Algebra Maths By Amiya

Amiya, QUESTIONS
1. If 3   4  5  0 and
   7    0 have at least one root common then a+c=?

2. If    √3 then   ?
  

3. What would be sum of all integrals values of x for which    19  89 would be a perfect
square.

4. What would be remainder when  

  1 is divided by              1

5. What would be remainder when    1 is divided by      1

a. 2x - 2 b. 3x+1 c. x-1 d. x+1 e. NoT

6. What would be remainder when 

  1 is divided by            1
a. x-1 b. x^3+x^2+x+2 c. - x^3 - x^2 - x - 2 d. x^2
x^ -1 e. NoT

7. What would be remainder when x^99 + x^98 + x^97 +...... +x^2 + x + 1 divided by x^3 -1

8. If        0 , forr all real a,b,c and if   0 then,, which one would always correct
a. a<0 b. a+b+c<0 c. a>0 d. a+b+c > 0 e. NoT

9. If x^3 - 3(b^2)x + 2*c^3 is divisible by xx-a and x-b

b then which one could be true for real a,b,c
a. a = b = - c b. a = -2b
2b = 2c c. a = - 2b = - 2c d. -aa = b = c d. NoT

10. If axes lines are as per convention rule (Y

(Y-
vertical and X-horizontal)
horizontal) then the graph has
equation y=ax^2 + bx + c then which one
would be a possible value of a+b+c

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a. 3 b. -3 c. 0
d. can have more than one value
e. NoT

11. If x^3 - 149x - 33 =0 has roots a,b & c , then what would be equation a three degree
polynomial whose roots are a+b , b+c & c+a

12. Which option is showing all roots of x^4 - x^3 -49x^2 +169x - 120 =0
a. 1,3,-5,8 b. 1, 2,3,-7 c. 1,3,5,-8 d. NoT

13. What is the value of greatest integer equal to or less than 1.000 001"
 
a. 1 b. 2 c. 3 d. More than 3 e. NoT

14. If roots of equation  2"    4"  2  0 is differ by 3 then what would be

difference of values of "a"
a. 1 b. 1.5 c. 2 d. 2.5 e. NoT

6"7 8"9 7 6 8:

15. How many integral values would satisfy ≤0
6
"  7 6;8<"
a. 2 b. 4 c. 9 d. 11 e. NoT

16. If a, b, c be in H.P. and a and c have different magnitude but same sign, then the roots of
the equation    2    0 would be
a. Real b. Real but irrational c. Imaginary d. Equal d. NoT

17. If equation        0 has no real roots, and , ,  ∈ J KL  0 then

   " is always
a. Positive b. Non negative c. 0 d. Non-positive e. Negative

18. If        0 and 7   24  25  0 have a common root, and a, b, c are the

sides of ∆ABC and then ∠C is
a. 30° b. 45° c. 60° d. 90° e. NoT

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19. If one root of    2       0 ^_ V  1"  `U , where a, b, V & U are real &
rational numbers and U is not a perfect square then among options which equation has one
root 2V
a.    2       0 b.    2       0
c.    2       0 d.    2       0
e. NoT

20. If     3; then  ;  X 
?


21. If     2 , then 2 ∗ Z ;  a  0 , where k is a constant then what is the digital sum of
k
a. 2 b. 5 c. 7 d. 8 e. NoT

22. What is the coefficient of   ^K   1"    2"    3" …    20"

20
a. -1 b. 1 c. 101 d. -101 e. NoT

23. If the graph is of f x", which is a polynomial in x

with no imaginary roots then f -2"?
a. 12 b. 24
c. -12 d. -24 e. 48

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24. If the graph is of f x", which is a polynomial in x with no imaginary roots and coefficient
of highest power of x is 1, then f -3"? [Note- All Roots are visible in the graph]

a. 30 b. -750 c. -30 d. 750 e. Can't be determined

25. If |3*x^2 - 11x  6|  n has n solutions for n being a natural number, then minimum
value of n*x^2 - n1"*x  n2 is
a. 5/8 b. 71/16 c. 8/5 d. 16/71 e. NoT ??"

26. Total number of integral solution x,y,z" of ||  |h|  |i|  210 if
   h   i   | ∗ h|  |h ∗ i|  |i ∗ |  0
a. 27 b. 210 c. 216 d. infinity e. NoT ???"

27. Number of different real roots of   4"    8"  32 is

a. 0 b. 1 c. 2 d. 4 e. NoT

28. Sum of all real roots of   4"    8"  32 is

a. 12 b. 12 c. 6 d. 6 e. NoT

29. Sum of all imaginary roots of   4"    8"  32 is

a. 12 b. 12 c. 6 d. 6 e. NoT

30. Product of all real roots of   4"    8"  32 is

a. 36 b. 36 c. 2160 d. 2160 e. NoT

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31. Product of all imaginary roots of   4"    8"  32 is
a. 60 b. 60 c. 2160 d. 2160 e. NoT

32. If x-2yz 0 ; x3y-3z0 then for integral values of x,y, z which can be a value of xyz
a. 44 b. 144 c. 244 d. 344 e. NoT

33. If x^3 - 4x^2 -10 , then   7   13   5   3  1 ?

a. 0 b. √3 c.√3 d.± √3 e. NoT ???"

34. If M is maximum value and "m" is the minimum value of    4  25 for the close
interval [0,3] then Mm  ?
a. 47 b. 46 c. 21 d. 0 e. NoT

35. If M is maximum value and "m" is the minimum value of    for the close interval [-
0.5,1] then Mm  ?

a. -2 b. √3 c. 0 d. e. NoT
√

36. If one of the root of      6  0 is 2√3 and both roots of        0 are

equal then what is the value of c
a. First equation is wrong b. 27 c. 3/4 d. 27/4 e. NoT

37. For how many integral values of K , all roots of    4   20   a has no imaginary
roots
a. 3 b. 32 c. 33 d. 34 e. NoT

38. If f x"  2x 2-x" then total number of solution of f f f x"""" x/4 for 0 ≤  ≤ 2
a. 1 b. 2 c. 6 d. 8

39. A robotic dog is standing on an experiment floor which is in the shape of co-ordinates
having path in the form of yk or xm, where k and m are integers. In one command dog
can only move one integral distance and can only move towards either positive X-axis or
negative Y- axis. If current position of Dog is 12,7" then how many different command
we can give so that Dog would reach 20,-1".
a. 48620 b. 31824 c. 12870 d. 3432 e. NoT

40. For how many integral value of "b" origin and point 1,1" lie on the same side of line
a^2*x  a*b*y  90 , ∀ , ∈ J
a. 13 b. 12 c. 11 d. 10

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41. There are how many ordered pair integral solution for x^2  y^2  127

42. No of total number of integral solution of  

√ √w √<
a. 4 b. 8 c. 5 d. 10 e. NoT

43. If N 7x  3  9y2  11z - 5 , where all variables are natural numbers and
2000N3000, then what would be sum of digits of N
a. 11 b. 20 c. 29 d. 38

44. If ratio of sum of nth term of two A.Ps is 12n3": 37n11" , then what would be ratio
of 16th term of both the A.Ps
a. 121:256 b. 125: 386 c. 1:2 d. 16:17

45. If f xy"  f x"  f y" and f 2"  8 then f 8" ?

a. 8 b. 16 c. 32 d. 512 e. Cant Say or NoT

6 7
46. If y "  2y z{  
then f 2"  ?
a. 2 b. 6.5 c. -11.5 d. 11.5 e. Cant Say or NoT

47. If f xy"  f x-y" and f 2" 8 then f 8" ?

a. 8 b. 32 c. 512 d. Cant Say or NoT

48. Total number of positive even solutions of x^2 - y^2 528 ; consider negative evens as
even number ,
a. 12 b. 8 c. 4 d. 2 e. NoT

49. Total number of whole number of solutions for 2x  2y  5z  20

a. 36 b. 37 c. 18 d. NoT

50. Total number of whole number of solutions for 2  2h  5i ≤ 10

a. 36 b. 28 c. 18 d. NoT

51. Total number of integral solution of    h     ∗ h   49

a. 8 b. 10 c. 12 d. 15

52. If     √3 then       ?

a. 6 b. 0 c. 2 d. 3

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53. If N is total number of integral solutions of |x*y*z|100, then
a. | ≤ 36 b. 37 ≤ | ≤ 144 c. 145 ≤ | ≤ 288 d. 289 ≤ |

54. What would be least sum of positive integral solution triplet of 15x - 10 y - 9z  0 & 40x -
35y z 0
a. 23 b. 33 c. 47 d. 7

55. There are how many real roots of equation x^24 - 1 is ______ ?
a. 1 b. 2 c. 3 d. More than 3

56. Number of real roots of x^12 - x^6  10 is

a.2 b. 4 c. 6 d. NoT

€ €
57. If y   h"  y " ∗ y h"  y z   { ∗ y z   h{ then f pi" ?
a. 1 b. 0 c. -1 d. Not or Cant Be determined

58. If y "  1          . . . . . 
 then what would be remainder if y 

" is
divided by f x" for x>1
a. 8 b. 10 c. 11 d. NoT

59. If a, b & c are roots of equation x^3 - 6*x^2 + 10*x^2 - 1 = 0 , then a^3 + b^3 + c^3 =?
a. 6 b. 16 c. 39 d. NoT

60. What would be the difference of maximum and minimum value of sum of all coefficients of
expression 2*x100" 2*x99" 2*x 98".... 2*xn" for integral K ≤ 100
a. 102! b. 100! c. Not Defined d. NoT or Cant Say

61. If 1     ƒ & 1     K then among options x ?

…8† …6†
a. ±„ b. ±√ƒ  K c. ±„ d. ±√K  ƒ
† †

62. If for positive x, y & z, 3x2yz  15 then what would be average of all possible integral
values of x^2*y^2*z
a. 108 b. 109 c. 54.5 d. 55 e. Can't Say or NoT

63. What would be minimum value of _^K   ‡_    ˆ K   ‡_‰    _‰    ‡ˆ  
a. 5 b. 6 c. 7 d. NoT

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64. f x"  x^3 - 3x^2  9*x - 5 has
a. 2 Real positive Roots & 1 negative Root
b. 1 Real Positive Root & 2 Imaginary Roots
c. 3 Real Positive Roots
d. NoT

65. What would be nature of roots of x^3 5x^2 25x125 0

a. 3 Real negative Roots
b. 1 Real Negative Root & 2 Imaginary Roots
c. 3 Real Positive Roots
d. 1 Real Positive Root & 2 Imaginary Roots

66. Number of real roots of x^42x^3 3x^2 2x 1  0

a. 0 b. 2 c. 4 d.NoT

67. If f x" is a n degree

egree polynomial then among option which could
be value of n
a. 2 b. 4 c. 5 d. 6

68. Which option would represent the graph

8
" 6"7 9 7 6 8:
69. For how many integral value of x , 6

 0
a. 3 b. 4 c. 5 d. Infinite e. NoT

70. If f x" x-38" x-36" x-34"......

34"...... xx-2*k" ; where k is an integer ≤19 then,
What would be maximum constant term of f x". in terms of a^b"* n!""

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71. If 3   1 ; 9    11 &  2    3 then what would be range for all values" of
Š8‹
Œ

72. If 4+i is one root of a quadratic equation x^2 + bx + c=0 , then what would be sum of roots of
equation x^2 + (b+c)x + b- c =0
a. 9 b. -9 c. 25 d. -25 e. CnD or NoT

73. What is the value of a^5  b^5 ?

I. a^4  b^4  17 II. a*b  2

a. Question can be answered with the help of statement I alone,

b. Question can be answered with the help of any one statement independently,
c. Question can be answered with the help of both statements together,
d. Question cannot be answered even with the help of both statements together.

 
74. How many real values of x satisfy the relationŽ   , where [x] refers to greatest
integer less than or equal to x.

a. 1 b. 2 c. 3 d. 4

75. If abc9 then for positive a, b & c what would be maximum value of
66a3b2c3abbc2caabc
a. 27 b. 216 c. 125 d. 64 e. NoT

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76. Match the equation to Graphs
1. h  ƒ^K    3  1,5  
" 2. h  ƒ     3  1,5  "
3. h  ƒ^K    3  1,5  
" 4. h  ƒ     3  1,5  "

a. 1-II , 2-III, 3-I, 4-IV b. 1-IV , 2-III, 3-I, 4-II

c. 1-I , 2-II, 3-III, 4-IV d. 1-IV , 2-I, 3-III, 4-II

77. If n is a least non-negative

negative number then, Total number of imaginary roots of equation

  
   <     ⋯ … . .  † is ___???
a. 106 b. 104 c. 58 d. Cannot be determined
e. NoT

78. Which one is always wrong for two numbers

a. AM  GM b. AM > GM c. AM > GM^2 d. More than 1
e.NoT

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 Š8‹
79. What is the value of ?
Œ
I.         3 ∗ ∗  ∗  II.      2 ∗ ∗ 

a. Question can be answered with the help of statement I alone,

b. Question can be answered with the help of statement II alone,
c. Question can be answered with the help of both statements together,
d. Question cannot be answered even with the help of both statements together.

80. How many roots of f x" and g x" are common, where y "     2   9  18 &
“ "  2   7   3  18
a. 0 b. 1 c. 2 d. 3

81. A carpenter makes two types of chairs GENERAL and STYLISH. Each type of GENERAL
chair requires 2 hours finishing and 4 hours of polishing; and each model of STYLISH
requires 8 hours of finishing and 6 hours of polishing. The carpenter has 2 finishers and 4
polishers. Each finishers works for 46 hours a week and each polisher works 72 hours a week.
If his Profit on GENERAL chair is Rs.300/- and on STYLISH is Rs.800 /- . Assume
whatever produced in a week is sold in the market. Then what would be maximum profit he
could earn in a week.
a.16200 b. 15800 c. 16100 d. 9800 e. NoT

82. If for positive integral x. y & z, x+2y+3z=16 and the value of   1" ∗ h  2" ∗ i  3"
is maximum then x+y+z=?
a. 6 b. 9 c. 8.5 d. 13

83. Number of nonnegative integral solutions of x  y*z4

a. Less than 10 b. Between 11 to 50 c. Between 51 to 100
d. Between 101to 999 e. More than 1000

84. Total number of integral solution of x^2  17y"* 17-y"

a. 6 b. 10 c. 12 d. 14 e. NoT

85. If |x||y||z|n has 102 integral solutions for a particular value of n then minimum
value of x*y*z  ?

a. minus infinity b. -3 c.  z{ d. -4 e. NoT

86. There are how many positive integral ordered" solutions of wxyz  40 which has
exactly 2 odd integers in its solution set
a. C 20,17" b. 6*C 20,17" c. c 39,36" d. NoT ???"

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87. There are how many INTEGRAL ORDERED PAIR solutions of `  ∗ h"  10
a. 18 b. 9 c. 4 d. 8 NoT

88. There are how many Non-Negative INTEGRAL ORDERED PAIR solutions of xy100 ,
such that –—˜ , h, 100"  1
a. 70 b. 60 c. 61 d. 40 e. NoT

89. How many statements are always correct if they are in determined form"
I. a^n"* b^n"  a*b"^n II. a^n"* a^m" a^ mn"
III. a^n"^m  a^ m*n" IV a^0  1

a. Only one statement is always true

b. Only two statements are always true
c. Only three statements are always true
d. All statements are always true
e. None are always true

90. There are how many integral solutions of 3|x|  5|y| 75

a. 16 b. 24 c. 20 d. 22

91. If HCF a,b"  13 & ab 13000 then how many ordered pair of a.b" be possible
a. 20 b. 200 c. 400 d. 2 e. Not

92. If x*y*z*w  100 has N integral ordered" solutions and |x|*|y|*|z|*|w|  100 has M
integral ordered" solutions then |M-N| ?
a. 300 b. 0 c. 900 d. 700 e. NoT ???"

93. For many integral value of K;    3 ∗    10  ™ has two positive and 1 negative
roots
a. 28 b. 30 c. 31 d. infinite e. NoT

94. For nonnegative x, y, z , if   ∗ h ∗ i   108 , then min value of xyz ?

a. 27 b. 18 c.3 d. 6 e. NoT

95. What is the minimum value of |2x-3|  |4-x|  |5-x|

a. 4 b. 6 c. 5 d. 0


Š Œ › Š7 8;Œ 7 6<› 7 ž
96. If ‹  š  œ  16 then z‹7 8;š7 6<œ7 { ?
a. 2 b. -2 c. 1 d. -1 e. NoT

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97. There are how many integral values satisfy    h   25  0
a. 64 b. 70 c. 69 d. 50

98. After solving a quadratic function f x" Ram get roots as 3 and 4 and Shyam get roots as -
3, 4 but they did one mistakes, one of them used wrong value of constant term and other
used wrong coefficient of x. Then what would be value of f 1" if graph of equation does
not touch or cut the x-axis.
a. -18 b. 18 c. -12 d. 12 e. NoT

99. If P x" x^100  x^99  x^98 ....x^2x1 , Q x"x^3 -1 then R x" is remainder
when P x" is divided by Q x" then R 2"?
a. 232 b. 2^101 - 1 c. 2 d. 1 e. NoT

100. Analyse the statements

I. If a function is odd then it must be cross from origin
II. If a function is even then it must not cross from origin

a. Only one statement is always true

b. Both are always true
c. Either one is always true
d. Both are always false
e. NoT

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Algebra Maths By Amiya, QUESTIONS &
Solutions
1. If 3   4  5  0 and    7    0 have at least one root common then a+c=?

Sol : - Since 3   4  5  0 has D<0 , means imaginary roots so if any equation has common
root the both roots are same so equation => a= 21/4 and c=35/4, a+c =14

2. If    √3 then   ?
  

Sol:    √3 ⇒     0 ⇒    0 ⇒  ;  0
   ¡  7X
1 1 1 1 1 1
¢ ;  £ ¢ <  £  0 ⇒       0 ⇒     ¢   £  √3
 ;  <    

3. What would be sum of all integrals values of x for which    19  89 would be a perfect
square.

Sol : -19 , for -8 and -11

Let    19  89  a  ⇒    19  89  a  "  0 ,should have integral roots
therefore D should be a perfect square
⇒ 19"  89  a  "  ¤‰¥y‰ˆ _¦§ ¥‰ ⇒ 5  4 ∗ a   K _ h"
⇒ K  4 ∗ a  "  5 ⇒ K  2a" K  2a"  5
⇒ n-2k = 5, n + 2 k = 1 or n-2k = 1, n + 2 k = 5
⇒ K  ±3, a  ± 1,
For a  ± 1,, equation becomes
⇒    19  88  0 ⇒   11"   8"  0 ⇒    11,  8.

4. What would be remainder when  

  1 is divided by              1

Sol: 0,

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             1 is factor of    1
since    1    1"              1"
and    1 is factor of  
  1    "  1^336

5. What would be remainder when    1 is divided by      1

a. 2x - 2 b. 3x+1 c. x-1 d. x+1 e. NoT

Sol: [c]
     1 is factor of    1 so if we check for    1 we would get an idea for      1,
   1 would give remainder x-1 for    1 , [put x^3 = 1 in the expression and get
remainder] since x-1 has lower degree than      1 , so remainder would be same x-1

6. What would be remainder when 

  1 is divided by            1
a. x-1 b. x^3+x^2+x+2 c. - x^3 - x^2 - x - 2 d. x^2 -1 e. NoT
Sol: [c]
           1 is factor of   1 so if we check for   1 we would get an idea for
           1.

  1 would give remainder    1 for   1 , [put x^5 = 1 in the expression and get
remainder]
Since remainder x^4 -1 has not lower degree than give divisor            1
so, we have to dived x^4 -1 by            1 which gives remainder         
2"

7. What would be remainder when x^99 + x^98 + x^97 +...... +x^2 + x + 1 divided by x^3 -1

Sol: 33 ∗    33 ∗   34 , put x^3 = 1 in given expression and get answer, check concept video
of similar type question- https://www.youtube.com/watch?v=67jg1SVb6Eg

8. If        0 , for all real a,b,c and if   0 then, which one would always correct
a. a<0 b. a+b+c<0 c. a>0 d. a+b+c > 0 e. NoT

Sol: [a]
Since not equal to 0 , means wont cut x-axis and c is negative it shows parabola is downward so
a<0.

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9. If x^3 - 3(b^2)x + 2*c^3 is div
divisible by x-a and x-b
b then which one could be true for real a,b,c
a. a = b = - c b. a = -2b
2b = 2c c. a = - 2b = - 2c d. -aa = b = c d. NoT

Sol: [c]
Since x^3 - 3(b^2)x + 2*c^3 = f(x) say, is divisible by x-a and x-b
so f(a) = 0 and f(b) = 0 , by f(b) = 0 , we would get b=c
and by f(a) = 0 , we would get  "  2"  0 , by simplifying the expression.
so we get a=b=c or a=-2b=-2c

10. If axes lines are as per convention rule (Y

(Y-
vertical and X-horizontal)
horizontal) then the graph has
equation y=ax^2 + bx + c then w which one
would be a possible value of a+b+c

a. 3 b. -3 c. 0
d. can have more than one value
e. NoT

So;: [b]
By seeing the graph we can say
c = -2 , a>0 (graph is upward) -b/2a
b/2a = 2/3 (value of x at minima)
and b^2 - 4ac / 4a = 10/3 (minima)
after solving we would get a=3 , b=
b=- 4 , c = -2 so, a+b+c= -3

Alternate - check y 1"       3

11. If x^3 - 149x - 33 =0 has roots a,b & c , then what would be equation a three degree
polynomial whose roots are a+b , b+c & c+a
Sol: Since a+b+c = 0 , so a+b=--c , b+c=-a , c+a =-b
b , means equation whose roots are a+b , b+c &
c+a has same roots but in opposite sign of given equation, so required equation would be   
149  33  0 or x^3 - 149x - 33 =0 or K*(x^3 - 149x - 33)=0 )=0 , K is any real number.

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12. Which option is showing all roots of x^4 - x^3 -49x^2 +169x - 120 =0
a. 1,3,-5,8 b. 1, 2,3,-7 c. 1,3,5,-8 d. NoT
Ans: [c] just check by options

13. What is the value of greatest integer equal to or less than 1.000 001"
 
a. 1 b. 2 c. 3 d. More than 3 e. NoT

Sol: [b]
https://www.facebook.com/MathsByAmiya/photos/627653000620153/
or
use - https://www.facebook.com/MathsByAmiya/photos/540943792624408/

14. If roots of equation  2"    4"  2  0 is differ by 3 then what would be

difference of values of "a"
a. 1 b. 1.5 c. 2 d. 2.5 e. NoT

Sol : [b] 1.5 By Solving we get a= 3 and 3/2 so difference is 1.5

6"7 8"9 7 6 8:

15. How many integral values would satisfy ≤0
6
"  7 6;8<"
a. 2 b. 4 c. 9 d. 11 e. NoT

Ans: [c]9 ;±4, ±3, ±2 , 1,0,5 ,

16. If a, b, c be in H.P. and a and c have different magnitude but same sign, then the roots of
the equation    2    0 would be
a. Real b. Real but irrational c. Imaginary d. Equal d. NoT

Sol: [c]
ŠŒ
a, b, c are in H.P ⇒   Š8Œ
ŠŒ 
D of    2    0 is 4   4   4[zŠ8Œ{  ]
ŠŒ Š6Œ 
 4  Ž 1    4  zŠ8Œ{  0 ; imaginary roots
Š8Œ"7

17. If equation        0 has no real roots, and , ,  ∈ J KL  0 then

   " is always
a. Positive b. Non negative c. 0 d. Non-positive e. Negative

Sol: [a]

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 Since y "        has no real roots so for all real values f x" would have same
sign
So f 0"*f 1"     " > 0

18. If        0 and 7   24  25  0 have a common root, and a, b, c are the

sides of ∆ABC and then ∠C is
a. 30° b. 45° c. 60° d. 90° e. NoT

Sol: [d]
7   24  25  0 has both roots imaginary so both roots are same with
     
Š ‹ Œ
So ;     ; its a ratio of right angled triangle at C

19. If one root of    2       0 ^_ V  1"  `U , where a, b, V & U are real &

rational numbers and U is not a perfect square then among options which equation has one
root 2V
a.    2       0 b.    2       0
c.    2       0 d.    2       0
e. NoT

Sol: [a]
If one root is V  1"  `U then another is V  1"  `U and let ¨ be third root.
So sum of roots
V  1"  `U + V  1"  `U + ¨ = -2
⇒ 2 V = ¨

So equation whose one root is 2 V i.e. ¨, we can get equation by putting x = -x in giiven
equation
"  2 "  "    0
⇒   2       0

20. If     3; then  ;  X ?

Sol:     3 ;     7  7 ;     ©  47

    3 ;        18
1 1 1 1
 ;  ;  ¢    £ ¢    £  ¢  £  843
   

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21. If     2 , then 2 ∗ Z ;  a  0 , where k is a constant then what is the digital sum of
k
a. 2 b. 5 c. 7 d. 8 e. NoT

Sol: [c]

    2 ⇒    2  4  0 ⇒ x2"    2  4"  0 ⇒   8  0
⇒   8
2 ∗ Z ;  a  0 ⇒ a  
2 ∗ 8"  2 ∗ 8 so, Digital sum of k7

22. What is the coefficient of   ^K   1"    2"    3" …    20"

20
a. -1 b. 1 c. 101 d. -101 e. NoT

Sol: [b]
Maximum power of x in the given expression  123...20  210
To get co-efficient
efficient of x^204 we have to find the constant term of expression having
power x^6 or remove factors of x^6" , which are

1. x^1-1" x^5-5" 5" product of constant term  5

2. x^2-2" x^4-4" 4" product of constant term  8
3. x^1-1" x^2-2" x^3-3" 3" product of constant term  -6
4. x^6 - 6 ; constant term  -6

Required coefficient  58

58-6-6  1

23. If the graph is of f x", which is a polynomial in x

with no imaginary roots then f -2"?
a. 12 b. 24
c. -12 d. -24 e. 48

Sol: [a]
According to wavy curve method
y "    1"›ª›† ∗   1"«šš ∗   2"«šš
y 0"  0  1"›ª›† ∗ 0  1"«šš ∗ 0  2"«šš
y 0"  2«šš  2 ‡¥L^K“ ˆ‡ ¦§‰_ˆ^‡K"
so power of x-2" is 1 so
y "    1"›ª›† ∗   1"«šš ∗   2"

y 2"  2  1"›ª›† ∗ 2  1"«šš ∗ 2  2"

 1 ∗ 3«šš ∗ 4
4
Now check options for odd power of -3 , only 12 satisfies.
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24. If the graph is of f x", which is a polynomial in x with no imaginary roots and coefficient
of highest power of x is 1, then f -3"? [Note- All Roots are visible in the graph]

a. 30 b. -750 c. -30 d. 750 e. Can't be determined

Sol: [d] 750

According to wavy curve method [Same sign of graph - odd times repetition of roots ,
opposite sign of graph - even times repeti
repetition
tion of roots; so in above, graph has same sign
in both side of -2
2 and opposite sign around 2 and 3, so -22 would be repeated even
number of times] Hence

y "    2"›ª›† ∗   2"«šš ∗   3"«šš

y 0"  0  2"… ∗ 0  2 2"†8
∗ 0  3"¬8
 96
2…8†8
∗ 3¬8
 96 ⇒ ƒ  K  3 & a  0
so ,
y "    2" ∗   2""
∗   3"
or y "    2" ∗   2" ∗   3"

y 1"  1  2" ∗ 1  2""

∗ 1  3"
y 1"  1  2" ∗ 1  2" ∗ 1  3"
 18
 162; cant be since y 1"  96
So
y "    2" ∗   2"" ∗   3"

y 3"  3  2" ∗ 3 3  2" ∗ 3  3"

 750

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25. If |3*x^2 - 11x  6|  n has n solutions for n being a natural number, then minimum
value of n*x^2 - n1"*x  n2 is
a. 5/8 b. 71/16 c. 8/5 d. 16/71 e. NoT ??"

Solution [b]
|3*x^2 - 11x  6|  n has n solutions > n4 , by graph
Now minimum value of 4*x^2 - 5x  6  71/16

26. Total number of integral solution x,y,z" of |x|^3  |y|^3 |z|^3  210 if x^2  y^2 
z^2 - |x*y| - |y*z| - |z*x| 0
a. 27 b. 210 c. 216 d. infinity e. NoT ???"

Ans: [NoT] -No integral Solutions possible

We cant use, |x|^3  |y|^3 |z|^3  3*|x|*|y|*|z| [since x^2  y^2  z^2 - |x*y| - |y*z| -
|z*x| 0]
3*|x|*|y|*|z|  210
|x|*|y|*|z|  70  2*5*7
having total integral solution,  3^3*2^3  216
Because since x^2  y^2  z^2 - |x*y| - |y*z| - |z*x| 0 means |x||y||z| , and it cant
hold for integral values of |x|^3  |y|^3 |z|^3  210

27. Number of different real roots of   4"    8"  32 is

a. 0 b. 1 c. 2 d. 4 e. NoT

28. Sum of all real roots of   4"    8"  32 is

a. 12 b. 12 c. 6 d. 6 e. NoT

29. Sum of all imaginary roots of   4"    8"  32 is

a. 12 b. 12 c. 6 d. 6 e. NoT

30. Product of all real roots of   4"    8"  32 is

a. 36 b. 36 c. 2160 d. 2160 e. NoT

31. Product of all imaginary roots of   4"    8"  32 is

a. 60 b. 60 c. 2160 d. 2160 e. NoT

Sol: 27. b"1 , 28. b" -12 29. b"-12 30. a" 36 31. a" 60
Take x6  y [ Why x6 , since its mean of given expressions x4 &
x8]
so, h  2"  h  2"  32
2 h   24 ∗ h   16"  32

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 h   24 ∗ h   0 ⇒ h   0 ‡¥  24
so real h  0 for imaginary h  ±2√6 ^
­‡,   6  0 ,   6 &  6 So,   6  ±2√6 ^ ,   6 ± 2√6 ^

You have all roots now solve questions

­§ƒ ‡y ¥‰ ® ¥‡‡ˆ_  6"  6"  12 ­§ƒ ‡y ¥‰ ® ¥‡‡ˆ_  12

32. If x-2yz 0 ; x3y-3z0 then for integral values of x,y, z which can be a value of xyz

a. 44 b. 144 c. 244 d. 344 e. NoT

Sol: [b] x:y:z  3:4:5 so summation would be multiple of 12

33. If x^3 - 4x^2 -10 , then   7   13   5   3  1 ?

a. 0 b. √3 c.√3 d.± √3 e. NoT ???"

x^3 - 4x^2 -1 is a factor of   7   13   5   3  1 , so if x^3 - 4x^2 -10 , then

  7   13   5   3  1  0

34. If M is maximum value and "m" is the minimum value of    4  25 for the close
interval [0,3] then Mm  ?
a. 47 b. 46 c. 21 d. 0 e. NoT

Sol: [b]
An quadratic gives minina at x-b/2a for a>0 , here x-4/2a2
So we get minima at x2 which is 21 and at extreme values of graph we have to to find
maxima of the given range
at x0 , we will get 25
at x3 , we will get 22
So Mm  2521  46

35. If M is maximum value and "m" is the minimum value of    for the close interval [-
0.5,1] then Mm  ?

a. -2 b. √3 c. 0 d. e. NoT
√

 
Sol: [e] <   √

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 y "     
y′ "  3   1  0 for maxima &
minina"
1
±
√3
y "  6
°°

So at we get minima since f'' x" is

√

positive" & at  we get maxima since f'' x"

√
is negative"

Our range is [-0.5

0.5 ,1] ,For M, maxima value at x
x is out of this range so we have to
√
calculate maxima at x-0.5
0.5 which is 3/8


and for m, minima at x , our minina is  √

√
 
So Mm   
< √

36. If one of the root of      6  0 is 2√3 and both roots of        0 are

equal then what is the value of c
a. First equation is wrong b. 27 c. 3/4 d. 27/4 e. NoT

Sol:
Another root of      6  0 is √3 ; [by product of roots  c/a]
so   3√3 [by summation of roots  -b/a]
‹  ;
       0 ; since roots are same, then   z{ 


37. For how many integral values of K , all roots of    4   20   a has no imaginary
roots
a. 3 b. 32 c. 33 d. 34 e. NoT

Sol: [c]
Let f x"     4   20
20 
Diff wrt to x
f' x" 4   12   40
at f' x" 0
4   12   40  0
⇒ 4 x-5"x x2"  0
⇒ x-2 , 0 or 5
f" x"  12   24  40

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 So curve of f x" has local maxima and minima at x
x-2
2 , 0 or 5 ; by substituting values in
f'' x" we can find a rough curve of f x"
For   2 , f'' x">0 ; so at   2 we would have local minima
For x0 , f'' x"0 ; so at x0 we would have local maxima
For x5 , f'' x">0 ; so at x5 we would have local minima
f -2"  -32
32 ; f 0"0 ; f 5"  -375

The graph of f x"     44   20  looks like

Note: If we add any positive in this curve it would
move upward and if we subtract a positive quantity
it would move downward.

This graph shows that 4 roots of f x" are -

1 negative , 2 at 0's and 1 is pos
positive. real
roots are values of curve where the touch xx-axis"

Now think about f x"  k

By this graph if k0 , graph would move downward so, we have only two real roots rest
would be imaginary.
If x>32 , graph would move upward but at x
x-2 it would be above
bove x axis so again, we
have only two real roots rest would be imaginary.

So for 0 ≤  ≤ 32 we have 4 real roots.

So total integral values of x is 33

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38. If f x"  2x 2-x"
x" then total number of solution of f f f x"""" x/4 for 0 ≤  ≤ 2
a. 1 b. 2 c. 6 d. 8

Sol: [d]
Concept :-:-
If a function is symmetrical about a line then fof or fofof... would also be symmetrical about
that line.
In other word if f x"k has n solution along symmetrical line then f f x""k would have
2^2 and f f f x"""k would have 2^3 sol
solutions and so on.

f x" 2x 2-x"x"  y say" is symmetrical along x1

and y  x/4
so f x" 2x 2-x"x"  x/4 has 2 solutions for 0 ≤  ≤ 2
So, f f f x"""" x/4 would have 2^3 8 solutions

Learn by graph

Check the graphs all are symmetrical about the same line x1

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 Check the number of solutions, point of intersections

A robotic dog is standing on an experiment floor which is nothing but a in the shape of
co-ordinates
ordinates axis with all four quadrants, origin and axis line. If current position can only
move on the integral graph path integral lines which is created by integral values of x
and y axis or lines in the form of yk or xm, where k and m are integers". And in one
command dogs only move one integral distance. If current posit
position
ion of Dog is 3,7" then
among options which could not be number of command to reach 7,3"

39. A robotic dog is standing on an experiment floor which is in the shape of co-ordinates
co
having path in the form of yk or xm, where k and m are integers. In one command
com dog
can only move one integral distance and can only move towards either positive X-axis
X or
negative Y- axis. If current position of Dog is 12,7" then how many different command
we can give so that Dog would reach 20, 20,-1".
a. 48620 b. 31824 c. 12870 d. 3432 e. NoT

Sol: C 16,8" 12870

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40. For how many integral value of "b" origin and point 1,1" lie on the same side of line
a^2*x  a*b*y  90 , ∀ , ∈ J
a. 13 b. 12 c. 11 d. 10

Sol: [c]
a^  ab  9 > 0 ⇒ b^2 - 4*9 > 0 ⇒ b ∈ 6,6" so total integral
gral values of b is -5 to 5  11

41. There are how many ordered pair integral solution for x^2  y^2  127

Sol: A number in the form of 4*n3 cant be written as sum of two perfect squares, so no
solution..

42. No of total number of integral solution of  

√ √w √<
a. 4 b. 8 c. 5 d. 10 e. NoT

Ans: [c]

Explanation of above solution in details

 
√ √w √<

⇒  
√ √w ∗√
By analysing the above equation we can say that √ ,& `h should have √3 as a term
Take √  ƒ ∗ √3& `h  K ∗ √3 , where m & n are positive integers, so we get

⇒  
…∗√ †∗√ ∗√

⇒  
… † 
and for positive integral roots of above equation we have 5 TNF 4^2""** solution
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yAmiya/photos/631921553526631/
** to know how,, make equation and work on factors of RHS constant"

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43. If N 7x  3  9y2  11z - 5 , where all variables are natural numbers and
2000N3000, then what would be sum of digits of N
a. 11 b. 20 c. 29 d. 38

Sol: By the normal method of two equations or CRT we would get the least value of
N479
and general term for N LCM 7,9,11"*m  479  693*m  479
For given conditions m 3 . N  693*3  479  2558

44. If ratio of sum of nth term of two A.Ps is 12n3": 37n11" , then what would be ratio
of 16th term of both the A.Ps
a. 121:256 b. 125: 386 c. 1:2 d. 16:17

Sol: [b]
Š8
š Š8š ³´… ´ µ« 
µ›¶…
∗
8

Ratio of 16th term of A.Ps  ±8
²  ±8²  ³´… ´ µ« 
µ›¶…  ;∗
8

 <

45. If f xy"  f x"  f y" and f 2"  8 then f 8" ?

a. 8 b. 16 c. 32 d. 512 e. Cant Say or NoT

Sol: [c]
If f xy"  f x"  f y" , then f x"  c*x & f 2"  c*28 > c4
then f 8"  4*8  32

6 7
46. If y "  2y z{  
then f 2"  ?
a. 2 b. 6.5 c. -11.5 d. 11.5 e. Cant Say or NoT

Sol: [b]
 7 6; ∗7 6;
By putting x 1/x and solving , we will get y "  
⇒ y 2"  
 6.5

47. If f xy"  f x-y" and f 2" 8 then f 8" ?

a. 8 b. 32 c. 512 d. Cant Say or NoT

Sol: [a] It is a constant function , so 8

48. Total number of positive even solutions of x^2 - y^2 528 ; consider negative evens as
even number ,
a. 12 b. 8 c. 4 d. 2 e. NoT

Ans: [d]

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49. Total number of whole number of solutions for 2x  2y  5z  20
a. 36 b. 37 c. 18 d. NoT

Ans: c"

50. Total number of whole number of solutions for 2  2h  5i ≤ 10

a. 36 b. 28 c. 18 d. NoT

Ans: c"

51. Total number of integral solution of    h     ∗ h   49

a. 8 b. 10 c. 12 d. 15

Sol: c"
x^2 1" y^21"50 ⇒ x^2 1" y^21"5*10 and 1*50 where factors are 1 more
than perfect squares"
  ±2 , h  ±3 &   ±3, h  ±2 ;   0, h  ±7 &   ±7, h  0
So 12

52. If     √3 then       ?

a. 6 b. 0 c. 2 d. 3

Sol:    √3 ⇒    1 _‡,       1  1  0



53. If N is total number of integral solutions of |x*y*z|100, then

a. | ≤ 36 b. 37 ≤ | ≤ 144 c. 145 ≤ | ≤ 288 d. 289 ≤ |

Ans: [c] N 288

54. What would be least sum of positive integral solution triplet of 15x - 10 y - 9z  0 & 40x -
35y z 0
a. 23 b. 33 c. 47 d. 7

Sol: [b] 33
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55. There are how many real roots of equation x^24 - 1 is ______ ?
a. 1 b. 2 c. 3 d. More than 3
Ans: [b]

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56. Number of real roots of x^12 - x^6  10 is
a.2 b. 4 c. 6 d. NoT

Ans: [d] NoT ; 0

€ €
57. If y   h"  y " ∗ y h"  y z   { ∗ y z   h{ then f pi" ?
a. 1 b. 0 c. -1 d. Not or Cant Be determined

Ans: -1 , f x"  cos x

58. If y "  1          . . . . . 
 then what would be remainder if y 

" is
divided by f x" for x>1
a. 8 b. 10 c. 11 d. NoT

Ans: [c]
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59. If a, b & c are roots of equation x^3 - 6*x^2 + 10*x^2 - 1 = 0 , then a^3 + b^3 + c^3 =?
a. 6 b. 16 c. 39 d. NoT
Sol:(c)
a+b+c = 6 ; ab + bc + ca = 10 & abc = 1
(a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)
a^2 + b^2 + c^2 = 16
a^3 + b^3 + c^3 - 3abc = (a+b+c)( a^2 + b^2 + c^2 -(ab + bc + ca))
a^3 + b^3 + c^3 - 3*1 = 6*(16 - 10)
a^3 + b^3 + c^3 = 39

60. What would be the difference of maximum and minimum value of sum of all coefficients of
expression 2*x100" 2*x99" 2*x 98".... 2*xn" for integral K ≤ 100
a. 102! b. 100! c. Not Defined d. NoT or Cant Say

Sol: [a]
Max sum  102! and min sum  0 for coefficient sum put x1"

61. If 1     ƒ & 1     K then among options x ?

…8† …6†
a. ±„ †
b. ±√ƒ  K c. ±„ †
d. ±√K  ƒ

Sol: [c]
1   ƒ ƒK ƒK
1    
⇒ 1   
 ⇒  
 ⇒   ±„
1   K K K

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62. If for positive x, y & z, 3x2yz  15 then what would be average of all possible integral
values of x^2*y^2*z
a. 108 b. 109 c. 54.5 d. 55 e. Can't Say or NoT

Sol: If 3x2yz  15 then max x^2*y^2*z"  108 so x^2*y^2*z can have all integral
values from 1 to 108 so average of all would be 1081"/2  54.5

63. What would be minimum value of _^K   ‡_    ˆ K   ‡_‰    _‰    ‡ˆ  
a. 5 b. 6 c. 7 d. NoT
Ans : 7 at   45°

64. f x"  x^3 - 3x^2  9*x - 5 has

a. 2 Real positive Roots & 1 negative Root
b. 1 Real Positive Root & 2 Imaginary Roots
c. 3 Real Positive Roots
d. NoT

Sol: [b]
For x0 , x^3 - 3x^2  9*x - 5 -5 negative"
and for x 1 we will get a positive outcome so, it has some real solution crossing X-axis" in
between 0 to 1
If the curve cuts the X-axis more than once so it should have local maxima or minima for this
f ' x" 0 should have real roots . if imaginary roots means it does not have local or global
maxima or minima so wont cut x-axis again
f x"  x^3 - 3x^2  9*x - 5
f ' x"  3*x^2 - 6x  9
f ' x"  3*x^2 - 6x  9  0 ,
its discriminant is less than 0, so we are not getting any real point of local minima or
maxima.
And at x0 its negative and for x1 its positive so its an ascending curve moving negative
to positive" , so cuts X-axis only once.
So, Only one real positive roots and rest two are imaginary roots.

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65. What would be nature of roots of x^3 5x^2 25x125 0
a. 3 Real negative Roots
b. 1 Real Negative Root & 2 Imaginary Roots
c. 3 Real Positive Roots
d. 1 Real Positive Root & 2 Imaginary Roots

Ans : [b] concept of above question

66. Number of real roots of x^42x^3 3x^2 2x 1  0

a. 0 b. 2 c. 4 d.NoT

Ans: [a]
x^42x^3 3x^2 2x 1  x^2 x  1"^2
Since x^2 x  1" has no real roots so its square also has no real roots
Alternative
x^42x^3 3x^2 2x 1  0
x^2  x"^2  x1"^2  x^2 0
sum of three real squares cant be zero , so no real solutions

67. If f x" is a n degree polynomial then among option which could

be value of n
a. 2 b. 4 c. 5 d. 6

Ans : [c]; same as question no 23

68. Which option would represent the graph

Ans : [b]
Hint check f 0" then f 0" to f 1" a smaller value of x" is increasing or decreasing and
f 0" to f -1"
1" a smaller value of x" is increasing or decreasing and match with graph or
factorise the options if possible"

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 8
" 6"7 9 7 6 8:
69. For how many integral value of x , 6

 0
a. 3 b. 4 c. 5 d. Infinite e. NoT

Ans: [b]

70. If f x" x-38" x-36" x-34"...... x-2*k" ; where k is an integer ≤19 then,
What would be maximum constant term of f x". in terms of a^b"* n!""

Sol: For maximum constant term  x-38" x-36" x-34".... x-4" and x0
 -38" -36" -34"...... -4"  2^18"* 19!"

71. If 3   1 ; 9    11 &  2    3 then what would be range for all values" of

Š8‹
Œ

Ans: R - -3,2"
https://www.youtube.com/watch?vKXyo9LjbUSc&index29&listPLqP8FM58Y3oHq
BeDbdKkUpz0aRhAXlWbY

72. If 4+i is one root of a quadratic equation x^2 + bx + c=0 , then what would be sum of roots of
equation x^2 + (b+c)x + b- c =0
a. 9 b. -9 c. 25 d. -25 e. CnD or NoT

Sol: [e] Cannot be determined

We cannot find b and c since we dont have second root, conjugate pair only occurs if b
and c are real, but it is not mentioned here.

73. What is the value of a^5  b^5 ?

I. a^4  b^4  17 II. a*b  2

a. Question can be answered with the help of statement I alone,

b. Question can be answered with the help of any one statement independently,
c. Question can be answered with the help of both statements together,
d. Question cannot be answered even with the help of both statements together.

Ans: [d]

 
74. How many real values of x satisfy the relationŽ   , where [x] refers to greatest
integer less than or equal to x.

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 a. 1 b. 2 c. 3 d. 4

Sol: c
 
LetŽ    a, where k is an integer. Thus, x  3*k, x also being an integer

 
Next, Ž   a ⟹ a ≤  a  1 ^y a ≥ 0

¬
The left had relation gives a ≤ which could be true only for k ≤ 0


The right hand relation gives,  a  1. This boils down to 3 ∗ a  5a  5 ^. ‰. a > 2.5.
Thus, the only acceptable k are -2 , -1 and 0. Thus, only 3 values of x satisfy the relation
i.e. -6, -3 and 0.

75. If abc9 then for positive a, b & c what would be maximum value of
66a3b2c3abbc2caabc
a. 27 b. 216 c. 125 d. 64 e. NoT

Sol: [c]
66a3b2c3abbc2caabc"  a1" b2" c3"

a1" b2" c3"  15

then max a1" b2" c3"  5*5*5  125

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76. Match the equation to Graphs
1. h  ƒ^K    3  1,5  
" 2. h  ƒ     3  1,5  "
3. h  ƒ^K    3  1,5  
" 4. h  ƒ     3  1,5  "

a. 1-II , 2-III, 3-I, 4-IV b. 1-IV , 2-III, 3-I, 4-II

c. 1-I , 2-II, 3-III, 4-IV d. 1-IV , 2-I, 3-III, 4-II

Ans: [d]

77. If n is a least non-negative

negative number then, Total number of imaginary roots of equation

  
   <     ⋯ … . .  † is ___???
a. 106
b. 104
c. 58
d. Cannot be determined
e. NoT

SOl: [b]

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 Here n2, By desecrates rule, no positive or negative roots, only 2 equal roots which are
0, so rest 104 roots would be imaginary.

78. Which one is always wrong for two numbers

a. AM  GM b. AM > GM c. AM > GM^2 d. More than 1
e.NoT

Ans : [a] GM is only defines for positives and AM cannot be less than GM

Š8‹
79. What is the value of ?
Œ
I.         3 ∗ ∗  ∗  II.      2 ∗ ∗ 

a. Question can be answered with the help of statement I alone,

b. Question can be answered with the help of statement II alone,
c. Question can be answered with the help of both statements together,
d. Question cannot be answered even with the help of both statements together.

Ans : [d]

80. How many roots of f x" and g x" are common, where y "     2   9  18 &
“ "  2   7   3  18
a. 0 b. 1 c. 2 d. 3
Sol: [b]

81. A carpenter makes two types of chairs GENERAL and STYLISH. Each type of GENERAL
chair requires 2 hours finishing and 4 hours of polishing; and each model of STYLISH
requires 8 hours of finishing and 6 hours of polishing. The carpenter has 2 finishers and 4
polishers. Each finishers works for 46 hours a week and each polisher works 72 hours a week.
If his Profit on GENERAL chair is Rs.300/- and on STYLISH is Rs.800 /- . Assume
whatever produced in a week is sold in the market. Then what would be maximum profit he
could earn in a week.
a.16200 b. 15800 c. 16100 d. 9800 e. NoT

Sol: [e] 13800

Let GENERAL = x and STYLISH = y
Max Z = 300*x + 800y
constraint to
2 ∗   8 ∗ h ≤ 46*2
4 ∗   6 ∗ h ≤ 72*4

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, h ≥ 0 & »Kˆ‰“‰¥
By solving we would get x= 46 & y =0 then maximum profit = 13800

82. If for positive integral x. y & z, x+2y+3z=16 and the value of   1" ∗ h  2" ∗ i  3"
is maximum then x+y+z=?
a. 6 b. 9 c. 8.5 d. 13

Sol: [b] , By ¼½ ≥ –½ ; x=4, y=3, & z=2

83. Number of nonnegative integral solutions of x  y*z4

a. Less than 10 b. Between 11 to 50 c. Between 51 to 100
d. Between 101to 999 e. More than 1000

Ans: [e] infinity

84. Total number of integral solution of x^2  17y"* 17-y"

a. 6 b. 10 c. 12 d. 14 e. NoT

Ans: [c]

85. If |x||y||z|n has 102 integral solutions for a particular value of n then minimum
value of x*y*z  ?

a. minus infinity b. -3 c.  z{ d. -4 e. NoT

Sol: [d] for 2,2,-1

86. There are how many positive integral ordered" solutions of wxyz  40 which has
exactly 2 odd integers in its solution set
a. C 20,17" b. 6*C 20,17" c. c 39,36" d. NoT ???"

Sol:[b]
2a-1  2b-12c2d 40
abcd 21
so positive integral solution  C 20,3"
so required answer  C 4,2"*C 20,17" 6*C 20,17" for ordered pair"

87. There are how many INTEGRAL ORDERED PAIR solutions of `  ∗ h"  10
a. 18 b. 9 c. 4 d. 8 NoT

Ans [a]
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88. There are how many Non-Negative INTEGRAL ORDERED PAIR solutions of xy100 ,
such that –—˜ , h, 100"  1
a. 70 b. 60 c. 61 d. 40 e. NoT

Sol: [c]
Total - Coprime integral solution  101 - Euler 100"  61

89. How many statements are always correct if they are in determined form"
I. a^n"* b^n"  a*b"^n II. a^n"* a^m" a^ mn"
III. a^n"^m  a^ m*n" IV a^0  1

a. Only one statement is always true

b. Only two statements are always true
c. Only three statements are always true
d. All statements are always true
e. None are always true

Ans : only II, III & IV are always true , I does not hold if both a & b are negatives

90. There are how many integral solutions of 3|x|  5|y| 75

a. 16 b. 24 c. 20 d. 22

Ans : [c]

91. If HCF a,b"  13 & ab 13000 then how many ordered pair of a.b" be possible
a. 20 b. 200 c. 400 d. 2 e. Not

Sol: [c] : Hint : Euler of 1000  400

92. If x*y*z*w  100 has N integral ordered" solutions and |x|*|y|*|z|*|w|  100 has M
integral ordered" solutions then |M-N| ?
a. 300 b. 0 c. 900 d. 700 e. NoT ???"

Sol: e" 800

x*y*z*w  100 , then N C 5,2"* C 5,2"* C 4,0" C 4,2"C 4,4"" 800
& |x|*|y|*|z|*|w|  100 , then M  C 5,2"* C 5,2"*2^4  1600
then |M-N|  800

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101. For many integral value of K;    3 ∗    10  ™ has two positive and 1 negative
roots
a. 28 b. 30 c. 31 d. infinite e. NoT

Sol: [b]
   3 ∗    10  0 , ℎ _ ˆℎ¥‰‰ ¥‡‡ˆ_, 2,5 & 0 , ^y “¥ ¤ℎ ƒ‡¿‰_ §¤‰ ¥L till local
minima in positive X, then it would have two positive and 1 negative solutions
À‡ ® ƒ^K^ƒ ‡y    3 ∗    10 ^_ ¤¤¥‡  30.04 so K can take 1 to 30 total 30
integral value of X

102. For nonnegative x, y, z , if   ∗ h ∗ i   108 , then min value of xyz ?

a. 27 b. 18 c.3 d. 6 e. NoT

Ans : [d] ¼½ ≥ –½

103. What is the minimum value of |2x-3|  |4-x|  |5-x|

a. 4 b. 6 c. 5 d. 0

Ans : [b]6 for x4


Š Œ › Š7 8;Œ 7 6<› 7 ž
104. If ‹  š  œ  16 then z‹7 8;š7 6<œ7 { ?
a. 2 b. -2 c. 1 d. -1 e. NoT

Ans : [c] By Ratio concept

105. There are how many integral values satisfy    h   25  0

a. 64 b. 70 c. 69 d. 50

Ans : [c] answer has to be odd, now check options.

A circle covers equal space in all four quadrants if centre is origin so it has 4*n  1
including origin" integral values in its area.
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106. After solving a quadratic function f x" Ram get roots as 3 and 4 and Shyam get roots
as -3, 4 but they did one mistakes, one of them used wrong value of constant term and
other used wrong coefficient of x. Then what would be value of f 1" if graph of equation
does not touch or cut the x-axis.
a. -18 b. 18 c. -12 d. 12 e. NoT
Sol: [d]
f x" would be either x^2 - 7x - 12 or x^2 -x12
Since D0 it means f x" x^2 -x12 so at f 1 " 1-112  12
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107. If P x" x^100  x^99  x^98 ....x^2x1 , Q x"x^3 -1 then R x" is remainder
when P x" is divided by Q x" then R 2"?
a. 232 b. 2^101 - 1 c. 2 d. 1 e. NoT

Sol: [e] 234

P x" x^100  x^99  x^98 ....x^2x1
&
Q x"x^3 -1
put x^3  1 in P x" we would get R x"  33* x^2x1"  x1
now R 2"  33*721 234
Concept- http://youtu.be/67jg1SVb6Eg

108. Analyse the statements

I. If a function is odd then it must be cross from origin
II. If a function is even then it must not cross from origin

a. Only one statement is always true

b. Both are always true
c. Either one is always true
d. Both are always false
e. NoT

Ans: [e] Both are not always true

I is wrong for f x"1/x

II is wrong for f x"x^2

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