ATHEMATICS

FUNDAMENTAL OF
ATHEMATICS
SSIGNMENT
by

BBandana Gupta
B.Tech, IIT Kharagpur

www.BandanaGupta.com
Fundamental of Mathematics by Bandana Gupta
PART - I : SUBJECTIVE QUESTIONS
Section (A) : Graphs of polynomials
A-1. Draw the graphs of following :
(i) y = 2x3 + 9x2 – 24x + 15 (ii) y = – 3x4 + 4x3 + 12x2 – 2
x 3 – 4x
(iii) y= (iv) y = x 4 – 2x 2 + 5
4
A-2_. Find the values of  for which equation 3x4 – 8x3 – 6x2 + 24x =  has
(i) 1 solution (ii) 2 solution (iii) 3 solution (iv) 4 solution.

Section (B) : Rational Inequalities and Modulus Function

B-1. Solve the following inequalities :
(i) x 2 – 7x + 10 > 0 (ii) x2 – 4x + 3 < 0
2x 1 x2 2x  3
(iii) 
2
2x  5x  2 x 1. (iv)
x2
>
4x  1
.

B-2. Find all real values of x which satisfy x2 – 3x + 2 > 0 and x2 – 3x – 4  0.

B-3. Solve the following inequalities :

x 4 ( x  1)2 ( x  2)
(i) (x – 1) (x + 1) (x – 4)  0
2 3
(ii) 0
( x  3 )3 ( x  4 )

( x  2) ( x 2  2x  1)
(iii) 0
 4  3x  x 2

B-4. Solve the following linear equations

(i) |x|2 – |x| + 4 = 2 x2  3x + 1 (ii) | |x – 1| – 2|= |x – 3|

B-5. Solve the simultaneous equations |x + 2| + y = 5, x – |y| = 1

B-6. Solve the following equations-

(i) |x – 3| = 2x –1 (ii) x |x| = 9
(iii) |x – 3| = x 1 (iv) |x + 1| + |x – 4|2 + x2 =0

B-7. Solve the following inequalities :

3x | x  3 | x
(i) 2 1 (ii) >1
x 4 x2
(iii) |x 2 + 3x| + x2 – 2  0 (iv) |x + 3| > |2x – 1|

Section (C) : Graphical Transformations

C-1. Draw the graph of the followings :

| sin x |
(i) y= (ii) y = | x  1|  2
sin x
   
(iii) y = tan  2x   (iv) y = sin  | x |  
 3  3
C-2. If f(x) = sin(x), then draw the graph of the followings :
(i) y = f(|x|) (ii) y = f(–|x|) (iii) |y| = f(x)

BandanaGupta Corp.Office: 601, Sapphire - II, Sector - 104, Hazipur, Noida (UP) 201 304
Phone : +91-9-9518-2-9518 | CIN U80903UP2018PTC110645 2
Classes Private Limited Website : www.BandanaGupta.com | E-mail : contact@BandanaGupta.com
Fundamental of Mathematics by Bandana Gupta
C-3. Draw the graph of followings
(i) y = – |x + 2| (ii) y = | | x – 1 | – 2|
(iii) y = |x + 2| + |x – 3| (iv) |y| + x = – 1

C-4. Draw the graphs of the following curves :

1 y
(i) y= – (ii) | x | 1 = – 1
| 2x  1 |
| x 2 – 1|
(iii) |y – 3| = |x – 1| (iv) y= nx
( x 2 – 1)

C-5. Draw the graph of y = log1/2 (1 – x).

Section (D) : Irrational inequality, Logarithmic equation and logarithmic inequality

D-1. Solve the following inequalities :
(i) x– 1 | x | < 0 (ii) x2  x  1  x

D-2. Solve the following equations

(i) log10 (x²  12 x + 36) = 2 (ii) log4 log3 log2 x = 0

 1 
(iii) log3  log 9 x   9 x  = 2x. (iv) 2log4 (4 – x) = 4 – log2 (– 2 – x).
 2 

(v) log 102 x + log10x 2 = log 102 2  1 (vi) log4 (log2x) + log2 (log4x) = 2
log x  5
(vii) x 3 = 105 + log x (viii)_ log 1 ( x  1)  log 1 ( x  1)  log (7  x )  1
1
2 2 2

D-3. Solve the following inequalities :

x 1
(i) log0.5 (x + 5)2 > log1/2 (3x – 1) 2. (ii) log1/2 log3 0
x 1

1
(iii) log(3x2  1) 2 < (iv) logx² (2 + x) < 1
2

Section (E) : Greatest Integer, Fractional Part and Signum Function

E-1_. Find the value(s) of x, if {x}, [x] & x are in A.P. (where [ . ] and { . } denotes greatest integer and
fractional part function respectively) :

E-2. Solve the following equations (where [ . ] and { . } denotes greatest integer and fractional part function
respectively) :
(i) 4[x] = x + {x} (ii) | 2x – 1 | = 3 [x] + 2{x}

E-3. Solve the equation 2x + 3 [x] – 4 {–x} = 4 for x (where [x] and {x} denote integral and fractional part of x)

E-4. Draw the graph of y = [x 2 – 2x] , 0  x  2 (where [ . ] denotes greatest integer function).
E-5_. Find the number of integers for which sgn(x2 – 2x – 8) = –1

BandanaGupta Corp.Office: 601, Sapphire - II, Sector - 104, Hazipur, Noida (UP) 201 304
Phone : +91-9-9518-2-9518 | CIN U80903UP2018PTC110645 3
Classes Private Limited Website : www.BandanaGupta.com | E-mail : contact@BandanaGupta.com
Fundamental of Mathematics by Bandana Gupta
Section (F) : Trigonometric equations & inequations

F-1. Find the general solution of trigonometric equations

(i) sin x + cos x =1 (ii) sin9 = sin
(iii) cot – tan = 2. (iv) cosec = cot + 3.

(v) tan + tan2 + 3 tan tan2 = 3.

F-2. Solve the following inequalities :

(i) sin 3x < sin x. (ii) tan2x – (1 + 3 ) tan x + 3 <0

PART - II : ONLY ONE OPTION CORRECT TYPE

Section (A) : Graphs of polynomials
A-1. Number of distinct real roots of the equation – x 3 + 6x 2 – 18x + 4 = 0 is
(A) 0 (B) 1 (C) 2 (D) 3

A-2. Number of distinct real values of x satisfying the equation x 3 – 5x2 + 7x – 3 = 0 is

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

A-3. The equation x 4 – 2x2 = a has 4 distinct real roots, then

(A) a  (–1, 1) (B) a  [–1, 0] (C) a  (–1, –1/2) (D) a  (–1, 0)

Section (B) : Rational Inequalities and Modulus Function

x2
B-1. The complete set of values of x which satisfy the inequations : 5x + 2 < 3x + 8 and < 4 is
x 1
(A) (– , 1) (B) (2, 3) (C) (– , 3) (D) (– , 1)  (2, 3)

x 4  3 x 3  2x 2
B-2. The complete solution set of the inequality  0 is:
x 2  x  30
(A) ( ,  5)  (1, 2)  (6, )  {0} (B) ( ,  5)  [1, 2]  (6, )  {0}
(C) ( ,  5]  [1, 2]  [6, )  {0} (D) none of these

B-3. The number of the integral solutions of x 2 + 9 < (x + 3) 2 < 8x + 25 is :

(A) 1 (B) 2 (C) 3 (D) none

B-4. Number of positive integral values of x satisfying the inequality

( x  4)2013 . ( x  8)2014 ( x  1)
 0 is
x 2016 ( x  2)3 . ( x  3)5 . ( x  6) ( x  9)2012

(A) 0 (B) 1 (C) 2 (D) 3

2 1 2x  1
B-5. Number of non-negative integral values of x satisfying the inequality    0 is
2
x  x 1 x  1 x3  1
(A) 0 (B) 1 (C) 2 (D) 3

B-6. Solutions of |4x + 3| + |3x – 4| = 12 are

7 3 5 2 11 13 3 7
(A) x = – , (B) x = – , (C) x = – , (D) x = – ,
3 7 2 5 7 7 7 5

B-7. The minimum value of f(x) = |x – 1| + |x – 2| + |x – 3| is equal to

(A) 1 (B) 2 (C) 3 (D) 0

BandanaGupta Phone : +91-9-9518-2-9518 | CIN U80903UP2018PTC110645

Corp.Office: 601, Sapphire - II, Sector - 104, Hazipur, Noida (UP) 201 304
4
Classes Private Limited Website : www.BandanaGupta.com | E-mail : contact@BandanaGupta.com
Fundamental of Mathematics by Bandana Gupta
B-8_. Number of solutions of the equations |2x2 + x – 1| = |x2 + 4x + 1|
(A) 1 (B) 2 (C) 3 (D) 4

3x 2 10 x  3
B-9. Number of real solution(s) of the equation x  3 = 1 is :
(A) exactly four (B) exactly three
(C) exactly two (D) exactly one

B-10. If |x2 – 2x – 8| + |x2 + x – 2| = 3 | x + 2|, then the set of all real values of x is
(A) [1, 4]  {–2} (B) [1, 4] (C) [–2, 1]  [4,) (D) (–, –2]  [1, 4]

B-11. The complete set of real ' x ' satisfying ||x – 1| – 1|  1 is:
(A) [0, 2] (B) [ 1, 3] (C) [ 1, 1] (D) [1, 3]

B-12. The set of all real numbers x for which x 2 – |x + 2| + x > 0 is

(A) (– , – 2 )  (2, ) (B) (– , – 2 )  ( 2 , )
(C) (– , – 1)  (1, ) (D) ( 2 , )

B-13_. Solution set of the inequalities |x2 + x – 2|  0 and |x2 – x + 2|  0 is

(A) x  [–2, 1] (B) [–2, –1] (C) {–2, 1} (D) {–2, –1, 1, 2}

Section (C) : Graphical Transformations

C-1. Given the graph of y = f(x), is

which of the following is graph of y = f(1– x) ?

(A) (B)

(C) (D)

  3 
C-2. The number of roots of the equation cot x = + x in  , is ,
2  2 
(A) 3 (B) 2 (C) 1 (D) infinite

C-3. Let y = f(x) has following graph

BandanaGupta Corp.Office: 601, Sapphire - II, Sector - 104, Hazipur, Noida (UP) 201 304
Phone : +91-9-9518-2-9518 | CIN U80903UP2018PTC110645 5
Classes Private Limited Website : www.BandanaGupta.com | E-mail : contact@BandanaGupta.com
Fundamental of Mathematics by Bandana Gupta

then graph of y = – f(|x|)

(A) (B)

(C) (D)

C-4. Graph of y = f(x) is given below :

1
then graph of y = is best represented by
f ( x)

(A) (B)

(C) (D)

BandanaGupta Corp.Office: 601, Sapphire - II, Sector - 104, Hazipur, Noida (UP) 201 304
Phone : +91-9-9518-2-9518 | CIN U80903UP2018PTC110645 6
Classes Private Limited Website : www.BandanaGupta.com | E-mail : contact@BandanaGupta.com
Fundamental of Mathematics by Bandana Gupta
  3 
C-5. Number of solutions of equation 4 tan x + 2x =  belonging to interval   ,  are
 2 2 
(1) 1 (2) 2 (3) 3 (4) 4

C-6. Number of roots of equation 3|x|– |2 – |x|| = 1 is

(A) 0 (B) 2 (C) 4 (D) 7

C-7. Number of solutions of equation x + 1 = x · 2x are

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

Section (D) : Irrational inequality, Logarithmic equation and logarithmic inequality

D-1. Complete set of values of x satisfying the inequality x – 3 < x 2  4 x  5 is

(A) (– , – 5] U [1, ) (B) (– 5, 3] (C) [3, 5) (D) (– 5, 3)
1
D-2. Set of all real values of x satisfying the inequation 4  x2  is
x

(A) [–2, 0)  [ 2  3 , 2  3 ] (B) (–, –2]  ( 2  3 , )

(C) (0, 2  3 ]  [ 2  3 , ) (D) (–, 2  3 ]  [ 2  3 , )

D-3. Number of real solutions of the equation log10 x  = log10 x 2 is :

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

log( x – 3 ) ( x 2 – 4 x  3 )
D-4. Number of integral solutions of inequality  1   1 are
 10 
(A) 0 (B) 1 (C) 2 (D) 3

D-5. Solution set of the inequality, 2  log2 (x 2 + 3x)  0 is :

(A) [ 4, 1] (B) [4, 3)  (0, 1] (C) ( 3)  (1, ) (D) ( 4)  [1, )
D-6. If log0.5 log5 (x 2 – 4) > log0.51, then ‘x’ lies in the interval
(A) (– 3, – 5 )  ( 5 , 3) (B) (– 3, – 5)( 5,3 5)

(C) ( 5 , 3 5 ) (D) 

D-7. If the solution set of the inequality log

0. 9
log5 ( x 2  5  x ) > 0 contains ‘n’ integral values, then n equals to
(A) 7 (B) 8 (C) 6 (D) 10
D-8. Exhaustive set of values of x satisfying log|x| (x2 + x + 1)  0 is
(A) (–1, 0) (B) (–, –1)  (1, ) (C) (–) – {–1, 0, 1} (D) (–, –1)  (–1, 0)  (1, )

D-9. Complete solution set of the inequality 2

log10 x  1 > log10x – 1 is

 1  1
(A) (10, ) (B)  0,   (10,  ) (C)  0,  (D) None of these
 10   10 

Section (E) : Greatest Integer , Fractional Part and Signum Function

1 n  151
E-1. Let f(n) =    , where [.] denotes the greatest integer function, then the value of
 2 100   f (n) is
n 1
(A) 101 (B) 102 (C) 104 (D) 103

BandanaGupta Corp.Office: 601, Sapphire - II, Sector - 104, Hazipur, Noida (UP) 201 304
Phone : +91-9-9518-2-9518 | CIN U80903UP2018PTC110645 7
Classes Private Limited Website : www.BandanaGupta.com | E-mail : contact@BandanaGupta.com
Fundamental of Mathematics by Bandana Gupta
E-2. The number of solutions of the equation 2{x} 2 – 5 {x} + 2 = 0 is (where {.} denotes the fractional part
function)
(A) no solution (B) 1 (C) 2 (D) infinite

E-3. Number of solutions of the equation [2x] – 3 {2x} = 1 (where [ . ] and { . } denotes greatest integer and
fractional part function respectively)
(A) 1 (B) 2 (C) 3 (D) 0

E-4. Graph of y = {x} + {– x} in the interval [–1, 2] is (where {.} denotes fractional part function)

(A) (B)

(C) (D)

E-5. Number of integers satisfying the equation [x] + [2x] + [3x] = {x} + {2x} + 2x 2 (where [.] and {.} denotes
greatest integer function and fractional part function respectively) are
(A) 0 (B) 1 (C) 2 (D) 3

E-6. The number of solutions of equation [ x]  2x  4, is (where [.] represents greater integer function)
(A) 2 (B) 4 (C) 5 (D) none of these

 x 2  5x  4 
 
E-7. The complete solution set of the equation sgn  { x}  = – 1, where {·} is fractional part function, is
 
(A) (1, 4) (B) [1, 4] (C) (– , 1)  (4, ) (D) (1, 2)  (2, 3)  (3, 4)

E-8. Number of integral solutions of the equation sgn(sinx) = 1 in [0, 10] is

(A) infinite (B) 6 (C) 7 (D) 8

Section (F) : Trigonometry equation & Inequality

sin 2 2x  4 sin 4 x – 4 sin 2 x cos 2 x 1
F-1. The least positive value of x satisfying 2 2
= is
4 – sin 2x – 4 sin x 9

   
(A) (B) (C) (D)
3 8 12 6
F-2. If a + b = 3 – cos 4 and a – b = 4 sin 2, then ab is always less than or equal to
1 2 3
(A) (B) 1 (C) (D)
2 3 4

 
F-3. If x  0 ,  then the number of solutions of the equation sin 7x + sin 4x + sin x = 0 is:
 2

(A) 3 (B) 5 (C) 6 (D) None

BandanaGupta Corp.Office: 601, Sapphire - II, Sector - 104, Hazipur, Noida (UP) 201 304
Phone : +91-9-9518-2-9518 | CIN U80903UP2018PTC110645 8
Classes Private Limited Website : www.BandanaGupta.com | E-mail : contact@BandanaGupta.com
Fundamental of Mathematics by Bandana Gupta
F-4. All solutions of the equation 2 sin + tan = 0 are obtained by taking all integral values of m and n are:
2 2
(A) 2n + , n  (B) n & 2m  ± , n, m 
3 3
 
(C) n & m  ± , n, m  (D) n & 2m  ± , n, m 
3 3

F-5. The general solution of the equation 2cos2x = 3.2cos2x  4 is

(A) x = 2nn  (B) x = nn  (C) x = nn  (D) x = n/2n 

   2
F-6. The general solution of the equation tan x + tan  x   + tan  x   = 3 is
 3 3

n  n  n 
(A)  n (B)  n (C)  n (D) none
4 12 3 6 3 12

F-7. The solution of inequality cos 2x  cos x is

    2 2 
(A) x  2n – , 2n  (B) x  2n – , 2n 
 3 3   3 3 

 2   2 
(C) x  2n, 2n   (D) x  2n – , 2n
 3   3 
F-8. The solution of inequality 4tanx – 3.2tanx + 2  0 is
   
(A) x  n, n   ; n  (B) x  n, n –  ; n 
 4  4

   
(C) x  n, n   ; n  (D) x  n, n –  ; n 
 6  6

PART - III : MATCH THE COLUMN

1. If y = f(x) has following graph, then match the column.

(A) y = |f(x)| (p)

(B) y = f(|x|) (q)

(C) y = f(– |x|) (r)

(D) y  | f ( |x| ) | (s)

BandanaGupta Corp.Office: 601, Sapphire - II, Sector - 104, Hazipur, Noida (UP) 201 304
Phone : +91-9-9518-2-9518 | CIN U80903UP2018PTC110645 9
Classes Private Limited Website : www.BandanaGupta.com | E-mail : contact@BandanaGupta.com
Fundamental of Mathematics by Bandana Gupta
2. Let f(x) = x3 – 3x + 2, then match the graphs in Column-II with corresponding expressions in column-I
Column-I Column-II

(A) y = f(|x|) (p)

(B) y = f(–|x|) (q)

(C) |y| = f(x) (r)

(D) |y| = f(|x – 2|) (s)

3. For all real value of 

Column I Column II
1 
(A) A = sin2  + cos4  (p) A   , 1
4 
1 3 
(B) A= (q) A   ,1
3 sin x  4 cos x  2 4 
(C) A = cos6 x + sin6 x (r) A  [2 2 , )
 1 1
(D) A = tan2  + 2 cot2  (s) A  R –  , 
 7 3

4. Match the general solution of following trigonometric equations

Column-I Column-II

1 n 
(A) sin 2 = (p)  ;n
2 2 8
n 
(B) sin 2 + cos 2 = 0 (q)  ;n
2 12
1 n 
(C) cos 4 = (r)  ;n
2 2 6
n 
(D) 3 cot 2 2 = 1 (s)  ( 1)n ;n
2 12

BandanaGupta Corp.Office: 601, Sapphire - II, Sector - 104, Hazipur, Noida (UP) 201 304
Phone : +91-9-9518-2-9518 | CIN U80903UP2018PTC110645 10
Classes Private Limited Website : www.BandanaGupta.com | E-mail : contact@BandanaGupta.com
Fundamental of Mathematics by Bandana Gupta
ANSWERS A-2_. (i)   {– 19} (ii)  (– 19, 8)  (13, )
(iii)  {8, 13} (iv)  (8, 13)
PART - I
Section (B) :
Section (A) : B-1. (i) x  (– , 2)  (5, ) (ii) x  (1, 3)
A-1. (iii) x  (2, 1)  ( 2/3,  1/2)
1 
(iv) x (– , – 2)   , 1  (4, )
4 

B-2. x  [–1, 1)  (2, 4]

B-3. (i) x  (–, –1]  {1}  [4, )

(ii) x  (–4, –1)  (–1, 0)  (0, 2)  (3, )
(i) (iii) x  (–, –2]  {1}

B-4. (i) x = – 3, 3 (ii) x  [1, )

B-5. x = 2, y = 1

4
B-6. (i) (ii) 3 (iii) 2, 5 (iv) 
3

B-7. (i) x  (–, –4]  [–1, 1]  [4, )

(ii) x  (–5, –2)  (–1, )
 2 1 
(iii) x    ,     ,  
 3  2 

(ii)  2 
(iv) x    , 4 
 3 
Section (C) :

C-1. (i)

(iii)

(ii)

(iv)
(iii)

BandanaGupta Corp.Office: 601, Sapphire - II, Sector - 104, Hazipur, Noida (UP) 201 304
Phone : +91-9-9518-2-9518 | CIN U80903UP2018PTC110645 11
Classes Private Limited Website : www.BandanaGupta.com | E-mail : contact@BandanaGupta.com
 Fundamental of Mathematics by Bandana Gupta
(iv)

(iii)

C-2.

(iv)

(i)

C-4. (i)
(ii)

(iii)
(ii)

C-3. (i)

(iii)

(ii)
(iv)

BandanaGupta Corp.Office: 601, Sapphire - II, Sector - 104, Hazipur, Noida (UP) 201 304
Phone : +91-9-9518-2-9518 | CIN U80903UP2018PTC110645 12
Classes Private Limited Website : www.BandanaGupta.com | E-mail : contact@BandanaGupta.com
Fundamental of Mathematics by Bandana Gupta
PART - II
Section (A) :
C-5. A-1. (B) A-2. (B) A-3. (D)

Section (B) :
B-1. (D) B-2. (B) B-3. (D)

Section (D) : B-4. (D) B-5. (D) B-6. (C)

B-7. (B) B-8_. (D) B-9. (B)

D-1. (i) [– 1, ( 5 – 1)/2) (ii) (–)
B-10. (A) B-11. (B) B-12. (B)
D-2. (i) x = 16 or x =  4 (ii) 8
B-13_. (C)
1 1
(iii) {1/3} (iv) {– 4} (v) ,
20 5 Section (C) :
(vi) x = 16 (vii) {10 5 ,103} (viii)_ x = 3 C-1. (B) C-2. (A) C-3. (D)
D-3. (i) (–, –5)(–5, –1)  (3, ) C-4. (A) C-5. (B) C-6. (B)
(ii) [2, ) (iii) ( ,  1)  (1, )

C-7. (B)
(iv) x  (– 2, –1)  (– 1, 0)  (0, 1)  (2, )
Section (D) :
Section (E) :
D-1. (A) D-2. (A) D-3. (C)
1
E-1_. x = 0, 3/2 E-2. (i) { 0 } (ii)   D-4. (A) D-5. (B) D-6. (A)
4
D-7. (B) D-8. (D) D-9. (B)
3  Section (E) :
E-3.   E-4.
2
E-1. (C) E-2. (D) E-3. (C)
E-5_. 5
E-4. (D) E-5. (C) E-6. (B)
Section (F) :
E-7.(D) E-8. (B)

  Section (F) :
F-1. (i) x = n + (–1)n – ; n 
4 4
F-1. (D) F-2. (B) F-3. (B)
m ( 2m  1)
(ii) , m  or , m  F-4. (B) F-5. (B) F-6. (C)
4 10
F-7. (B) F-8. (A)
 1  2
(iii)  n   , n  (iv) 2n + , n 
 4 2 3 PART - III
1. (A)  (r) ; (B)  (p) ; (C)  (q) ; (D)  (s)
 1 
(v)  n   , n 
 3  3
2. (A)  (s) ; (B)  (q) ; (C)  (p) ; (D)  (r)
F-2. (i) ( (8n –1)/4, 2n)  ((8n + 1)/4,
(8n + 3)/4)(+2n,(8n+5)/4),(n  Z) 3. (A)  (q) ; (B)  (s) ; (C)  (p) ; (D)  (r)
(ii) ((4n + 1)/4,  (3n + 1)/3), (n  Z)
4. (A)  (s) ; (B)  (p) ; (C)  (q) ; (D)  (r)

BandanaGupta Corp.Office: 601, Sapphire - II, Sector - 104, Hazipur, Noida (UP) 201 304
Phone : +91-9-9518-2-9518 | CIN U80903UP2018PTC110645 13
Classes Private Limited Website : www.BandanaGupta.com | E-mail : contact@BandanaGupta.com