Consortium of Medical Engineering and Dental Colleges of Karnataka (COMEDK-2007)

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Tamsyn Bell
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1 Consortium of Medical Engineering and Dental Colleges of Karnataka (COMEDK-007) MATHEMATICS. Log'f/ is equal to 7 Log In the group (G 5 ), where G = {3, 6, 9, }; 5 is multiplication modulo 5, the identity element is A group (G *) has 0 elements. The minimum num er of elements of G, which are their own inverses is If ii and 6 are vectors such that la+ 6 = la -6, therj. the angle between a and 6 is x + 5. is equal to X -6x+8 3 (x- (x - ) (x- (x (x- (x ' 3 + (x, (x-

2 6. If a = i + 3 J-k, b = i + J- 5 k, c = 3 [ + 5 J-k,.then a vector perpendicular to a and in the plane contaii:,ring b and c is +]-3 k - ]-97k - - ]-97k - - ]+97k ' ' 7. OA and BO are two vectors of magnitudes 5 and 6 respectively. If!BOA= 60, then OA OB is equal to A vector perpendicular to the plane containing the points A (, -,, B (, 0, -, C (0,, is s +4]+4k +]-k 4i+B]-4k 3i+]+k = (3n-(3n+ n 6n+3 n 6n-4 n+l 6n+4 6n+4 0. The ninth term of the expansion ( 3 x -; x j is - 5x 9 5x 9 56-x x 8

3 . A = [;r l 0 B +: ;J andb is the inverse of A, th the value ofa is 0 4 J 5.. If A = [; : : is singular, then the possible values of are 0,, - 0, +, - 0, 5, - 5 0, 4-4: 3. If A=[ l then A adj(a) is equal to l) [H ] [; ; ;] [ H] [H ] 4. Iff:R R is defined by f(x)=jxj,then, r ) ( x = l x l f- (x)=_!_ X The functjon r (x ). does not exist. X p q 5. The value of P x q is p q X (x-p)(x- q)(x+p+q) pq(x-p)(x-q). x(x-p) (f-q) (p - q) ( X -t- q) ( X - p).

4 6. The number of common tangents to the circles x + y = 4 and x + y - 6x - Sy - 4 = 0 is, 4 3 ) 3 7. If 3x + y + k = 0 is a tangent to the circle x + y = 0, the values of k a.re, ± 5 ±9 ± 7. ± 0 8. The negation of the proposition "If is prime, then 3 is odd" is is prime and 3 is not odd If is not prime then 3 is not odd If is not prime then 3 is odd. is not prime and 3 is odd. 9. The equation to two circles which touch the Y-axis at (0, and make an intercept of 8 units on X-axis are x + y ± 6x - l0y + 9 = 0 x + y ± l0x -6y + 9 i::: 0 x + y + lox ± 6y + 9 = 0 x + y -Bx± loy + 9;:: 0 0. The orthocentre of the triangle with vertices A(O, 0), B(O, ½), C(-5, 0) is (-½, ¾) (0, 0) (½, ¾) 4 ) (-5, ½)

5 . x + y -6x -6y + 4 = 0, x + y -x -4y + 3 = 0, x + y + k x + y + = 0 If the Radical centre of the above three circles exists, then which of the following cannot be the value of k? 4 5. If the circles x + y -x -y -7 = O and x + y + 4x + y + k = O cut orthogenally, then. the length of the common chord of the circles is 8,. y The co-ordinates of the foot of the perpendicular drawn from the point (3, on the line x+y-7 =0 is (, 5) ( ;) (, -5) (-5, 4. The area enclosed by the pair of lines xy = 0, the line x -4 = O and y + 5 = 0 is 0 sq. units. 0 sq. units 0 sq. units. 5 4 sq. umts.. ' 5. If the area of the auxillary circle of the ellipse : + ; = (a > b) is twice the area of the ellipse, then the eccentricity of the ellipse is 3.! 3

6 6. A graph G has 'm' vertices of odd degree and 'n' vertices of even degree. Then which ' of " the following statements is necessarily true? m + n is an even number m + n is an odd number m + is an odd number n + is an even number 7. If pis any point on the ellipse + L, and S and S' are the foci, then PS+ PS'= s [c - 5]. 8. The value of m os 3 is x y. _.. _. If 36 - k = s a hyperbola, then which of the followmg statements can be true? (3, lies on the hyperbola (-3, lies on the hyperbola (5, lies on the hyperbola 30. The focus of the parabola is (.! - 3' (0, lies on the hyperbola (. ¾] (½, ¾)

7 3. '., n. The solution of Tan x + Cot x = - s Sin 7.5 o +Sin 7.5 o is equal to Tan 45 Sin 45 o.. The conjugate of the complex number (l+i)... is -i l+i --i Cos 9o 0 Cos 3o 0 -i -l+i 34. ABC is a triangle with [A = 30 BC = 0 ems The area of the circum-circle of the triangle is A B 5 sq. ems. 00,r sq. ems. 00,r sq.' ems. 5 sq. ems. 35. If Sin30 =Sin.0, how many solutions exist such that -n < 0 < n?

8 36. The imaginary part of ii is, The amplitude of ( + i )5. is -3n 3n n -5n ABC is a tringle. G is the centroid.dis the mid point of BC. If A= (, and G = (7, 5), then the point D is ( :, 6) ( ( 8, ;) 4 ) ( ' ) 39.. Tan(x - Lim ---- is equal to X x- - 4 ) dy. If y = Log x, then dx s g x -Log Log x ) Log Lo gx Log Log x X

9 -(l+x) dy 4. If Sec -- = a then - is.dx -y ' y+l x-l. y-l x+l x- y+l x- y- 3x. 3x d y. 4. If y=cos - -Sm - then - is ' dx. 9y -3 -y. 4 ) - 9y 43. If the function f( x ) = of k is 0 - -Cosx k x for x :#:0 for x =0 ½ t 4 ) is conti uous at x = O, then the value 44. If, w, w are the cube roots of unity then ( +w) ( +w ) ( +w 4 ) ( + w 8 ) is equal to 0 w dy. 45. If x x = yy then dx s 4 ) w X y _{'_ l+logx 3 )!+Logy 4 ) +Log(;)

10 . : i 46. The point on the cw-ve y = x, the tangent' at which makes an angle 450 with X-axis is (½' ¼) (½' ½) (¼' ½) 4 ) (½' -½) 47. The length of the subtangent to the curve x y = a 4 at (-a, a) is a ½ ½ a 48. The number of positive divisors of 5 is The remainder obtained when 54 is divided by 4 is, Which of the following is not a group with respect to the given operation? The set of odd integers under additon. The set of even integers under addition. {, -} under multiplication. { 0 } under addition.

11 5. The range in which y =.:..x + 6x - 3 is increasing is.... X >3 X <3 5 < X < 6 7 <x < 8 n/( ) 5. The value of the integral f Sm x -Cos x dx is. 0. ' 00! (00) OA and OB are two roads enclosing an angle of 0. X and Y start from 'O' at the same time. X travels along OA with a speed of 4 km/hour and Y travels along OB with a speed of 3 km/hour. The rate at which the shortest distance between X and Y is increasing after hour is 37 km/hour 37 km/hour A Ji3 km/hour ' 3 knvhour 54. If 3 k J x f (3x )dx = ft f (t )dt, then the value of k is ½ } 9 ½ 55. The value of J l+cosbx dx is Tan Bx +C 8 Tan..x 8 +C Tan4x + C Tan4x 4 +C

12 60. The order and de ee of the differential equation [ + (: j ] 3! are respectively 56. The value of J e x (x 5 +5x 4 + dx is e x -x 5 +e x +C 5x 4 -e x e x X 5 ' e x +l.x 5 +C 57. The value of J x dx is X - Log (-- x+l ) +_C x- Log( -- x- ) +C x+l Log(x -l)+c. x+log ( X -- ) +C x+l 58. The area bounded by the curve x = 4 -y and the Y-axis is 3 sq. units 6 sq. units 6 3 sq. units 3 3 sq. units 59. The differential equation of the family of straight lines whose slope is equal toy-intercept is (x+l) d y + y =O dx (x+l) d y - y =O dx x+l x- -= dx y+l dx y- d y d y,, 3, 5, 5

QUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola)

QUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola) QUESTION BANK ON CONIC SECTION (Parabola, Ellipse & Hyperbola) Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q. Two mutually perpendicular tangents