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Target Centum Practising Package +2 GENERAL MATHEMATICS All Practice papers is for 100 marks Duration to finish the paper is 1 Hr 30 Minutes Applications of Matrices & Determinants, Complex Numbers Section A Choose the best option : 10 x 1 = 10 1. If A is a scalar matrix with scalar k 0, of order 3, then A 1 is (a) ( 1 / k 2 ) I (b) ( 1 / k 3 ) I (c) ( 1 / k ) I (d) k I 2. If the matrix 1 3 2 has an inverse then the value of k is 1 k 3 1 4 5 (a) 4 (b) 4 (c) 4 (d) 4 3. If A is a 1x3 matrix then the rank of AA T is (a) 2 (b) 3 (c) 1 (d) 0 4. If A is a square matrix of order n then adj A is (a) A 2 (b) A n (c) A n 1 (d) A 5. If A and B are any two matrices such that AB = O and A is non-singular then (a) B = O (b) B is singular (c) B is non singular (d) B = A 6. The equations 2x + y + z = l, x 2y + z = m and x + y 2z = n when l + m + n = 0 has (a) trivial solution (b) infinitely many solution (c) No solution (d) a non-zero unique solution 7. The inverse of the matrix 3 1 is 5 2 (a) 2 1 (b) 2 5 (c) 3 1 (d) 3 5 5 3 1 3 5 3 1 2 8. The rank of a diagonal matrix of order n is (a) 1 (b) 0 (c) n (d) n 2 9. The system of equations ax + y + z = 0, x + by + z = 0 and x + y + cz = 0 has a non trivial solution then 1 / ( 1 a ) + 1 / ( 1 b ) + 1 / ( 1 c ) = (a) 1 (b) 2 (c) 1 (d) 0 10. In a system of 3 linear non homogenous equations with three unknowns, if = 0 and x = 0, y 0 and z = 0 then the system has (a) unique solutions (b) two solutions (c) no solution (d) infinitely many solutions 11. The value of [ ( 1 + i 3 ) / 2 ] 100 + [ ( 1 i 3 ) / 2 ] 100 is (a) 2 (b) 0 (c) 1 (d) 1

12. If x 2 + y 2 = 1 then the value of ( 1 + x + iy ) / ( 1 + x iy ) is (a) x iy (b) 2x (c) 2iy (d) x + iy 13. If z represents a complex number then arg ( z ) + arg ( z ) is (a) /4 (b) / 2 (c) (d) 0 14. The value of i + i 22 + i 23 + i 24 + i 25 is (a) i (b) i (c) 1 (d) 1 15. The equation having 4 3i and 4 + 3i as roots is (a) x 2 + 8x + 25 = 0 (b) x 2 + 8x 25 = 0 (c) x 2 8x + 25 = 0 (d) none 16. If is a cube root of unity then the value of ( 1 + 2 ) 4 + ( 1 + 2 ) 4 is (a) 0 (b) 32 (c) 16 (d) 32 17. If ( m 5 ) + i ( n + 4 ) is the complex conjugate of (2m + 3 ) + i (3n 2 ) then ( n, m ) is (a) ( ½, 8 ) (b) ( ½, 8 ) (c) ( ½, 8 ) (d) ( ½, 8 ) 18. If is the cube root of unity then the value of ( 1 ) ( 1 2 ) ( 1 4 ) ( 1 8 ) is (a) 9 (b) 9 (c) 16 (d) 32 19. If z lies in the third quadrant then z lies in the (a) 1 st quadrant (b) 2 nd quadrant (c) 3 rd quadrant (d) 4 th quadrant 20. The least positive integer n such that [ ( 1 + i ) / ( 1 i ) ] n = 1 is (a) 1 (b) 2 (c) 4 (d) 8 Section B Answer any 5 questions from the following : 5 x 6 = 30 21. Find the inverse of the matrix 8 1 3 5 1 2 10 1 4 22. If A = 1 2 and B = 0 1 verify that ( AB ) 1 = B 1 A 1. 1 1 1 2 23. Find the rank of the matrix 4 2 1 3 6 3 4 7 2 1 0 1 24. Find the adjoint of the matrix A = 1 2 and verify A ( adj A ) = ( adj A ) A = A I. 3 5 25. Solve by matrix inversion method : 2x y = 7, 3x 2y = 11. 26. P represents the variable complex number z, find the locus of P if arg [ ( z 1 ) / ( z + 1 ) ] = / 3. 27. Prove that the points representing the complex numbers 2i, 1 + i, 4 + 4i and 3 + 5i on the Argand diagram are the vertices of a rectangle. 28. Find the square root of 7 + 24i. 29. Solve the equation x 4 8x 3 + 24 x 2 32 x + 20 = 0 if 3 + i is a root. Section C Answer any 5 questions from the following : 5 x 10 = 50 30. Solve using matrix inversion method : 2x y + z = 7, 3x + y 5z = 13, x + y + z = 5 31. A bag contains 3 types of coins namely Re.1, Rs.2 and Rs.5. There are 30 coins amounting to Rs.100 in total. Find the number of coins in each category.

32. Verify whether the given system of equations is consistent. If it is consistent, solve them. 2x + 5y + 7z = 52, x + y + z = 9, 2x + y z = 0 33. Investigate for what values of, the simultaneous equations x + y + z = 6, x + 2y + 3z = 10, x + 2y + z = have (i) no solution (ii) unique solution (iii) an infinite number of solutions. 34. For what values of k, the system of equations kx + y + z = 1, x + ky + z = 1, x + y + kz = 1 have (i) unique solution (ii) more than one solution (iii) no solution 35. Solve x 9 + x 5 x 4 1 = 0. 36. If n is a positive integer, prove that ( 3 + i ) n + ( 3 i ) n = 2 n + 1 cos ( n / 6 ). 37. If and are the roots of x 2 2x + 2 = 0 and cot = y + 1 show that [ ( y + ) n ( y + ) n ] / ( ) = sin n / sin n. 38. If x = cos + i sin and y = cos + i sin, prove that x m y n + 1/x m y n = 2 cos ( m + n ). VECTOR ALGEBRA Section A Choose the best option : 20 x 1 = 20 1. If a + b + c = 0, a = 3, b = 4, c = 5 then the angle between a and b is (a) / 6 (b) 2 / 3 (c) 5 / 3 (d) / 2 2. The vectors 2 i + 3 j + 4 k and a i + b j + c k are perpendicular when (a) a = 2, b = 3, c = 4 (b) a = 4, b = 4, c = 5 (c) a = 4, b = 4, c = 5 (d) a = 2, b = 3, c = 4 3. If the projection of a on b and projection of b on a are equal then the angle between a + b and a b is (a) / 2 (b) / 3 (c) / 4 (d) 2 / 3 4. a and b are two unit vectors, is the angle between them, then ( a + b ) is a unit vector if (a) = / 3 (b) = / 4 (c) = / 2 (d) = 2 / 3 5. If a is a non-zero vector and m is a non-zero scalar then m a is a unit vector if (a) m = 1 (b) m = a (c) a = 1 / m (d) a = 1 6. If a and b include an angle 120 and their magnitude are 2 and 3 then a. b is (a) 3 (b) 3 (c) 2 (d) 3 / 2 7. If a + b = a b, then (a) a is parallel to b (b) a is r to b (c) a = b (d) none 8. The work done by the force of magnitude 5 and direction along 2 i + 3 j + 6 k acting on a particle which is displaced from ( 4, 4, 4 ) to ( 5, 5, 5 ) is (a) 11 (b) 55 (c) 55 / 7 (d) 11 / 7 9. If a b = 0 where a and b are non zero vectors then (a) = 90 (b) = 45 (c) = 60 (d) = 0 10. If a b = a. b then (a) = 90 (b) = 45 (c) = 60 (d) = 0 11. If a = 3, b = 4 and a. b = 9 then the value of a b is (a) 3 / 4 (b) 14 (c) 63 (d) 43 12. If a = 2, b = 7 and a b = 3 i 2 j + 6 k then

(a) = / 4 (b) = / 6 (c) = / 2 (d) = / 3 13. If [ a x b, b x c, c x a ] = 64 then [ a, b, c ] is (a) 32 (b) 8 (c) 128 (d) 0 14. The value of [ i + j, j + k, k + i ] is equal to (a) 0 (b) 1 (c) 2 (d) 4 15. The point of intersection of the lines ( x 6 ) / 6 = ( y + 4 ) / 4 = ( z 4 ) / 8 and ( x + 1 ) / 2 = ( y + 2 ) / 4 = ( z + 3 ) / 2 is (a) (0, 0, 4 ) (b) ( 1, 0, 0 ) (c) ( 0, 2, 0 ) (d) ( 1, 2, 0 ) 16. If u = a x ( b x c ) + b x ( c x a ) + c x ( a x b ), then (a) u is a unit vector (b) u = a + b + c (c) u = 0 (d) u 0 17. If a x ( b x c ) = ( a x b ) x c for non coplanar vectors a, b, c then (a) a is parallel to b (b) b is parallel to c (c) c is parallel to a (d) a + b + c = 0 18. The vector equation of the plane in parametric form passing through the point with P.V. a and parallel to the vectors u and v is (a) r = a + t ( b a ) + s ( c a ) (b) r = a + t u + s v (c) [ r, a, u ] = [ r, a, v ] (d) [ r a, u, v ] = 0 19. The point of intersection of the line r = ( i k ) + t ( 3 i + 2 j + 7 k ) and the plane r. ( i + j k ) = 8 is (a) ( 8, 6, 22 ) (b) ( 4, 3, 11 ) (c) ( 4, 3, 11 ) (d) ( 8, 6, 22 ) 20. The centre and the radius of the sphere x 2 + y 2 + z 2 6x + 8y 10z + 1 = 0 are (a) ( 3, 4, 5 ), 49 (b) ( 6, 8, 10 ), 1 (c) ( 3, 4, 5 ) 7 (d) ( 6, 8,10),7 Section B Answer any 5 questions from the following: 5 x 6 = 30 21. Show that the points whose position vectors 4 i 3 j + k, 2 i 4 j + 5 k, i j form a right angled triangle. 22. The constant forces 2 i 5 j + 6 k, i + 2 j k and 2 i + 7 j act on a particle which is displaced from the position ( 4, 3, 2 ) to ( 6, 1, 3 ). Find the work done. 23. (a) Find the unit vector perpendicular to the plane containing 2 i + j + k and i + 2 j + k. (b) Find the area of the parallelogram whose diagonals are represented by 2 i + 3 j + 6 k and 3 i 6 j + 2 k. 24. Forces 2 i + 7 j, 2 i 5 j + 6 k, i + 2 j k act at a point P ( 4, 3, 2 ).Find the moment of the resultant of the forces acting at P about the Q ( 6, 1, 3 ). 25. Find the vector and Cartesian equations of the sphere on the join of the points A and B having position vectors 2 i + 6 j 7 k and 2 i 4 j + 3 k respectively as a diameter. 26. Find the centre and radius of the sphere r 2 r. ( 4 i + 2 j 6 k ) 11 = 0. 27. Find the equation of the plane passing through the intersection of the planes 2x 8y +4z = 3 and 3x 5y +4z +10 = 0 and perpendicular to the plane 3x y 2z 4 = 0 28. Find the angle between the line ( x 2 ) / 3 = ( y + 1 ) / ( 1 ) = ( z 3 ) / ( 2 ) and the plane 3x + 4y + z + 5 = 0. Section C Answer any 5 questions from the following : 5 x 10 = 50

29. Prove that the altitudes of a triangle are concurrent using vector methods. 30. Prove that sin ( A + B ) = sin A cos B + cos A sin B using cross product of vectors. 31. If a = 2 i + 3 j k, b = 2 i + 5 k, c = j 3 k verify that a ( b c ) = ( a. c ) b ( a. b ) c. 32. Verify that ( a b ) ( c d ) = [ a b d ] c [ a b c ] d if a = i + j + k, b = 2 i + k, c = 2 i + j + k, d = i + j + 2 k. 33. Show that the lines ( x 1 ) / 3 = ( y 1 ) / ( 1 ) = ( z + 1 ) / 0 and ( x 4 ) / 2 = y / 0 = ( z + 1 ) / 3 intersect and hence find the point of intersection. 34. Find the shortest distance between the skew lines ( x 6 ) / 3 = ( y 7 ) / ( 1 ) = z 4 and x / ( 3 ) = ( y + 9 ) / 2 = ( z 2 ) / 4. 35. Find the vector and Cartesian equations of the plane passing through the points ( 2, 2, 1 ), ( 3, 4, 2 ) and ( 7, 0, 6 ). 36. Find the vector and Cartesian equations of the plane through the point ( 2, 1, 3 ) and parallel to the lines ( x 2 ) / 3 = ( y 1 ) / 2 = ( z 3 ) / ( 4 ) and ( x 1 ) / 2 = ( y + 1 ) / ( 3 ) = ( z 2 ) / 2. 37. Find the vector and Cartesian equations of the plane through the points ( 1, 1, 1 ) and ( 1, 1, 1 ) and perpendicular to the plane x + 2y + 2z = 5. ANALYTICAL GEOMETRY Section A Choose the best option : 20 x 1 = 20 1. The length of the latus rectum of the parabola y 2 4x + 4y + 8 = 0 is (a) 8 (b) 6 (c) 4 (d) 2 2. The directrix of the parabola y 2 = x + 4 is (a) x = 15 / 4 (b) x = 15 / 4 (c) x = 17 / 4 (d) x = 17 / 4 3. The line 2x + 3y + 9 = 0 touches the parabola y 2 = 8x at the point (a) ( 0, 3 ) (b) ( 2, 4 ) (c) ( 6, 9 / 2 ) (d) ( 9 / 2, 6 ) 4. The focus of the parabola x 2 = 16y is (a) ( 4, 0 ) (b) ( 0, 4 ) (c) ( 4, 0 ) (d) ( 0, 4 ) 5. The common tangent for the parabolas y 2 = 4ax and y 2 = -4ax is (a) x = 0 (b) x + a = 0 (c) y = 0 (d) y - a = 0 6. The vertex of the parabola x 2 = 8y 1 is (a) ( 0, 1 ) (b) ( 0, 1 / 8 ) (c) ( 0, 1 / 8 ) (d) ( 0, 1 ) 7. The length of the semi latus rectum of the parabola ( y + 2 ) 2 = 8 ( x + 1 ) is (a) 8 (b) 4 (c) 4 (d) 8 8. The axis of the parabola whose vertex is ( 1, 4 ) and focus is ( 2, 4 ) is (a) x = 1 (b) x = 2 (c) y = 0 (d) y = 4 9. The equation of the directrix of the parabola y 2 = 8x (a) x = 2 (b) x = 2 (c) y = 2 (d) y = 2 10. The equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 represents a parabola if B 2 4AC (a) = 0 (b) > 0 (c) < 0 (d) 0 11. The distance between the vertices of the ellipse 9x 2 + 5y 2 = 180 is (a) 8 (b) 6 (c) 18 (d) 12 12. The equation of the major axis of the ellipse ( x 3 ) 2 / 36 + ( y + 3 ) 2 / 27 = 1 is

(a) x = 3 (b) x = 3 (c) y = 3 (d) y = 3 13. The sum of the distance of any point on the ellipse 4x 2 + 9y 2 = 36 from the foci is (a) 6 (b) 4 (c) 2 (d) 3 14. The eccentricity of the ellipse 5x 2 + 9y 2 = 45 is (a) 3 / 2 (b) 2 / 3 (c) 14 / 3 (d) none 15. For an ellipse, centre is ( 2, 3 ) and one of the focus is ( 3, 3 ) then the other focus is (a) ( 5 / 2, 3 ) (b) ( ½, 0 ) (c) ( 1, 3 ) (d) ( 1, 0 ) 16. For an ellipse, one of the foci is ( 5, 4 ) and one of the vertex is ( 1, 4 ) then the equation of the major axis is (a) x = 4 (b) x = 4 (c) y = 4 (d) y = 4 17. The length of semi latus rectum of the ellipse ( x + 2 ) 2 / 9 + ( y 5 ) 2 / 36 = 1 is (a) 3 (b) 8 (c) 3 / 2 (d) 2 / 3 18. The eccentricity of hyperbola whose latus rectum is equal to half of its conjugate axis is (a) 3 / 2 (b) 5 / 3 (c) 3 / 2 (d) 5 / 2 19. The line 5x 2y + 4k = 0 is a tangent to 4x 2 y 2 = 36 then k is (a) 4 / 9 (b) 2 / 3 (c) 9 / 4 (d) 81 / 16 20. One of the foci of the rectangular hyperbola xy = 18 is (a) ( 6, 6 ) (b) ( 3, 3 ) (c) ( 4, 4 ) (d) ( 5, 5 ) Section B Answer any 5 questions from the following: 5 x 6 = 30 21. Find the axis, vertex, focus, directrix, equation of the latus rectum and its length for the parabola ( x 4 ) 2 = 4 ( y + 2 ). 22. On lighting a rocket cracker it gets projected in a parabolic path and touches a maximum height of 4mts when it is 6 mts away from the point of projection. Finally it reaches the ground 12 mts away from the starting point. Find the angle of projection. 23. Find the equation of the ellipse given foci ( 3, 0 ) and length of LR is 32 / 5. 24. Find the equation of the ellipse whose vertices are ( 1, 4 ), ( 7, 4 ) and e = 1/3. 25. Find the equations of directrices, latus rectum and the length of latus rectum of the hyperbola 9x 2 36x 4y 2 + 32y + 8 = 0. 26. Find the equation of the hyperbola whose centre is ( 1, 4 ), one of the foci is ( 6, 4 ) and the corresponding directrix is x = 9 / 4. 27. A standard rectangular hyperbola has its vertices at ( 5, 7 ) and ( 3, 1 ). Find its equation and asymptotes. 28. Prove that the tangent at any point to the rectangular hyperbola forms with the asymptotes a triangle of constant area. 29. Find the equation of the tangent and normal to the parabola x 2 + x 2y + 2 = 0 at ( 1, 2 ) Section C Answer any 5 questions from the following : 5 x 10 = 50 30. Find the axis, vertex, focus, directrix, equation of LR and length of LR for the parabola y 2 8x + 6y + 9 = 0 and trace the curve. 31. The girder of a railway bridge is in the parabolic form with span 100ft. and the highest point on the arch is 10ft. above the bridge. Find the height of the bridge at 10 ft. to the left or to the right from the midpoint of the bridge.

32. A cable of suspension bridge hangs in the form of a parabola when the load is uniformly distributed horizontally. The distance between two towers is 1500ft., the points of support of the cable on the towers are 200ft. above the roadway and the lowest point on the cable is 70ft. above the roadway. Find the vertical distance to the cable ( parallel to the roadway ) from a pole whose height is 122ft. 33. Find the centre, foci, vertices and e of the ellipse 16x 2 + 9y 2 + 32x 36y = 92. 34. A ladder of length 15m moves with its ends always touching the vertical wall and the horizontal floor. Determine the equation of the locus of a point P on the ladder, which is 6m from the end of the ladder in contact with the floor. 35. The ceiling in a hallway 20ft wide is in the shape of a semi ellipse and 18ft high at the centre. Find the height of the ceiling 4ft from either wall if the height of the side walls is 12ft. 36. Find the centre, vertices, foci and e of the hyperbola 9x 2 18x 16y 2 64y 199 = 0 and also trace the curve. 37. Find the separate equations of the asymptotes of the hyperbola 3x 2 5xy 2y 2 + 17x + y + 14 = 0. 38. Find the equation of the hyperbola if its asymptotes are parallel to x + 2y 12 = 0 and x 2y + 8 = 0, centre is ( 2, 4 ) and it passes through ( 2, 0 ). DIFFERENTIAL CALCULUS APPLICATIONS Section A Choose the best option : 20 x 1 = 20 1. The point on the curve y = 2x 2 6x 4 at which the tangent is parallel to the x axis is (a) ( 5 / 2, 17 / 2 ) (b) ( 5 / 2, 17 / 2 ) (c) ( 5 / 2, 17 / 2 ) (d) none 2. If the volume of an expanding cube is increasing at the rate of 4 cm 3 / sec then the rate of surface area when the volume of the cube is 8 cc is (a) 8 cm 2 / sec (b) 16 cm 2 / sec (c) 2 cm 2 / sec (d) 4 cm 2 / sec 3. The value of a so that the curves y = 3 e x and y = ( a / 3 ) e x intersect orthogonally is (a) 1 (b) 1 (c) 1 / 3 (d) 3 4. The value of c in Lagrange s Mean Value Theorem for the function f ( x ) = x 2 + 2x 1 in the interval [ 0, 1 ] (a) 1 (b) 1 (c) 0 (d) ½ 5. The curve y = e x is (a) concave upward for x > 0 (b) concave downward for x > 0 (c) everywhere concave upward (d) everywhere concave downward 6. The least possible perimeter of a rectangle of area 100m 2 is (a) 10 (b) 20 (c) 40 (d) 60 7. Lim x ( x 2 / e x ) = (a) 2 (b) 0 (c) 1 (d) 8. Which of the following functions is increasing in ( 0, )? (a) e x (b) 1 / x (c) x 2 (d) x 2 9. The point of inflection of the curve y = x 4 is at (a) x = 0 (b) x = 3 (c) x = 12 (d) no where 10. If f ( x ) = x 2 4x + 5 on [ 0, 3 ] then the absolute maximum value is (a) 2 (b) 3 (c) 4 (d) 5

11. The function f ( x ) = x 2 5x + 4 is increasing in (a) (, 1 ) (b) ( 1, 4 ) (c) ( 4, ) (d) everywhere 12. The equation of the tangent to the curve y = x 3 / 5 at ( 1, 1 / 5 ) is (a) 5y + 3x = 2 (b) 5y 3x = 2 (c) 3x 5y = 2 (d) 3x + 3y = 2 13. The angel between the curves ( x 2 / 25 ) + ( y 2 / 9 ) = 1 and ( x 2 / 8 ) ( y 2 / 8 ) = 1 is (a) / 4 (b) / 3 (c) / 2 (d) / 6 14. When the volume of a sphere is increasing at the same rate as its radius, its surface area is (a) 4 / 3 (b) 4 (c) 1 / 2 (d) 1 15. If the normal to the curve x 2 / 3 + y 2 / 3 = a 2 / 3 makes an angle with the x axis then the slope of the normal is (a) cot (b) tan (c) tan (d) cot 16. For what value of x is the rate of increase of x 3 2x 2 + 3x + 8 is twice the rate of increase of x (a) ( 1 / 3, 3 ) (b) ( 1 / 3, 3 ) (c) ( 1 / 3, 3 ) (d) ( 1 / 3, 1 ) 17. If the velocity of a particle moving along a straight line is directly proportional to the square of its distance from a fixed point on the line. Then its acceleration is proportional to (a) s (b) s 2 (c) s 3 (d) s 4 18. The value of c of Lagrange s Mean Value theorem for f ( x ) = x when a = 1 and b = 4 is (a) 9 / 4 (b) 3 / 2 (c) 1 / 2 (d) 1 / 4 19. The angle between the parabola y 2 = x and x 2 = y at the origin is (a) 2 tan 1 ( 3 / 4 ) (b) tan 1 ( 4 / 3 ) (c) / 2 (d) / 4 20. lim x 0 [ ( a x b x ) / ( c x d x ) ] = (a) [ log ( a / b ) ] / [ log ( c / d ) ] (b) 0 (c) log ( ab / cd ) (d) Section B Answer any 5 questions from the following: 5 x 6 = 30 21. A boy, who is standing on a pole of height 14.7 m throws a stonevertically upwards. It moves in a vertical line slightly away from the pole andfalls on the ground. Its equation of motion in meters and seconds is x = 9.8 t 4.9t 2 (i) Find the time taken for upward and downward motions. (ii) Also find the maximum height reached by the stone from the ground. 22. Find the equations of the tangent and the normal to the curve y = x 2 4x 5 at x = 2. 23. Verify Rolle s theorem for the function f ( x ) = 4x 3 9x, 3/2 x 3/2. 24. Obtain the Maclaurin s series for the function f ( x ) = tan 1 x. 25. Verify Lagrange s mean value theorem for f ( x ) = 2x 3 + x 2 x 1, [ 0, 2 ]. 26. Evaluate lim x 1 x 1 / ( x 1 ). 27. Find the critical numbers and stationary points of x 3/5 (4 x) 28. Find the absolute maximum and absolute minimum values of f ( x ) = x 2 2x + 2,[ 0, 3 ] 29. Determine the points of inflection, if any, of the function f ( x ) = x 3 3x + 2. Section C Answer any 5 questions from the following : 5 x 10 = 50

30. If the curve y 2 = x and xy = k are orthogonal then prove that 8k 2 = 1. 31. Let P be a point on the curve y = x3 and suppose that the tangent line at P intersects the curve again at Q. Prove that the slope at Q is four times the slope at P. 32. Find the condition for the curves ax 2 + by 2 = 1, a 1 x 2 + b 1 y 2 = 1 to intersect orthogonally. 33. Evaluate lim x 0 + x sin x. 34. Obtain the expansion of tanx using maclaurin s series 35. Show that the volume of the largest right circular cone that can be inscribed in a sphere of radius a is ( 8 / 27 ) x ( volume of the sphere ). 36. A closed ( cuboid ) box with a square base is to have volume of 2000 cc. The material for the top and the bottom of the box is to cost Rs.3 per square cm and the material for the sides is to cost Rs.1.50 per square cm. If the cost of the materials is to be the least, find the dimensions of the box. 37. Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r. 38. Find the points of inflection and determine the intervals of convexity and concavity of the Gaussion curve y = e x 2. Important VIQs( Very Important Questions ) Differential Calculus - II 1. Trace the following curves (a) y = x 3 + 1 (b) y = x 3 (c) Semi cubical parabola (d) y 2 = ( x - a) ( x - b ) 2 2. Page No. 72 Example 6.5 3. Page No. 73 Example 6.7 4. Exercise 6.1 Q. No. 3 5. Example 6.7 6. Examples 6.15, 6.18, 6.19, 6.20, 6.22 7. Exercise 6.3 [ Q.No. 3 (i) (iii), Q.No. 4 (i) (ii) (iii), Q. No. 5 (i) (iv) SMS your NAME, SCHOOL to 8122235773 to get additional IMPORTANT QUESTIONS