Inequalities Practice Exercise

BASIC APPLICATION
USEFUL CONCEPTS PART - I

1. Solve he following inequations :

( x  1)( x  4)
(A) ( x  3)(x  1)(x  2)(x  9)  0 (B) 0 (C) x 2  16  0
( x  3)
(D) ( x  1)(2 x  5)  0 (E) ( x  1)(3x  1)  0 (F) (2 x  3)(x  1)(x  5)  0
(G) x( x  3)(x  1)  0 (H) x( x  1)(x  3)  0 (I) x(2 x  5)(3x  2)(x  9)  0
x( x  1)( x  2)
(J) 0 (K) x 2  17  0 (L) x( x 2  5 x  6)  0
x5
(M) ( x 2  5x  6)( x 2  3x  2)  0 (N) x 3  5x 2  4x  0 (O) x 3  7 x 2  14x  8  0
( x  1)( x  2)(5  x) (2 x  3)(7 x  2)(5  3x)
(P) ( x  2)(3  x)  0 (Q) 0 (R) 0
(2 x  5) (3  x)
( x  1) x( x 2  4) x 2  5x  6 x 1
(S) 0 (T) 0 (U) 2
x7 x  3x  2
2 x4
2. Solve the following :
(A) 3x 2  7 x  4  0 (B) 3x 2  7 x  6  0 (C) x 2  3x  5  0
3x  2
(D) x 2  4 x  3  0 (E) x 3  6 x 2  11x  6  0 (F)  2, x  1
x 1
x 2  x  12 x 2  4x  7 2 x 2  6 x  11
(G) 2 (H)  (I)  1
x 2  3x  3 x 2  7x  12 3 x3
x 1 x3 x 1
(J)  (K)  1 x (L) 3x 2  11x  2  0
4x  5 4x  3 x3
3. Solve the following :
x 2  3x  4 1
(A) 2 x 2  x  15  0 (B)  x 2  3x  2  0 (C) 
x5 3
x 2  2x  5 1 x 2  2 x  24
(D)  (E) 4 (F) x 4  2x 2  63  0
3x 2  2 x  5 2 x 2  3x  4
INEQUATIONS WITH INTEGRAL POWERS ON FACTORS
4. Solve the following :
(A) ( x  1)( x  2)( x  3) 2  0 (B) (x  1)(x  3)(x  4)2  0
( x  1)( x  2) 2
(C) 0 (D) ( x  1) 3 ( x  5) 4 ( x  1)13 ( x  2)  0
( x  4)
( x  1) 5 ( x  1)( x  4) 2 ( x  3) 4 ( x  1)( x  8)( x  6) 2 ( x  5) 4
(E) 0 (F) 0
( x  8)(2 x  3)( x  6) 2 ( x  3) 2 ( x  1)
( x  1) 2 ( x  1) 3
(G) ( x  1)( x  2) 2 ( x  3)  0 (H) 0
x 4 ( x  2)
5. Solve the following system of inequalities :
(A) x 2  9  0 & x 2  3x  2  0 (B) 2x  7  5  x,11 5x  1
x
(C) 2x 1  3 & 3x  2  5 & 5x  4  10 (D) x 2  3x  2  0 & 0
x 1
x 1 6x 1
(E) x 2  9  0 & x 2  6 x  8  0 (F)  , 
2x  1 4 4x  1 2
x( x  1)(x  3) (5  x)(x  2)
(G)  0& 0 (H) x( x  4)  0 & x 2  5 x  6  0
( x  4) ( x  8)
(I) 6 x 2  7 x  3  0 & 2 x 2  7 x  4  0
x 5
6. Find the smallest integral x satisfying the inequality 0
x  5x  14
2

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 x 2 (3x  4) 2 ( x  2) 2
7. Find the number of positive integral solutions of  0.
( x  5) 2 (2 x  7) 6
( x  2)(x  4)(x  7) x 2  3x  1
8. (i) 1 (ii) ( x 2  x  1)( x 2  x  7)  5 (iii)  3  3
( x  2)(x  4)( x  7) x2  x 1
(iv) Find all integers satisfying : (5 x  1)  ( x  1) 2  (7 x  3)
1 1
9. Prove that the solution of simultaneous system 7a2  12a  8  0  4a2  1  0 will be a  ( , )
2 2
 7  45
10. Prove that the solution of simultaneous system a2  7a  1  0  a2  8a  4  0 will be a  ( ,4  2 3 )
2
3a 16
11. Prove that the solution of simultaneous system 9a 2  16a(a  1)  0   1  2a 2  3a  1  0 will be a  [  ,1)
2(a  1) 7
 3  x  3 x 
12. Solve for x : x  x  2
 x  1  x 1 
USEFUL CONCEPTS PART – II
BASIC APPLICATION OF ABSOLUTE VALUE (MOD)
1. Solve for x :
(A) | x | = 5 (B) | x | = - 2 (C) | x – 1 | = 3 (D) | x + 2 | = 5
(E) | 7 + x | = 2 (F) | x – 3 | = 3 (G) | 2x – 5 | = 1 (H) | 3x – 5 | = 3
(I) | 2x + 5 | = 3 (J) | 2 – x | = 5 (K) | 5x2 – 3 | = 2 (L) |x – 1| + |x – 3| =2
x4
(M) | x | + | x + 5 | = 5 (N) |x – 1| + | x – 4| = 2 (O) 3 (P) | x 2  4x |  5
x2
2. Solve for x :
(A) | x | > 2 (B) | x – 1| > 3 (C) | x – 2| < 1 (D) | x + 1| ≥ 2
x3
(E) | x – 1| < 5 (F) | 2x – 3 | > 7 (G) | 3x + 5| < 2 (H) 1
x5
x4
(I) 1 (J) | x 2  4x |  5
x2
3. Solve the following equations and inequalities :
(A) | x + 1 | - 2 = 2 (B) | 3x – 4 | = 1/2 (C) | x + 2 | = 2 (3 – x)
(D) | x – 3 | > - 1 (E) | x | = - 3x – 5 (F) | 3x – 2.5 | ≥ 2
(G) | x – 2 | ≤ | x + 4 | (H) x2 + | x – 1| = 1 (I) | x2 – x – 6| = x + 2
(J) | 4 – 3x | ≤ ½ (K) x2 – 2 | x | - 3 = 0 (L) | 2x – x2 – 3| = 1
x 2  | x | 12 x 1 2x  5
(M)  2x (N) | x  1 |  (O)  1
x3 2 | x 3|
(P) | x 2  2 x  3 |  3x  3 (Q) | x | x 3  0 (R) ( x 2  5 x  6) 2  5 | x 2  5 x  6 | 6  0
| x  2 | x
(S) | x 2  4 x  3 | 2 x  5  0 (T) | x  1 |  | x  2 |  | x  3 |  2 (U) 2
x
(V) | x 2  1 |  x  1  0 (W) | x 2  6 x  8 |  4  x (X) x 2  | 3x  2 |  x  0
1
(Y) x 2  | 5x  3 |  x  2 (Z) ( x  1)(| x | 1)  
2
4. Solve system of equations.
(A) | x | 2 | y | 3 & 5 y  7 x  2 (B) y  2 | x | 3  0 & | y |  x  3  0
5. Solve of x :
4
(A)  | x 1| (B) | 7x  12 |  | 7x  11|  1 (C) (| x | 5)(| x | 7)  0
| x  1 | 2
x 2  3x  1
(D) || x  1 | 2 |  1 (E) 3 (F) | x  1  x 2 |  | x 2  3x  4 |
x2  x 1
6. Find the least integral value of x which satisfies the equation | x  3 | 2 | x  1 |  4
7. Solve the inequality | x 2  3x |  x 2  2  0
x 2  8 x  12 x 2  8 x  12
8. Solve the equation 
x 2  10 x  21 x 2  10 x  21

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9. Solve for x :
x 2  5x  4
(A) | x 2  9 |  | x 2  4 |  5 (B) 1
x2  4
(C) (| x  1 | 3 |)(| x  2 | 5)  0 (D) | x 3  1 |  x 2  x  1
10. Solve the equation
x x2
(A) | x | 4  x || 2 x  4 (B) | x| 
x 1 | x 1|
| x| 1
11. Solve the inequation 1  
1 | x | 2
12. Solve the inequality | x  1 |  | x |  | 2 x  3 |  2 x  4
13. Solve for x ; 2| x  2|  | 2 x1  1 |  2 x1  1
| x 2  2 x |  y  1
14. Solve the system of equations  
 x 2  | y |  1 
15. For a < 0, determine all roots of x 2  2a | x  a | 3a 2  0. Also calculate the roots if a > 7.
16. Find the number of negative integral solutions of x 2 2 x1  2| x3|2  x 2 2| x3|4  2 x1
9(2 | x | 2)
17. Solve for x : ( x 2  2 | x |)(2 | x | 2)  2 0
x 2| x|
USEFUL CONCEPTS PART - III
1. Solve for x :
(i) [x] = 3.9 (ii) [x] < 3.9 (iii) [x] > 3.9 (iv) [x] > 5 (v) 2.1 < [x] < 5
(vi) [x] ≤ 5 (vii) -9 ≤ [x] < -5.3 (viii) [x] ≥ -5.3 (ix)[x] ≤ -5.3 (x) {x} = 0
2. Solve for x :
(A) [ x ]  3 (B) [x2 ]  5 (C) [ x]  [x]  0 (D) x  [ x]  1 / 2
(E) [ x]  3[ x]  2  0
2
(F) [ x]  5[ x]  6  0
2
(G) [ x]  3[ x]  2  0
2
(H) [3x]  1
(I) [3x]  [3x]  0 (J) [ x ]  9
2
(K) [| x |]  3 (L) | [ x] |  2
(M) 4[ x]  x  {x} (N) [ x]  n(n  l ) (O) [ x]  n(n  l ) (P) 2.3  [ x]  3
(Q) 2  [ x]  5.3
3. If [ x]2  [ x  2] , where [x] = the greatest integer less than or equal to x, then x must be such that
(A) x  2,  1 (B) x  [2, 3) (C) x  [1, 0) (D)none of these

4. Let y = 2[x] + 3 = 3[x – 2] + 5 then[x +y] equals

(A) 10 (B) 15 (C) 12 (D) None of these
5[ x]  4
5. Find the complete set of values of x satisfying (i)  3 (ii) [ x]2  3[ x]  2  0
11[ x]  7
6. If [x] denotes the integral part of x for real x, then the value of
1 1 1  1 1  1 3   1 199 
 4    4  200    4  100    4  200  .........  4  200  is
         
(A) 44 (B) 46 (C){51} (D) 50
1
7. Solution set of 5 {x} = x + [x] & [x] – {x} = will contain
2
(A) no value (B) 1 value (C) 2 values (D) None of these
100
1 n 
8. Let f(n) =  
 2 100  where [x]s denotes the integral part of x. Then the value of
 n 1
f(n) is 
(A) 50 (B) 51 (C) 1 (D) None of these
9. The solutions of the equation 2x – 2 [x] = 1
1 1 1
(A) x  n  , n  N (B) x  n  , n  N (C) x  n  , n  l (D) n  x  n 1, n  l
2 2 2

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  x   x   x  31
10. If 0 < x < 1000 and          x, where [x] is the greatest integer less than or equal to x, the number of possible
 2   3   5  30
values of x is
(A) 34 (B) 32 (C) 33 (D) None of these
11. Let R = the set of real numbers, Z  the set of integers, N = the set of natural numbers. If S be the solution set of the
equation ( x) 2  [ x]2  ( x  1) 2  [ x  1]2 , where (x) = the least integer greater than or equal to x and [x] = the greatest
integer less than or equal to x, then
(A) S = R (B) S = R - Z (C) S = R – N (D) None of these
12. The number of solutions of |[x] – 2x| = 4, where [x] is the greatest integer ≤ x, is
(A) 2 (B) 4 (C) 1 (D) infinite
13.   
4  [ x]2  [ x]2  1  7 has {[ ]→ integral part}
(A) no solution (B) two solutions (C) 4 solutions (D) None of these
14. No. of pairs of (x, y) satisfying [x]2 + {y}2 = 3 will ([ ] & { } are integral & fractional parts)
(A) 0 (B) 1 (C) finitely many (D) infinitely many
15. If [x] = the greatest integer less than or equal to x, and (x) = the least integer greater than or equal to x, and [x] 2 + (x)2 > 25
then x belongs to
(A) [3, 4] (B) (-∞, 4] (C) (,  4]U[4,  ) (D) None of these
16. The solution set of ( x) 2  ( x  1) 2  25, where (x) is the nearest integer than or equal to x, is
(A) (2, 4) (B) (5,  4]U(2, 3] (C) (4,  3]U(3, 4] (D) None of these
17. Number of solutions of [x] + [x] = 2x in x  [1, 1] will be
(A) zero (B) one (C) two (D) None of these
18. [x + [x + [x]]] ≤ 3 then value of x lie in the interval
(A) (-∞, 0) (B) (-∞, 1) (C) (-∞, 2) (D) None of these
19. Solve for (x, y) (i) [| x |]  [| y |]  1 (ii) [ x] 2  [ y ] 2  1 (iii) [| x |] [| y |]  2
20. Solve the equation | 2x  1 |  3[ x]  2{x}
21. Prove that if x, y  [0, 1) then prove that [2x] + [2y] ≥ [x] + [y] + [x + y]
22. Prove that [x + y] ≤ [x] + [y] + 1
23. Let [a] denote the greatest integer less than or equal to a. Given that the quadratic equation x 2  [a 2  5a  b  4]x  b  0
has roots – 5 and 1, find all the values of a.
24. Prove that [ x]  [5x]  [10x]  [20x]  36k  35(k  l ) cannot have any real solutions.
25. For every positive integer n, prove that 4n  1  n  n  1  4n  2 Hence or otherwise, prove that
[ n  n  1]  [ 4n  1] . Where [x] denotes the greatest integer not exceeding x. s
 1  2  1
26. Prove that (i) [ x]   x     x    [3x] (ii) [ x]   x    [2 x]
 3   3  2
[ x ]  3 [ x]  1
27. Solve   1; (where [ ] denotes integral part)
[ x]  2 [ x]  3
28. Solve the equation x 2  4 x  [ x]  3  0
 x   x  1
29. Prove for every real + ve x,  2    2   [ x] . Hence or otherwise prove that
   
 n  1  n  2   n  4   n  8 
 2    4    8    16   ............. till ∞.  n  n  N
       
30. Solve the following equations for x.

(A) x 2  3  {x} (B) x 2  [ x]  4  0

31. Solve for x 3x  x[ x]  x 2  3[ x]  0

1 1 1
32. Prove that inequality 2 n  2  1   .........  2 n  1 n  N , Hence or otherwise calculate
2 3 n

 1 1 1 
1    ........  
 2 3 3600 

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33. Consider two integers x, y such that 1 < x, y < 9, A, B, C are A.M., G.M., H.M. of x & y. A two digit number N is formed
by using digits x & y. A number M is formed by interchanging the position of digits x, y in N.
N N N N
Let    16,    20,   A  2    B then show that N – M + 10 = ABC
A B A B
1 1 1
34. Find all x satisfying   {x} 
[ x] [ 2 x] 3

USEFUL CONCEPTS – IV
System of Equations and Inequalities

 yx1  0 x  2 y  6  0  uv  2 | x  1 |  | y  2 | 1
1.  2.  3.  4.  .
| y | x  1  0 | x  3 |  y  0 | 3u  v | 1  y  3 | x  1 |

 1 1 1  1
 2 2
 2 2  3x  2 y  3  x  y  x  1
x  y  41 x  y  41 
5.  6.  7.  1 1 1
8.  x
 yx  1
  xy  9
  2     2
 9 x 4y 2 4  x  y

 x(x  y)  9  x 2  xy  28 x 2  y 2  25  2xy x 2  y 2  68
9.  10.  2 11.  12. 
y(x  y)  16 y  xy  12
 
 y(x  y)  10 
 xy  8

 xy  1
13.  3 3 14. x4  y4  82 ,xy=3
x  y  7

System of Irrational Equations

 x y 5 2x  1 y2 x 5  y
x  y  x  y  20       x 2  y  5
15.  2 2 16.  y  x  2 17.  y  2  2 x  1  2 18.  y x 2 19.  2 .
 x  y  136    2  y  x  7
 xy  5  xy  2 2
 x  y  15

USEFUL CONCEPTS – V
Exponential equation
1. solve :  3  2   3  2
x x
 10 2. Solve :  3 2 2   3 2 2  6
x x
.

2 2 2
 2 x 1  2 x 1
Solve (5  2 6 ) ( x 3)  (5  2 6 ) ( x 3)  10
2 2
3. Solve : (2  3 ) x  (2  3 ) x  . 4.
2 3
5. 64.9 x  84.12 x  27.16 x  0 6. 6.91/ x  13.61/ x  6.41/ x  0
x 1 x 2  2 x 11
0 . 2 x  0 .5
9
(0.04) x 5  9  5
7.  8.      
5 25  3  25   3

9. 9 |3 x 1|  38 x  2 . 10. 4 x x 2 5
 12.2 x 1 x 2 5
8  0
x 1 / 2 x 1 / 2 2 x 1 3 3 / x x  0.5
11. 4 3
x
3 2 . 12. 64 1/ x
2  12  0 . 13. 9  2
x
 2 x 3.5  3 2 x 1
2
1 2 2
1 2
2 2
1 2
3 x 1
14. 2x  3x  3x  2x . 15. 9x  36.3 x 3  0 x
16. 5 . 8  500 .
x

x 5 x 17
6 6 1
(6 x 1 ) 4
2 2 2 2
17. 32 x 7  0.25.128 x 3 18. 2x .3 x  5
19. 2
sin x
 4.2 cos x
6.
6
Solve 12 5 x  2  ( 24 3 ) 3 x
2
20.

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21. 5 x 1 10 x. 2 x.5 x 1 22. 7log x  5log x 1  3.5log x 1 13.7log x 1

23. 22 log 4 x 1  7log 4x  7log 4 x 1  3.4log 4 x 24. 72x

2
5 x 6
  2 3 log 2 49
.

25. 3log tan x  2.3 log cot x + 1 = 1. 26. 32 x 1  3 x  2  1  6.3 x  32( x 1) .

64  x 23x  3 12  0
26
27. x 28. 51log 4 x  5log1 / 4 x1  .
5
Exponential Inequalities
2 x 3
1
29. 2 x  2 x1  3  0 30. 4
 x0.5
 7.2  x  4  0 31. 2 2 x 1  21.  20
2
| x  2|
 1  2 | x|
32. (0.3) 2  4  6 ..... 2 x  (0.3) 72 , x  N 33. 9 x  3 x2  3 x  9 34.   9
 3
1 2
x 
1 2 x 1 2 X 2
 0.12 X 3 .
2
35.    . 36. 9 x  3x  2  9  3x . 37. 0.1
4X

 3 27
38. 9.4 1/ X  5.6 1/ X  4.9 1/ X . 39. 4  3  2
X X 1
40. 2  2
X |X |
2 2.
2 3 x
2 x 1  1 1
41.  2. 42. 343 x  35   6  0.
2 x 1  1  3
43. 2 x  2  2 x 1  2 x 1  2 x 2  9. 44. 4 x 1  16 x  2 log 4 8

45. x 2 2 2 x  9( x  2)2 x  8 x 2  ( x  2)2 2 x  9 x 2 2 x  8 x  16.

USEFUL CONCEPTS – VI
IRRATIONAL EQUATIONS AND INEQUALITIES
IRRATIONAL EQUATIONS

1. (X2  1) 2X  1  0 2. (X 2  4) X  1  0 3. (9  X 2 ) 2  X  0
3
4 X 3 4
4. 3 X  23 X2  3 5.  2 6. 2  X  2
3
X 2 5 2X 3
3 x4
7. 2 X1 5 8.  X8 9. LOG2 X  3 LOG2 X  2
X1 1 X2
3X 2X X1 X1 3
10. 3 4 11.   12. X  X  1  5
2X 3X X1 X1 2
13. 21  2X  7  X 14. 4 X  6  X  1 15. 4  2X  X 2  X  2
13  X 2 2  19  2 X
16. 1 17. 1 18. X2  4X  6  2X2  8X  12
X1 X
19. 2X  4  X  5  1 20. 2X  5  8  X  1 21. 15  X  3  X  6
22. X  5  2X  7  2 X 23. X  1  4X  13  3X  12 24. X  5  X  3  2
X X1 X 1X 5
25. 2 3 26.  
X1 X 1X X 2

IRRATIONAL INEQUALITIES
X2 X3
27. (X  1) X 2  X  2  0 28.  1 29. 0
1  2X X2
2 X 2  15X  17
30. 3X  10  6  X 31. 0 32. X 2  X  1
10  X
33. 2 X  1  X 34. X  18  2  X 35. X  24  5X
36. 2X  1  X  2 37. (X  6)(1  X)  3  2X 38. X  1  7  X

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 1  1  4X 2
39. X  2  X 40. 4  X  X 2  2x 41. 3
X
X2
|X2||X|
42. X 2  4x  X  3 43. 0 44.  1   3 X
4  X3 3
X5
45. 0 46. * 2 x 1  x 47. * 2 x 2  3x  5  x  1
log 2 (x  4)  1

Objective Exercise
1. The equation | x 1|  a  4 can have real solutions for x if ‘a’ belongs to the interval
(A) ( ,4] (B) ( ,4] (C) (4, ) (D) ( ,0)
2. The value of [sin x]  [1 sin x]  [2  sin x] in x (,3 / 2) can be ( [.] is the greatest integer function) can
be
(A) 0 (B) 1 (C) 2 (D) 3
3. The equation (0.4)x-1 = (6.25)6x-5 has
(A) no solution (B) one solution (C) two solutions (D) more than two solutions
4. The number of real roots of the equation x2 + x + 3 + 2 sin x = 0, x  [-,] is
(A) 2 (B) 3 (C) 4 (D) none of these
5. The number of roots of the equation x3  x2  2x  sin x  0 in [-2, 2] is
(A) 1 (B) 2 (C) 3 (D) none of these

6. The number of real solutions of the equation 2 x / 2  ( 2 1)x  (5  2 2 )x / 2 is

(A) 1 (B) 2 (C) 4 (D) infinite

7. The solution set of the in equation 5 x  (2 3 ) 2 x  13x is

(A) [2, ) (B) {2} (C) (-, 2] (D) [0, 2]

8. If x2 – ax + 1 – 2a2 > 0 for all x  R, then
(A) a  (2 / 3,2 / 3) (B) a  [2 / 3,2 / 3] (C) a  (2 / 3,1) (D) a  (0,2 / 3)

9. Let 2 sin2 x + 3 sin x – 2 > 0 and x2 – x – 2 < 0 (x is measured in radians). Then, x lies in the interval

(A)  / 6,5 / 6 (B)  1,5 / 6 (C) (-1,2) (D)  / 6,2

10. The set of all values of x, measured in radians, satisfying the two inequalities

2 cos2 x  2  3cos x and x 2  4x  12 is

(A) (2, / 3)  ( / 3,5 / 3) (B) ( / 3,  / 2)  (3 / 2,2)

(C) (-2,6) (D) (-2, 1/2)

11. The set of values of ‘a’ for which x2  2ax  a2  6a  0 in [1,2], is

(A) (4  15 ,4  15 ) (B) (5  21,5  21) (C) (5  21,4  15 ) (D) (4  15 ,5  21)

12. The sum of all the real roots of the equation |x-2|2 + |x-2| - 2 = 0 is
(A) 4 (B) 1 (C) 3 (D) 2
13. Let α and β be the roots of equation sin x  3 sin | x | 2  0, where x [2 , 2 ] , then sum of roots of
2

equation is
(A) 3 (B) – 3 (C) 0 (D) 2

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14. Let a, b, c be three distinct positive real numbers, then the number of real roots of ax2 + 2b|x| + c
= 0 is
(A) 0 (B) 1 (C) 2 (D) 4
15. The number of the real solutions of the equation x2 – 3 |x| + 2 = 0 is:

(A) 2 (B) 4 (C) 1 (D) 3

16. The real number x when added to its inverse gives the minimum value of the sum at x equals to:

(A) 2 (B) 1 (C) – 1 (D) - 2

2
7 x  7
17. The number of real roots of 32 x = 9 is equal to:

(A) zero (B) 2 (C) 1 (D) 4

18. If a, b, and c are distinct positive numbers, then the expression (b+c–a) (c+a–b) (a+b–c)–abc is

(A) positive (B) negative (C) non-positive (D) non-negative.

19. The number of real solutions of x 2  4 x  3  x 2  9  4 x 2 14 x  6 is

(A) one (B) two (C) three (D) None of these
2 15 2
20. The roots of the equation (a  b ) x  (a  b ) x 15  2a , where a 2  b  1 are

(A)  2,  3 (B)  4 ,  14 (C)  3,  5 (D)  6,  20

21. Let a, b, c be positive numbers. The following system of equations in x, y and z

x2 y2 z2 x2 y2 z2 2
y2 z2
2
 2
 2
 1, 2
 2
 2
1 and  x2  2
 1 has
a b c a a c a b c2

(A) no solution (B) unique solution

(C) infinitely many solutions (D) finitely many solutions
22. Let M and m be respectively, the maximum and the minimum of n arbitrary real numbers
x1 , x2 .........xn . Furthur let M’ and m’ denote the maximum and minimum, respectively of the

following numbers.
x1  x 2 x1  x 2  x 3 x  x 2  ....  x n
x1 , , ....... 1 Then
2 3 n

(A) m  m'  M  M' (B) m  m'  M'  M (C) m'  m  M'  M (D) M'  m  M  M'
x 1 x 1
23. If x < 0, y < 0, x + y +  and (x  y )   , then x and y are
y 2 y 2
1 1 1 1
(A)  , (B)  , (C) 1,1 (D) none of these
4 4 2 2
24. The natural number x satisfying both inequations x + 3 < 4 + 2x and 5x – 3 < 4x – 1 is
(A) 2 (B) 3 (C) 4 (D) 1
1 1
x x
2 x 1
25. If 4  2x
3 2 3 2 then =?
1 3 5
(A) (B) (C) (D) 1
2 2 2
 x   2x  9x
26. If [sin x]     , then the number of solutions in the interval (30, 40) is
 2   5  10
(A) 0 (B) 1 (C) 2 (D) infinite

8/8
 x 1 ( x  1) 2
27. The equation  | x  1 | is valid for x belonging to
x | x|

(A) {x : x  0} (B) {x : x > 0}{-1} (C) {-1,1} (D) {x : x 1}

x x2
28. The equation  | x | has
x 1 | x 1 |

(A) exactly one solution (B) exactly two solutions

(C) at most two solutions (D) infinite number of solutions

For all ‘x’, x  2ax  (10  3a)  0 , then the interval in which ‘a’ lies is
2
29.

(A) a  5 (B)  5  a  2 (C) a  5 (D) 2  a  5

30. Solution of the equation (x 10)  (x  2)  6 are

(A) 0 (B) 6 (C) 4 (D) None of these

31. The roots of the equation (3  x)4  (2  x)4  (5  2x)4 are

(A) all real (B) all imaginary (C) two real & two imaginary (D) None of these

32. If x and y are real, then the equation x 2  4x cos( xy)  4  0

(A) is satisfied by only one value of x (B) is satisfied by two values of x

(C) has infinite ordered pairs (x, y) as solutions (D) All above

The set of all real numbers x for which x  | x  2 |  x  0 is

2
33.

(A) (,  2)  (2, ) (B) (  ,  2 )  ( 2 ,  )

(C) (,  1)  (1, ) (D) ( 2 ,  )

Answers–Key
(USEFUL CONCEPTS-I)

1. (A) x  (,2]U[1, 3]U[9, ) (B) x  (,4) U (1, 3) (C) x  (,  4) U(4, )

5 1 3
(D) x  (1, ) (E) x  (,  1) U ( ,) (F) x  (1, ) U (5, )
2 3 2

2 5
(G) x  (,3) U(0, 1) (H) x [1, 0] U[3, ) (I) x  ( , 0) U ( , 9)
3 2

(J) x  (,5) U(1, 0) U(2, ) (K) x  ( 17 , 17 ) (L) x  (, 0) U(2, 3)

(M) x  (,  1) U(1, 2) U(6, ) (N) x [4,  1]U[0, ) (O) x  (, 1) U(2, 4)
5 2 3 5
(P) x  (2, 3) (Q) x  (, ) U (1, 2) U (5, ) (R) x  (, ) U ( , ) U (3, )
2 7 2 3

(S) x  (,7) U[2, 0]U[1, 2] (T) x  (,2) U(1, 2]U[3, ) (U) x  (9,  4)

9/9
  4 2
2. (A) x  1,  (B) x  ( ,3) (C) x  (, ) or x  R
 3 3

(D) x  (, 1]U[3, ) (E) x  (, 1) U(2, 3) (F) x  (0, 1)

(G) x  (,  1) U(6, ) (H) x  (3, 1) U(3, 4) (I) x  (,  8) U(3, 1)

5 3 11  97 11  97
(J) x  ( , ) (K) x  (4,  3) U(1, ) (L) x  ( , )
4 4 6 6
 5  7
3. (A) x  (,  3] U  ,   (B) x  (1, 2) (C) x  (,  5) U (1, )
2  3

5  2 
(D) x  (5,  1) U ( , 3) (E) x    , U [ 4, ) (F) x [3, 3]
3  3 

4. (A) x  (, 1) U(2, 3) U(3, ) (B) x  (1, 3) (C) x  (4, 1)

3
(D) x  (1, 1) U(2, ) (E) x  (8,  4) U (4,  1) U (1, )
2

(F) x  (,  1) U(1, 3) U(3, 5) U(5, 6) U(6, 8) (G) x  [1, 3]

(H) x [1, 0) U(0, 2)

6. x=-6 7. One
8.. (i) (,  7) U (4, - 2) (ii) (2, - 1) U (2, 3) (iii) (,  2) U (1, ) (iv) x = 3

12. x R  {1, 1}
[USEFUL CONCEPTS – II]

1. (A)  5 (B)  (C) 4, -2 (D) 3, -7 (E) -5, -9

2 8 1
(F) 0, 6 (G) 2, 3 (H) , (I) -1, -4 (J) -3, 7 (K)  1,  (L) [1, 3] (M) [-5, 0]
3 3 5
(N)  (O)  1,
5
(P) -1, 5
2

2. (A) (,  2) U(2, ) (B) (,  2) U(4, ) (C) (1, 3)

7
(D) (,  3]U[1, ) (E) (-4, 6) (F) (,  2) U(5, ) (G) ( ,  1)
3
(H) (4, 5) U (5, ) (I) (,  3] (J) (1, 5)
3 7 4
3. (A) -5, 3 (B) , (C) (D) (, )
2 6 3
5 1 3
(E)  (F) (, ] U [ , ) (G) [1, ) (H) 0, 1
2 6 2
7 3
(I) -2, 2, 4 (J)  , (k )  3 (L)  (M) (, 3)
6 2
1
(N) (, ] U[3, ) (O) (2, 3) U (3, ) (P) (2, 5) (Q) 0, -1
3
5  13
(R) 1, 4, (S)  4,  1  3 (T) 2 (U) (, 0) U(1, )
2
(V) -1 (W) [1, 3]U{4} (X) (,  2  2 ] U[1,  3 , )
1 1
(Y) ( 5, 3  2 2 ) (Z) ,1 
2 2
11 23
4. (A) (1,  1), ( , ) (B) (2, 1), (0, -3), (-6, 9)
19 19

10/10
 11
5. (A) 5 ,2  5 (B) ( , ] (C) [7,  5]U[5, 7] (D) 
7
3 2 1
(E) (,  2) U(1, ) (F) (, ] 6. -1 7. (, - ] U[ , )
2 3 2
8 5
8. [2, 3) U[6, 7) 9. (A) [3,  2]U[2, 3] (B) [0, ] U[ , ) (C) (7,  2) U (3, 4) (D) (0, 2)
5 2
10. (A) 0 (B) (1, ) U{0} 11. [-1, 1]
3 1 5 1 5 
12. (  ,  ) 13. [1, )U{3} 14. (0, 1), (1, 0),  , 
2  2 2 

15. For a < 0, a (1  2 ), a( 6  1) & for a > 7, a( 2  1), a(1 6 ) 16. None
17. x  (1, 0) U(0, 1) U(3,  2) U(2, 3)
[USEFUL CONCEPTS - III]
1. (i) x  (ii) x < 4 (iii) x ≥ 4 (iv) x ≥ 6
(v) 3 ≤ x < 5 (vi) x < 6 (vii) -9 ≤ x < -5 (viii) x ≥ -5
(xi) x < -5 (x) xl

2. (A) 9 ≤ x < 16 (B) x  ( 6 , 5 ] U[ 5 , 6 ) (C) x l (D) n + 1/2

(E) 1 ≤ x < 3 (F) (, 2) U[4, ) (G) [1, 3) (H) -1/3 ≤ x < 0
(I) 0 (J) x  3 U x , 4 (K) (4,  3]U[3, 4)
(L) [2,  1) U[2, 3) (M) 0 (N) x ≥ n (O) x < n
(P) [-2, 4) (Q) [-1, 6)
3. (d) 4. (b) 5. (i)  (ii) [1, 3) 6. (d)
7. (b) 8. (b) 9. (a) (b) (c)
10. (c) 11. (b) 12. (b)
13. (a) 14. (a) 15. (c)
16. (b) 17. (d) 18. (c)
53 5 53 5
20. ½ 23. (1,
2
] U[
2
,6) 27. (, 0) U[5, )

1 21
28.  30. (a) (b)  2 , 6 32. x < 3 ≠ 1
2
19 29 97
33. 118 34. , ,
6 12 24

USEFUL CONCEPT:-I V
1. (0, 1) 2. {(0,-1)} 3.  1 / 4, 7 / 4, 3 / 4,5 / 4 
4. C,4  C in C  1, 2; C, C  2 if C  0,1 5.  5,  4, 4,5
 8 12 
6. 4,5, 5,4 7.  ,  8. 1,3 / 2, (2 / 3
 13 5 
  9  16   9 16 
9.  , ,  ,  10.  7,3, (7,3) 11. {( 3,  2), (3,2)}
 5 5   5 5 
12. {(-2, -4),(-4, -2),(2,4), (4,2)} 13. {(-1, -2), (2, 1)} 14. {(-3, -1), (-1, -3), (1,3), (3, 1)}
 5 1 
15. {(6,10), (10, 6)} 16. {(1,4),(4,1)} 17.  , 
 3 3 

 15 4 15    4 15 15 
18.  ; , ,  19. {(-6, -1), (-3, 2), (9, -4), (2,3)}
   
17 
 17 17   17 

11/11
 USEFUL CONCEPT:- V

1. x =  2 2. x =  2 3. x = 0, 1
4. x =  2,  2 5. {1, 2} 6. {-1, 1} 7. x = - 1
9 
8. {-7/2, 2} 9. {2/7} 10.  ,3 11. {3/2}
4 

12. 3 log 62 , 3 13. {3/2} 
14.  3 , 3  
15.  2 ,1,1, 2 
16. {3} 17. {1/9, 9} 18. {1, 3}
 2k  1     13 ,1
19.    20. x =   21. {2} 22. {100}
 2    3 
23. {25} 24. {3/2} 25. {9}
26. { log 3 (2  11 / 3 ) } 27. {3} 28. {1/16,1} 29. (0, 1)
30. (2,)
 (log 23  2) 
31.  ,   32. {1, 2, 3, 4, 5, 6, 7} 33. (2,) 34. (2, 6)
 2 
 83   1 ,0 
35. (-, -1)  (0, 2) 36. log 3 ,  37. R 38.   
 19   2 

39. (, log 23 ] 40. (-, log 2 ( 2  1)] 

 1 , 
 2  41. {1/16 , 1}

42. (,1  (log 3 5) / 3) 43. (  , 2 ) 44. (,0) (log 4 3,   ) 45. [-1,0] U [2,3]

USEFUL CONCEPT:- VI
 27 
1. {1/2, 1} 2. {-1, 2} 3. {-3, 2} 4.  ,1
 8 
5. {8, 27} 6. {1} 7. {3} 8. {9}
 3 1 5
9. {2} 10.  ,  11.   12.{8}
 2 2 3
13. {28} 14. {19} 15. {3} 16. {2}
17. {5} 18. {2} 19. {20} 20. {10}
21. {-1} 22. {4} 23. {-1} 24. {-1}
 4 1 
25.   26. {1/3} 27. 2,  28.  ,2
 3 2 
 17 
29. 0,2  9, 30. (4, 6] 31.   ,   1,10 32. (1 / 2,
 2
33. 1,2  (2,) 34. [-18, -2] 35. (3, 24/5) 36. (5,]
8 
37. [1,6] 38. (,3) 39. [-2,2] 40.  , 
3 
 1   1
41.  .0   0, 
 2   2
9
2

42.  ,0   , 


43.  1, 3 4  44. [-2,2]


45. {5}  4  2 ,  46. [1/2, ∞) ≠ 1 47. [5/2, 3)

12/12
 (Objective)
1. B 7. C 13. C 19. A 25. B 31. C
2. A 8. A 14. A 20. B 26. B 32. C
3. B 9. D 15. B 21. D 27. B 33. B
4. D 10. A 16. B 22. B 28. D
5. A 11. C 17. B 23. A 29. B
6. A 12. A 18. B 24. D 30. A

13/13