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B.Mathematics Entrance Exam Question Paper |

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Re: B.Mathematics Entrance Exam Question Paper
As you want the question paper of Indian Statistical Institute B. Mathematics Entrance Exam, so I am giving you some questions of that paper: Instructions. MI(A) is a multiple choice examination. In each of the following questions, exactly one of the choices is correct. Please tick the correct answer. You get four marks for each correct answer, one mark for each unanswered question, and zero marks for each incorrect answer. You have two hours to answer these questions. 1. De¯ne an = (12 + 22 + : : : + n2)n and bn = nn(n!)2. Recall n! is the product of the ¯rst n natural numbers. Then, (a) an < bn for all n > 1 (b) an > bn for all n > 1 (c) an = bn for in¯nitely many n (d) none of the above. 2. The last digit of (2004)5 is: (a) 4 (b) 8 (c) 6 (d) 2 3. If n is a positive integer such that 8n + 1 is a perfect square, then (a) n must be odd (b) n cannot be a perfect square (c) 2n cannot be a perfect square (d) none of the above. 4. The coe±cient of a3b4c5 in the expansion of (bc + ca + ab)6 is: (a) 12! 3!4!5! (b) 6! 3! (c) 33 (d) 3 ¢ ¡ 6! 3!3! ¢ 5. If log10 x = 10log100 4, then x equals (a) 410 (b) 100 (c) log10 4 (d) none of the above. 6. Let C denote the set of all complex numbers. De¯ne A = f(z;w) jz;w 2 C and jzj = jwjg B = f(z;w) jz;w 2 C, and z2 = w2g: Then, (a) A = B (b) A ½ B and A 6= B (c) B ½ A and B 6= A (d) none of the above. 7. The set of all real numbers x such that x3(x + 1)(x ¡ 2) ¸ 0 is: (a) the interval 2 · x < 1 (b) the interval 0 · x < 1 (c) the interval ¡1 · x < 1 (d) none of the above. 8. Let z be a non-zero complex number such that z 1+z is purely imaginary. Then (a) z is neither real nor purely imaginary (b) z is real (c) z is purely imaginary (d) none of the above. 9. In the interval (0; 2¼), the function sin( 1 x3 ) (a) never changes sign (b) changes sign only once (c) changes sign more than once, but ¯nitely many times (d) changes sign in¯nitely many times. 10. lim x!0 (ex¡1) tan2 x x3 (a) does not exist (b) exists and equals 0 (c) exists and equals 2 3 (d) exists and equals 1. 11. Let f1(x) = ex; f2(x) = ef1(x) and generally fn+1(x) = efn(x) for all n ¸ 1. For any ¯xed n, the value of d dxfn(x) is: (a) fn(x) (b) fn(x)fn¡1(x) (c) fn(x)fn¡1(x) : : : f1(x) (d) fn+1(x)fn(x) : : : f1(x)ex. 12. Let f(x) = a0 + a1jxj + a2jxj2 + a3jxj3; where a0; a1; a2; a3 are constants. Then (a) f(x) is di®erentiable at x = 0 whatever be a0; a1; a2; a3 (b) f(x) is not di®erentiable at x = 0 whatever be a0; a1; a2; a3 (c) f(x) is di®erentiable at x = 0 only if a1 = 0 (d) f(x) is di®erentiable at x = 0 only if a1 = 0; a3 = 0. 13. If f (x) = cos(x) ¡ 1 + x2 2 , then (a) f(x) is an increasing function on the real line (b) f(x) is a decreasing function on the real line (c) f(x) is increasing on the interval ¡1 < x · 0 and decreasing on the interval 0 · x < 1 (d) f(x) is decreasing on the interval ¡1 < x · 0 and increasing on the interval 0 · x < 1. 14. The area of the region bounded by the straight lines x = 1 2 and x = 2, and the curves given by the equations y = loge x and y = 2x is (a) 1 loge 2 (4 + p2) ¡ 5 2 loge 2 + 3 2 (b) 1 loge 2 (4 ¡ p2) ¡ 5 2 loge 2 (c) 1 loge 2 (4 ¡ p2) ¡ 5 2 loge 2 + 3 2 (d) none of the above. 15. The number of roots of the equation x2 +sin2 x = 1 in the closed interval [0; ¼ 2 ] is (a) 0 (b) 1 (c) 2 (d) 3 16. The number of maps f from the set f1; 2; 3g into the set f1; 2; 3; 4; 5g such that f(i) · f(j) whenever i < j is (a) 60 (b) 50 (c) 35 (d) 30 17. Let a be a real number. The number of distinct solutions (x; y) of the system of equations (x ¡ a)2 + y2 = 1 and x2 = y2, can only be (a) 0; 1; 2; 3; 4 or 5 (b) 0, 1 or 3 (c) 0; 1; 2 or 4 (d) 0; 2; 3; or 4 18. The set of values of m for which mx2 ¡6mx+5m+1 > 0 for all real x is (a) m < 1 4 (b) m ¸ 0 (c) 0 · m · 1 4 (d) 0 · m < 1 4 . 19. A lantern is placed on the ground 100 feet away from a wall. A man six feet tall is walking at a speed of 10 feet/second from the lantern to the nearest point on the wall. When he is midway between the lantern and the wall, the rate of change in the length of his shadow is (a) 3:6 ft./sec. (b) 2:4 ft./sec. (c) 3 ft./sec. (d) 12 ft./sec. 20. Let n ¸ 3 be an integer. Assume that inside a big circle, exactly n small circles of radius r can be drawn so that each small circle touches the big circle and also touches both its adjacent small circles. Then, the radius of the big circle is: (a) r cosec ¼ n (b) r(1 + cosec 2¼ n ) (c) r(1 + cosec ¼ 2n) (d) r(1 + cosec ¼ n). 21. The digit in the units' place of the number 1! + 2! + 3! + : : : + 99! is (a) 3 (b) 0 (c) 1 (d) 7. 22. The value of lim n!1 13+23+:::+n3 n4 is: (a) 3 4 (b) 1 4 (c) 1 (d) 4. 23. The function x(® ¡ x) is strictly increasing on the interval 0 < x < 1 if and only if (a) ® ¸ 2 (b) ® < 2 (c) ® < ¡1 (d) ® > 2. 24. For any integer n ¸ 1, de¯ne an = 1000n n! . Then the sequence fang (a) does not have a maximum (b) attains maximum at exactly one value of n (c) attains maximum at exactly two values of n (d) attains maximum for in¯nitely many values of n. 25. The equation x3y + xy3 + xy = 0 represents (a) a circle (b) a circle and a pair of straight lines (c) a rectangular hyperbola (d) a pair of straight lines. 26. Let P be a variable point on a circle C and Q be a ¯xed point outside C. If R is the mid-point of the line segment PQ, then the locus of R is (a) a circle (b) an ellipse (c) a line segment (d) segment of a parabola. 27. Let d1; d2; : : : ; dk be all the factors of a positive integer n including 1 and n. If d1 + d2 + : : : + dk = 72, then 1 d1 + 1 d2 + ¢ ¢ ¢ + 1 dk is: (a) k2 72 (b) 72 k (c) 72 n (d) none of the above. 28. A subset W of the set of real numbers is called a ring if it contains 1 and if for all a; b 2 W, the numbers a ¡ b and ab are also in W. Let S = ©m 2n j m; n integersª and T = np q j p; q integers, q oddo. Then: (a) neither S nor T is a ring (b) S is a ring T is not a ring. (b) T is a ring S is not a ring. (d) both S and T are rings. Instructions. All questions carry equal marks. You have two hours to solve these problems. Answer as many questions as you can. Credit will be given to a partially correct answer. Do not feel discouraged if you cannot solve all the questions. 1. Find the sum of all distinct four digit numbers that can be formed using the digits 1, 2, 3, 4, 5, each digit appearing at most once. 2. Consider the squares of an 8 £ 8 chessboard ¯lled with the numbers 1 to 64 as in the ¯gure below. If we choose 8 squares with the property that there is exactly one from each row and exactly one from each column, and add up the numbers in the chosen squares, show that the sum obtained is always 260. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 3. An isosceles triangle with base 6 cms. and base angles 30o each is inscribed in a circle. A second circle, which is situated outside the triangle, touches the ¯rst circle and also touches the base of the triangle at its midpoint. Find its radius. 4. Let an = 1 : : : 1 with 3n digits. Prove that an is divisible by 3an¡1. 5. If a circle intersects the hyperbola y = 1=x at four distinct points (xi; yi); i = 1; 2; 3; 4, then prove that x1x2 = y3y4. 6. Show that the function f (x) de¯ned below attains a unique minimum for x > 0. What is the minimum value of the function? What is the value of x at which the minimum is attained? f(x) = x2 + x + 1 x + 1 x2 for x 6= 0: Sketch on plain paper the graph of this function. 7. Let S = f1; 2; : : : ; ng. Find the number of unordered pairs fA;Bg of subsets of S such that A and B are disjoint, where A or B or both may be empty. 8. Find the maximum value of x2 + y2 in the bounded region, including the boundary, enclosed by y = x 2 ; y = ¡x 2 and x = y2 + 1. 9. How many real roots does x4 + 12x ¡ 5 have? 10. Find the maximum among 1; 21=2; 31=3; 41=4; : : :. 11. For real numbers x; y and z, show that jxj + jyj + jzj · jx + y ¡ zj + jy + z ¡ xj + jz + x ¡ yj: The hints given below should be used only if necessary. A candidate is expected to ¯nd the answer himself/herself. It should also be noted that there are ways to answer the given questions which are di®erent from the ones sketched in the hints. Indeed, a student should feel encouraged if (s)he ¯nds a di®erent way to solve some of these problems. All the Best! Hints for MI(A) Sample Questions. Q.1 (b). Take the nth root of an and bn and use A.M.¸ G.M. Q.2 (a). As 2004 = 2000 +4, the last digits of (2004)5 and 45 are equal. Q.3 (c) If 8n + 1 = m2, then 2n is a product of two consecutive integers. Q.4 (d) Use binomial expansion of (bc + a (b + c))6. Q.5 (b) Let y = log10 x. Then log10 y = log100 4. Hence y = 2. Q.6 (c) z2 = w2 ) z = §w ) B µ A. But jij = 1 and i2 6= 1. Q.7 (d) Check for `test points' ¡1, and 1. Q.8 (a) Check that (b) and (c) are obviously false, then check that (a) is true. Q.9 (d) sin ¡ 1 x3 ¢ changes sign at the points (n¼)¡1 3 for all n ¸ 1. Q.10 (d) Observe that (ex¡1) tan2 x x3 = (ex¡1) x ¢ sin2 x x2 ¢ 1 cos2 x . Q.11 (c) Use induction and chain rule of di®erentiation. Q.12 (c) Amongst 1; jxj; jxj2; jxj3, only jxj is not di®erentiable at 0. Q.13 (d) Look at the derivative of f. Q.14 (c) Compute the integral 2 R1=2 2xdx ¡ 2 R1=2 log xdx. Q.15 (b) Draw graphs of y = cos x and y = §x and ¯nd the number of points of intersections. Q.16 (c) Compute the number of maps such that f(3) = 5, f(3) = 4 etc.. Alternatively, de¯ne g : f1; 2; 3g ! f1; 2; : : : ; 7g by g (i) = f (i)+(i ¡ 1). Then, g is a strictly increasing function and its image is a subset of size 3 of f1; 2; : : : 7g. Q.17 (d) Draw graphs of (x + y)(x ¡ y) = 0 and (x ¡ a)2 + y2 = 1. Q.18 (d) Calculate the discriminant (b2 ¡ 4ac) of the given quadratic. Q.19 (b) Show that the height function is 60 t . Q.20 (d) Let s be distance between the centre of the big circle and the centre of (any) one of the small circles. Then there exists a right angle triangle with hypoteneuse s, side r and angle ¼ n. Q.21 (a) The unit digit of all numbers n! with n ¸ 5 is 0. Q.22 (b) Use the formula for n Pi=1 i3. Q.23 (a) Di®erentiate. Q.24 (c) Find out the ¯rst values of n for which an+1 an becomes < 1. Q.25 (d) The equation is xy(x2 + y2 + 1) = 0. Q.26 (a) Compute for C = ©x2 + y2 = 1ª and Q = (a; 0) for some a > 1. Q.27 (c) Multiply the given sum by n. Q.28 (d) Verify using the given de¯nition of a ring. Hints for MI(B) Sample Questions. Q.1 The answer is 399960. For each x 2 f1; 2; 3; 4; 5g, there are 4! such numbers whose last digit is x. Thus the digits in the unit place of all the 120 numbers add up to 4! (1 + 2 + 3 + 4 + 5). Similarly the numbers at ten's place add up to 360 and so on. Thus the sum is 360 (1 + 10 + 100 + 1000). Q.2 Let the chosen entries be in the positions (i; ai), 1 · i · 8. Thus a1; : : : ; a8 is a permutation of f1; : : : ; 8g. The entry in the square corresponding to (i; j)th place is i + 8 (j ¡ 1). Hence the required sum is 8 Pi=1 (i + 8 (aj ¡ 1)). Q.3 Radius is 3p3 2 . Use trigonometry. Q.4 Observe that an = an¡1 ¡1 + t + t2¢ where t = 103n Q.5 Substitute y = 1 x in the equation of a circle and clear denominator to get a degree 4 equation in x. The product of its roots is the constant term, which is 1. Q.6 The function f (x) ¡ 4 is a sum of squares and hence non-negative. So the minimum is 4 which is attained at x = 1. Q.7 The number is 3n+1 2 . An ordered pair (A;B) of disjoint subsets of S is determined by 3 choices for every element of S (either it is in A, or in B or in neither of them). Hence such pairs are 3n in number. An unordered pair will be counted twice in this way, except for the case A and B both empty. Hence the number is 1 + 3n¡1 2 . Q.8 Answer is 5. The maximum is attained at points (2; 1) and (2;¡1). Q.9 Answer is 2. Let f be the given polynomial. Then f (0) is negative and f is positive as x tends to §1. Hence it has at least 2 real roots. Since the derivative of f is zero only at 3 p¡3, it cannot have more than two real roots. Q.10 Maximum is 3 p3. Either check the maximum of the function x 1 x , or compare 3 p3 with npn. Q.11 Rewrite the given inequality in terms of the new variables ® = x + y ¡ z, ¯ = y + z ¡ x, ° = x + z ¡ y, and use the triangle inequality.
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Re: B.Mathematics Entrance Exam Question Paper
As per your demand I will help you here to get the B.Mathematics exam paper so that you easily solve the paper without any stress. Here is the paper B.Mathematics entrance exam. Part A. Choose the correct option and explain your reasoning briefly. Each problem is worth 3 points. 1. The word MATHEMATICS consists of 11 letters. The number of distinct ways to rearrange these letters is (A) 11! (B) 11!3 (C) 11!6 (D) 11!8 2. In a rectangle ABCD, the length BC is twice the width AB. Pick a point P on side BC such that the lengths of AP and BC are equal. The measure of angle CPD is (A) 75◦ (B) 60◦ (C) 45◦ (D) none of the above 3. The number of θ with 0 ≤ θ < 2π such that 4 sin(3θ 2) = 1 is (A) 2 (B) 3 (C) 6 (D) none of the above 4. Given positive real numbers a1,a2,...,a2011 whose product a1a2 •••a2011 is 1, what can you say about their sum S = a1 a2 ••• a2011 ? (A) S can be any positive number. (B) 1 ≤ S ≤ 2011. (C) 2011 ≤ S and S is unbounded above. (D) 2011 ≤ S and S is bounded above. 5. A function f is defined by f(x) = ex if x < 1 and f(x) = loge(x) ax2 bx if x ≥ 1. Here a and b are unknown real numbers. Can f be differentiable at x = 1? (A) f is not differentiable at x = 1 for any a and b. (B) There exist unique numbers a and b for which f is differentiable at x = 1. (C) f is differentiable at x = 1 whenever a b = e. (D) f is differentiable at x = 1 regardless of the values of a and b. 6. The equation x2 bxc = 0 has nonzero real coefficients satisfying b2 > 4c. Moreover, exactly one of b and c is irrational. Consider the solutions p and q of this equation. (A) Both p and q must be rational. (B) Both p and q must be irrational. (C) One of p and q is rational and the other irrational. (D) We cannot conclude anything about rationality of p and q unless we know b and c. 7. When does the polynomial 1 x ••• xn have x − a as a factor? Here n is a positive integer greater than 1000 and a is a real number. (A) if and only if a = −1 (B) if and only if a = −1 and n is odd (C) if and only if a = −1 and n is even (D) We cannot decide unless n is known. 1 Part B. Attempt any 7 problems. Explain your reasoning. Each problem is worth 7 points. 1. In a business meeting, each person shakes hands with each other person, with the exception of Mr. L. Since Mr. L arrives after some people have left, he shakes hands only with those present. If the total number of handshakes is exactly 100, how many people left the meeting before Mr. L arrived? (Nobody shakes hands with the same person more than once.) 2. Show that the power of x with the largest coefficient in the polynomial (1 2x3 )20 is 8, i.e., if we write the given polynomial as ∑i aixi then the largest coefficient ai is a8. 3. Show that there are infinitely many perfect squares that can be written as a sum of six consecutive natural numbers. Find the smallest such square. 4. Let S be the set of all 5-digit numbers that contain the digits 1,3,5,7 and 9 exactly once (in usual base 10 representation). Show that the sum of all elements of S is divisible by 11111. Find this sum. 5. It is given that the complex number i − 3 is a root of the polynomial 3x4 10x3 Ax2 Bx − 30, where A and B are unknown real numbers. Find the other roots. 6. Show that there is no solid figure with exactly 11 faces such that each face is a polygonhaving an odd number of sides. 7. To find the volume of a cave, we fit X, Y and Z axes such that the base of the cave isin the XY-plane and the vertical direction is parallel to the Z-axis. The base is the regionin the XY-plane bounded by the parabola y2 = 1 − x and the Y-axis. Each cross-sectionof the cave perpendicular to the X-axis is a square. (a) Show how to write a definite integral that will calculate the volume of this cave. (b) Evaluate this definite integral. Is it possible to evaluate it without using a formula for indefinite integrals? 8. f(x) = x3 x2 cx d, where c and d are real numbers. Prove that if c > 13 , then fhas exactly one real root. 9. A real-valued function f defined on a closed interval [a, b] has the properties that f(a) = f(b) = 0 and f(x) = f′(x) f′′(x) for all x in [a,b]. Show that f(x) = 0 for all xin [a, b]
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