MATHEMATICS Code No. 13 INSTRUCTIONS

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1 DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO COMBINED COMPETITIVE (PRELIMINARY) EXAMINATION, 00 Serial No. MATHEMATICS Code No. A Time Allowed : Two Hours Maximum Marks : 00 INSTRUCTIONS. IMMEDIATELY AFTER THE COMMENCEMENT OF THE EXAMINATION, YOU SHOULD CHECK THAT THIS TEST BOOKLET DOES NOT HAVE ANY UNPRINTED OR TORN OR MISSING PAGES OR ITEMS, ETC, IF SO, GET IT REPLACED BY A COMPLETE TEST BOOKLET.. ENCODE CLEARLY THE TEST BOOKLET SERIES A, B, C OR D AS THE CASE MAY BE IN THE APPROPRIATE PLACE IN THE RESPONSE SHEET.. You, have to enter your Roll Number on this Your Roll No. Test Booklet in the Box provided alongside. DO NOT write anything else on the Test Booklet.. This Booklet contains 00 items (questions). Each item comprises four responses (answers). You will select one response which you want to mark on the Response Sheet. In case you feel that there is more than one correct response, mark the response which you consider the best. In any case, choose ONLY ONE response for each item. 5. In case you find any discrepancy in this test booklet in any question(s) or the Responses, a written representation explaining the details of such alleged discrepancy, be submitted within three days, indicating the Question No(s) and the Test Booklet Series, in which the discrepancy is alleged. Representation not received within time shall not be entertained at all. 6. You have to mark all your responses ONLY on the separate Response Sheet provided. See directions in the Response Sheet. 7. All items carry equal marks. Attempt ALL items. Your total marks will depend only on the number of correct responses marked by you in the Response Sheet. 8. Before you proceed to mark in the Response Sheet the response to various items in the Test Booklet, you have to fill in some particulars in the Response Sheet as per instructions sent to you with your Admit Card and Instructions. 9. While writing Centre, Subject and Roll No. on the top of the Response Sheet in appropriate boxes use ONLY BALL POINT PEN. 0. After you have completed filling in all your responses on the Response Sheet and the examination has concluded, you should hand over to the Invigilator only the Response Sheet. You are permitted to take away with you the Test Booklet. DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO BKU-9-A [Turn over

2 BKU-9-A ROUGH WORK

3 . Rank of the matrix (D). Test whether the matrix equation is consistant? If so, solution is x = 5 y 5 z Inconsistant Consistant, x =, y =, z = No solution, exist (D) Consistant, x =, y =, z = 0 7. Inverse of the matrix (D) If A = then A =...? (D) BKU-9-A [Turn over

4 5. Value of the determinant zero 8 8 (D) 6 6. Limit of n x a x a n, as x a 0 zero 0 n a n 7. Taylor s series of tan x, in powers of x : (D) x n a x x x x + x +... x x x +... (D) x Differentiation of a x, w.r.t. x, is : a x a x a x log a (D) x a x 9. Split into parts such that the continued product of the first, square of the second and cube of the third may be minimum., 8,, 6,, 9, (D), 6, 5 0. The radius of curvature of y = e x at x = 0 is : () / (D) () /. x The asymptotes of the curve ay = are : x a x = 0, a x = a; a x = 0, a (D) a = 0; x = 0 BKU-9-A

5 . Curvature of the circle x + y 5x + 6y 6 = 0 is : (5) / (5) / 5 (D) 5. Equation of the tangent at (, ) to the circle x + y = 5 : x + y + 5 = 0 x + y + 5 = 0 x + y 5 = 0 (D) x + y = 0. Equation of the normal to the parabola y = x at (, ) is : x + y = 0 x + y + = 0 x + y = 0 (D) x y + 8 = 0 5. Points of inflexion are : Maximum points Neither Maximum nor Minimum Minimum points (D) Double points π/ 6. The value of 0 7 cos θ dθ = π 05 (D) 8π 0 7. The value of π 0 dθ = + cos θ π π 5 π (D) π 5 8. Volume generated by solid of revolution of y = ax about x axis by its Latus rectum; is : πa (πa) π (D) πa a BKU-9-A 5 [Turn over

6 9. Surface area generated by revolving y = sin x, between 0 and π about x axis is : π π π (D) π/ 0. Area of the region bounded by the curve y = x x, and the x axis between x = and x = is : 7/55 55/7 5/7 (D) 55/7 π /. Value of π/ x sin x dx is : 0 (D). If x y + y x = c, then (dy/dx) (partial diffn w.r.t. x)/p.d. w.r.t. y gives : yx y /xy x (yx y + y x log x)/(xy x + x y log x) (yx y + y x log x) (D) xy x + x y log x u v. If x y + u + v = and x + y u v = then =...? and =...? x x xu; xv u v ; x y x x ; u v (D) xu; xv x + y u u. If u = sin then x + y =...? xy x y sin u tan u cot u (D) tan u 5. Test the series for convergence convergent divergent absolute convergent (D) oscillatory 6. Test the convergence of (n + ). n (n + ) convergent oscillatory divergent (D) absolute convergent BKU-9-A 6

7 dy 7. Singular solution of y = x + is : dx (dy/dx) y = x y = x y = x (D) y = x 8. Rewrite the equation p = sin(y px) in CLAIRAUT S equation : y = px + sin p y = cx + sin c y + px + sin p = 0 (D) x = py + sin p dy 9. Solve yp + (x y)p x = 0, here p =. dx (y + x c) (x + y c) = 0 (y x c) (x + y c) = 0 y + x = x + y (D) (y + x) = φ(x + y ) 0. Solution of dy dx dx = is : (y x + c) (y 5x + c) = 0 (y + x + c) (y + 5x + c) = 0 (y + x) (y + 5x) = 0 (D) (y x) (y 5x) = 0. Particular solution of (D + ) y = x, is : x + x x + x x x + 7 (D) x. In which the region of the curve y (a + x) = x (a x) does not lie? x > 0 0 < x < a x a and x > a (D) a < x < a. Particular integral of D y = x sin x : (Ax + B) (x + ) cos x sin x ( x) sin x (D) ( x) sin x cos x. The curve 9y = x ( x ) is symmetrical about y-axis x-axis y = x (D) both the axis BKU-9-A 7 [Turn over

8 5. Equation of parabola whose focus is (, ) and its directrix y = 0 : (y + ) = (x ) (x ) = (y + ) (x x) + 6y + = 0 (D) x(x ) = 9(y ) 6. The axis of the parabola y y + 8x = 0 is : y = x = x = (D) y = 7. 6x y x y = 0 represents : an ellipse a circle a parabola (D) a hyperbola 8. The line x + y = c, is a tangent to the parabola y = 6x, then c is equal to : (D) 9. The point of intersection of the tangent at t = t and t = t to the parabola y = 8x, is : (6t, 8t) (8t, 6t ) (t, t) (D) (t, t ) 0. The length of the latus rectum of the parabola y x + y + 8 = 0 is : 8 6 (D). The directrix of the parabola y = x +, is : 5 x = 7 x = (D) 5 x = 7 x =. The length of the latus rectum of the parabola whose vertex is (, ) and the directrix x = is : 6 (D) 8. The focus of the parabola x = 6y, is : (, 0) (0, ) (, 0) (D) (0, ). The vertex of the parabola x = 8y, is :, 0, 0 8 8, 0 (D) BKU-9-A 8

9 5. The line x + y + 9 = 0 touches the parabola y = 8x at the point : (0, ) (, ) 9 9 6, (D), 6 6. The tangents at the end of any focal chord to the parabola y = x, intersect on the line : x = 0 (x + ) = 0 y + = 0 (D) y = 7. The angle between the two tangents drawn from the point (, ) to y = 6x is : 5º 0º 60º (D) 90º 8. The equation of a sphere with centre at (,, 5) which touches the xoy plane, is (x + ) + (y + ) + (z + 5) = 5 (x ) + (y ) + (z 5) = 5 x + y + z + x + 6y + 0z + 6 = 0 (D) x + y + z = 5 x y 9. The length of the semi-major and the length of minor axis of the ellipse + = 69 are : 6; ; ; 6 (D) ; 50. The distance between the foci of the ellipse 9x + 5y = 80, is : 6 8 (D) 5. The length of major and semi-minor axes of an ellipse are 8, and their corresponding equation are y = 6 and x =, then the equation of ellipse is : (x + ) (y 6) + = 6 (x + ) (y 6) = 6 (x + ) (y 6) + = 6 (x + ) (y 6) (D) = 6 5. The straight line x y + c = 0 is a tangent to the ellipse x + 8y =, if c is : ± ±6 6 (D) ± BKU-9-A 9 [Turn over

10 5. The sum of the distance of any point on the ellipse x + 9y = 6, from ( 5, 0) and ( 5, 0) is : 8 6 (D) 8 5. Two tangents from (5, ) to the ellipse x + 7y = are : x y = ; x + 9y = x = y; x + 9y + = 0 x + 9y = 0; x y = 0 (D) x y = ; x 9y + = Equation of chord of contact of tangents from the point (, ) to the ellipse x + 5y = 0 is : x + 5y = 5 x + 5y 5 = 0 x 5y = 5 (D) x = 5y 56. Equation of the tangent and normal to the parabola x + x y + = 0 at (, ) : x y = 0 x y + = 0 x + y + 8 = 0 x + y 8 = 0 x + y + = 0 (D) x = y x + y + 8 = 0 x + y = The point (,, ) is one end of the diameter of a sphere x + y + z x y + z = 0; the other end is : (0,, 8) (, 0, 8) (8, 0, ) (D) (0, 8, ) 58. Supplies are dropped from a helicopter and distance fallen in time t seconds is given by x = ½ gt where g = 9.8 m/sec. Velocity and acceleration of the supplies after it has fallan for seconds : 9.6 m/sec ; 9.8 m/sec 9.6 ; ; 9.8 (D) 8 ; Angle between the asymptotes of the hyperbola x y x 6y 9 = 0 is : π π (D) π π 60. Eccentricity of ellipse is less than : zero one.5 (D) x y 6. If + a b =, represents ellipse b must be equal to : a ( + e ) a /e a (e ) (D) a ( e ) BKU-9-A 0

11 6. Which of the following is not a binary operation on R? a * b = ab a * b = a b a * b = ab (D) a * b = a + b 6. A monoid becomes a group if it also satisfies the : Closure axiom Associative axiom Identity axiom (D) Inverse axiom 6. Which of the following is not a group? (Z n, + n ) (Z, +) (Z,.)(D) (R, +) 65. In the set of integers with operation * defined by a * b = (a + b ab), the value of * ( * 5) is : (D) The order of [7] in [Z 9, + 9 ] is : 9 6 (D) 67. In the multiplicative group of cube root of unity, the order of w is : (D) 68. The value of []+ ([5] + [6]) is : [0] [] [] (D) [] 69. In the set of real numbers R, an operation * is defined by a * b = a + b of ( * ) * 5 is : (D) 50. Then the value 70. Which of the following is correct? An element of a group can have more than one inverse If every element of a group is its own inverse then the group is abelian The set of all real matrices forms a group under matrix multiplication (D) (a * b) = a * b for all a, b G BKU-9-A [Turn over

12 7. The order of i, in the multiplicative group of unity is : FOUR THREE TWO (D) ONE 7. In the multiplicative group of n th roots of unity, the inverse of w k is; (k < n) : w /k w w n k (D) w n/k 7. In the set of integers under the operation defined by a b = a + b, the identity element is : zero one a (D) b 7. If S be a non-empty set and o be a binary operation on S defined by x o y = x; x, y S then o is : Non-commutative but associative Commutative Non-commutative (D) Commutative and associative 75. The Rolle s constant for the function y = x on [, ] is : Zero Two (D) 76. The C of the Lagranges Mean value theorem for the function f(x) = x + x ; a = 0; b = is : 0 (D) x 77. The value of C in Rolle s theorem for the function f (x) = cos on [ π, π] is : 0 π π/ (D) π/ 78. The value of C of Lagranges Mean value theorem for f (x) = x when a =, and b = is : 9 (D) BKU-9-A

13 79. x Lt =...? x x e 0 (D) 80. The area bounded by the line y = x, then x-axis, the ordinates x =, x = is : / 5/ / (D) 7/ g(x).f(a) g(a).f(x) 8. If f(a) = ; f '(a) = ; g(a) = ; g'(a) = then the value of Lt x a x a to : 5 5 (D) is equal 8. Which of the following function is increasing in (0, )? e x x x (D) x 8. The function f(x) = x 5x +, is increasing in : (, ) (, ) (, ) (D) everywhere 8. The function f(x) = x, is decreasing in : (, ) (, 0) (0, ) (D) (, ) 85. The function y = tan x x, is : π an increasing function in 0, π a decreasing function in 0, π π increasing in 0, and decreasing in, π π π (D) decreasing in 0, and increasing in, π BKU-9-A [Turn over

14 86. In a given semi-circle of diameter cm a rectangle is to be inscribed. The maximum area of the rectangle is : 8 (D) The least possible perimeter of a rectangle of 00 m is : (D) If f(x) = x x + 5, on [0, ] then the absolute maximum value is : (D) The curve y = e x is : concave upward for x > 0 concave downward for x > 0 everywhere concave upward (D) everywhere concave downward 90. Which of the following curves is concave down? y = x y = x y = e x (D) y = x + x 9. The point of inflexion of the curve y = x, is at : x = 0 x = x = (D) nowhere 9. The curve y = ax + bx + cx + d, has a point of flexion at x =, then : a + b = 0 a + b = 0 a + b = 0 (D) a + b = u 9. If u = x y, then, is equal to : x yx y u log x u log y (D) xy x x + y 9. If u = sin and f = sin u, then f is a homogeneous function of degree : x + y 0 (D) BKU-9-A

15 95. If u = then, x + y u u x + y, is equal to : x y u u u (D) u 96. The curve y (x ) = x ( + x), has : an asymptote parallel to x-axis an asymptotes parallel to both axes an asymptote parallel to y-axis (D) no asymptotes 97. If x = r cos θ, y = r sin θ, then sec θ cos θ r is equal to : x sin θ (D) cosec θ x + y 98. If u = log xy 0 then u u x + y, is : x y u u (D) u 99. Air is being pumped into spherical balloon so that its volume increases at a rate of 00 cm /sec. How fast is the radius of the balloon is increasing when diameter is 500 m? 5π cm/s 5 cm/s 5π cm/sec (D) 5π cm/s 00. An asymptote to the curve y (a + x) = x (a x) is : x = a x = a/ x = a/ (D) x = 0 BKU-9-A 5 [Turn over

16 ROUGH WORK BKU-9-A 6 60

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