MATHEMATICS - careerpoint.ac.in 39 (1).pdf · chapter wise questions asked in IIT-JEE / JEE...
Transcript of 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
♦ Answers ♦ Solutions
158 159
11. Limits 171‐180
♦ Answers ♦ Solutions
174 175
Cont....
Chapter Page No.
12. Continuity & Differentiability 181‐204
♦ Answers ♦ Solutions
188 189
13 Differentiation 205‐214
♦ Answers ♦ Solutions
208 209
14. Tangent & Normal 215‐224
♦ Answers ♦ Solutions
217 218
15. Monotonicity 225‐234
♦ Answers ♦ Solutions
228 229
16. Maxima & Minima 235‐252
♦ Answers ♦ Solutions
240 241
17. Indefinite Integration 253‐262
♦ Answers ♦ Solutions
255 256
18. Definite Integration 263‐304
♦ Answers ♦ Solutions
274 275
19. Area under the curve 305‐328
♦ Answers ♦ Solutions
310 311
20. Differential Equation 329‐346
♦ Answers ♦ Solutions
333 334
21. Point & Straight Lines 347‐368
♦ Answers ♦ Solutions
353 354
22. Circle 369‐398
♦ Answers ♦ Solutions
376 377
23. Parabola 399‐418
♦ Answers ♦ Solutions
404 405
Cont...
Chapter Page No.
24. Ellipse 419‐432 ♦ Answers
♦ Solutions
422 423
25. Hyperbola 433‐442
♦ Answers ♦ Solutions
436 437
26. Vectors 443‐482
♦ Answers ♦ Solutions
455 457
27. 3D Geometry 483‐498
♦ Answers ♦ Solutions
489 490
28. Trigonometric ratio & Identities 499‐506
♦ Answers ♦ Solutions
501 502
29. Trigonometric Equations 507‐520
♦ Answers ♦ Solutions
511 512
30. Inverse Trigonometric Functions 521‐528
♦ Answers ♦ Solutions
523 524
31. Properties of Triangles 529‐550
♦ Answers ♦ Solutions
534 535
32. Height & Distance 551‐560
♦ Answers ♦ Solutions
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