? if x = 0  x Check the continuity or f given by 2 2 ( x − 9 ) / ( x − 2 x − 3) if 0 < x < 5 and x ≠ 3 f ( x) =  at the point 3. 3. 8. b ≠ 1) x → 0 b x −1  x + 1 if  Check the continuity of the function f given below at 1 and at 2. 1. f ( x ) =  2 x if 1 + x 2 if  x + 2 if − 1 < x≤3 lim lim  Find f ( x ) and f ( x ) . where f ( x ) =  2 . 1. 4. 9. x→0 x x →0− x lim  e x − 1  Complete   x → 0  1 + x −1  Show that Complete lim a x − 1 (a > 0. if 3 < x < 5 x → 3+ x → 3−  x lim sin x Find x → 0 x cos 2 x x ≤1 1< x < 2 x≥2 lim  x   x → 0  1 + x − 1 − x  lim x − 3 Show that = −1 x → 3 x −3 lim  1 + x − 1 + x 2    x → 0  1 − x 2 − 1 − x  9.30pm ) iv) cos x sin x + cos x 10 . 3 ax + b c + d ≠ 0) ii) x + 2 x 4 + 3x 6 ( x > 0 ) ( cx + d 1 − cos 2 x I) ii) Tan -1 ( log x ) iii) log sin -1 ( e x ) 1 + cos 2 x v) cot -1 (cosec 3x) Find the derivatives of the following functions. 1. 4 x − 9 if 1 < x < 2  3 x + 4 if x ≥ 2  cos ax − cos bx if x ≠ 0  x2 Show that f ( x ) =   1 ( b2 − a 2 ) if x = 0  2 where a and b are real constants. 6. 7. is continuous at 0. 13. 2. lim x lim x = 1 and = −1( x ≠ 0) . DIFFERENTIATION VERY SHORT ANSWER TYPE QUESTIONS 2M. Find the derivatives of the following functions f ( x ) 2. 3. Solved problems : 1.  4 − x2 if x ≤ 0   x − 5 if 0 < x ≤ 1 Check the continuity of f given oy f ( x ) =  2 at the point 0. 5 I) ( Vanithatv – GUIDE education and Career Live show daily at 8. 3. 5. 1. 2. 1. 1 and 2.12. b > 0. 8. i) If y = ax n +1 + bx n then x 2 y '' = n(n + 1) y 1. dx 2  2x  Find the second order derivative of y = Tan −1  . dx dy . 2. then find 4. dx dy . find 3. y = a (1 + cos t ) find 8. Vanithatv – GUIDE education and Career Live show daily at 8.  2x   2x  ii) f ( x ) = Tan −1  . If e x+ y = xy then find dy . dy . 7. i) f ( x ) = e x . If y = ( sin ( log x ) ) . y = a sin 3 t . y = e−k /2 x d2 y h 2 − ab = dx 2 (hx + by )2 (a cos nx +b sin nx) then prove that y "+ k y '+  n  2 + k2  y=0 4  Solved problems : 1. 1. If f ( x ) = x e x sin x. If x 3 + y 3 − 3axy = 0. 2   1− x  9. dx dy . 1. find 7. then find f ' ( x ) dy dx 2.  1 + x2 − 1  tan x i) Tan −1  ii) ( log x )    x   iii) ( x x ) x iv) 20log (tan x ) 2x + 3 If y = then find y " 4x + 5 Find the second order derivatives of the following functions f ( x ) . we shall find 6. dx d2y . If x = a (t − sin t ). g ( x ) = Sin −1  2  2   1− x   1+ x  Find the derivatives of the following functions. y = 2 sin t + sin 2t then find 10. If x = 2cos t + cos 2t + 1.. ) Differentiate f ( x ) with respect to g ( x ) for the following. 6. i) If ax 2 + 2hxy + by 2 = 1 then prove that iv) If 8. 1. 5. dx SHORT ANSWER TYPE QUESTIONS 4 M. ii) e x sin x cos 2 x i) log (4x 2 − 9) Prove the following.30pm 11 . g ( x ) = x 4. If x = a cos3 t . π  If y = (tan x)sin x  0 < x <  2  5. If y = x x ( x > 0). a−x  ii) Tan−1    1 + ax  1   1  iv) Sec −1  2   0 < x <  2  2x −1   i) Cos −1 ( 4 x 3 − 3 x ) iii) Tan −1 1 − cos x 1 + cos x ( v) sin Tan−1 ( e − x ) 2. y = 3sin t − 2sin 3 t ii) x = a (cos t + t sin t ). Establish the following. i) y x = x sin y 1 If f ( x ) = log x ( x > 0). LONG ANSWER TYPE QUESTIONS 7 M. If 1 − x 2 + 1 − y 2 = a ( x − y ) then ii) If f ( x ) = sin −1 x−β x−β and g ( x ) = Tan−1 then α −β α −x f ' ( x) = g '( x) ( β < x < α ) Vanithatv – GUIDE education and Career Live show daily at 8. i) (sin x)log x + x sin x ii) (sin x) x + x sin x 2. 2. dy 1− y2 = dx 1 − x2 dy y 1 − log x log y  ii) x log y = log x then =  dx x  log 2 x  dy y ( x log y − y ) iii) x y = y x then = dx x ( y log x − x) Find the derivatives of the following functions. C os -1   (a > 0.   1 + cos x  2.  1+ x2 −1  i) f ( x ) = Tan −1   . 2. 3. Differentiate f ( x ) with respect to g ( x ) for the following. 4. 1.1.30pm 12 . 1. dx i) x = 3cos t − 2cos 3 t . then prove that f ' ( x ) = x 1. i) x 4 + 4 ii) x + 1 v) cot x vi) sec 3 x Find the derivatives of the following functions. iii) sin 2x vii) cos 2 x iv) tan 2x dy for the following functions. Find the derivatives of the following functions. b > 0)  a + b cos x   cos x  Tan -1  3. y = a (sin t − t cos t ) Find 3. b > 0)  a + b sin x   b + a cos x  2. Establish the following  yx y −1 + y 2 log y  dy i) If x y + y x = a b then =− y x −1  dx  x log x + xy  i) 2. 1− x x i) ( x > 0) 1+ x x Find the derivatives of the following functions f ( x ) from the first principles. 1. g ( x ) = Tan −1 x   x   Find the derivative of the function y defined implicitly by each of the following equations.  b + a sin x  Sin -1  1.  ( a > 0. find the percentage error in its volume Show that the relative error in the nth power of a number is n times the relative error in that number. then show that dy sin 2 (a + y ) = (a is not a multiple of π ) . 1+ x2 + 1 − x2  dy  for 0 < x < 1. find . dx  1 + x 2 − 1 − x 2  cos x dy 2. y = x 2 + 3x + 6. 14. dx sin a 10. dx dy log x 1. 2 The displacement s of a particle traveling in a straight line in t seconds is given by s = 45t + 11t 2 − t 3 Find the time when the particle comes to rest The radius of a circular plate is increasing in length at 0. If sin y = x sin (a + y). Find the slope of the tangent to the curve y = .01 2. = dx (1 + log x )2 1. when x = 5. 13. 12. y = x 2 + 2 x . 10. 16.1 5.1 1 3. y = e x when x = 0.01 cm/sec. then show that . when the radius is 12 cm State the points at which the following functions are increasing and the points at which they are decreasing 2 ii) x 3 ( x − 2 ) i) x 3 − 3 x 2 Determine the intervals in which the functions are increasing and the intervals in which they are decreasing ln t i) ii) 25 − 4x 2 t If an error of 3% occurs in measuring the side of a cube.0175   180  x −1 7. x ≠ 2 at x = 10 x−2 8. Find the velocity 1 and acceleration when t = seconds. 19. 17. y = when x = 2. ∆x = −0. 20. 15. APPLICATIONS OF DIFFERENTIATION VERY SHORT ANSWER QUESTIONS 1. Find an approximate value of 3 123 Show that the length of the subnormal at any point on the curve y 2 = 4ax is a constant. Find the slope of the normal to the curve x = a cos 3 θ . when x = 10. 5) Find the equations of tangent and normal to the curve y = x3 + 4 x 2 at (-1. 18.30pm 13 . 11. If y = Tan −1  2. Find the slope of the tangent to the curve y = x 3 − 3x + 2 at the point whose x-coordinate is 3 9. Find the approximate value of sin 60°1'  = 0. ∆x = 0. If the increase in the side of a square is 1% find the percentage of change in the area of the square.  π  6. ∆x = 0. If x y = e x− y . Vanithatv – GUIDE education and Career Live show daily at 8. 3) At time t the distance s of a particle moving in a straight line is given by s = −4t 2 + 2t . ∆x = 0. y = a sin 3 θ at θ = π 4 Find the equations of tangent and normal to the curve xy = 10 at (2. find .002 x 4. What is the rate at which the area is increasing. If y = x tan x + ( sin x ) . SHORT ANSWER QUESTIONS 1. Find the lengths of subtangent.30pm 14 . high walks at a uniform rate of 4 miles per hour away from a lamp 20 ft. When the slant height of the water is 4 cm. normal. 11. 2. Find the rate at which the length of his shadow increases. then show that AP : BP is a constant.) 4. 7.B. Show that the area of a rectangle inscribed in a circle is maximum when it is a square. x 2 + y 2 = 5 Find the dimensions of the right circular cylinder with the greatest volume that can be inscribed in a sphere of radius a. per hour away from a lamp post of 450 cm. 4. then show that the length AB is a constant If the tangent at any point P on the curve x m y n = a m+ n ( mn ≠ 0 ) meets the coordinate axes in A. at a uniform rate of 2cm3 . then the height of the cylinder is 2R Show that the semi-vertical angle of the right circular cone of maximum voume and of given slant height is Tan −1 2 . find the rate of decrease of the slant height of the water given that the vertical angle of the funnel is 120° 3. A ball is dropped from the same height (64 m. 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. high walks at a uniform rate of 12 km. subtangent and subnormal./ sec through a tiny hole at the vertex at the bottom. Water is dripping out from a conical funnel. high. y1 ) on the curve 2 3 −1 1 x + y = a is yy1 2 + xx1 2 = a 2 2 3 2 3 If the tangent at any point on the curve x + y = a intersects the coordinate axes in A and B. Show that when the curved surface of right circular cylinder inscribed in a sphere of radius R is maximum. 3. find the lengths of tangent. 6. Show that the curves y 2 = 4 ( x + 1) and y 2 = 36 ( 9 − x ) intersect orthogonally 16. A man 180 cm. 2π ] Vanithatv – GUIDE education and Career Live show daily at 8. from the light. −1 Show that the tangent at P ( x1 . m + n ≠ 0 ) . How fast is the height of the sand – cone increasing when the height is 4 cm. (1 mile = 5280 ft. high. A light is at the top of a pole 64 m. 12. At any point t on the curve x = a (t + sin t). 8. 10. subnormal at a point t on the curve x = a(cos t + t sin t). Find the rate at which the length of his shadow increases. Show that the condition for the orthogonality of the curves ax 2 + by 2 = 1 and a1 x 2 + b1 y 2 = 1 is 1 1 1 1 − = − . m th power of the length of the subtangent varies as the nth power of length of the subnormal. 5. high. 13. A man 6 ft./ sec.) from a point 20 m. 5. Find the rectangle of maximum perimeter that can be inscribed in a circle π Show that f (x) = sin x (1 + cosx) has a maximum value at x = 3 m+n m− n 2 n Show that at any point on the curve x = a y ( a > 0. y = a (1 – cos t). how fast is the shadow of the ball moving along the ground 2 seconds later ? LONG ANSWER QEUSTIONS 1. Assuming that the ball falls according to the law s = 5t 2 . y = a(sin t – t cos t) Find the angle between the curves y 2 = 4 x. 14. Find the maximum and minimum value of 2 sin x + sin 2x over [ 0. Sand is poured from a pipe at the rate of 12 cc. 9. 2. a b a1 b1 15.
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