TEST CODE: MIII (Objective type) 2010 SYLLABUS

Transcription

1 TEST CODE: MIII (Objective type) 200 SYLLABUS Algebra Permutations and combinations. Binomial theorem. Theory of equations. Inequalities. Complex numbers and De Moivre s theorem. Elementary set theory. Simple properties of a group. Functions and relations. Algebra of matrices. Determinant, rank and inverse of a matrix. Solutions of linear equations. Eigenvalues and eigenvectors of matrices. Coordinate geometry Straight lines, circles, parabolas, ellipses and hyperbolas. Elements of three dimensional coordinate geometry straight lines, planes and spheres. Calculus Sequences and series. Power series. Taylor and Maclaurin series. Limits and continuity of functions of one or more variables. Differentiation and integration of functions of one variable with applications. Definite integrals. Areas using integrals. Definite integrals as limits of Riemann sums. Maxima and minima. Differentiation of functions of several variables. Double integrals and their applications. Ordinary linear differential equations. SAMPLE QUESTIONS Note: For each question there are four suggested answers of which only one correct.. If a, b are positive real variables whose sum a constant λ, then the minimum value of ( + /a)( + /b) (A) λ /λ (B) λ + 2/λ (C) λ + /λ 2. Let x be a positive real number. Then (A) x 2 + π 2 + x 2π > xπ + (π + x)x π (B) x π + π x > x 2π + π 2x (C) πx + (π + x)x π > x 2 + π 2 + x 2π 3. Suppose in a competition matches are to be played, each having one of 3 dtinct outcomes as possibilities. The number of ways one can predict the outcomes of all matches such that exactly 6 of the predictions turn out to be correct (A) ( 6 ) 2 5 (B) ( ) 6 (C) 3 6

2 . A club with x members organized into four committees such that (a) each member in exactly two committees, (b) any two committees have exactly one member in common. Then x has (A) exactly two values both between and 8 (B) exactly one value and th lies between and 8 (C) exactly two values both between 8 and 6 (D) exactly one value and th lies between 8 and Let X be the set {, 2, 3,, 5, 6, 7, 8, 9, 0}. Define the set R by R = {(x, y) X X : x and y have the same remainder when divided by 3}. Then the number of elements in R (A) 0 (B) 36 (C) 3 (D) Let A be a set of n elements. The number of ways, we can choose an ordered pair (B, C), where B, C are djoint subsets of A, equals (A) n 2 (B) n 3 (C) 2 n (D) 3 n. 7. Let ( + x) n = C 0 + C x + C 2 x C n x n, n being a positive integer. The value of ( + C ) ( 0 + C ) (... + C ) n C C 2 C n (A) ( ) n n + (B) n + 2 n n n! (C) ( ) n n (D) n + (n + ) n. n! 8. The value of the infinite product P = n3 n 3 + (A) (B) 2/3 (C) 7/3 9. The number of positive integers which are less than or equal to 000 and are divible by none of 7, 9 and 23 equals (A) 85 (B) 53 (C) 60 2

3 0. Consider the polynomial x 5 + ax + bx 3 + cx 2 + dx + where a, b, c, d are real numbers. If ( + 2i) and (3 2i) are two roots of th polynomial then the value of a (A) 52/65 (B) 52/65 (C) /65 (D) /65.. The number of real roots of the equation ( x 2 ) + x 2 cos = 2 x + 2 x 6 (A) 0 (B) (C) 2 (D) infinitely many. 2. Consider the following system of equivalences of integers. x 2 mod 5 x mod 2. The number of solutions in x, where x 35, to the above system of equivalences (A) 0 (B) (C) 2 (D) The number of real solutions of the equation (9/0) x = 3 + x x 2 (A) 2 (B) 0 (C). If two real polynomials f(x) and g(x) of degrees m ( 2) and n ( ) respectively, satfy f(x 2 + ) = f(x)g(x), for every x R, then (A) f has exactly one real root x 0 such that f (x 0 ) = 0 (B) f has exactly one real root x 0 such that f (x 0 ) = 0 (C) f has m dtinct real roots (D) f has no real root. 5. Let X = Then, (A) X < (B) X > 3/2 (C) < X < 3/2 (D) none of the above holds. 3

4 6. The set of complex numbers z satfying the equation represents, in the complex plane, (A) a straight line (3 + 7i)z + (0 2i)z + 00 = 0 (B) a pair of intersecting straight lines (C) a point (D) a pair of dtinct parallel straight lines. 7. The limit lim n e 2πik n n k= e 2πi(k ) n (A) 2 (B) 2e (C) 2π (D) 2i. 8. Let ω denote a complex fifth root of unity. Define for 0 k. Then b k = jω kj, j=0 b k ω k equal to k=0 (A) 5 (B) 5ω (C) 5( + ω) (D) Let a n = ( ) ( ), n. Then lim a n 2 n + n (A) equals (B) does not ext (C) equals π (D) equals Let X be a nonempty set and let P(X) denote the collection of all subsets of X. Define f : X P(X) R by Then f(x, A B) equals (A) f(x, A) + f(x, B) (B) f(x, A) + f(x, B) f(x, A) = (C) f(x, A) + f(x, B) f(x, A) f(x, B) (D) f(x, A) + f(x, A) f(x, B) { if x A 0 if x / A.

5 ( ) x 3x 2. The limit lim equals x 3x + (A) (B) 0 (C) e 8/3 (D) e /9 ( n 22. lim n n n + + n n n ) equal to 2n (A) (B) 0 (C) log e 2 (D) 23. Let cos 6 θ = a 6 cos 6θ+a 5 cos 5θ+a cos θ+a 3 cos 3θ+a 2 cos 2θ+a cos θ+a 0. Then a 0 (A) 0 (B) /32. (C) 5/32. (D) 0/ The set {x : x + x > 6} equals the set (A) (0, 3 2 2) ( , ) (B) (, 3 2 2) ( , ) (C) (, 3 2 2) ( , ) (D) (, 3 2 2) ( , 3 2 2) ( , ) 25. Suppose that a function f defined on R 2 satfies the following conditions: f(x + t, y) = f(x, y) + ty, f(x, t + y) = f(x, y) + tx and f(0, 0) = K, a constant. Then for all x, y R, f(x, y) equal to (A) K(x + y). (B) K xy. (C) K + xy. 26. Consider the sets defined by the real solutions of the inequalities A = {(x, y) : x 2 + y } B = {(x, y) : x + y 6 }. Then (A) B A (B) A B (C) Each of the sets A B, B A and A B non-empty 5

6 27. If f(x) a real valued function such that for every x R, then f(2) 2f(x) + 3f( x) = 5 x, (A) 5 (B) 22 (C) (D) If f(x) = 3 sin x, then the range of f(x) 2 + cos x (A) the interval [, 3/2] (B) the interval [ 3/2, ] (C) the interval [, ] 29. If f(x) = x 2 and g(x) = x sin x + cos x then (A) f and g agree at no points (B) f and g agree at exactly one point (C) f and g agree at exactly two points (D) f and g agree at more than two points. 30. For non-negative integers m, n define a function as follows n + if m = 0 f(m, n) = f(m, ) if m = 0, n = 0 f(m, f(m, n )) if m = 0, n = 0 Then the value of f(, ) (A) (B) 3 (C) 2 (D). 3. A real 2 2 matrix M such that ( M 2 = 0 0 ε ) (A) exts for all ε > 0 (B) does not ext for any ε > 0 (C) exts for some ε > 0 (D) none of the above true 6

7 32. The eigenvalues of the matrix X = are (A),, (B),, (C) 0,, (D) 0,,. 33. Let x, x 2, x 3, x, y, y 2, y 3 and y be fixed real numbers, not all of them equal to zero. Define a matrix A by x 2 + y 2 x x 2 + y y 2 x x 3 + y y 3 x x + y y x 2 x + y 2 y x y2 2 x 2 x 3 + y 2 y 3 x 2 x + y 2 y A =. x 3 x + y 3 y x 3 x 2 + y 3 y 2 x y3 2 x 3 x + y 3 y Then rank(a) equals x x + y y x x 2 + y y 2 x x 3 + y y 3 x 2 + y 2 (A) or 2. (B) 0. (C). (D) 2 or If M a 3 3 matrix such that [ 0 2 ]M = [ 0 0 ] and [ 3 5 ]M = [ 0 0 ] then [ ]M equal to (A) [ 2 2 ] (B) [ 0 0 ] (C) [ 2 0 ] (D) [ ]. 35. Let λ, λ 2, λ 3 denote the eigenvalues of the matrix 0 0 A = 0 cos t sin t. 0 sin t cos t If λ + λ 2 + λ 3 = 2 +, then the set of possible values of t, π t < π, (A) Empty set (B) { π } (C) { π, π } 36. The values of η for which the following system of equations has a solution are x + y + z = x + 2y + z = η x + y + 0z = η 2 (D) { π 3, π }. 3 (A) η =, 2 (B) η =, 2 (C) η = 3, 3 (D) η =, 2. 7

8 37. Let P, P 2 and P 3 denote, respectively, the planes defined by a x + b y + c z = α a 2 x + b 2 y + c 2 z = α 2 a 3 x + b 3 y + c 3 z = α 3. It given that P, P 2 and P 3 intersect exactly at one point when α = α 2 = α 3 =. If now α = 2, α 2 = 3 and α 3 = then the planes (A) do not have any common point of intersection (B) intersect at a unique point (C) intersect along a straight line (D) intersect along a plane. 38. Angles between any pair of main diagonals of a cube are (A) cos / 3, π cos / 3 (B) cos /3, π cos /3 (C) π/2 39. If the tangent at the point P with co-ordinates (h, k) on the curve y 2 = 2x 3 perpendicular to the straight line x = 3y, then (A) (h, k) = (0, 0) (B) (h, k) = (/8, /6) (C) (h, k) = (0, 0) or (h, k) = (/8, /6) (D) no such point (h, k) exts. 0. Consider the family F of curves in the plane given by x = cy 2, where c a real parameter. Let G be the family of curves having the following property: every member of G intersects each member of F orthogonally. Then G given by (A) xy = k (B) x 2 + y 2 = k 2 (C) y 2 + 2x 2 = k 2 (D) x 2 y 2 + 2yk = k 2. Suppose the circle with equation x 2 +y 2 +2fx+2gy +c = 0 cuts the parabola y 2 = ax, (a > 0) at four dtinct points. If d denotes the sum of ordinates of these four points, then the set of possible values of d (A) {0} (B) ( a, a) (C) ( a, a) (D) (, ). 8

9 2. The polar equation r = a cos θ represents (A) a spiral (B) a parabola (C) a circle 3. Let Then V = V 2 = V 3 = ( ) , ( ) , ( ) (A) V 3 < V 2 < V (B) V 3 < V < V 2 (C) V < V 2 < V 3 (D) V 2 < V 3 < V.. If a sphere of radius r passes through the origin and cuts the three co-ordinate axes at points A, B, C respectively, then the centroid of the triangle ABC lies on a sphere of radius (A) r (B) r 3 (C) 2 3 r (D) 2r Consider the tangent plane T at the point (/ 3, / 3, / 3) to the sphere x 2 + y 2 + z 2 =. If P an arbitrary point on the plane x 3 + y 3 + z 3 = 2, then the minimum dtance of P from the tangent plane, T, always (A) 5 (B) 3 (C) (D) none of these. 6. Let S denote a sphere of unit radius and C a cube inscribed in S. Inductively define spheres S n and cubes C n such that S n+ inscribed in C n and C n+ inscribed in S n+. Let v n denote the sum of the volumes of the first n spheres. Then lim n v n (A) 2π. (B) 8π 3. (C) 2π 3 (9 + 3). (D) π. 3 9

10 7. If 0 < x <, then the sum of the infinite series 2 x x3 + 3 x + (A) log + x x (C) (B) x + log( + x) x x + log( x) (D) x + log( x). x 8. Let {a n } be a sequence of real numbers. Then lim n a n exts if and only if (A) lim a 2n and lim a 2n+2 exts n n (B) lim a 2n and lim a 2n+ ext n n (C) lim a 2n, lim a 2n+ and lim a 3n ext n n n 9. Let {a n } be a sequence of non-negative real numbers such that the series a n convergent. If p a real number such that the series a n n p n= diverges, then (A) p must be strictly less than 2 (B) p must be strictly less than or equal to 2 (C) p must be strictly less than or equal to but can be greater than 2 (D) p must be strictly less than but can be greater than or equal to Suppose a > 0. Consider the sequence Then a n = n{ n ea n a}, n. (A) (C) lim a n does not ext (B) lim a n = e n n lim a n = 0 n 5. Let {a n }, n, be a sequence of real numbers satfying a n for all n. Define A n = n (a + a a n ), for n. Then lim n(an+ A n ) equal to n (A) 0 (B) (C) (D) none of these. 0

11 52. In the Taylor expansion of the function f(x) = e x/2 about x = 3, the coefficient of (x 3) 5 (A) e 3/2 5! (B) e 3/ ! (C) e 3/ ! 53. Let σ be the permutation: , I be the identity permutation and m be the order of σ i.e. Then m m = min{positive integers n : σ n = I}. (A) 8 (B) 2 (C) 360 (D) Let A = and B = Then (A) there exts a matrix C such that A = BC = CB (B) there no matrix C such that A = BC (C) there exts a matrix C such that A = BC, but A = CB (D) there no matrix C such that A = CB. 55. A rod of length 0 feet slides with its two ends on the coordinate axes. If the end on the x-ax moves with a constant velocity of 2 feet per minute then the magnitude of the velocity per minute of the middle point of the rod at the instant the rod makes an angle of 30 with the x-ax (A) 9/2 (B) 2 (C) / 9 (D) 2/ Let the position of a particle in three dimensional space at time t be (t, cos t, sin t). Then the length of the path traversed by the particle between the times t = 0 and t = 2π (A) 2π. (B) 2 2π. (C) 2π

12 57. Let n be a positive real number and p be a positive integer. Which of the following inequalities true? (A) n p > (n + )p+ n p+ p + (C) (n + ) p < (n + )p+ n p+ p + (B) n p < (n + )p+ n p+ p The smallest positive number K for which the inequality holds for all x and y sin 2 x sin 2 y K x y (A) 2 (B) (C) π 2 (D) there no smallest positive value of K; any K > 0 will make the inequality hold. 59. Given two real numbers a < b, let d(x, [a, b]) = min{ x y : a y b} for < x <. Then the function satfies f(x) = d(x, [0, ]) d(x, [0, ]) + d(x, [2, 3]) (A) 0 f(x) < 2 for every x (B) 0 < f(x) < for every x (C) f(x) = 0 if 2 x 3 and f(x) = if 0 x (D) f(x) = 0 if 0 x and f(x) = if 2 x Let f(x, y) = { e /(x 2 +y 2 ) if (x, y) = (0, 0) 0 if (x, y) = (0, 0). Then f(x, y) (A) not continuous at (0, 0) (B) continuous at (0, 0) but does not have first order partial derivatives (C) continuous at (0, 0) and has first order partial derivatives, but not differentiable at (0, 0) (D) differentiable at (0, 0) 2

13 6. Consider the function x {5 + y }dy if x > 2 0 f(x) = 5x + 2 if x 2 Then (A) f not continuous at x = 2 (B) f continuous and differentiable everywhere (C) f continuous everywhere but not differentiable at x = (D) f continuous everywhere but not differentiable at x = Let w = log(u 2 + v 2 ) where u = e (x2 +y) and v = e (x+y2). Then w x x=0,y=0 (A) 0 (B) (C) 2 (D) 63. Let p > and for x > 0, define f(x) = (x p ) p(x ). Then (A) f(x) an increasing function of x on (0, ) (B) f(x) a decreasing function of x on (0, ) (C) f(x) 0 for all x > 0 (D) f(x) takes both positive and negative values for x (0, ). 6. The map f(x) = a 0 cos x + a sin x + a 2 x 3 differentiable at x = 0 if and only if (A) a = 0 and a 2 = 0 (B) a 0 = 0 and a = 0 (C) a = 0 (D) a 0, a, a 2 can take any real value. 65. f(x) a differentiable function on the real line such that lim f(x) = and x lim f (x) = α. Then x (A) α must be 0 (B) α need not be 0, but α < (C) α > (D) α <. 3

14 66. Let f and g be two differentiable functions such that f (x) g (x) for all x < and f (x) g (x) for all x >. Then (A) if f() g(), then f(x) g(x) for all x (B) if f() g(), then f(x) g(x) for all x (C) f() g() (D) f() g(). 67. The length of the curve x = t 3, y = 3t 2 from t = 0 to t = (A) (B) 8(5 5 + ) (C) 5 5 (D) 8(5 5 ). 68. Given that e x2 dx = π, the value of e (x2 +xy+y 2) dxdy (A) π/3 (B) π/ 3 (C) 2π/3 (D) 2π/ Let I = Then I equals x + x 2 + x 3 x x + x 2 + x 3 + x dx dx 2 dx 3 dx. (A) /2 (B) /3 (C) / (D). 70. Let D = {(x, y) R 2 : x 2 + y 2 }. The value of the double integral (x 2 + y 2 ) dxdy D (A) π (B) π 2 (C) 2π (D) π 2 7. Let g(x, y) = max{2 x, 8 y}. Then the minimum value of g(x, y) as (x, y) varies over the line x + y = 0 (A) 5 (B) 7 (C) (D) 3.

15 72. Let 0 < α < β <. Then k= /(k+α) /(k+β) + x dx equal to (A) log e β α (B) log e + β + α (C) log e + α + β (D). 73. If f continuous in [0, ] then [n/2] lim n n f j=0 ( ) j n (where [y] the largest integer less than or equal to y) (A) does not ext (B) exts and equal to 2 (C) exts and equal to (D) exts and equal to 0 0 /2 0 f(x) dx f(x) dx f(x) dx. 7. The volume of the solid, generated by revolving about the horizontal line y = 2 the region bounded by y 2 2x, x 8 and y 2, (A) 2 2π (B) 28π/3 (C) 8π 75. If α, β are complex numbers then the maximum value of α β + αβ αβ (A) 2 (B) (C) the expression may not always be a real number and hence maximum does not make sense 76. For positive real numbers a, a 2,..., a 00, let Then 00 p = a i and q = i= i<j 00 a i a j. (A) q = p2 2 (B) q 2 p2 2 (C) q < p2 2 5

16 77. The differential equation of all the ellipses centred at the origin (A) y 2 + x(y ) 2 yy = 0 (B) xyy + x(y ) 2 yy = 0 (C) yy + x(y ) 2 xy = 0 (D) none of these. 78. The coordinates of a moving point P satfy the equations dx dt = tan x, dy dt = sin2 x, t 0. If the curve passes through the point (π/2, 0) when t = 0, then the equation of the curve in rectangular co-ordinates (A) y = /2 cos 2 x (B) y = sin 2x (C) y = cos 2x + (D) y = sin 2 x. 79. Let y be a function of x satfying If y(0) = 0 then y() equals dy dx = 2x3 y xy (A) /e 2 (B) /e (C) e /2 (D) e 3/ Let f(x) be a given differentiable function. Consider the following differential equation in y f(x) dy dx = yf (x) y 2. The general solution of th equation given by x + c (A) y = f(x) (C) y = f(x) x + c (B) y 2 = f(x) x + c (D) y = [f(x)]2 x + c. 8. Let y(x) be a non-trivial solution of the second order linear differential equation d 2 y dy + 2c + ky = 0, dx2 dx where c < 0, k > 0 and c 2 > k. Then (A) y(x) as x (B) y(x) 0 as x (C) lim y(x) exts and finite x ± (D) none of the above true. 6

17 82. The differential equation of the system of circles touching the y-ax at the origin (A) x 2 + y 2 2xy dy dx = 0 (C) x 2 y 2 2xy dy dx = 0 (B) x2 + y 2 + 2xy dy dx = 0 (D) x2 y 2 + 2xy dy dx = Suppose a solution of the differential equation (xy 3 + x 2 y 7 ) dy dx =, satfies the initial condition y(/) =. Then the value of dy dx when y = (A) 3 (B) 3 (C) 6 5 (D) Consider the group G = {( a b 0 a ) : a, b R, a > 0 } with usual matrix multiplication. Let {( ) } b N = : b R. 0 Then, (A) N not a subgroup of G (B) N a subgroup of G but not a normal subgroup (C) N a normal subgroup and the quotient group G/N of finite order (D) N a normal subgroup and the quotient group omorphic to R + (the group of positive reals with multiplication). 85. Let G be the group {±, ±i} with multiplication of complex numbers as composition. Let H be the quotient group Z/Z. Then the number of nontrivial group homomorphms from H to G (A) (B) (C) 2 (D) 3. 7