Fundamental of Mathematics Exercise

Fundamentals of Mathematics  Marked questions are recommended for Revision. SUBJECTIVE QUESTIONS Section (A) : Representation of sets, Types of sets, subset and power set A-1. Which of the following collections is not a set ? (i) The collection of natural numbers between 2 and 20 (ii) The collectihon of numbers which satisfy the equation x2 –5x + 6 = 0 (iii) The collection of prime numbers between 1 and 100. (iv) The collection of all intelligent women in Jalandhar. A-2. Write the set A = {x : x is a positive prime < 10} in the tabular form A-3. Which of the following is the empty / non-empty set (i) {x : x is a real number and x2 – 1 = 0} (ii) {x : x is a real number and x2 + 1 = 0} (iii) {x : x is a real number and x2 – 9 = 0} (iv) {x : x is a real number and x2 = x + 2} A-4. Which of the folowing sets is an finite / infinite set ? (i) Set of divisors of 24 (ii) Set of all real number which lie between 1 and 2 (iii) Set of all humman beings living in India. (iv) Set of all three digit natural numbers A-5. If A = {x : –3 < x < 3, x  Z} then find the number of subsets of A. A-6. Find Power set of the set A = {, {}}. Section (B) : Operations on sets, Law of Algebra of sets B-1. Sets A and B have 3 and 6 elements respectively. What can be the minimum number of elements in AB? B-2. Given the sets A = {1, 2, 3}, B = {3, 4}, C = {4, 5, 6}, then find A  (B  C). B-3. Let A = {x : x  R, –1 < x < 1} , B = {x : x  R, x  0 or x  2} and A  B = R – D, then find set D B-4. Find the smallest set A such that A  {1, 2} = {1, 2, 3, 5, 9} B-5. If A = {2, 3, 4, 8, 10}, B = {3, 4, 5, 10, 12}, C = {4, 5, 6, 12, 14} then find (A  B)  (A  C). B-6. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 5}, B = {6, 7}, then show that A  B is same as set A. B-7. If A = {x : x = 4n + 1, n  5, n  N} and B {3n : n  8, n  N}, then find (A – (A – B)). B-8. If aN = {ax : x  N} and bN  cN = dN, where b, c  N, b  2, c  2 are relatively prime, then find relation between d,b and c. Section (C) : Cardinal number Problems C-1. C-2. Let n(U) = 700, n(A) = 200, n(B) = 300 and n(A  B) = 100, then find n(A'  B') In a group of 1000 people, there are 750 people, who can speak Hindi and 400 people, who can speak Bengali. Fundamentals of Mathematics (i) (ii) (iii) C-3. Find number of people who can speak Hindi only. Number of people who can speak Bengali only is Number of people who can speak both Hindi and Bengali is A class has 175 students. The following data shows the number of students obtaining one or more subjects : Mathematics 100, Physics 70, Chemistry 40, Mathematics and Physics 30, Mathematics and Chemistry 28, Physics and Chemistry 23, Mathematics & Physics & Chemistry 18. How many students have offered Mathematics alone ? C-4. 31 candidates appeared for an examination, 15 candidates passed in English, 15 candidates passed in Hindi, 20 candidates passed in Sanskrit. 3 candidates passed only in English. 4. candidates passed only in Hindi, 7 candidates passed only in Sanskrit. 2 candidates passed in all the three subjects How many candidates passed only in two subjects ? C-5. In a survery, it was found that 21 persons liked product A, 26 liked product B and 29 liked product C. If 14 persons liked products A and B, 12 liked products C and A, 13 persons liked products B and C and 8 liked all the three products then (i) Find the number of persons who liked the product C only (ii) The number of persons who like the products A and B but not C Section (D) : Graphs of polynomial D-1. D-2. Draw the graph of following function (i) y = 4x3 – 30x2 + 72x – 55 (iii) y = x4 – 8x3 + 22x2 – 24x + 8.5 (v) y = x4 – 4x3 + 8x2 – 8x – 21 (ii) (iv) (vi) y = x3 + x2 + x – 3 y = x4 – 6x2 – 8x + 13 y = x4 + 2x2 + 4x + 1 Find the number of solution of the following equation x4 – 6x2 – 8x – 3 = 0 D-3. Find the range of '' for which equation x3 + x2 – x – 1 –  = 0 has 3 real solution. Section (E) : Rational inequaties, Modulus & Graphical transformations E-1. Solve the following rational in equalities (x – 1)(x + 2) (i) <0 (x – 3)(x + 3) (ii) (1– x)3 (x + 2)4 0 (x + 9)2 (x – 8) (iii) (x 2 – 3x + 1)3 0 (x − 1)(x + 2) (iv) x(2x – 3 x ) 0 (x + x + 1)(x – 1) (v) (x – 1)(x – 2)(x – 3) 1 (x + 1)(x + 2)(x + 3) (vi) 1< 2 3x 2 – 7x + 8 2 x2 + 1 E-2. Find the number of positive integral value of x satisfying the inequality E-3. If 1 < (i) E-4. E-5. (3 x – 5 x )(x – 2) 0 (x 2 + 5x + 2) x –1  7 then find the range of x+2 x (ii) x2 Define and plot (i) y = |x – 2| + 3 |x – 3| (iii) y = |x – 1| + |x – 4| – 2 |x + 1| Solve for x (i) |x + 1| = 4x + 3 (ii) |x + 1| = |x + 3| (iii) 7|x – 2| – |x – 7| = 5 (iii) 1 x (ii) y = ||x – 2| – 3| + |x| Fundamentals of Mathematics ||x – 1| – 2| = 6x + 8 |2x2 – 3x + 1| = |x2 + x – 3| (iv) (v) E-6. Solve for x (i) 2|x+1| + 2|x| = 6 and x (ii) x2 + x + 1 + |x – 3|  |x2 + 2x – 2| (iii) |2x – 4| – 2|x2 + x – 3| + 2|x – 1||x + 1| = 0 E-7. Solve the following in equalities (i) |x + 7| > 5 E-8. E-9. (ii) (iii) (x + 2) < |x2 + 3x + 5| (iv) (v) |x – 6|  x2 – 5x + 9 (vi) (vii) (|x – 1| – 3) (|x + 2| – 5) < 0 (viii) Find the number of solution of the following equation (i) |||x – 1| – 2| – 1| = 1 |x + 3| < 10 2x – 1 >2 x –1 x 2 – | x | –12  2x x–3 |x – 1| + |x – 2| + |x – 3|  6 (ii) 2|(x – 1) (x – 5)| = (x – 3)2 If graph of y = (x – 1)(x – 2) is (0,2) 1 2 –1/4 then draw the graph of the following (i) y = |(x – 1) (x – 2)| (iii) y = (|x| – 1) (|x| – 2) (v) |y| = |(|x| – 1) (|x| – 2)| E-10. (ii) (iv) Let graph of y = f(x) is (–6,2) (–1,1) (–4,0) (6,2) (0,2) (–2,0) (–3,–1) (1,1) (2,1) (7,0) (3,0) (5,0) (4,–1) Now draw the graph of following (i) y = 2f (–x) (ii) y =f (|x + 1|) (iii) y = – f(|x| + 1) (iv) |y + 1| = f(2x – 1)  x  [–1,3] (8,–1) |y| = (x – 1) (x – 2) y = |(|x| – 1) (|x| – 2)| Fundamentals of Mathematics Section (F) : Irrational inequality, logarithmic equation & logarithimic inequality F-1. Solve the following inequlities (i) x – 1 < x – 3 (iii) (v) 3 x–3 > x 2 + 4x + 9 > x + 2 (iv) 4 – x < (x – 4)(x – 6) > 2 (vi) 1 1 1 3 – < – x 2 x2 x | x+2| – | x| (ix) 0 8 – x3 (vii) F-2. (ii) 7–x 2x – x2 x 2 + 3x + 5 < x2 + x + 1 1 2x 2 + 7x – 4  2 x+4 (viii) Find the value of (i) (log105)(log1020) + (iii) log0.75 log2 (v)  1   49    (log102)2 1 0.125 (ii) 3 (iv) 5 (vi) 7 log 1+log7 2 + 5 – log1/ 5 7 5 1 log7 5 5 2 + 1 ( − log10 0.1) + 9log3 7 − 8log2 5 log3 5 log5 7 +3 F-3. Let log102 = a and log103 = b determine the following in term of a and b (i) log4100 + 2log27100 (ii) log144 45 F-4. Prove that 1 1 1 (i) + + =1 log bc (abc) log ca abc log ab (abc) F-5 loga b = b logb a (iv) Solve the following equations : (i) logx(4x – 3) = 2 (iii) log2 (log8 (x2 – 1)) = 0 a 21/ 4 ( 2 2 – 3log27 (a ) +1)3 – 2a 74log49 a – a – 1 = a2 + a + 1 (ii) log2(x – 1) + log2(x – 3) = 3 (iv) 4log2 x − 2x − 3 = 0 F-6. Solve the following equations log10 x + 7 log2 (9 – 2x ) (log x +1) (i) =1 (ii) = 10 10 x 4 3–x (iii) (log10(100x))2 + (log10(10x))2 = 14 + log10(1/x) (iv) log105 + log10(x + 10) – 1 = log10(21x – 20) – log10(2x – 1) (v) 52x = 32x + 2.5x + 2.3x F-7 Solve the following inequalities (i) log5 (3x – 1) < 1 (iii) log3(x + 1) + log3(x + 7)  3 F-8. (ii) (log.5x)2 + log.5x – 2  0 (iv) log1/2 log3(x2 + 5) + 1  0 Solve the following inequalities (i) |log3x| – log3x – 3 < 0 (iii) log x −1 (x–2) > 0 x –1 1 log3 (9 – 3 x ) – 3 (iv) logx (x3 – x2 – 2x) < 3 (ii) x −5 F-9 Solve the following inequalities (i) 15x – 25.3x – 9.5x + 225  0 log5 3 −7 (ii) (log210) (log280) – (log25) (log2160) = 4 log (iii) a log3 7 −5  3x – 2 (ii) 8.  x x 3 –2  2 > 1 +   3  x Fundamentals of Mathematics Section (G) : Greatest integer function, fractional part & signum function G-1. Solve for x (where [•] denotes greatest integer function and {•} represent fractional part function) (i) [2x] = 1 (ii) {x}2 + [x] = 2 (iii) 6{x}2 – 5{x} + 1 = 0 (iv) 6[x]2 – 5[x] – 1 = 0 G-2 Solve the following equations (where [•] denotes greatest integer function and {•} represent fractional part function) (i) 2[x] + 3{x} = 4x – 1 (ii) 4[x] = x + {x} (iii) [x] + 2{–x} = 3x G-3. Solve the following equations (where [•] denotes greatest integer function and {•} represent fractional part function and sgn represents signum function) (i) [x] + |x – 2|  0 and x  [–1,3] (ii) [2x] – 2x = [x + 1] (iii) [x2] + 2 [x] = 3x, 0  x  2 G-4. Solve the following inequalities (where [•] denotes greatest integer function and {•} represent fractional part function) 1 (i) [x + [x]] < 0 (ii) [2x2 – x] < 1 (iii) {x} < 2 G-5 Solve the following equations (i) sgn ({[x]}) = 0 G-6 (ii) sgn(x2 – 2x – 8) = –1  x 2 − 5x + 4  (iii) sgn  = – 1 {x}   Find the number of solution of equation (where sgn represent signum function) (i) sgn (x) = |x| (ii) sgn (x2 – 1) = (x + 1)2 Section (H) : Trigonometric Equations H-1. Solve the following equation    (i) 5 cos2 + 2cos2 + 1 = 0, – <  < 2 2 2 (ii) sin7 + sin4 + sin = 0 , 0     (iii) tan + sec  = 3 , 0    2 H-2. Find the most general solution of the following (i) sin6x = sin4x – sin2x (ii) sec4x – sec2x = 2     (iii) tan  + x  + tan  – x  = 2 4 4     H-3. Solve the following inequations (i) (sinx – 2) (2sinx–1) < 0 (iii) sinx + 3 cosx  1 (v) tan2x > 3 H-4 (ii) (2cosx – 1) (cosx)  0 (iv) cos2x + sinx  2 Find the number of solution of the following equation x (i) |sinx| = (ii) n|x| = sinx 10 1 H-5. Solve the inequation 2 cos2 x y2 – y + 1 1 2 Fundamentals of Mathematics Marked questions are recommended for Revision. * Marked Questions may have more than one correct option. OBJECTIVE QUESTIONS 1. 2. Which of the following are true ? (A) [3, 7]  (2, 10) (B) (0, )  (4, ) (C) (5, 7]  [5, 7) (D) [2, 7]  (2.9, 8) (C) A  (B – C) (D) A – (B  C) The shaded region in the given figure is (A) A  (B  C) (B) A  (B  C) 3. Consider the following statements : 1. N  (B  Z) = (N  B)  Z for any subset B of R, where N is the set of positive integers, Z is the set of integers, R is the set of real numbers. 2. Let A = {n  N : 1  n  24, n is a multiple of 3}. There exists no subset B of N such that the number of elemets in A is equal to the number of elements in B. Which of the above statements is/are correct ? (A) 1 only (B) 2 only (C) Both 1 and 2 (D) Neither 1 nor 2 4. Which of the following venn-diagrams best represents the sets of females, mothers and doctors ? (A) (B) (C) (D) 5. In a college of 300 students, every student reads 5 newspapers and every newspaper is read by 60 students. The number of newspaper is(A) at least 30 (B) at most 20 (C) exactly 25 (D) exactly 30 6. In a town of 10,000 families it was found that 40% families buy newspaper A, 20% families buy newspaper B and 10% families buy newspaper C, 5% families buy A and B, 3 % buy B and C and 4% buy A and C. If 2% families buy all the three news papers, then number of families which buy newspaper A only is (A) 3100 (B) 3300 (C) 2900 (D) 1400 7. Let A1, A2 and A3 be subsets of a set X. Which one of the following is correct ? (A) A1  A2  A3 is the smallest subset of X containing elements of each of A 1, A2 and A3 (B) A1  A2  A3 is the smallest subset of X containing either A1 or A2  A3 but not both (C) The smallest subset of X containing A1  A2 and A3 equals the smallest subset of X containing both A1 and A2  A3 only if A2 = A3 (D) None of these 8. Let A, B, C be distinct subsets of a universal set U. For a subset X of U, let X ' denote the complement of X in U. Consider the following sets : 1. ((A  B)  C)  B) = B  C 2. (A   B )  (A  B  C) = (A  (B  C)) Which of the above statements is/are correct ? (A) 1 only (B) 2 only (C) Both 1 and 2 (D) Neither 1 nor 2 Fundamentals of Mathematics 9. Let U be set with number of elements in U is 2009. Consider the following statements :   If A, B are subsets of U with n (A  B) = 280, then n(A  B) = x13 + x32 = y13 + y32 for some positive integers x1, x2 y1, y2   If A is a subset of U with n (A) = 1681 and out of these 1681 elements, exactly 1075 elements belong to a subset B of U  then n (A – B) = m2 + p1 p2 p3 for some positive integer m and distinct primes p1, p2, p3 Which of the statements given above is / are correct ? (A) I only (B) II only (C) Both I and II (D) Neither I nor II. 10. Consider the following statements : 1. If A = {(x, y)  [R × R : x3 + y3 = 1] and B = {(x, y)  [R : x – y = 1]}, then A  B contains exactly one elements. 2. If A = {(x, y)  [R × R : x3 + y3 = 1] and B = {(x, y)  [R : x + y = 1]}, then A  B contains exactly two elements. Which of the above statements is/are correct ? (A) 1 only (B) 2 only (C) Both 1 and 2 (D) Neither 1 and 2 11. In a class of 42 students, the number of students studying different subjects are 23 in Mathematics, 24 in Physics, 19 in Chemistry, 12 in Mathematics and Physics 9 in Mathematics and Chemistry, 7 in Physics and Chemistry and 4 in all the three subjects. The number of students who have taken exactly one subject is (A) 15 (B) 30 (C) 22 (D) 27 12. In an examination of a certain class, at least 70% of the students failed in Physics, at least 72% failed in Chemistry, at least 80% failed in Mathematics and at least 85% failed in English. How many at least must have failed in all the four subjects ? (A) 5% (B) 7% (C) 15% (D) Cannot be determined due to insufficient data 13.* A and B are two sets such that n(A) = 3 and n(B) = 6, then (A) minimum value of n(A  B) = 6 (B) minimum value of n(A  B) = 9 (C) maximum value of n(A  B) = 6 (D) maximum value of n(A  B) = 9 14. 15. The number of solution of equation x3 – 21x –20 = ex is/are (A) 0 (B) 1 (C) 2 If a  0 then complete set of solution of (A) (–, –|a|)  (|a|, ) (C) (–,–a)  [a,) 16. x 2 – 2x + 2|a|  0 is x 2 – a2 (B) (–,–a)  (a,) (D) (–|a|, |a|) The complete set of solution of equation 1– (A) a + b = 8 (B) a + b = 3 (D) 3 1 |x| = is {a,b} then 2 1+ | x | (C) a + b = 0 (D) a + b = – 3 17. The smallest integral value of a such that |x + a – 3| + |x –2a| = |2x – a –3| is true  x  R is (A) 0 (B) 1 (C) 2 (D) 3 18. Number of positive integral solution of the equation |x 2 – 3x – 3| > |x2 + 7x – 13| is/are (A) 0 (B) 1 (C) 2 (D) 3 19.  2 | x | –2  If (x2 – 2|x|) (|2x| – 2) – 9  2   0 then  x – 2 | x | (A) x  (–, – 1]  (0,1]  (2,3] (B) x  (–, – 1]  (2,3]  {1} (C) x  (–,3] (D) x  [–3 – 2)  [–1,0)  (0,1]  (2,3] Fundamentals of Mathematics 20. Number of solution of pair of equations y = ||x| – 2| – 2| and y = (A) 1 (B) 2 (C) 3 x+2 equals to 2 (D) 4 21*. For making graph of equations |y| = |f(|x|)| through y = f(x) which order of step is right among the following order of Step  : y = f(|x|) (replace x by |x|) Step  : y = |f(x)| (take modulus of R.H.S) Step  : |y| = f(x) (replace y by |y|) (A) , ,  (B) , ,  (C) , ,  (D) , ,  22. The sum of all the integral values of a {where a  [–10, 10)} such that the graph of the function f(x) = ||x – 2| – a| – 3 has exactly three x-intercepts is (A) 10 23. (B) 5 (C) 3 (D) 0 (C) y = |x| (D) y = 2 Let graph of y = f(x) is (1, 1) (–1, –1) and graph of y = g(x) is (1, 1) (–1, –1) The graph of y = |f(x) + g(x)| is same as (A) y = |f(x)| + |g(x)| (B) y = 2|x| 24. Complete set of solution of inequation (A) (–,1) 25. 26. (B) (–,1] Complete set of solution of inequation b, c  R+) then a + b + c + d equation (A) 4 (B) 3 Complete set of solution of a inequation (A) (–,1) (B) (–,1] 3 2– x – 2 – x < 2 is (C) (1,) 3x2 + 5x + 7 – (D) [1,) 3x2 + 5x + 2 > 1 is (–a, – b]  [–c,d) (where a, (C) 2 x + 2 x –1 + (C) [1, ) (D) 1 x – 2 x –1 > 3 2 (D) (1, ) 27. If a, b, c are distinct positive number but no one among them is equal to one and logba logca + logab logcb + logac logbc = 3, then value of abc is (A) 2 (B) 3 (C) 0 (D) 1 28. Let , , are two real solution of equation (log10x)2 + log10x2 = (log102)2 –1, then (A) 20 (B) 3 (C) 10 (D) 1 1 equal to  Fundamentals of Mathematics 29. Let a, b, c, d are positive integer such that logab = 3/2 and logcd = 5/4. If a – c = 9, then value of (b – d) is equal to (A) 20 (B) 93 (C) 10 (D) 1 30. The values of a for which the equation 2(log3x)2 – |log3x| + a = 0 posses four real solution 1 1 (A) –2 < a < 0 (B) 0 < a < (C) 0 < a < 5 (D) – < a < 0 8 8 31. If log 1 2 x 2 + 6x + 9 < – log2(x + 1) then complete set of values of x is 2(x + 1) (A) (–1, 1 + 2 2 ) 32. 35. 36. (C) (–1, ) (D) ( 1– 2 2, 1 + 2 2 ) x log2   2 (B) 4 Complete set of solution of equation (A) [1, 2)  (2, 3] 34. ) The least positive integer x, which satisfies the inequality log (A) 3 33. ( (B) 1– 2 2,2 (x2 – 10x + 22) > 0 is equal to (C) 7 (D) 8 log0.3 (x – 2) 0 |x| (B) [1, 3] (C) (2, 3] (D) {1} The solution set of the inequality |9x – 3x+1 + 15| < 2.9x – 3x is (A) (–,1) (B) (1,) (C) (–,1] (D) [2, ) The complete set of values of x satisfying the equation x2 . 2x+1 + 2|x–3| + 2 = x2.2|x–3|+ 4 + 2x–1 is 1  1 1   1 1 (A) [3,) (B)  – ,   (3,) (C)  –  –  (D)  – ,   [3,) 2   2 2  2 2   x       x    x    x  If f(x) = {x} +  x +  + x +  + x +  ........+  x +  , then values of 2  2  2  2   1 + 99x   1 + x    1 + 2x    1 + 3x        [f( 3 )] is (where [•] denotes greatest integer function and {•} represent fractional part function) (A) 5050 (B) 4950 (C) 17 (D) 73 37. The number of solution of the equation sgn ({x}) = |1– x| is/are (where {•} represent fractional part function and sgn respresent signum function) (A) 4 (B) 3 (C) 2 (D) 1 38. The complete set of solution of Inequality x 2 – 5x + 6 sgn(x)  0 is (where sgn respresent signum xsgn(x – 1) + 1 function) (A) (–, –1]  [0,2]  [3,) (C) (–,2]  [3,) 39. (B) (–, 0]  [2,) (D) (–, –1]  [0,)  3   3  2 3  4 sin  + sin  3 +   + sin  3 +    is equal to      3 –3 4 (A) (B) (C) 4 4 3 1 sin3 40.* If (m + 2) sin + (2m – 1) cos = 2m + 1 then 3 2m (A) tan = (B) tan = 2 4 m +1 (C) tan = (D) 2m m2 – 1 –4 3 (D) tan = 4 3 Fundamentals of Mathematics  and x = X cos + Ysin , y = Xsin – Ycos such that x2 + 4xy + y2 = aX2 + bY2, 2 where a, b are constants then  (A) a = –1, b = 3 (B)  = /4 (C) a = 3, b = – 1 (D)  = 3 41. * Let 0    42. Let (1 + tan 1º) (1 + tan2º) ............(1 + tan45º) = 2k then k equals to (A) 21 (B) 22 (C) 23 (D) 24 The number of solution of 2 cosx = |sinx| where x  [0,4] is/are (A) 2 (B) 3 (C) 4 (D) 1 43. 44. * If the equation sin (x2) – sin(x2 + 2x) = 0 is solved for positive roots, then in the increasing sequence of positive root (A) first term is −1 + 7 2 (C) third term is 1 45. 46. (B) first term is (D) third term is −1 + 3 2 −1 + 11 2 In (0, 6), the number of solutions of the equation tan + tan 2 + tan 3 = tan .tan2.tan3 is /are (A) 15 (B) 17 (C) 20 (D) 12  n  If 2tan2x – 5 secx – 1 = 0 has 7 different roots in 0, , n  N, then the greatest value of n is 2   (A) 15 (B) 13 (C) 14 (D) 16 47. The number of integral values of a for which the equation cos 2x + a sin x = 2a − 7 possesses a solution. (A) 0 (B) 1 (C) 3 (D) 5 48. If the arithmetic mean of the roots of the equation 4cos 3x – 4cos2x – cos( + x) – 1 = 0 in the interval [0, 315] is equal to k , then the value of k is (A) 10 (B) 20 (C) 50 (D) 80 49. Number of solution of sinx cosx – 3 cosx + 4 sinx – 13 > 0 in [0,2] is equal to (A) 0 (B) 1 (C) 2 (D) 4 50. The solution of 5 − 2 sin x  6 sin x – 1 is (A) [ (12n – 7)/6,  (12n + 7)/6] (n  Z) (C) [ (2n – 7)/6,  (2n + 1)/6] (n  Z) (B) [ (12n – 7)/6,  (12n + 1)/6] (n  Z) (D) [ (12n – 7)/3,  (12n + 1)/3] (n  Z) Fundamentals of Mathematics * Marked Questions may have more than one correct option. Marked questions are recommended for Revision. PART - I : (ADVANCED) / PROBLEMS (PREVIOUS YEARS) 1.* 2*. cos4 x sin4 x 1 + = , then 3 2 5 2 (A) tan2x = 3 1 (C) tan2 x = 3 If For 0 <  < (A)  , the solution(s) of 2  4 (B) sin8 x cos8 x 1 + = 8 27 125 sin8 x cos8 x 2 (D) + = 8 27 125 (B) 6  m =1 (m − 1)   m    cos ec   + cos ec   + =4  4 4      6 (C)  12 (D) The maximum value of the expression 4. The positive integer value of n > 3 satisfying the equation 5. The number of values of  in the interval  – ,  such that   5  2 2 tan = cot 5 as well as sin 2 = cos 4 is 6. Let (x0, y0) be the solution of the following equations (2x) n2 = (3y) n3 3 nx = 2 ny . Then x0 is 7. 8.* 2   1 6 (B) Let P = { : sin  – cos  = (A) P  Q and Q – P   (C) P  Q 5 12 1 is sin  + 3sin  cos  + 5cos2  3. (A) 2 is(are) 1 3 (C) 1 1 = +     2  sin   sin   sin n  n   1 2 2 cos } and Q = { : sin  + cos  = (B) Q  P (D) P = Q n 1 is  3   n    for n = 0, ±1, ± 2 and (D) 6 2 sin } be two sets. Then    Let ,   [0, 2] be such that 2cos(1 – sin) =  tan + cot  sin2 cos – 1, tan(2 – ) > 0 and 2 2  3 . Then  cannot satisfy 2   4 (A) 0 <  < (B) < < 2 2 3 –1 < sin < – (C) 3 4 < < 2 3 (D) 3 <  < 2 2 Fundamentals of Mathematics 9. 10.* 11.   1 1 1 1 4– 4– 4– ....  is The value of 6 + log3    3 2 3 2 3 2 2 3 2   If 3x = 4x – 1 , then x = 2log3 2 (A) 2log3 2 – 1 (B) 2 2 – log2 3 (C) 1 1– log4 3 (D) 2log2 3 2log2 3 – 1 For x  (0, ), the equation sinx + 2 sin 2x – sin 3x = 3 has (A) infinitely many solutions (C) one solution (B) three solutions (D) no solution 12. The number of distinct solutions of the equation 5 cos2 2x + cos4 x + sin4 x + cos6 x + sin6x = 2 in the interval [0, 2] is 4 13. Let – 14. Let S = x  (– , ) : x  0,   < <– . Suppose 1 and 1 are the roots of the equation x2 – 2x sec  + 1 = 0 and 2 and 6 12  are the roots of the equation x2 + 2x tan  – 1 = 0. If 1 > 1 and 2 > 2 , then 1 + 2 equals (A) 2(sec – tan ) (B) 2sec  (C) – 2tan  (D) 0     . The sum of all distinct solutions of the equation 2 3 sec x + cosec x + 2(tan x – cot x) = 0 in the set S is equal to (A) – 7 9 2 9 (B) – (C) 0 (D) 5 9 13 15. The value of  1 is equal to   (k − 1)    k  k = 1 sin +  sin +  6  4 6  4 ( (A) 3 − 3 16. 17. (B) 2 3 − 3 ( ) ) (C) 2 3 − 1 ( (D) 2 2 + 3 ) Let  and  be nonzero real numbers such that 2(cos  – cos ) + cos  cos  = 1. Then which of the following is/are true?     (A) 3 tan   – tan   = 0 (B) tan   – 3 tan   = 0 2 2 2 2           (C) tan   + 3 tan   = 0 (D) 3 tan   + tan   = 0 2 2 2 2 1 2 log2 (log2 9) The value of ((log 2 9) )  ( 7) 1 log4 7 is ___________. Fundamentals of Mathematics PART - II : (MAIN) / AIEEE PROBLEMS (PREVIOUS YEARS) 1. If A, B and C are three sets such that A  B = A  C and A  B = A  C, then (1) A = C 2. (2) B = C (1) A is false and B is true (3) both A and B are false Let cos( + ) = (1) 4. (4) A = B Let A and B denote the statements A : cos  + cos  + cos  = 0 B : sin  + sin  + sin  = 0 If cos ( – ) + cos ( – ) + cos ( – ) = – 3. (3) A  B =  56 33 3 , then : 2 (2) both A and B are true (4) A is true and B is false  4 5 and let sin( – ) = , where 0  ,   . Then tan 2 = 4 5 13 (2) 19 12 If A = sin2 x + cos4 x, then for all real x : 3 13 (1)  A  1 (2)  A 1 4 16 (3) 20 7 (3) 1  A  2 (4) 25 16 (4) 3 13 A 4 16 5. Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can formed such that Y  X, Z  X and Y  Z is empty, is : (1) 52 (2) 35 (3) 25 (4) 53 6. In a PQR, if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then the angle R is equal to : (1) 5 6 (2)  6 (3)  4 (4) 3 4 7. Let A and B two sets containing 2 elements and 4 elements respectively. The number of subsets of A × B having 3 or more elements is (1) 256 (2) 220 (3) 219 (4) 211 8. The expression 9. tan A cot A can be written as : + 1 − cot A 1 − tan A (1) sinA cosA + 1 (2) secA cosecA + 1 (3) tanA + cotA (4) secA + cosecA If X = {4n – 3n – 1 : n  N} and Y = {9(n – 1) : n  N}, where N is the set of natural numbers, then X  Y is equal to (1) X (2) Y (3) N (4) Y – X Fundamentals of Mathematics 10. Let fk (x) = (1) 1 (sinkx + coskx) where x  R and k  1. Then f4(x) – f6(x) equals k 1 4 (2) 1 12 (3) 1 6 (4) 1 3 11. Let A and B be two sets containing four and two elements respectively. Then the number of subsets of the set A × B, each having at least three elements is: (1) 219 (2) 256 (3) 275 (4) 510 12. If the angles of elevation of the top of a tower from three collinear points A, B and C, on a line leading to the foot of the tower, are 30º, 45º and 60º respectively, then the ratio, AB : BC , is 3 :1 (1) 3: 2 (2) (3) 1 : 3 (4) 2 : 3 13. If 0  x < 2, then the number of real values of x, which satisfy the equation cosx + cos2x + cos3x + cos4x = 0, is (1) 5 (2) 7 (3) 9 (4) 3 14. If 5(tan2 x – cos2x) = 2cos2x + 9, then the value of cos4x is : −3 1 2 (1) (2) (3) 3 9 5 15. 17. 6 7 1 4 (3) 2 9 (4) 4 9 x ( x – 6) + 6 = 0}. Then S : (2) contains exactly four elements. (4) contains exactly one element      1 If sum of all the solutions of the equation 8 cosx.  cos  + x  .cos  − x  −  = 1 in [0, ] is k, then k 6 6     2  (2) 20 9 (3) 2 3 (4) 13 9 PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR. If the angles of elevation of the top of the tower at P, Q and R are respectively 45º, 30º and 30º, then the height of the tower (in m) is : (1) 100 3 19. (2) Let S = {x  R : x  0 and 2| x – 3| + (1) contains exactly two elements. (3) is an empty set. is equal to : 8 (1) 9 18. 7 9 Let a vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P be a point on the ground such that AP = 2AB. If BPC = , then tan is equal to (1) 16. (4) − (2) 50 2 Let S = {x  R : x  0 and 2| x – 3| + (1) contains exactly two elements. (3) is an empty set. (3) 100 (4) 50 x ( x – 6) + 6 = 0}. Then S : (2) contains exactly four elements. (4) contains exactly one element Fundamentals of Mathematics 20. In a class 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible 5 opted Chemistry course. Then the number of student who did not opt for any of the three courses is : (1) 38 21. (2) 42 The value of cos (1) 23. (4) 1 3   The sum of all values of   0,  satisfying sin2 2 + cos4 2 = is : 4 2   (1)  22. (3) 102 1 1024 (2)  22 . cos  23 (2)  2 . .... . cos 1 2 (3)  210 . sin  210 3 8 (4) 5 4 1 512 (4) 1 256 is : (3) If sin4  + 4 cos 4  + 2 = 4 2 sin cos  ; ,   [0, ] , then cos( + ) – cos( – ) is equal to (1) – 2 (2) 0 (3) 2 (4) –1 Fundamentals of Mathematics EXERCISE # 1 Section (A) : A-1. (iv) A-2. {2,3,5,7} A-3. (i) non-empty (ii) empty (iii) non-empty (iv) non-empty A-4. (i) Finite (ii) Infinite (iii) Finite (iv) Finite A-5. 32 A-6. P(A) = {, {}, {{}}, {, {}}} = {, {}, {{}}, A} B-2. {1, 2, 3, 4} 60 C-4. Section (B) : B-1. 6 B-3. {x : 1  x < 2} (3) {x : 1  x  2} (4) {x : 1 < x < 2} B-4. {3, 5, 9} B-5. {3, 4, 10} B-7. {9, 21} (ii) 250 B-8. d = bc 150 C-3. Section (C) : C-1. 300 C-2. (i) 600 C-5. (i) 12 (ii) 6 (iii) Section (D) : (2,1) • D-1. (i) (1,0) (ii) •(0,–3) • (3,–1) • (0,–55) y (0, 8.5) (2,1/2) • (iii) • • (1,–1/2) • (iv) • (–1,16) (0,13) x (2,-11) • (3,–1/2) y (0,1) (–1,0) (v) D-2. 2 (0,–21) (1,–24) D-3.  –32   27 ,0    (vi) 15 Fundamentals of Mathematics Section (E) : E-1. (i) (–3,–2)  (1,3) (ii) {–2}  [1, 8) (iii) (iv) (–,1) – {0} (v) (–3,–2)  (–1,) (vi) (ii)  25   4 ,    (iii) E-2. 2 E-3. (i) 5   – ,– 2    (i)  –4x + 11 x  (– ,2)  y =  –2x + 7 x  [2,3)  4x – 11 x  [3, )  E-4.   3+ 5 3– 5  –2,    1,  2  2    [1,6]  2   – 5 ,0    (2,3) (3,1) (2,5) (5,5) (ii)  –2x – 1 x  (– ,–1)  1 x  [–1,0)  x  [0,2)  2x + 1  5 x  [2,5)   2x – 5 x  [5,  ) (–1,1) (0,1) (–1,7) –2 –1 0 1 2 3 4 5 (1,–1) (4,–7) (iii) (i) – 2 3 (v) 2, 1 7 3 E-6. (i) –2, 1 E-7. (i) (–,–12) (–2,) E-5. (ii) –2 (iii) 1 13 , 3 4 (ii) [3,) (iii) [–1,1]  [2,) (ii) (–13,7) (iv) – 9 7 Fundamentals of Mathematics E-8. (iii) R (iv) (v) (vii) (–,1] [3,) (–7,–2)  (3,4) (vi) (viii) (i) 5 3   4 ,1  (1, )   (–,3) [0,4] (ii) 4 (0,2) (0,2) E-9. – (i) 1 4 (1,0) (2,0) (ii) (1,0) (2,0) (0,–2) (0,2) (0,2) (–1,0) (–1,0) (iii) 1 – 4 (–2,0) – (iv) (2,0) (–2,0) (–1,0) (0,2) (v) (1,0) (–1,0) (–2,0) (2,0) (0,–2) (0,4) (–6,4) (6,4) (–2, 2) (–1, 2) (–7,0) E-10. (i) (–5, 0) (–8,–2) (2, 0) (–3, 0) (4, 0) (3,–2) (–4,–2) (–1, 2) (–7, 2) (–3, 1) (5, 2) (0, 1) (1, 1) (–2, 1) (ii) (–8, 0) (–9, –1) (–6, 0) (–4, 0) (–5, –1) (2, 0) 1 4 (4, 0) (3, –1) (6, 0) (7, –1) (1,0) (2,0) Fundamentals of Mathematics (iii) (–7, 1) (–6, 0) (–3, 1) (3, 1) (–2, 0) (–4, 0) (7, 1) (4, 0) (2, 0) (6, 0) (1, –1) (–1, –1) (5, –2) (–5, –2) 1   2 ,1   3   2 ,0    (0,0) (1,0) (0,–2) (1,–2) (iv) (2,–1) 1   2 ,–3    Section (F) : F-1. F-2. (i) (5, ) (vi) (–,–2) (vii) (5,7]  1 x   0,   3 (i) 1 (ii) 2 (iii) 1 (v) 1 7+ 196 (vi) 0 (ii) 1  2b + 1– a  2  2b + 4a  (ii) –72 (iv) x 1 8   (viii) x  (–,–4)   ,  2 7 (iii) R (iv) (v) (ix) x  [–1,2) F-3. (i) 1 4 + a 3b F-5 (i) 3 (ii) 5 (iii) ±3 (iv) 3 F-6. (i) 0 (ii) 10, 10–4 (iii) 10, 10–9/2 (iv) 10, 3/2 F-7 1  (i)  ,2  3  F-8.  1  (i)  3 / 2 ,   3   F-9 (i) R 1  (ii)  ,4  2  (–,2)  (8, ) (v) (iv) (–, –2]  [2, ) (iii) [2,) (ii) [log3 9/10,2) (iii) (5, ) (iv) (2,) (ii) (0, log 3 3 ) 2 Section (G) : G-1. 1  (i) x   ,1 2  (ii) 2 (iii) x  G-2 (i) x  (ii) {0} (iii) {0,2/5, –1/5} G-3. (i) no solution (ii) –1, – 1 2 G-4. (i) x  (–) G-5 (i) x  R 1 1  n + ,n +  3 2 n  (iii) 0,1  −1  (ii) x   ,1  2  (ii)x  (–2, 4) (iv) x  [1,2) (iii) x  1  n,n + 2   n  (iii) (1, 2)  (2, 3)  (3, 4) G-6 (i) 2 (ii) 2 1 Fundamentals of Mathematics Section (H) :  − , 3 3 H-1. (i) H-2. (i) x = H-3. (i) x  H-4 (ii) 0,   3 2 4 8 , , , , , , 4 2 4 9 9 9 (iii)  6 n  or x = m± where m,nI 4 6  (ii) x = (2n+1) where n   10 (iii) x = m, m 5    6 + 2n, 6 + 2n   n    (iii) x  [– +2n, 2n+ ] nI 2 6 (i) 11 (ii) 6  −  − +2n, + 2n]  [ +2n, + 2n] 2 2 3 3        (v) x   – 2 + 2n,– 3 + 2n    3 + 2n, 2 + 2n     n  (ii) x  [ nI (iv) x R H-5. y= 1 and x = nwhere n   2 EXERCISE # 2 1. (A) 2. (D) 3. (A) 4. (D) 5. (C) 6. (B) 7. (A) 8. (B) 9. (C) 10. (C) 11. (C) 12. (B) 13.* (AD) 14. (C) 15. (A) 16. (C) 17. (B) 18. (A) 19. (D) 20. (C) 21*. (B) 22. (C) 23. (A) 24. (A) 25. (A) 26. (C) 27. (D) 28. (C) 29. (B) 30. (B) 31. (A) 32. (D) 33. (C) 34. (B) 35. (D) 36. (D) 37. (D) 38. (A) 39. (C) 40.* (CD) 41. * (BC) 42. (C) 43. (C) 44. * (BC) 45. (B) 46. (A) 47. (D) 48. (C) 49. (A) 50. (B) EXERCISE # 3 PART-I (CD) 3. 2 (ACD) 9. (4) 10.* (C) (BC)/ BONUS 1.* (AB) 8.* 15. 2*. 16. (n = 7) 5. 3 6. (C) 7. (D) (ABC) 11. (D) 12. 8 13. (C) 14. (C) 17. (8) 4. PART-II 1. (2) 2. (2) 3. (1) 4. (1) 5. (2) 6. (2) 7. (3) 8. (2) 9. (2) 10. (2) 11. (1) 12. (1) 13. (2) 14. (4) 15. (3) 16. (1) 17. (4) 18. (3) 19. (1) 20. (1) 21. (2) 22. (3) 23. (1)