mathematics - careerpoint.ac.in 39 (1).pdf · chapter wise questions asked in iit-jee / jee...

MATHEMATICS

Years IIT-JEE CHAPTERWISE SOLVED PAPERS

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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

Content Development for their efforts to make this wonderful book.

Career Point Ltd.

CONTENTS

Chapter Page No.

1. Logarithm and their Properties 1‐4 ♦ Answers

♦ Solutions

2 3

2. Theory of Equations 5‐22 ♦ Answers

♦ Solutions

10 11

3. Sequences & Series 23‐42 ♦ Answers

♦ Solutions

29 30

4. Complex Numbers 43‐70 ♦ Answers

♦ Solutions

50 51

5 Permutation & Combination 71‐80 ♦ Answers

♦ Solutions

75 76

6. Binomial Theorem 81‐92 ♦ Answers

♦ Solutions

84 85

7. Probability 93‐124 ♦ Answers

♦ Solutions

103 105

8. Determinants 125‐140 ♦ Answers

♦ Solutions

129 130

9. Matrices 141‐152

♦ Answers ♦ Solutions

145 146

10. Functions 153‐170


158 159

11. Limits 171‐180


174 175

Cont....

Chapter Page No.

12. Continuity & Differentiability 181‐204


188 189

13 Differentiation 205‐214


208 209

14. Tangent & Normal 215‐224


217 218

15. Monotonicity 225‐234


228 229

16. Maxima & Minima 235‐252


240 241

17. Indefinite Integration 253‐262


255 256

18. Definite Integration 263‐304


274 275

19. Area under the curve 305‐328


310 311

20. Differential Equation 329‐346


333 334

21. Point & Straight Lines 347‐368


353 354

22. Circle 369‐398


376 377

23. Parabola 399‐418


404 405

Cont...

Chapter Page No.

24. Ellipse 419‐432 ♦ Answers

♦ Solutions

422 423

25. Hyperbola 433‐442


436 437

26. Vectors 443‐482


455 457

27. 3D Geometry 483‐498


489 490

28. Trigonometric ratio & Identities 499‐506


501 502

29. Trigonometric Equations 507‐520


511 512

30. Inverse Trigonometric Functions 521‐528


523 524

31. Properties of Triangles 529‐550


534 535

32. Height & Distance 551‐560


553 554

33. Mathematical Induction 561‐572

♦ Solutions

563

34. Miscellaneous 573‐578 ♦ Answers

♦ Solutions

575 576

35. Model Test Papers 579‐608 ♦ Practice Test‐1 [Paper‐1]

♦ Practice Test‐1 [Paper‐2] ♦ Practice Test‐2 [Paper‐1] ♦ Practice Test‐2 [Paper‐2]

579 585 591 599

,,

1 Logarithm and their Properties

Chapter

ONLY ONE CORRECT ANSWER

1. The least value of the expression 2 log10 x – logx (0.01), for x > 1, is : [1980] (A) 10 (B) 2 (C) – 0.01 (D) None of these 2. 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. The equation 45–xlog)x(log

43

22

2x

+ = 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 : [1990, 2M] (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 : [2001] (A) 3 (B) 1 (C) 2 (D) 0 6. Let (x0, y0) be the solution of the following

equations (2x)ln 2 = (3y)ln 3 3ln x = 2ln y Then x0 is [2011]

(A) 61 (B)

31

(C) 21 (D) 6

ONE OR MORE THAN ONE CORRECT ANSWERS

1. If 3x = 4x – 1, then x = [2013]

(A) 12log2

2log2

3

3

− (B)

3log22

2−

(C) 3log1

1

4− (D)

13log23log2

2

2

−

ANALYTICAL & DESCRIPTIVE QUESTIONS

1. Solve for x the following equation : log(2x + 3) (6x2 + 23x + 21)

= 4 – log(3x + 7) (4x2 + 12x + 9) [1987, 3M]

2. The value of

−−−+ ...

2314

2314

2314

231log6

23

is [2012]

TOPIC- WISE JEE Advanced Questions with Solutions

2

ANSWERS

Only One Correct Answer

1. (D) 2. (A) 3. (B) 4. (C) 5. (B) 6. (C)

One or More than One Correct Answers

1. (A,B,C)

Analytical & Descriptive Question

1. x = 41

− 2. 4

3 LOGARITHM AND THEIR PROPERTIES

SOLUTIONS

Only One Correct Answer

1. Here, 2 log10 x – logx (10)(–2) = 2 log10x + 2 logx 10

= 2 log10 x + 2xlog

1

10

= 2

+xlog

1xlog10

10 …(1)

using, A.M. ≥ G.M., we get

2

xlog1xlog10

10 + ≥

2/1

1010 xlog

1.xlog

⇒ log10 x + xlog

1

10 ≥ 2 …(2)

or 2 log10 x – logx (0.01) ≥ 4 ∴ least value is 4. 2. log0.3 (x – 1) < log0.09 (x – 1) Here, x – 1 > 0 and log(0.3) (x – 1) < 2)3.0(log (x – 1)

⇒ x > 1 and log0.3 (x – 1) < 21 log0.3 (x – 1)

⇒ x > 1 and log(0.3) (x – 1) < 0 ⇒ x > 1 and x – 1 > 1 ⇒ x > 1 and x > 2 x ∈ (2, ∞)

3. 45–xlog)x(log

43

22

2x

+ = 2

⇒ 43 (log2 x)2 + log2 x –

45 = logx 2

⇒ 43 (log2 x)2 + log2 x –

45 =

xlog21

2

⇒ 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, – 31

⇒ log2 x = 1, – 2, – 31

⇒ x = 2, 3/121 ,

41

4. Let x = log2 7 ⇒ 2x = 7. Which is only possible for irrational number.

5. log4 (x – 1) = log2 (x – 3) = 2/14log (x – 3)

⇒ log4 (x – 1) = 2 log4 (x – 3) ⇒ log4 (x – 1) = log4 (x – 3)2 ⇒ (x – 3)2 = x – 1 ⇒ x2 + 9 – 6x = x – 1 ⇒ x2 – 7x + 10 = 0 ⇒ x2 – 5x – 2x + 10 = 0 ⇒ x(x – 5) – 2(x – 5) = 0 ⇒ (x – 2) (x – 5) = 0 ⇒ x = 2, or x = 5 Hence, x = 5 [x = 2 makes log (x – 3) undefined]. Therefore, (B) is the answer. 6. (2x)ln 2 = (3y)ln 3 ln 2 (ln 2 + ln x) = ln 3 (ln 3 + ln y) ln 2 . ln x – ln 3 ln y = (ln 3)2 – (ln 2)2 .....(1) 3ln x = 2ln y ln x . ln 3 = lny . ln 2

ln y = ln x 23

nnl

l .....(2)

Solving (1) & (2)

ln x = – ln 2 ⇒ x = 21

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

TOPIC- WISE JEE Advanced Questions with Solutions

4

x = 12log2

2log2

3

3

−

= 3log2

2

2−

= 3log1

1

4−

Analytical & Descriptive Question 1. log(2x + 3) (6x2 + 23x + 21) = 4 – log(3x + 7) (4x2 + 12x + 9) ⇒ log(2x + 3) (2x + 3).(3x + 7) = 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 + y2 – 3 = 0

⇒ 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, – 41 .

∴ x = – 2, – 4, – 41

But, log exists only when, 6x2 + 23x + 21 > 0. 4x2 + 12x + 9 > 0, 2x + 3 > 0 and 3x + 7 > 0

⇒ x > – 23

∴ x = – 41 is the only solution.

2. Let x = ...23

1423

1423

14 −−−

x2 = 4 – 23

1 x

23 x2 + x – 12 2 = 0

x = 26

212.23.411 ++−

x = 26171+− =

238

6 + log3/2

94 = 6 + log3./2

2

23 −

= 6 – 2 = 4

mathematics - careerpoint.ac.in 39 (1).pdf · chapter wise questions asked in iit-jee / jee...

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