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Nbhm m. a. and m.sc. scholarship test 2007

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MD Kutubuddin Sardar
MD Kutubuddin Sardar

This document provides instructions for a mathematics scholarship test consisting of 45 multiple-choice questions across 3 sections: Algebra, Analysis, and Geometry. The instructions specify that candidates should answer each question in the provided answer booklet rather than on the question paper. Various mathematical terms and notation are defined for reference. The questions cover a wide range of topics in higher mathematics including algebra, analysis, geometry, and complex analysis.

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NATIONAL BOARD FOR HIGHER MATHEMATICS M. A. and M.Sc. Scholarship Test September 22, 2007 Time Allowed: 150 Minutes Maximum Marks: 45 Please read, carefully, the instructions on the following page 1
INSTRUCTIONS TO CANDIDATES • Please ensure that this question paper booklet contains 11 numbered (and printed) pages. The reverse of each printed page is blank and can be used for rough work. • There are three parts to this test: Algebra, Analysis and Geometry. Each part consists of 15 questions adding up to 45 questions in all. • Answer each question, as directed, in the space provided for it in the answer booklet, which is being supplied separately. This question paper is meant to be retained by you and so do not answer questions on it. • In certain questions you are required to pick out the qualifying state- ment(s) from multiple choices. None of the statements, or more than one statement may qualify. Write none if none of the statements qual- ify, or list the labels of all the qualifying statements (amongst (a),(b) and (c)). • Points will be awarded in the above questions only if all the correct choices are made. There will be no partial credit. • N denotes the set of natural numbers, Z - the integers, Q - the rationals, R - the reals and C - the field of complex numbers. Rn denotes the n- dimensional Euclidean space. The symbol ]a, b[ will stand for the open interval {x ∈ R | a < x < b} while [a, b] will stand for the corresponding closed interval; [a, b[ and ]a, b] will stand for the corresponding left- closed-right-open and left-open-right-closed intervals respectively. The symbol I will denote the identity matrix of appropriate order. 2
Section 1: Algebra 1.1 Let A be the matrix A = 1 √ 2 − √ 2 −1 Compute the matrix B = 3A − 2A2 − A3 − 5A4 + A6 . 1.2 How many elements of order 2 are there in the group (Z/4Z)3 ? 1.3 Consider the permutation π given by n = 1 2 3 4 5 6 7 8 9 10 π(n) = 5 7 8 10 6 1 2 4 9 3 Find the order of the permutation π. 1.4 Consider the system of simultaneous equations 2x −2y −2z = a1 −2x +2y −3z = a2 4x −4y +5z = a3 Write down the condition to be satisfied by a1, a2, a3 for this system NOT to have a solution. 1.5 Write down a polynomial of degree 4 with integer coefficients which has√ 3 + √ 5 as a root. 1.6 A finite group G acts on a finite set X, the action of g ∈ G on x ∈ X being denoted by gx. For each x ∈ X the stabiliser at x is the subgroup Gx = {g ∈ G : gx = x}. If x, y ∈ G and if y = gx, then express Gy in terms of Gx. 1.7. Write down the last two digits of 91500 . 3
1.8 A permutation matrix A is a nonsingular square matrix in which each row has exactly one entry = 1, the other entries being all zeros. If A is an n×n permutation matrix, what are the possible values of determinant of A? 1.9 Let V be the vector space of all polynomials of degree at most equal to 2n with real coefficients. Let V0 stand for the vector subspace V0 = {P ∈ V : P(1)+P(−1) = 0} and Ve stand for the subspace of polynomials which have terms of even degree alone. If dim(U) stands for the dimension of a vector space U, then find dim(V0) and dim(V0 ∩ Ve). 1.10 Let a, b, m and n be integers, m, n positive, am + bn = 1. Find an integer x (in terms of a, b, m, n, p, q) so that x ≡ p (mod m) x ≡ q (mod n) where p and q are given integers. 1.11 In the ring Z/20Z of integers modulo 20, does the equivalence class 17 have a multiplicative inverse? Write down an inverse if your answer is yes. 1.12 Let R[x] be the ring of polynomials in the indeterminate x over the field of real numbers and let J be the ideal generated by the polynomial x3 − x. Find the dimension of the vector space R[x]/J . 1.13 In the ring of polynomials R = Z5[x] with coefficients from the field Z5, consider the smallest ideal J containing the polynomials, p1(x) = x3 + 4x2 + 4x + 1 p2(x) = x2 + x + 3. Which of the following polynomials q(x) has the property that J = q(x)R? (a) q(x) = p2(x) (b) q(x) = x − 1 (c) q(x) = x + 1 1.14 In how many ways can 20 indistinguishable pencils be distributed among four children A,B,C and D ? 4
1.15 Let w = u+iv and, z = x+iy be complex numbers such that w2 = z2 +1. Then which of the following inequalities must always be true? (a) x ≤ u (b) y2 ≤ v2 (c) v2 ≤ y2 5
Section 2: Analysis 2.1 Evaluate: lim x→0 sin x x 1 x2 . 2.2 Evaluate: lim n→∞ 1 n2 n k=1 √ n2 − k2. 2.3 Pick out the uniformly continuous functions from the following and, in such cases, given ε > 0, find δ > 0 explicitly as a function of ε so that |f(x) − f(y)| < ε whenever |x − y| < δ. (a) f(x) = √ x, 1 ≤ x ≤ 2. (b) f(x) = x3 , x ∈ R. (c) f(x) = sin2 x, x ∈ R. 2.4 Which of the following functions are differentiable at x = 0? (a) f(x) = x2 , if x is rational 0, if x is irrational. (b) f(x) = |x|x. (c) f(x) = x2 sin 1 x , if x = 0 0, if x = 0. 2.5 Find the coefficient of x7 in the Maclaurin series expansion of the func- tion f(x) = sin−1 x. 2.6 Compute f(x) = lim n→∞ n2 x(1 − x2 )n where 0 ≤ x ≤ 1. 6
2.7 Which of the following series are convergent? (a) ∞ n=1 2n2 + 3 5n3 + 7 . (b) ∞ n=1 (n + 1)n nn+ 3 2 . (c) ∞ n=1 1 n sin 1 n . 2.8 Find the interval of convergence of the series: ∞ n=1 log(n + 1) √ n + 1 (x − 5)n . 2.9 Evaluate: π 2 0 sin2 x dx sin x + cos x . 2.10 Examine for maxima and minima: f(x, y) = x2 + 5y2 − 6x + 10y + 6. 2.11 Find the point(s) on the parabola 2x2 + 2y = 3 nearest to the ori- gin.What is the shortest distance? 2.12 Let S be the triangular region in the plane with vertices at (0, 0), (1, 0) and (1, 1). Let f(x, y) be a continuous function. Express the double integral S f(x, y) dA in two different ways as iterated integrals (i.e. in the forms β α δ(x) γ(x) f(x, y) dy dx and b a d(y) c(y) f(x, y) dx dy.) 2.13 Let ω = 1 be a seventh root of unity. Write down a polynomial equation of degree ≤ 6 satisfied by ω. 7
2.14 Let z = x + iy. Which of the following functions are analytic in the entire complex plane? (a) f(x, y) = ex (cos y − i sin y). (b) f(x, y) = e−x (cos y − i sin y). (c) f(x, y) = min{2, x2 + y2 }. 2.15 Let C denote the boundary of the square whose sides are given by the lines x = ±2 and y = ±2. Assume that C is described in the positive sense, i.e., anticlockwise. Evaluate: C cos z dz z(z2 + 8) . 8
Section 3: Geometry 3.1 Let A be the point (0, 4) in the xy-plane and let B be the point (2t, 0). Let L be the mid point of AB and let the perpendicular bisector of AB meet the y-axis at M. Let N be the mid-point of LM. Find the locus of N (as t varies). 3.2 Let (a1, a2), (b1, b2) and (c1, c2) be three non-collinear points in the xy- plane. Let r, s and t be three real numbers such that (i)r + s + t = 0, (ii) ra1 + sb1 + tc1 = 0 and (iii) ra2 + sb2 + tc2 = 0. Write down all the possible values of r, s and t. 3.3 Consider the equation 2x + 4y − x2 − y2 = 5. Which of the following does it represent? (a) a circle. (b) an ellipse. (c) a pair of straight lines. 3.4 Write down the equations of the circles of radius 5 passing through the origin and having the line y = 2x as a tangent. 3.5 Two equal sides of an isoceles triangle are given by the equations y = 7x and y = −x. If the third side passes through the point (1, −10), pick out the equation(s) which cannot represent that side. (a) 3x + y + 7 = 0. (b) x − 3y − 31 = 0. (c) x + 3y + 29 = 0. 3.6 Let m = 0. Consider the line y = mx + a m and the parabola y2 = 4ax. Pick out the true statements. (a) The line intersects the parabola at exactly one point. (b) The line intersects the parabola at two points whenever |m| < 2 √ a. (c) The line is tangent to the parabola only when |m| = 2 √ a. 3.7 Consider the circle x2 + (y + 1)2 = 1. Let a line through the origin O meet the circle again at a point A. Let B be a point on OA such that OB/OA = p, where p is a given positive number. Find the locus of B. 9
3.8 Let a > 0 and b > 0. Let a straight line make an intercept a on the x-axis and b on the line through the origin which is inclined at an angle θ to the x- axis, both in the first quadrant. Write down the equation of the straight line. 3.9 What does the following equation represent? 12x2 + 7xy − 10y2 + 13x + 45y − 35 = 0. 3.10 Find the coordinates of the centre of the circumcircle of the triangle whose vertices are the points (4, 1), (−1, 6) and (−4, −3). 3.11 Let A and B be the points of intersection of the circles x2 +y2 −4x−5 = 0 and x2 + y2 + 8y + 7 = 0. Find the centre and radius of the circle whose diameter is AB. 3.12 Ten points are placed at random in the unit square. Let ρ be the min- imum distance between all pairs of distinct points from this set. Find the least upper bound for ρ. 3.13 Let K be a subset of the plane. It is said to be convex if given any two points in K, the line segment joining them is also contained in K. It is said to be strictly convex if given any two points in K, the mid-point of the line segment joining them lies in the interior of K. In each of the following cases determine whether the given set is convex (but not strictly convex), strictly convex or not convex. (a) K = {(x, y) | x2 + y2 ≤ 1}. (b) K = {(x, y) | |x| + |y| ≤ 1}. (c) K = {(x, y) | x 2 3 + y 2 3 ≤ 1}. 3.14 Consider the set K = {(x, y) | |x| + |y| ≤ 1} in the plane. Given a point A in the plane, let PK(A) be the point in K which is closest to A. Let B = (1, 0) ∈ K. Determine the set S = {A | PK(A) = B}. 10
A B C D E a b c d e 3.15 Let A, B, C, D and E be five points on a circle and let a, b, c, d and e be the angles as shown in the figure above. Which of the following equals the ratio AD/BE? (a) sin(a+d) sin(b+e) . (b) sin(b+c) sin(c+d) . (c) sin(a+b) sin(b+c) . 11

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Nbhm m. a. and m.sc. scholarship test 2007

  • 1. NATIONAL BOARD FOR HIGHER MATHEMATICS M. A. and M.Sc. Scholarship Test September 22, 2007 Time Allowed: 150 Minutes Maximum Marks: 45 Please read, carefully, the instructions on the following page 1
  • 2. INSTRUCTIONS TO CANDIDATES • Please ensure that this question paper booklet contains 11 numbered (and printed) pages. The reverse of each printed page is blank and can be used for rough work. • There are three parts to this test: Algebra, Analysis and Geometry. Each part consists of 15 questions adding up to 45 questions in all. • Answer each question, as directed, in the space provided for it in the answer booklet, which is being supplied separately. This question paper is meant to be retained by you and so do not answer questions on it. • In certain questions you are required to pick out the qualifying state- ment(s) from multiple choices. None of the statements, or more than one statement may qualify. Write none if none of the statements qual- ify, or list the labels of all the qualifying statements (amongst (a),(b) and (c)). • Points will be awarded in the above questions only if all the correct choices are made. There will be no partial credit. • N denotes the set of natural numbers, Z - the integers, Q - the rationals, R - the reals and C - the field of complex numbers. Rn denotes the n- dimensional Euclidean space. The symbol ]a, b[ will stand for the open interval {x ∈ R | a < x < b} while [a, b] will stand for the corresponding closed interval; [a, b[ and ]a, b] will stand for the corresponding left- closed-right-open and left-open-right-closed intervals respectively. The symbol I will denote the identity matrix of appropriate order. 2
  • 3. Section 1: Algebra 1.1 Let A be the matrix A = 1 √ 2 − √ 2 −1 Compute the matrix B = 3A − 2A2 − A3 − 5A4 + A6 . 1.2 How many elements of order 2 are there in the group (Z/4Z)3 ? 1.3 Consider the permutation π given by n = 1 2 3 4 5 6 7 8 9 10 π(n) = 5 7 8 10 6 1 2 4 9 3 Find the order of the permutation π. 1.4 Consider the system of simultaneous equations 2x −2y −2z = a1 −2x +2y −3z = a2 4x −4y +5z = a3 Write down the condition to be satisfied by a1, a2, a3 for this system NOT to have a solution. 1.5 Write down a polynomial of degree 4 with integer coefficients which has√ 3 + √ 5 as a root. 1.6 A finite group G acts on a finite set X, the action of g ∈ G on x ∈ X being denoted by gx. For each x ∈ X the stabiliser at x is the subgroup Gx = {g ∈ G : gx = x}. If x, y ∈ G and if y = gx, then express Gy in terms of Gx. 1.7. Write down the last two digits of 91500 . 3
  • 4. 1.8 A permutation matrix A is a nonsingular square matrix in which each row has exactly one entry = 1, the other entries being all zeros. If A is an n×n permutation matrix, what are the possible values of determinant of A? 1.9 Let V be the vector space of all polynomials of degree at most equal to 2n with real coefficients. Let V0 stand for the vector subspace V0 = {P ∈ V : P(1)+P(−1) = 0} and Ve stand for the subspace of polynomials which have terms of even degree alone. If dim(U) stands for the dimension of a vector space U, then find dim(V0) and dim(V0 ∩ Ve). 1.10 Let a, b, m and n be integers, m, n positive, am + bn = 1. Find an integer x (in terms of a, b, m, n, p, q) so that x ≡ p (mod m) x ≡ q (mod n) where p and q are given integers. 1.11 In the ring Z/20Z of integers modulo 20, does the equivalence class 17 have a multiplicative inverse? Write down an inverse if your answer is yes. 1.12 Let R[x] be the ring of polynomials in the indeterminate x over the field of real numbers and let J be the ideal generated by the polynomial x3 − x. Find the dimension of the vector space R[x]/J . 1.13 In the ring of polynomials R = Z5[x] with coefficients from the field Z5, consider the smallest ideal J containing the polynomials, p1(x) = x3 + 4x2 + 4x + 1 p2(x) = x2 + x + 3. Which of the following polynomials q(x) has the property that J = q(x)R? (a) q(x) = p2(x) (b) q(x) = x − 1 (c) q(x) = x + 1 1.14 In how many ways can 20 indistinguishable pencils be distributed among four children A,B,C and D ? 4
  • 5. 1.15 Let w = u+iv and, z = x+iy be complex numbers such that w2 = z2 +1. Then which of the following inequalities must always be true? (a) x ≤ u (b) y2 ≤ v2 (c) v2 ≤ y2 5
  • 6. Section 2: Analysis 2.1 Evaluate: lim x→0 sin x x 1 x2 . 2.2 Evaluate: lim n→∞ 1 n2 n k=1 √ n2 − k2. 2.3 Pick out the uniformly continuous functions from the following and, in such cases, given ε > 0, find δ > 0 explicitly as a function of ε so that |f(x) − f(y)| < ε whenever |x − y| < δ. (a) f(x) = √ x, 1 ≤ x ≤ 2. (b) f(x) = x3 , x ∈ R. (c) f(x) = sin2 x, x ∈ R. 2.4 Which of the following functions are differentiable at x = 0? (a) f(x) = x2 , if x is rational 0, if x is irrational. (b) f(x) = |x|x. (c) f(x) = x2 sin 1 x , if x = 0 0, if x = 0. 2.5 Find the coefficient of x7 in the Maclaurin series expansion of the func- tion f(x) = sin−1 x. 2.6 Compute f(x) = lim n→∞ n2 x(1 − x2 )n where 0 ≤ x ≤ 1. 6
  • 7. 2.7 Which of the following series are convergent? (a) ∞ n=1 2n2 + 3 5n3 + 7 . (b) ∞ n=1 (n + 1)n nn+ 3 2 . (c) ∞ n=1 1 n sin 1 n . 2.8 Find the interval of convergence of the series: ∞ n=1 log(n + 1) √ n + 1 (x − 5)n . 2.9 Evaluate: π 2 0 sin2 x dx sin x + cos x . 2.10 Examine for maxima and minima: f(x, y) = x2 + 5y2 − 6x + 10y + 6. 2.11 Find the point(s) on the parabola 2x2 + 2y = 3 nearest to the ori- gin.What is the shortest distance? 2.12 Let S be the triangular region in the plane with vertices at (0, 0), (1, 0) and (1, 1). Let f(x, y) be a continuous function. Express the double integral S f(x, y) dA in two different ways as iterated integrals (i.e. in the forms β α δ(x) γ(x) f(x, y) dy dx and b a d(y) c(y) f(x, y) dx dy.) 2.13 Let ω = 1 be a seventh root of unity. Write down a polynomial equation of degree ≤ 6 satisfied by ω. 7
  • 8. 2.14 Let z = x + iy. Which of the following functions are analytic in the entire complex plane? (a) f(x, y) = ex (cos y − i sin y). (b) f(x, y) = e−x (cos y − i sin y). (c) f(x, y) = min{2, x2 + y2 }. 2.15 Let C denote the boundary of the square whose sides are given by the lines x = ±2 and y = ±2. Assume that C is described in the positive sense, i.e., anticlockwise. Evaluate: C cos z dz z(z2 + 8) . 8
  • 9. Section 3: Geometry 3.1 Let A be the point (0, 4) in the xy-plane and let B be the point (2t, 0). Let L be the mid point of AB and let the perpendicular bisector of AB meet the y-axis at M. Let N be the mid-point of LM. Find the locus of N (as t varies). 3.2 Let (a1, a2), (b1, b2) and (c1, c2) be three non-collinear points in the xy- plane. Let r, s and t be three real numbers such that (i)r + s + t = 0, (ii) ra1 + sb1 + tc1 = 0 and (iii) ra2 + sb2 + tc2 = 0. Write down all the possible values of r, s and t. 3.3 Consider the equation 2x + 4y − x2 − y2 = 5. Which of the following does it represent? (a) a circle. (b) an ellipse. (c) a pair of straight lines. 3.4 Write down the equations of the circles of radius 5 passing through the origin and having the line y = 2x as a tangent. 3.5 Two equal sides of an isoceles triangle are given by the equations y = 7x and y = −x. If the third side passes through the point (1, −10), pick out the equation(s) which cannot represent that side. (a) 3x + y + 7 = 0. (b) x − 3y − 31 = 0. (c) x + 3y + 29 = 0. 3.6 Let m = 0. Consider the line y = mx + a m and the parabola y2 = 4ax. Pick out the true statements. (a) The line intersects the parabola at exactly one point. (b) The line intersects the parabola at two points whenever |m| < 2 √ a. (c) The line is tangent to the parabola only when |m| = 2 √ a. 3.7 Consider the circle x2 + (y + 1)2 = 1. Let a line through the origin O meet the circle again at a point A. Let B be a point on OA such that OB/OA = p, where p is a given positive number. Find the locus of B. 9
  • 10. 3.8 Let a > 0 and b > 0. Let a straight line make an intercept a on the x-axis and b on the line through the origin which is inclined at an angle θ to the x- axis, both in the first quadrant. Write down the equation of the straight line. 3.9 What does the following equation represent? 12x2 + 7xy − 10y2 + 13x + 45y − 35 = 0. 3.10 Find the coordinates of the centre of the circumcircle of the triangle whose vertices are the points (4, 1), (−1, 6) and (−4, −3). 3.11 Let A and B be the points of intersection of the circles x2 +y2 −4x−5 = 0 and x2 + y2 + 8y + 7 = 0. Find the centre and radius of the circle whose diameter is AB. 3.12 Ten points are placed at random in the unit square. Let ρ be the min- imum distance between all pairs of distinct points from this set. Find the least upper bound for ρ. 3.13 Let K be a subset of the plane. It is said to be convex if given any two points in K, the line segment joining them is also contained in K. It is said to be strictly convex if given any two points in K, the mid-point of the line segment joining them lies in the interior of K. In each of the following cases determine whether the given set is convex (but not strictly convex), strictly convex or not convex. (a) K = {(x, y) | x2 + y2 ≤ 1}. (b) K = {(x, y) | |x| + |y| ≤ 1}. (c) K = {(x, y) | x 2 3 + y 2 3 ≤ 1}. 3.14 Consider the set K = {(x, y) | |x| + |y| ≤ 1} in the plane. Given a point A in the plane, let PK(A) be the point in K which is closest to A. Let B = (1, 0) ∈ K. Determine the set S = {A | PK(A) = B}. 10
  • 11. A B C D E a b c d e 3.15 Let A, B, C, D and E be five points on a circle and let a, b, c, d and e be the angles as shown in the figure above. Which of the following equals the ratio AD/BE? (a) sin(a+d) sin(b+e) . (b) sin(b+c) sin(c+d) . (c) sin(a+b) sin(b+c) . 11