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# Past year iit entrance mathematics problems

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APEX INSTITUTE was conceptualized in May 2008, keeping in view the dreams of young students by the vision & toil of Er. Shahid Iqbal. We had a very humble beginning as an institute for IIT-JEE / Medical, with a vision to provide an ideal launch pad for serious JEE students . We actually started to make a difference in the way students think and approach problems. We started to develop ways to enhance students IQ. We started to leave an indelible mark on the students who have undergone APEX training. That is why APEX INSTITUTE is very well known of its quality of education

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### Past year iit entrance mathematics problems

1. 1. Complex Numbers Entrance QuestionsQ1. The number of complex numbers z such that |z – 1| = |z + 1| = |z – i| equals AIEEE–2010 (a) 0 (b) 1 (c) 2 (d) 4Q2. If z – = 2, then the maximum value of |z| is equal to AIEEE–2009 z (a) 3 +1 (b) 5 +1 (c) 2 (d) 2+ 2 1Q3. The conjugate of a complex number is . Then the complex number is AIEEE–2008 i –1 1 1 1 1 (a) (b) – (c) (d) – i –1 i –1 i 1 i 1Q4. If |z + 4| 3, then the maximum value of |z + 1| is AIEEE–2007 (a) 4 (b) 10 (c) 6 (d) 0 zQ5. If |z| = 1 and z 1, then all the values of lie on AIEEE–2007 1 – z2 (a) a line not passing through the origin (b) |z| = 2 (c) the x-axis (d) the y-axis 10 2k 2kQ6. The value of sin i cos is AIEEE–2006 k 1 11 11 (a) 1 (b) –1 (c) –i (d) iQ7. If w = + i , where 0 and z 1, satisfies the condition that w – w is purely real, then the 1– set of values of is IIT JEE–2006 (a) |z| = 1, z = 2 (b) |z| = 1 and z 1 (c) z= z (d) None of these zQ8. If w = and |w| = 1, then z lies on AIEEE–2005 i z– 3 (a) a circle (b) an ellipse (c) a parabola (d) a straight lineQ9. The locus of z which lies in shaded region is represented by IIT JEE–2005
2. 2. (a) z : |z + 1| > 2, | (z + 1) | < (b) z : |z – 1| > 2, | (z – 1) | < 4 4 (c) z : |z + 1| < 2, | (z + 1) | < (d) z : |z – 1| < 2, | (z – 1) | < 2 2Q10. If |z – 1| = |z| + 1, then z lies on 2 2 AIEEE–2004 (a) the real axis (b) an ellipse (c) a circle (d) imaginary axisQ11. If a, b, c are integers not all equal and is a cube root of unity ( 1), then minimum value of |a 2 +b +c | is equal to IIT JEE–2004 3 1 (a) 0 (b) 1 (c) (d) 2 2 n 2n 2 1Q12. If 1, , are the cube roots of unity, then n 2n is equal to AIEEE–2003 1 2n n 1 2 (a) 0 (b) 1 (c) (d) z –1Q13. If z is a complex number such that |z| = 1, z 1, then real part of is IIT JEE–2003 z 1 1 –1 2 (a) 2 (b) 2 (c) 2 (d) 0 z 1 z 1 z 1Q14. If is an imaginary cube root of unity, then (1 + – 2 7 ) equals AIEEE–2002 (a) 128 (b) –128 (c) 128 2 (d) –128 2Q15. For all complex numbers z1, z2 satisfying |z1| = 12 and |z2 – 3 – 4i| = 5, the minimum value of |z1 – z2| is equal to IIT JEE–2002 (a) 0 (b) 2 (c) 7 (d) 17
3. 3. Quadratic Equations Entrance QuestionsQ1. If and are the roots of the equation x2 – x + 1 = 0, then 2009 + 2009 is equal to AIEEE–2010 (a) –2 (b) –1 (c) 1 (d) 2Q2. If the roots of the equation bx2 + cx + a = 0 be imaginary, then for all real values of x, the expression 3b2x2 + 6bcx + 2c2 AIEEE–2009 (a) greater than 4ab (b) less than 4ab (c) greater than – 4ab (d) less than – 4abQ3. The quadratic equations x2 – 6x + a = 0, x2 – cx + 6 = 0 have one root in common. The other roots of the first and second equations are integers in the ration 4 : 3. Then, the common root is AIEEE–2008 (a) 2 (b) 1 (c) 4 (d) 3Q4. If the difference between the roots of the equation x2 + ax + 1 = 0 is less than 5 , then the set of possible values of a is AIEEE–2007 (a) (–3, 3) (b) (–3, ) (c) (3, ) (d) (– , –3)Q5. Let , be the roots of the equation x2 – px + r = 0 and , 2 be the roots of the equation x2 – 2 qx + r = 0. Then, the value of r is IIT JEE–2007 2 2 (a) (p – q)(2q – p) (b) (q – p)(2p – q) 9 9 2 2 (c) (q – 2p)(2q – p) (d) (2p – q)(2q – p) 9 9Q6. All the values of m for which both roots of the equation x2 – 2mx + m2 – 1 = 0 are greater than –2 but less than 4 lie in the interval AIEEE–2006 (a) m>3 (b) –1 < m < 3 (c) 1<m<4 (d) –2 < m < 0Q7. Let a, b, c be the sides of a scalene triangle. If the roots of the equation x2 + 2(a + b + c)x + 3 (ab + bc + ca) = 0, R are real, then IIT JEE–2006 4 5 1 5 4 5 (a) < (b) > (c) , (d) , 3 3 3 3 3 3
4. 4. Q8. The value of a for which the sum of the squares of the roots of the equation x2 – (a – 2)x – a – 1 = 0 assumes the least value is AIEEE–2005 (a) 0 (b) 1 (c) 2 (d) 3Q9. If both the roots of the equation x2 – 2kx + k2 + k – 5 = 0 are less than 5, then k AIEEE–2005 (a) (6, ) (b) (5, 6] (c) [4, 5] (d) (– , 4)Q10. If 1 – p is a root of x2 + px + 1 – p = 0, then its roots are AIEEE–2004 (a) 0, 1 (b) –1, 2 (c) 0, –1 (d) –1, 1Q11. If one root is square of the other root of the equation x2 + px + q = 0, then the relation between p and q is IIT JEE–2004 (a) p3 – (3p – 1)q + q2 = 0 (b) p3 – q(3p + 1) + q2 = 0 (c) p3 + q(3p – 1) + q2 = 0 (d) p3 + q(3p + 1) + q2 = 0Q12. If one root of (a2 – 5a + 3)x2 + (3a – 1)x + 2 = 0 is twice the other, then a is equal to AIEEE–2003 (a) 2/3 (b) –2/3 (c) 1/3 (d) –1/3Q13. If and 2 = 5 – 3, 2 = 5 – 3, then the equation having and as its root, is AIEEE–2002 (a) 3x2 + 19x + 3 = 0 (b) 3x2 – 19x + 3 = 0 (c) 3x2 – 19x – 3 = 0 (d) x2 – 16x + 1 = 0Q14. If b > a, then the equation (x – a)(x – b) – 1 = 0 has IIT JEE–2000 (a) both roots in (a, b) (b) both roots in (– , a) (c) both roots in (b, ) (d) one root in (– , a) and other in (b, ) Inequalities & LogarithmsQ1. For all x, x2 + 2ax + (10 – 3a) > 0, then the interval in which a lies is IIT JEE–2004 (a) a < –5 (b) –5 < a < 2 (c) a>5 (d) 2<a<5Q2. If 1, log3 31–x 2 , log3( * –1) are in AP, then x is equal to AIEEE–2002
5. 5. (a) log34 (b) 1 – log34 (c) 1 – log43 (d) log43Q3. The set of all real number’s x for which x2 – |x + 2| + x > 0 is IIT JEE–2002 (a) (– , –2) (2, ) (b) (– , – 2) ( 2, ) (c) (– , –1) (1, ) (d) ( 2, ) Sequences & SeriesQ1. A person is to count 4500 currency notes. Let an denotes the number of notes he counts in the nth minute. If a1 = a2 = ….= a10 = 150 and a10, a11…are in AP with common difference –2, then the time taken by him to count all notes, is AIEEE–2010 (a) 24 min (b) 24 min (c) 125 min (d) 135 min 2 6 10 14Q2. The sum of the infinity of the series 1 + + + + 4 + …. is AIEEE–2009 3 32 32 3 (a) 3 (b) 4 (c) 6 (d) 2Q3. The first terms of a geometric progression add upto 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then first term is AIEEE–2008 (a) 4 (b) –4 (c) –12 (d) 12Q4. In a geometric progression consisting of positive term, each term equals to the next two terms. Then, the common ratio of this progression equals AIEEE–2007 1 1 1 (a) (1 – 5) (b) 5 (c) 5 (d) 5 –1 2 2 2 a1 a2 ..... a p p2 a6Q5. Let a1, a2, , be terms of an AP. If = ,p q, then equals a1 a2 aq q2 a21 AIEEE–2007 7 2 11 41 (a) (b) (c) (d) 2 7 41 11Q6. If x = an , y = bn , z = c n where a, b, c are in AP and |a| < 1, |b| < 1, |c| < 1, then x, y, z n 0 n 0 n 0 are in AIEEE–2005 (a) AP (b) GP (c) HP (d) AGP
6. 6. Q7. Let Tr be the rth term of an AP whose first term is a and common difference d. If for some 1 positive integers m, n, m n Tm = 1 , Tn = , then a – d is equal to AIEEE–2004 n m 1 1 1 (a) 0 (b) 1 (c) (d) + mn m n nQ8. The sum of the first n terms of the series 12 + 2 22 + 32 + 2 42 + 52 + 2 + …. is (n + 1)2, when n 2 is even. When n is odd the sum is AIEEE–2004 n2 n n2 n n 1 (a) (n + 1) (b) (n – 1)2 (c) (n – 1) (d) 2 2 2 2Q9. An infinite GP has term x and sum S, then x belongs to IIT JEE–2004 (a) x < –10 (b) –10 < x < 0 (c) 0 < x < 10 (d) x > 10Q10. The value of 21/4 41/8 81/16…. is AIEEE–2002 3 (a) 1 (b) 2 (c) (d) 4 2 MatricesQ1. Consider the system of linear equations x1 + 2x2 + x3 = 3 2x1 + 3x2 + x3 = 3 3x1 + 5x2 + 2x3 = 1 The system has AIEEE–2010 (a) Infinite number of solutions (b) Exactly 3 solutions (c) A unique solution (d) No solutionQ2. The number of 3 × 3 non-singular matrices, with four entries as 1 and all other entries as 0, is AIEEE–2010 (a) less than 4 (b) 5 (c) 6 (d) at least 7 Directions (Q. No. 36 to 38) : For the following questions choose the correct answer from the codes (a), (b), (c), (d) defined as follows : (a) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I (b) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I
7. 7. (c) Statement I is true; Statement II is false (d) Statement I is false; Statement II is trueQ3. Let A be a 2 × 2 matrix with non-zero entries and let A2 = I is 2 × 2 identity matrix. Define Tr(A) = sum of diagonal elements of A and |A| = determinant of matrix A. AIEEE–2010 Statement-I Tr(A) = 0. Statement-II |A| = 1.Q4. Let A be 2 × 2 matrix. AIEEE–2009 Statement-I adj(adj A) = A Statement-II |adj A| = AQ5. Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity matrix. Denote by tr(A), the sum of diagonal entries of A. Assume that A2 = I. AIEEE–2008 Statement-I If A I and A –I, then det(A) = –1. Statement-II If A I and A –I, then tr(A) 0.Q6. Let A be a square matrix all of whose entries are integers. Then, which one of the following is true? AIEEE–2008 (a) If det(A) = 1, then A–1 need not exist (b) If det(A) = 1, then A–1 exists but all its entries are not necessarily integers (c) If det(A) 1, then A–1 exists and all its entries are non-integers (d) If det(A) = 1, then A–1 exists and all its entries are integers 5 5Q7. Let A = 0 5 If det(A2) = 25, then | | is AIEEE–2007 0 0 5 1 (a) 1 (b) (c) 5 (d) 52 5Q8. If A and B are 3 × 3 matrices such that A2 – B2 = (A – B) (A + B), then AIEEE–2006 (a) either A or B is zero matrix (b) either A or B is unit matrix (c) A=B (d) AB = BA 1 2 a 0Q9. Let A = and B = , a, b, , N then AIEEE–2006 3 4 0 b (a) there exists exactly one B such that AB = BA (b) there exists infinitely man B’s such that AB = BA
8. 8. (c) there cannot exist any B such that AB = BA (d) there exist more than but finite number of B’s such that AB = BAQ10. The system of equations ax + y + z = –1 x+ y+z= –1 x+y+ z= –1 has no solution if is AIEEE–2005 (a) –2 or 1 (b) –2 (c) 1 (d) –1 3 1 1 1Q11. If P = ,A= and Q = PAPT, then PTQ2005 P is equal to AIEEE–2005 2 2 0 1 1 3 – 2 2 1 2005 4 + 2005 3 6015 (a) (b) 0 1 2005 4 – 2005 3 1 2 3 1 1 2005 2– 3 (c) (d) 4 –1 2– 3 4 2+ 3 2005 0 0 –1Q12. If A = 0 –1 0 , then AIEEE–2004 –1 0 0 (a) A is zero matrix (b) A = (–1) I (c) A–1 does not exist (d) A2 = I 2Q13. If A = and det A3 = 125, then us equal to IIT JEE–2004 2 (a) 1 (b) 2 (c) 3 (d) 5 a bQ14. If A = and B2 = , then AIEEE–2003 b a (a) = a2 + b2, = ab (b) = a2 + b2, = 2ab (c) = a2 + b2, = a2 – b2 (d) = 2ab, = a2 + b2 0 1 0Q15. If A = and B = , then A2 = B for IIT JEE–2003 0 1 5 1 (a) =4 (b) = –1 (c) =1 (d) no
9. 9. Determinants a a 1 a –1 a 1 b 1 c –1Q1. Let a, b, c be such that (b + c) 0. If –b b 1 b –1 + = 0 then the a –1 b –1 c 1 c c –1 c 1 (–1) n+2 a (–1)n 1 b (–1)n c value of ‘n’ is AIEEE–2009 (a) zero (b) any even integer (c) any odd integer (d) any integerQ2. Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x = cy + bz, y = az + cx and z = bx + ay. Then, a2 + b2 + c2 + 2abc is equal to AIEEE–2008 (a) 1 (b) 2 (c) –1 (d) 0 1 1 1Q3. If D = 1 1 x 1 for xy 0, then D is divisible by AIEEE–2007 1 1 1 y (a) both x and y (b) x but not y (c) y but not x (d) neither x nor y 1 a2 x (1 b2 )x (1 c 2 )xQ4. If a + b + c = –2 and f(x) = (1+a 2 )x 1+b2 x 2 2 2 (1 + c 2 )x , then f(x) is a polynomial of degree (1+a 2 )x (1+b2 )x 1+c 2 x AIEEE–2005 (a) 0 (b) 1 (c) 2 (d) 3 log an log an 1 log an 2Q5. If a1, a2, a3,….. are in GP, then = log a log an log an is equal to AIEEE–2004 n 3 4 5 log an 6 log an 7 log an 8 (a) 0 (b) 1 (c) 2 (d) 4Q6. Given 2x – y + 2z = 2, x – 2y + z = –4, x + y + z = 4, then the value of such that the given system of equation has no solution, is IIT JEE–2004 (a) 3 (b) 1 (c) 0 (d) –3 n 2n 1 2Q7. Of 1, , are the cube roots of unity, then = n 2n 1 is equal to AIEEE–2003 2n n 1 2 (a) 0 (b) 1 (c) (d) 2 2 1 1+i+Q8. If ( 1) is a cubic roots of unity, then 1 – i –1 2 –1 equals AIEEE–2002 –i –1 –i –1
10. 10. (a) 0 (b) 1 (c) i (d)Q9. If the system of equations x + ay = 0, az + y = 0 and ax + z = 0 has infinite solutions, then the value of a is IIT JEE–2002 (a) 0 (b) –1 (c) 1 (d) no real values Binomial Theorem & Its ApplicationsQ1. Statement-I n n r 1 Cr = (n + 2)n–1 r 0 Statement-II n r 1 nCr xr = (1 + x)n + nx(1 + x)n–1 AIEEE–2008 r 0 (a) Statement–I is true, Statement–II is true; Statement–II is a correct explanation for Statement–I (b) Statement–I is true, Statement–II is true; Statement–II is not a correct explanation for Statement–I (c) Statement–I is true; Statement–II is false (d) Statement–I is false; Statement–II is true aQ2. In the expansion of (a – b)n, n 5, the sum of 5th and 6th term is zero, then is equal to b AIEEE–2007, IIT JEE–2001 n–5 n–4 5 6 (a) (b) (c) (d) 6 5 n–4 n–5 1Q3. If the expansion, in powers of x of the function is a0 + a1x + a2x2 + …, then an, 1 – ax 1 – bx is AIEEE –2006 a n – bn a n 1 – bn 1 bn 1 – a n 1 bn – a n (a) (b) (c) (d) b–a b–a b–a b–a 11 11 1 1Q4. If the coefficients of x7 in ax 2 and x–7 in ax – are equal, then bx bx 2 AIEEE–2005 (a) a+b=1 (b) a–b=1 (c) ab = –1 (d) ab = 1
11. 11. 30 30 30 30 30 30 30 30Q5. – + –….+ is equal to IIT JEE–2005 0 10 1 11 2 12 20 30 30 60 30 65 (a) (b) (c) (d) 11 10 10 55Q6. The coefficient of xn in the expansion of (1 + x)(1 – x)n is AIEEE–2004 (a) n–1 (b) (–1)n (1 – n) (c) (–1)n–1(n–1)2 (d) (–1)n–1xQ7. The coefficients of the middle term in the binomial expansion in powers of x of (1 + x)4 and of (1 – x)6 is the same, if AIEEE–2004 5 3 3 10 (a) – (b) (c) – (d) 3 5 10 3Q8. If n–1Cr = (k2 – 3)n Cr+1, then k belongs to IIT JEE–2004 (a) (– , –2] (b) [2, ) (c) – 3, 3 (d) ( 3 , 2]Q9. The number of integral terms in the expansion of ( 3 + 51/8)256 is AIEE–2003 (a) 32 (b) 33 (c) 34 (d) 35Q10. The coefficient of x24 in (1 + x2)12 (1 + x12)(1 + x24) is IIT JEE–2003 12 12 12 12 (a) (b) +1 (c) +2 (d) +3 6 6 6 6Q11. Let Tn denote the number of triangles which can be formed by using the vertices of regular polygon of n sides. AIEEE–2002 If Tn+1 – Tn = 21, then n is equal to (a) 5 (b) 7 (c) 6 (d) 4 m 10 20 pQ12. The sum , when = 0, if p < q is maximum for m is equal to i i m–i q IIT JEE–2002 (a) 5 (b) 10 (c) 15 (d) 20 n n nQ13. For 2 r n, +2 + is equal to IIT JEE–2000 r r –1 r–2 n 1 n 1 n 2 n 2 (a) (b) 2 (c) 2 (d) r –1 r 1 r r
12. 12. Mathematical InductionQ1. The remainder left out when 82n – (62)2n+1 is divided by 9 is AIEEE–2009 (a) 0 (b) 2 (c) 7 (d) 8Q2. Statement-I For every natural number n 2. 1 1 1 + + + > n. 1 2 n Statement-II For every natural number n 2. n n 1 < n + 1. AIEEE–2008 (a) Statement-I is true, Statement-II is true; Statement-II is a correct explanation for Statement-I (b) Statement-I is true, Statement-II is true; Statement-II is not a correct explanation for Statement-I (c) Statement-I is true; Statement-II is false (d) Statement-I is false; Statement-II is true 1 0 1 0Q3. If A = and I = , then which one of the following holds for all n 1, by the 1 1 0 1 principle of mathematical induction ? AIEEE–2005 (a) An = 2n–1 A + (n – 1)I (b) An = nA + (n – 1)I (c) An = 2n–1 A – (n – 1)I (d) An = nA – (n – 1)IQ4. Let S(k) = 1 + 3 + 5 + + (2k – 1) = 3 + k2. Then which of the following is true ? AIEEE–2004 (a) S(1) is correct (b) S(k) S(k + 1) (c) S(k) S(k+ 1) (d) Principle of mathematical induction can be used to prove the formula
13. 13. Permutations & CombinationsQ1. There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done, is AIEEE–2010 (a) 3 (b) 36 (c) 66 (d) 108 10 10 10Q2. Let S1 = j (j – 1)10Cj, S2 = j 10Cj and S3 = j 2 10Cj j 1 j 1 j 1 Statement-I S3 = 55 × 29. Statement-II S1 = 90 × 28 and S2 = 10 × 28.Q3. In a shop there are five types of ice-creams available. A child buys six ice-creams. Statement-I The number of different ways the child can buy the six ice-creams is 10C5. Statement-II The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6 A’s and 4 B’s in a row. AIEEE–2008 (a) Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I (b) Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I (c) Statement-I is true; Statement-II is false (d) Statement-I is false; Statement-II is trueQ4. How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent ? AIEEE–2008 6 8 6 7 8 (a) 7 C4 C4 (b) 8 C4 C4 (c) 6 7 C4 (d) 6 8 7C4Q5. The set S = {1, 2, 3,….., 12} is to be partitioned into three sets A, B, C of equal size. Thus, A B C = S, A B=B C=A C = . The number of ways to partition S is AIEEE–2007 (a) 12 /3 (4 )3 (b) 12 /3 (3 )4 (c) 12 /(4 )3 (d) 12 (3 )4Q6. The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is IIT JEE–2007 (a) 360 (b) 192 (c) 96 (d) 48
14. 14. Q7. At an election, a voter may vote for any number of candidates not greater than the number to be elected. If a voter votes for at least one candidate, then the number of ways in which he can vote, is AIEEE–2006 (a) 6210 (b) 385 (c) 1110 (d) 5070Q8. If the letters of the word SACHIN are arranged in all possible ways and these words are written in dictionary order, then the word SACHIN appears at serial number AIEEE–2005 (a) 600 (b) 601 (c) 602 (d) 603Q9. The number of ways of distributing 8 identical balls in 3 distinct boxes so that no box is empty, is AIEEE–2004 8 (a) 5 (b) (c) 38 (d) 21 3Q10. A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is AIEEE–2003 (a) 140 (b) 196 (c) 280 (d) 346Q11. The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is AIEEE–2003 (a) 65 (b) 30 (c) 54 (d) 57Q12. The number of arrangements of the letters of the word BANANA, which the two N’s do not appear adjacently is IIT JEE–2002 (a) 20 (b) 40 (c) 60 (d) 80 Sets, Relations & FunctionsQ1. Consider the following relations R = {(x, y)| x, y are real numbers and x = wy for some rational number w}; m p S= , m, n, p and q are integers such that n, q 0 and qm = pm. Then n q AIEEE–2010 (a) R is an equivalence relation but S is not an equivalence relation (b) Neither R nor S is an equivalence relation (c) S is an equivalence relation but R is not an equivalence relation (d) R and S both are equivalence relations
15. 15. Q2. If A, B, and C are three sets such that A B=A C and A B=A C, then AIEEE–2009 (a) A=C (b) B=C (c) A B= (d) A=BQ3. For real x, let f(x) = x3 + 5x +1, then AIEEE–2009 (a) f is one-one but not onto R (b) f is onto R but not one-one (c) f is one-one and onto R (d) f is neither one-one nor onto RQ4. Let f(x) = (x + 1)2 – 1, x –1 AIEEE–2009 Statement-I The set {x : f(x) = f –1(x)} = {0, –1} Statement-II f is a bijection.Q5. Let R be the real line. Consider the following subsets of the plane R × R S = {(x, y): y = x + 1 and 0 < x < 2} T = {(x, y) : x – y is an integer} Which one of the following is true? AIEEE–2008 (a) T is an equivalence relation on R but S is not (b) Neither S nor T is an equivalence relation on R (c) Both S and T are equivalence relations on R (d) S is an equivalence relation on R but T is notQ6. Let f : N Y be a function defined as f(x) = 4x +3 for some x N}. Show that f is invertible and its inverse is AIEEE–2008 y–3 3y 4 y 3 y 3 (a) g(y) = (b) g(y) = (c) g(y) = 4+ (d) g(y) = 4 3 4 4 –x 2 xQ7. The largest interval lying in – , for which the function f(x) = 4 + cos–1 –1 + 2 2 2 log(cos x) is defined, is AIEEE–2007 (a) [0, ] (b) – , (c) – , (d) 0, 2 2 4 2 2Q8. Let W denotes the words in the English dictionary. Define the relation R by R = {(x, y) W×W : the words x and y have at least one letter in common}. Then, R is AIEEE–2006 (a) reflexive, symmetric and not transitive (b) reflexive, symmetric and transitive (c) reflexive, not symmetric and transitive
16. 16. (d) not reflexive, symmetric and transitiveQ9. Let R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9, 12}. The relation is AIEEE–2005 (a) an equivalence relation (b) reflexive and symmetric (c) reflexive and transitive (d) only reflexive 2xQ10. Let F : (–1, 1) B be a function defined by f(x) = tan–1 , then f is both one-one and 1 – x2 onto when B is in the interval AIEEE–2005 (a) – , (b) – , (c) 0, (d) 0, 2 2 2 2 2 2 x, if x is rationalQ11. f(x) = and 0, if x is irrational 0, if x is rational g(x) = . Then , f – g is IIT JEE–2005 x, if x is irrational (a) one-one and into (b) neither one-one nor onto (c) many-one and onto (d) one-one and ontoQ12. Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a 8 relation on the set A = {1, 2, 3, 4}. The relation R is AIEEE–2004 (a) reflexive (b) transitive (c) not symmetric (d) a functionQ13. If f(x) = sin x + cos x, g(x) = x2 – 1, then g{f(x)} is invertible in the domain IIT JEE–2004 (a) 0, (b) – , (c) – , (d) [0, ] 2 4 4 2 2Q14. A function f from the set of natural numbers to integers defined by n –1 , n f(n) = 2 is AIEEE–2003 n – , n 2 (a) one-one but not onto
17. 17. (b) onto but not one-one (c) one-one and onto both (d) neither one-one nor ontoQ15. Domain of definition of the function f(x) = sin –1 (2x) for real valued x, is IIT JEE–2003 6 1 1 1 1 1 1 1 1 (a) – , (b) – , (c) – , (d) – , 4 2 2 2 2 9 4 4 5x – x 2Q16. The domain of definition of the function f(x) = log10 is AIEEE–2002 4 (a) [1, 4] (b) [1, 0] (c) [0, 5] (d) [5, 0]Q17. Suppose f(x) = (x + 1)2 for x –1. If g(x) is the function whose graph is reflection of the graph of f(x) w.r.t. the line y = x, then g(x) equals IIT JEE–2002 (a) – x – 1, x 0 (b) 1 , x > –1 2 x+1 (c) x + 1 , x –1 (d) x – 1, x 0 xQ18. Let f(x) = , x –1. Then, for what value of is f [f(x)] = x ? IIT JEE–2001 x+1 (a) 2 (b) – 2 (c) 1 (d) –1 log 2 x 3Q19. The domain of definition of f(x) = is IIT JEE–2001 x 2 3x 2 r R –3, (a) (b) (–2, ) (c) (d) –1, –2 –1, –2, –3 –1, –2Q20. Let f( ) = sin (sin + sin3 ). Then, f( ) IIT JEE–2000 (a) 0 only when 0 (b) 0 for all real (c) 0 for all real (d) 0 only when 0
18. 18. Limits, Continuity & DifferentiabilityQ1. If f : (–1, 1) R be a differentiable function with f(0) = –1 and f ’(0) = 1. Let g(x) = [f(2f(x) + 2)]2. Then g’(0) is equal to AIEEE–2010 (a) 4 (b) –4 (c) 0 (d) –2 f (3x) f (2x)Q2. Let f : R R be a positive increasing function with lim = 1. Then, lim is x f (x) x f (x) equal to AIEEE–2010 2 3 (a) 1 (b) (c) (d) 3 3 2 1Q3. Let f : R R be continuous function defined by f(x) = x AIEEE–2010 e 2e – x 1 Statement-I f(c) = , for some c R. 3 1 Statement-II 0 < f(x) , for all x R. 2 2 1Q4. Let f(x) = (x –1)sin x –1 , if x 1 0 , if x 1 Then which one of the following is true? AIEEE–2008 (a) f is differentiable at x = 1 but not at x = 0 (b) f is neither differentiable at x = 0 nor at x = 1 (c) f is differentiable at x = 0 and at x = 1 (d) f is differentiable at x = 0 but not at x = 1Q5. Let f : R R be function defined by f(x) = {x + 1, |x| + 1}. Then, which of the following is true? AIEEE–2007 (a) f(x) 1 for all x R (b) f(x) is not differentiable at x = 1 (c) f(x) is differentiable everywhere (d) f(x) is not differentiable
19. 19. 2x f (t )dt 2Q6. lim 2 equals IIT JEE–2007 x 4 x – 2 16 8 2 2 1 (a) f(2) (b) f(2) (c) f (d) 4 f(2) 2 xQ7. The set of points, where f(x) = is differentiable, is AIEEE–2006 1 x (a) (– , –1) (b) (– , ) (c) (0, ) (d) (– , 0) (0, ) 1 2 1 2 2 4 n 2Q8. lim 1 equals to AIEEE–2005 n n2 n2 n2 n2 n2 1 1 1 (a) tan1 (b) tan 1 (c) 1 (d) 1 2 2 2Q9. Let f be twice differentiable function satisfying f(1) = 1, f(2) = 4, f(3) = 9, then IIT JEE–2005 (a) f”(x) = 2, x (R) (b) f’(x) = 5 f”(x), for some x (1, 3) (c) there exists at least one x (1, 3) such that f”(x) = 2 (d) none of the above 1 – tan xQ10. Let f(x) = ,x ,x 0, . If f(x) is continuous in 0, , then f is 4x – 4 2 2 4 AIEEE–2004 (a) 1 (b) 1/2 (c) –1/2 (d) –1 2x a bQ11. If lim 1 = e2, then the values of a and b are AIEEE–2004 x x x2 (a) a R, b R (b) a = 1, b R (c) a R, b = 2 (d) a = 1, b = 2 1 1 – x xQ12. If f(x) = xe , x 0, then f (x) is AIEEE–2003 0 , x 0 (a) continuous as well as differentiable for all x (b) continuous for all x but not differentiable at x = 0 (c) neither differentiable nor continuous at x = 0 (d) discontinuous everywhere
20. 20. a – n nx – tan x sin nxQ13. If lim = 0, where n is non-zero real number, then a is equal to x 0 x2 IIT JEE–2003 n 1 1 (a) 0 (b) (c) n (d) n+ n n 1p 2 p 3p npQ14. lim is equal to AIEEE–2002 x np 1 1 1 1 1 1 (a) (b) (c) – (d) p 1 1– p p p –1 p 2 1/ x f (1 x)Q15. Let f : R R be such that f (1) = 3 and f’(1) = 6. Then, lim equals x 0 f (1) IIT JEE–2002 1/2 2 3 (a) 1 (b) e (c) e (d) eQ16. The left hand derivative of f(x) = [x] sin( x) at x = k, k an integer is IIT JEE–2001 (a) (–1)k(k – 1) (b) (–1)k–1 (k – 1) (c) (–1)k k (d) (–1)k–1 kQ17. Let f : R R be any function. Define g : R R by g(x) = |f (x)| for all x. then, g is IIT JEE–2000 (a) onto if f is onto (b) one-one if f is one-one (c) continuous if f is continuous (d) differentiable if f is differentiable DifferentiationQ1. Let y be an implicit of x defined by x2x – 2xx cot y – 1 = 0. Then, y’(1) equals AIEEE–2009 (a) –1 (b) 1 (c) log 2 (d) –log 2Q2. Let f(x) = x|x| and g(x) = sin x AIEEE–2009 Statement-I gof is differentiable at x = 0 and its derivative is continuous at that point. Statement-II gof is twice differentiable at x = 0. d 2xQ3. is equal to IIT JEE–2007 dy 2
21. 21. –1 –1 –3 d2y d2y dy (a) (b) 9 dx 2 dx 2 dx –2 –3 d2y dy d2y dy (c) (d) – dx 2 dx dx 2 dx dyQ4. If xm yn = (x + y)m + n, then is AIEEE–2006 dx x y x y (a) (b) xy (c) (d) xy y xQ5. If f ”(x) = –f(x), where f(x) is a continuous double differentiable function and g(x) = f ’(x). If 2 2 x x F(x) = f + g and F(5) = 5, then f(10) is IIT JEE–2006 2 2 (a) 0 (b) 5 (c) 10 (d) 25Q6. If y is a function of x and log(x + y) = 2xy, then the value of y’(0) is equal to IIT JEE–2004 (a) 1 (b) –1 (c) 2 (d) 0 d2y dyQ7. If y = (x + 1 x 2 )n, then (1 + x2) 2 +x is AIEEE–2002 dx dx (a) n2y (b) –n2y (c) –y (d) 2x2y Application of Derivatives 4Q1. The equation of the tangent to the curve y = x + , that is parallel to the x-axis, is x2 AIEEE–2010 (a) y=0 (b) y=1 (c) y=2 (d) y=3 k – 2x, if x –1Q2. Let f : R R be defined by f(x) = . If f has a local minimum at x = –1, 2x 3, if x –1 then a possible value of k, is AIEEE–2010 1 (a) 1 (b) 0 (c) – (d) –1 2Q3. Given, P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P’(x) = 0. If P(–1) < P(1), then in the interval [–1, 1] AIEEE–2009 (a) P(–1) is the minimum and P(1) is the maximum of P
22. 22. (b) P(–1) is not minimum but P(1) is the maximum of P (c) P(–1) is the minimum and P(1) is not the maximum of P (d) neither P(–1) is the minimum nor P(1) is the maximum of PQ4. The shortest distance between the line y – x = 1 and the curve x = y2 is AIEEE–2009 3 2 2 3 3 2 3 (a) (b) (c) (d) 8 8 5 4Q5. Suppose the cubic x3 – px + q has three distinct real roots where p > 0 and q > 0. Then, which one of the following holds ? AIEEE–2008 p p (a) The cubic has maxima at both and – 3 3 p p (b) The cubic has minima at and maxima at – 3 3 p p (c) The cubic has minima at – and maxima at 3 3 p p (d) The cubic has minima at both and – 3 3Q6. How many real solutions does the equation x7 + 14x5 + 16x3 + 30x – 560 = 0 have ? AIEEE–2008 (a) 5 (b) 7 (c) 1 (d) 3 3Q7. The total number of local maxima and local minima of the function f(x) = 2 x , – 3 x –1 x 2 / 3 , –1 x 2 is IIT JEE–2008 (a) 0 (b) 1 (c) 2 (d) 3Q8. A value of c for which the conclusion of Mean Value theorem holds for the function f(x) = loge x on the interval [1, 3] is AIEEE–2007 1 (a) 2 log3 e (b) loge 3 (c) log3 e (d) loge 3 2Q9. The function f(x) = tan–1 (sin x + cos x) is an increasing function in AIEEE–2007 (a) , (b) – , (c) 0, (d) – , 4 2 2 4 2 2 2
23. 23. Q10. The tangent to the curve y = ex drawn at the point (c, ec) intersects the line joining the points (c – 1, ec – 1) and (c + 1, ec + 1) IIT JEE–2007 (a) on the left of x = c (b) on the right of x = c (c) at no paint (d) at all points 3x 2 9x 17Q11. If x is real, the maximum value of is AIEEE–2006 3x 2 9x 7 17 1 (a) 41 (b) 1 (c) (d) 7 4Q12. A spherical iron ball 10 cm in radius is coated with a layer ice of uniform thickness that melts at a rate of 50 cm3/min. When the thickness of ice 15 cm, then the rate at which the thickness of ice decreases, is AIEEE–2005 5 1 1 1 (a) (b) (c) (d) 6 54 18 36Q13. The tangent at (1, 7) to curve x2 = y – 6 touches the circle x2 + y2 + 16x + 12y + c = 0 at IIT JEE–2005 (a) (6, 7) (b) (–6, 7) (c) (6, –7) (d) (–6, –7)Q14. The normal to the curve x = a(1 + cos ), y = a sin at ‘ ’ always passes through the fixed point AIEEE–2004 (a) (a, a) (b) (0, a) (c) (0, 0) (d) (a, 0)Q15. If f(x) = x3 + bx2 + cx + d and 0 < b2 < c, then in (– , ) IIT JEE–2004 (a) f(x) is strictly increasing function (b) f(x) has a local maxima (c) f(x) is strictly decreasing function (d) f(x) is boundedQ16. Let f(a) = g(a) = k and their nth derivatives f n(a), gn(a) exist and are not equal for some n. f (a) g ( x) – f (a) – g (a) f ( x) g (a ) Further, if lim = 4, then the value of k is equal to x a g ( x) f ( x) AIEEE–2003 (a) 4 (b) 2 (c) 1 (d) 0Q17. If f(x) = x2 + 2bx + 2c2 and g(x) = –x2 – 2cx + b2 such that min f(x) > g(x), then the relation between b and c is IIT JEE–2003 (a) no real values of b and c (b) 0<c<b 2 (c) |c| < |b| 2 (d) |c| > |b| 2
24. 24. Q18. The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 – 2 = 0 AIEEE–2002 (a) cut at right angled (b) touch each other (c) cut at an angle (d) cut at an angle 3 4Q19. The length of a longest interval in which the function 3 sin x – 4 sin3 x is increasing is IIT JEE–2002 3 (a) (b) (c) (d) 3 2 2Q20. If f(x) = xee(1 – x), then f(x) is IIT JEE–2001 1 (a) increasing on – ,1 (b) decreasing on R 2 1 (c) increasing on R (d) decreasing on – ,1 2Q21. Let f(x) = e x (x – 1)(x – 2)dx. Then, f decreases in the interval IIT JEE–2000 (a) (– , –2) (b) (–2, –1) (c) (1, 2) (d) (2, ) Indefinite Integrals sin x dxQ1. The value of 2 is AIEEE–2008 sin x – 4 (a) x – log cos x – +c (b) x + log cos x – +c 4 4 (c) x – log sin x – +c (d) x + log sin x – +c 4 4 dxQ2. equals AIEEE–2007 cos x 3 sin x 1 x 1 x (a) log tan +c (b) log tan – +c 2 2 12 2 2 12 x x (c) log tan +c (d) log tan – +c 2 12 2 12
25. 25. x 2 –1 dxQ3. The value of is IIT JEE–2006 x 3 2x 4 – 2x 2 1 2 1 2 1 (a) 2 2– +c (b) 2 2 + c x2 x4 x2 x4 1 2 1 (c) 2– (d) None of the above 2 x2 x4 2 log x – 1Q4. 2 dx is equal to AIEEE–2005 1 log x xe x x log x x (a) +c (b) +c (c) (d) +c 1 x2 log x 2 1 log x 2 c x 2 1 sin xQ5. If dx = Ax + B log sin(x – ) + c, then value of (A, B) is AIEEE–2004 sin(x – ) (a) (sin , cos ) (b) (cos , sin ) (c) (–sin , cos ) (d) (–cos , sin ) dxQ6. is equal to AIEEE–2004 cos x – sin x 1 x 1 x (a) log tan – +c (b) log cot +c 2 2 8 2 2 1 x 3 1 x 3 (c) log tan – +c (d) log tan +c 2 2 8 2 2 8 dxQ7. is equal to AIEEE–2002 x(x n 1) 1 xn 1 xn 1 (a) log n +c (b) log +c n x 1 n xn xn (c) log +c (d) None of the above xn 1
26. 26. Definite Integrals f (3x)Q1. Let p(x) be a function defined on R such that lim =1, p’(x) = p’(1 – x), for all x [0, 1], x f (x) 1 p(0) = 1 and p(1) = 41. Then, p (x) dx equals AIEEE–2010 0 (a) 41 (b) 21 (c) 41 (d) 42Q2. cot x dx, [ ] denotes the greatest integer function, is equal to AIEEE–2009 0 (a) (b) 1 (c) –1 (d) – 2 2 1 sin x 1 cos xQ3. Let = dx and J = dx. Then, which one of the following is true ? 0 0 x x AIEEE–2008 2 2 2 2 (a) I> and J < 2 (b) I> and J > 2 (c) I< and J < 2 (d) I< and J > 2 3 3 3 3 1 x log tQ4. Let f (x) = f (x) + f , where f (x) = dt. Then, f (e) equals AIEEE–2007 x 1 1 t 1 (a) (b) 0 (c) 1 (d) 2 2 aQ5. The value of x f ’(x) dx, a > 1, where [x] denotes the greatest integer not exceeding x is 1 AIEEE–2006 (a) [a] f (a) – {f (1) + f (2) +…..+ f ([a])} (b) [a] f ([a]) – {f (1) + f (2) +….+ f (a)} (c) a f ([a]) – {f (1) + f (2) +…..+ f (a)} (d) a f (a) – {f (1) + f (2) +…..+ f ([a])} – /2Q6. [(x + )3 + cos 2 (x + 3 )] dx is equal to AIEEE–2006 –3 / 2 4 4 (a) + (b) (c) –1 (d) 32 2 2 4 32 cos 2 xQ7. The value of dx, a > 0, is AIEEE–2005, IIT JEE–2001 – 1 + ax (a) 2 (b) /a (c) /2 (d) a
27. 27. 1 1Q8. If t 2 f (t) dt = 1 – sin x, x (0, /2), then f is IIT JEE–2005 sin x 3 (a) 3 (b) 3 (c) 1/3 (d) None of these /2Q9. If xf (sin x)dx = A f (sin x) dx, then A is equal to AIEEE–2004 0 0 (a) 0 (b) (c) /4 (d) 2 t2 2 5 4Q10. If f (x) is differentiable and x f(x)dx = t , then f equals IIT JEE–2004 0 5 25 (a) 2/5 (b) –5/2 (c) 1 (d) 5/2 1Q11. The value of the integral I = x (1 – x)n dx is AIEEE–2003 0 1 1 1 1 1 1 (a) (b) (c) – (d) + n 1 n 2 n 1 n 2 n 1 n 2 x2 1 2Q12. If f (x) = 2 e–t dt, then f (x) increases in IIT JEE –2003 x (a) (2, 2) (b) no value of x (c) (0, ) (d) (– , 0) 2Q13. [x 2 ] dx is AIEEE–2002 0 (a) 2– 2 (b) 2+ 2 (c) 2–1 (d) – 2 – 3 +5 1/ 2 1 xQ14. The integral [x] log dx equals IIT JEE–2002 –1/2 1– x (a) –1/2 (b) 0 (c) 1 (d) 2 log (1/2) ecos x sin x , 3Q15. If f (x) = |x| 2, then f (x) dx is equal to IIT JEE–2000 –2 2 , (a) 0 (b) 1 (c) 2 (d) 3
28. 28. Area of Curves 3Q1. The area bounded by the curves y = cos x and y = sin x between the ordinates x = 0 and x = 2 is AIEEE–2010 (a) (4 2 – 2) sq unit (b) (4 2 + 2)sq unit (c) (4 2 – 1) sq unit (d) (4 2 + 1)sq unitQ2. The area of the region bounded by the parabola (y – 2)2 = x – 1, the tangent to the parabola at the point (2, 3) and the x-axis is AIEEE–2009 (a) 6 sq unit (b) 9 sq unit (c) 12 sq unit (d) 3 sq unitQ3. The area of the plane region bounded by the curves x + 2y2 = 0 and x + 2y2 = 1 is equal to AIEEE–2008 4 5 1 2 (a) sq unit (b) sq unit (c) sq unit (d) sq unit 3 3 3 3Q4. The area enclosed between the curves y2 = x and y = | x | is AIEEE–2007 (a) 2/3 sq unit (b) 1 sq unit (c) 1/6 sq unit (d) 1/3 sq unitQ5. The parabolas y2 = 4x and x2 = 4y divide the square region bounded by the line x = 4, y = 4 and the coordinate axes. If S1, S2, S3 are respectively the areas of these parts numbered from top to bottom, then S1 : S2 : S3 is AIEEE–2005 (a) 2:1:2 (b) 1:1:1 (c) 1:2:1 (d) 1:2:3Q6. Let f (x) be a non-negative continuous functions. Such that the area bounded by the curve y = f (x), x-axis and the coordinates x = ,x= > is sin cos 2 . Then f 4 4 4 is AIEEE–2005 2 (a) 1– – 2 (b) 1– 2 4 4 (c) 2 –1 (d) – 2 1 4 4 1Q7. The area bounded by the curve y = (x + 1)2, y = (x – 1)2 and the line y = is AIEEE–2005 4 (a) 1/6 sq unit (b) 2/3 sq unit (c) 1/4 sq unit (d) 1/3 sq unit
29. 29. Q8. The area of the region bounded by the curve y = |x – 2|, x = 1, x = 3 and the axis is AIEEE –2004 (a) 4 sq unit (b) 2 sq unit (c) 3 sq unit (d) 1 sq unitQ9. The area of the region bounded by y = ax2 and x = ay2, a > 0 is 1, then a is equal to IIT JEE–2004 1 1 (a) 1 (b) (c) (d) None of these 3 3Q10. The area bounded by the curve y = 2x – x2 and the straight line y = –x is given by AIEEE–2002 (a) 9/2 sq unit (b) 43/6 sq unit (c) 35/6 sq unit (d) None of theseQ11. The area bounded by the curves y = | x | – 1 and y = –| x | + 1 is IIT JEE–2002 (a) 1 sq unit (b) 2 sq unit (c) 2 2 sq unit (d) 4 sq unit Differential EquationsQ1. Solution of the differential equation cos xdy = y(sin x – y)dx, 0 < x < , is AIEEE–2010 2 (a) sec x = (tan x + c)y (b) y sec x = tan x + c (c) y tan x = sec x + c (d) tan x = (sec x + c)y c xQ2. The differential equation which represents the family of curves y = c1e 2 , where c1 and c2 are arbitrary constants is AIEEE–2009 2 (a) y’ = y (b) y” = y’ y (c) yy” = y’ (d) yy” = (y’)2Q3. The differential equation of the family of circles with fixed radius 5 unit and centre on the line y = 2 is AIEEE–2008 (a) (x – 2) y’ = 25 – (y – 2) 2 2 2 (b) (x – 2) y’ = 25 – (y – 2) 2 2 (c) (y – 2) y’2 = 25 – (y – 2)2 (d) (y – 2)2 y’2 = 25 – (y – 2)2 dy x + yQ4. The solution of the differential equation = satisfying the condition y(1) = 1 is dx x AIEEE–2008 (a) y = x log x + x (b) y = log x + x (c) y = x log x + x2 (d) y = xe(x – 1)Q5. The differential equation of all circles passing through the origin and having their centres on the x-axis is AIEEE–2007