Years IIT-JEE
CHAPTERWISE SOLVED PAPERS
MATHEMATICS
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Preface
Whenever a student decides to prepare for any examination, her/his first and foremost curiosity
about the type of questions that he/she has to face. This becomes more important in the context of
competitive examinations where there is neck-to-neck race.
We feel great pleasure to present before you this book. We have made an attempt to provide
chapter wise questions asked in IIT-JEE / JEE Advanced from 1978 to 2016 along with solutions.
Solutions to the questions are not just sketch rather have been written in such a manner that the
students will be able to under the application of concept and can answer some other related
questions too.
We firmly believe that the book in this form will definitely help a genuine, hardworking student.
We have tried our best to keep errors out of this book. Comment and criticism from readers will be
highly appreciated and incorporated in the subsequent edition.
We wish to utilize the opportunity to place on record our special thanks to all team members of
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Career Point Ltd.
CONTENTS
Chapter
Page No.
1.
Logarithm and their Properties
♦ Answers
♦ Solutions
1‐4
2
3
2.
Theory of Equations
♦ Answers
♦ Solutions
5‐22
10
11
3.
Sequences & Series
♦ Answers
♦ Solutions
23‐42
29
30
4.
Complex Numbers
♦ Answers
♦ Solutions
43‐70
50
51
5
Permutation & Combination
♦ Answers
♦ Solutions
71‐80
75
76
6.
Binomial Theorem
♦ Answers
♦ Solutions
81‐92
84
85
7.
Probability
♦ Answers
♦ Solutions
93‐124
103
105
8.
Determinants
♦ Answers
♦ Solutions
125‐140
129
130
9.
Matrices
♦ Answers
♦ Solutions
141‐152
145
146
10.
Functions
♦ Answers
♦ Solutions
153‐170
158
159
11.
Limits
♦ Answers
♦ Solutions
171‐180
174
175
Cont....
Chapter
Page No.
12.
Continuity & Differentiability
♦ Answers
♦ Solutions
181‐204
188
189
13
Differentiation
♦ Answers
♦ Solutions
205‐214
208
209
14.
Tangent & Normal
♦ Answers
♦ Solutions
215‐224
217
218
15.
Monotonicity
♦ Answers
♦ Solutions
225‐234
228
229
16.
Maxima & Minima
♦ Answers
♦ Solutions
235‐252
240
241
17.
Indefinite Integration
♦ Answers
♦ Solutions
253‐262
255
256
18.
Definite Integration
♦ Answers
♦ Solutions
263‐304
274
275
19.
Area under the curve
♦ Answers
♦ Solutions
305‐328
310
311
20.
Differential Equation
♦ Answers
♦ Solutions
329‐346
333
334
21.
Point & Straight Lines
♦ Answers
♦ Solutions
347‐368
353
354
22.
Circle
♦ Answers
♦ Solutions
369‐398
376
377
23.
Parabola
♦ Answers
♦ Solutions
399‐418
404
405
Cont...
Chapter
,,
Page No.
24.
Ellipse
♦ Answers
♦ Solutions
419‐432
422
423
25.
Hyperbola
♦ Answers
♦ Solutions
433‐442
436
437
26.
Vectors
♦ Answers
♦ Solutions
443‐482
455
457
27.
3D Geometry
♦ Answers
♦ Solutions
483‐498
489
490
28.
Trigonometric ratio & Identities
♦ Answers
♦ Solutions
499‐506
501
502
29.
Trigonometric Equations
♦ Answers
♦ Solutions
507‐520
511
512
30.
Inverse Trigonometric Functions
♦ Answers
♦ Solutions
521‐528
523
524
31.
Properties of Triangles
♦ Answers
♦ Solutions
529‐550
534
535
32.
Height & Distance
♦ Answers
♦ Solutions
551‐560
553
554
33.
Mathematical Induction
♦ Solutions
561‐572
563
34.
Miscellaneous
♦ Answers
♦ Solutions
35.
Model Test Papers
♦ Practice Test‐1 [Paper‐1]
♦ Practice Test‐1 [Paper‐2]
♦ Practice Test‐2 [Paper‐1]
♦ Practice Test‐2 [Paper‐2]
573‐578
575
576
579‐608
579
585
591
599
Chapter
1
Logarithm and their Properties
ONLY ONE CORRECT ANSWER
1.
2.
The least value of the expression
[1980]
2 log10 x – logx (0.01), for x > 1, is :
(A) 10
(B) 2
(C) – 0.01
(D) None of these
If log0.3 (x – 1) < log0.09 (x – 1), then x lies in
the interval :
[1985, 2M]
(A) (2, ∞)
(B) (1, 2)
(C) (– 2, – 1)
(D) None of these
3
3.
ONE OR MORE THAN ONE CORRECT ANSWERS
The equation x 4
(log 2 x ) 2 + log 2 x –
5
4
=
2 has :
[1987, 2M]
(A) at least one real solution
(B) exactly three real solutions
(C) exactly one irrational solution
(D) complex roots
4.
The number log2 7 is :
(A) an integer
(B) a rational number
(C) an irrational number
(D) a prime number
5.
The number of solutions of
log4 (x – 1) = log2 (x – 3) is :
(A) 3
(B) 1
(C) 2
(D) 0
6.
1.
If 3x = 4x – 1, then x =
2 log 3 2
(A)
2 log 3 2 − 1
(C)
1
1 − log 4 3
[2013]
2
(B)
2 − log 2 3
(D)
2 log 2 3
2 log 2 3 − 1
ANALYTICAL & DESCRIPTIVE QUESTIONS
1.
2.
Solve for x the following equation :
log(2x + 3) (6x2 + 23x + 21)
= 4 – log(3x + 7) (4x2 + 12x + 9)
[1987, 3M]
The value of


 1

1
1
1
6 + log 3 
4−
4−
4−
... 
3 2
3 2
3 2 
23 2


is
[2012]
[1990, 2M]
[2001]
Let (x0, y0) be the solution of the following
equations
(2x)ln 2 = (3y)ln 3
3l n x = 2 l n y
[2011]
Then x0 is
1
1
(A)
(B)
6
3
1
(C)
(D) 6
2
2
TOPIC- WISE JEE Advanced
Questions with Solutions
ANSWERS
¾ Only One Correct Answer
1. (D)
2. (A)
3. (B)
¾ One or More than One Correct Answers
1. (A,B,C)
¾ Analytical & Descriptive Question
1. x = −
1
4
2. 4
4. (C)
5. (B)
6. (C)
3
LOGARITHM AND THEIR PROPERTIES
SOLUTIONS
Only One Correct Answer
1.
Here, 2 log10 x – logx (10)(–2)
= 2 log10x + 2 logx 10
1
= 2 log10 x + 2
log10 x
⇒ log2 x = 1, – 2, –
⇒ x = 2,
4.

1 
= 2 log10 x +

log10 x 

1
1/ 3
2
,
1
3
1
4
Let x = log2 7
⇒ 2x = 7.
Which is only possible for irrational number.
…(1)
using, A.M. ≥ G.M., we get
log10 x +
1
log10 x
2
⇒ log10 x +
5.
1/ 2

1 

≥  log10 x.
log10 x 

1
≥2
log10 x
⇒ log4 (x – 1) = 2 log4 (x – 3)
⇒ log4 (x – 1) = log4 (x – 3)2
⇒ (x – 3)2 = x – 1
…(2)
⇒ x2 + 9 – 6x = x – 1
⇒ x2 – 7x + 10 = 0
or 2 log10 x – logx (0.01) ≥ 4
2.
∴ least value is 4.
⇒ x2 – 5x – 2x + 10 = 0
log0.3 (x – 1) < log0.09 (x – 1)
Here, x – 1 > 0
⇒ (x – 2) (x – 5) = 0
⇒ x(x – 5) – 2(x – 5) = 0
⇒ x = 2, or x = 5
Hence, x = 5 [x = 2 makes log (x – 3) undefined].
Therefore, (B) is the answer.
and log(0.3) (x – 1) < log(0.3) 2 (x – 1)
⇒ x > 1 and log0.3 (x – 1) <
1
log0.3 (x – 1)
2
⇒ x > 1 and log(0.3) (x – 1) < 0
3 ln x = 2 ln y
x ∈ (2, ∞)
⇒
⇒
ln x . ln 3 = lny . ln 2
=
ln y = ln x
2
3
5
(log2 x)2 + log2 x –
= logx
4
4
2
3
5
1
(log2 x)2 + log2 x –
=
4
4
2 log 2 x
⇒ 3(log2 x)3 + 4(log2 x)2 – 5(log2 x) – 2 = 0
Put, log2 x = y
⇒ 3y3 + 4y2 – 5y – 2 = 0
⇒ (y – 1) (y + 2) (3y + 1) = 0
⇒ y = 1, – 2, –
(2x)ln 2 = (3y)ln 3
ln 2 . ln x – ln 3 ln y = (ln 3)2 – (ln 2)2 .....(1)
⇒ x > 1 and x > 2
3
5
(log 2 x ) 2 + log 2 x –
4
x4
6.
ln 2 (ln 2 + ln x) = ln 3 (ln 3 + ln y)
⇒ x > 1 and x – 1 > 1
3.
log4 (x – 1) = log2 (x – 3) = log 41/ 2 (x – 3)
1
3
ln3
ln 2
.....(2)
Solving (1) & (2)
ln x = – ln 2 ⇒ x =
1
2
One or More than One Correct Answers
1.
3x = 4x – 1
Take log3 both sides
x = (x – 1) log3 4
x = (x – 1) 2log3 2
x (1 – 2log3 2) = – 2log3 2
4
TOPIC- WISE JEE Advanced
x=
=
=
2 log 3 2
2 log 3 2 − 1
2.
2
2 − log 2 3
1
1 − log 4 3
Let x =
4−
x2 = 4 –
1
1
3 2
4−
1
3 2
x
x=
− 1 + 1 + 4.3 2 .12 2
6 2
log(2x + 3) (6x2 + 23x + 21)
= 4 – log(3x + 7) (4x2 + 12x + 9)
x=
8
−1 + 17
=
6 2
3 2
⇒ log(2x + 3) (2x + 3).(3x + 7)
4
6 + log3/2   = 6 + log3./2
9
=6–2=4
= 4 – log(3x + 7) (2x + 3)2
⇒ 1 + log(2x + 3) (3x + 7)
= 4 – 2 log(3x + 7) (2x + 3)
Put log(2x + 3) (3x + 7) = y
⇒ y+
2
–3=0
y
⇒ y2 – 3y + 2 = 0
⇒ (y – 1) (y – 2) = 0
⇒ y = 1 or y = 2
⇒ log(2x + 3) (3x + 7) = 1
or log(2x + 3) (3x + 7) = 2
⇒ 3x + 7 = 2x + 3
or (3x + 7) = (2x + 3)2
⇒ x=–4
or 3x + 7 = 4x2 + 12x + 9
4x2 + 9x + 2 = 0
4x2 + 8x + x + 2 = 0
(4x + 1) (x + 2) = 0
x = – 2, –
1
.
4
∴ x = – 2, – 4, –
4−
3 2 x2 + x – 12 2 = 0
Analytical & Descriptive Question
1.
3 2
Questions with Solutions
1
4
But, log exists only when, 6x2 + 23x + 21 > 0.
4x2 + 12x + 9 > 0,
2x + 3 > 0 and 3x + 7 > 0
⇒ x>–
3
2
∴ x=–
1
is the only solution.
4
3
 
2
−2
1
3 2
...