Objective Mathematics

97 Pages·2012·7.43 MB·English

IIT-JEE-2013
Objective Mathematics

{Mains & Advance}

Er.L.K.Sharma
B.E.(CIVIL), MNIT,JAIPUR(Rajasthan)
© Copyright L.K.Sharma 2012.
Er. L.K.Sharma an engineering graduate
from NIT, Jaipur (Rajasthan), {Gold
medalist, University of Rajasthan} is a well
known name among the engineering
aspirants for the last 15 years. He has
been honored with "BHAMASHAH
AWARD" two times for the academic
excellence in the state of Rajasthan. He is
popular among the student community
for possessing the excellent ability to
communicate the mathematical concepts
in analytical and graphical way.

He has worked with many IIT-JEE coaching institutes of Delhi and
Kota, {presently associated with Guidance, Kalu Sarai, New Delhi as
senior mathematics faculty}. He has been a senior mathematics
{IIT-JEE} faculty at Delhi Public School, RK Puram for five years.

He is actively involved in the field of online teaching to the
engineering aspirants and is associated with iProf Learning
Solutions India (P) Ltd for last 3 years. As a premium member of
www.wiziq.com (an online teaching and learning portal), he has
delivered many online lectures on different topics of mathematics at
IIT-JEE and AIEEE level.{some of the free online public classes at
wizIQ can be accessed at http://www.wiziq.com/LKS }.

Since last 2 years many engineering aspirants have got tremendous
help with the blog “mailtolks.blogspot.com” and with launch of the
site “mathematicsgyan.weebly.com”, engineering aspirants get the
golden opportunity to access the best study/practice material in
mathematics at school level and IIT-JEE/AIEEE/BITSAT level. The
best part of the site is availability of e-book of “OBJECTIVE
MATHEMATICS for JEE- 2013” authored by Er. L.K.Sharma,
complete book with detailed solutions is available for free download
as the PDF files of different chapters of JEE-mathematics.

© Copyright L.K.Sharma 2012.
Contents
1. Quadratic Equations 1 - 8
2. Sequences and Series 9 - 16
3. Complex Numbers 17 - 24
4. Binomial Theorem 25 - 30
5. Permutation and Combination 31 - 36
6. Probability 37 - 44
7. Matrices 45 - 50
8. Determinants 51 - 57
9. Logarithm 58 - 61
10. Functions 62 - 70
11. Limits 71 - 76
12. Continuity and Differentiability 77 - 82
13. Differentiation 83 - 88
14. Tangent and Normal 89 - 93
15. Rolle's Theorem and Mean Value Theorem 94 - 97
16. Monotonocity 98 - 101
17. Maxima and Minima 102 - 108
18. Indefinite Integral 109 - 113
19. Definite Integral 114 - 122
20. Area Bounded by Curves 123 - 130
21. Differential Equations 131 - 137
22. Basics of 2D-Geometry 138 - 141
23. Straight Lines 142 - 148
24. Pair of Straight Lines 149 - 152
25. Circles 153 - 160
26. Parabola 161 - 167
27. Ellipse 168 - 175
28. Hyperbola 176 - 182
29. Vectors 183 - 191
30. 3-Dimensional Geometry 192 - 199
31. Trigonometric Ratios and Identities 200 - 206
32. Trigonometric Equations and Inequations 207 - 212
33. Solution of Triangle 213 - 218
34. Inverse Trigonometric Functions 219 - 225
Multiple choice questions with ONE correct answer : 7. Total number of integral solutions of inequation
( Questions No. 1-25 )
x2(3x4)3(x2)4
1. If the equation | x – n | = (x + 2)2 is having exactly 0 is/are :
(x5)5(72x)6
three distinct real solutions , then exhaustive set of
values of 'n' is given by :
(a) four (b) three
 5 3  5 3 (c) two (d) only one
(a)  ,  (b)  ,2, 
 2 2  2 2
8. If exactly one root of 5x2 + (a + 1) x + a = 0 lies in the
 5 3  9 7
(c)  ,  (d)  ,2,  interval x(1,3) , then
 2 2  4 4 s
(a) a > 2 c
i
t
2. Let a , b , c be distinct real numbers , then roots of (b) – 12 < a < – a3
(x – a)(x – b) = a2 + b2 + c2 – ab – bc – ac , are : (c) a > 0 m
(a) real and equal (b) imaginary (d) noene of these
h
(c) real and unequal (d) real t
a
9. If both rooats of 4x2 – 20 px + (25 p2 +15p – 66) = 0 are
M less tmhan 2 , then 'p' lies in :
3. If 2x312x23x160 is having three po sitive r
e a
r(eaa) l4 roots , then '' musTt -beJ :(bE ) c8E ti v K .S h (a) 45 ,2 (b) (2,)
(c) 0 II (ed) 2 L .  4
b j r. (c) 1,  (d) (,1)
E  5
O
4. If a , b , c are distinct real numbers , then number of
real roots of equation 10. If x2 – 2ax + a2 + a – 3 0 xR , then 'a' lies in
(xa)(xb) (xb)(xc) (xc)(xa)
  1 (a) [3,) (b) (,3]
(ca)(cb) (ab)(ac) (bc)(ba)
(c) [–3 , ) (d) (,3]
is/are :
(a) 1 (b) 4
11. If x3 + ax + 1 = 0 and x4 + ax2 + 1 = 0 have a common
(c) finitely many (d) infinitely many root , then value of 'a' is
(a) 2 (b) –2
5. If ax2 + 2bx + c = 0 and ax2 + 2bx + c = 0 have a
1 1 1 (c) 0 (d) 1
a b c
common root and , , are in A.P. , then
a b c 12. If x2 + px + 1 is a factor of ax3 + bx + c , then
1 1 1
a , b , c are in : (a) a2 + c2 + ab = 0
1 1 1
(b) a2 – c2 + ab = 0
(a) A.P. (b) G.P.
(c) a2 – c2 – ab = 0
(c) H.P. (d) none of these
(d) a2 + c2 – ab = 0
6. If all the roots of equations
(a1)(1xx2)2 (a1)(x4x21) 13. If expression a2(b2c2)x2b2(c2a2)xc2(a2b2)
is a perfect square of one degree polynomial of x ,
are imaginary , then range of 'a' is :
then a2 , b2 , c2 are in :
(a) (,2] (b) (2,)
(a) A.P. (b) G.P.
(c) (2,2) (d) (2,)
(c) H.P. (d) none of these
[ 1 ] Mathematics for JEE-2013
e-mail: [email protected]
Author - Er. L.K.Sharma
www.mathematicsgyan.weebly.com
14. The value of  for which the quadratic equation 22. If real polynomial f (x) leaves remainder 15 and
x2 – (sin–2) x – (1 + sin) = 0 (2x + 1) when divided by (x – 3) and (x – 1)2
has roots whose sum of squares is least , is : respectively , then remainder when f (x) is divided
by (x – 3)(x – 1)2 is :
 
(a) (b)
4 3 (a) 2x – 1 (b) 3x2 + 2x – 4
(c) 2x2 – 2x + 3 (d) 3x + 6
 
(c) (d)
2 6
23. Let aR and equation 3x2 + ax + 3 = 0 is having
15. If cos,sin,sin are in G.P. , then roots of one of the root as square of the another root , then 'a'
is equal to :
x22(cot)x10 are :
(a) 2/3 (b) –3
(a) equal (b) real
(c) 3 (d) 1/3
(c) imaginary (d) greater than 1
24. If the quadratic equation
x2ax2 a2 (x + 1)2 + b2(2x2 – x + 1) – 5x2 – 3 = 0
16. If 3 2 holds  xR , then 'a'
x2x1 is satisfied for all xR, then number of ordered pairs
belongs to : (a , b) which are possible is/are :
(a) 0 s(b) 1
(a) [–2 , 1) (b) (–2 , 1) c
i
(c) R – [–2 , 2] (d) (–2 , 2) (c) finitely many t (d) infinitely many
a
m
25. The smallest value of 'k' for which both the roots of
17. The number of real solutions of the equation
the eqeuation x2 – 8kx + 16(k2 – k + 1) = 0 are real and
h
2x 2x4 4 is/are : distinct and have values at least 4 , is :
t
(a) 0 (b) 1 a (a) 1 a (b) 2
M m
(c) 2 (d) 4 (c) –1 (d) 3
r
e a
E v h
18. Laxe2t + b,x + bce =th 0eI ,rI otohTtes-n Jorfo eEoqtusca odtfr aitthice eeqquuLaa.ttiiooKnn.S 26. Lxet [ 1f ,(3x)] ,= w (hx e–re 3kk)(xR –, kth –en 3 c)o mbep lneeteg asteitv oef fvoarl uaelsl
ax2 – bx (x – 1) + c(x – 1)2 j= 0 are : . of 'k' belong to :
b r
E
  O    1 1  1
(a) , (b) , (a)  ,  (b) 0, 
1 1 1 1  2 2  3
1 1 1 1 1 
(c) , (d) , (c)  ,3 (d) 3,0
    3 
19. If the equation x510a3x2b4xc5 0 has 3 equal
27. Let Ay:4 y150, yN and A, then
roots , then :
total number of values of '' for which the equation
(a) b4 5a3 (b) 2c5a2b350
x23x0 is having integral roots , is equal to :
(c) c56a5 0 (d) 2b25a3c0
(a) 8 (b) 12
(c) 9 (d) 10
20. If a , b and c are not all equal and , are
the roots of ax2 + bx + c = 0 , then value of
28. Let ,,R and ln3 ,ln 3 ,ln3
(1 +  + 2)(12) is :
form a geometric sequence. If the quadratic equation
(a) zero (b) positive
x2x0 has real roots , then absolute value
(c) negative (d) non-negative
   
21. The equation x34(log2x)2(log2x)54  2 has : of     is not less than :
(a) exactly two real roots (b) no real root (a) 4 (b) 2 3
(c) one irrational root (d) three rational roots (c) 3 2 (d) 2 2
[ 2 ] Mathematics for JEE-2013
e-mail: [email protected]
Author - Er. L.K.Sharma
www.mathematicsgyan.weebly.com
29. Let a,b,cRand f(x)ax2bxc , where the 34. If all the four roots of the bi-quadratic equation
x412x3x2x810 are positive in nature ,
equation f(x)0 has no real root. If yk 0 is
then :
tangent to the curve y f(x) , where kR,then :
(a) value of  is 45
(a) a – b + c > 0 (b) c0 (b) value of  is 108
(c) value of 20
(c) 4a2bc0 (d) a2b4c0

(d) value of log 5log 25
30. Let a,b,c be the sides of a scalene  0.5 2
triangle and R. If the roots of the equation
35. Let , be the real roots of the quadratic
x22(abc)x3(abbcac)0 are real ,
then : equation x2axb0, where a,bR.
(a) maximum positive integral value of  is 2 If Ax:x240; xR and ,A, then
(b) minimum positive integral value of  is 2 which of the following statements are incorrect :
b
 2 2 (a) |a|2
(c) values of  lies in  ,
  2
 3 3
s
b c
(d) ,4/3 (b) |a|22 ti
a
(c) |a|4m
e
2
(dh) a 4b0
t
31. Let | a | < | b | and a , b are the real roots of equation a
a
x2||x||0. If 1||b, then the equationM m
r
e a
x2 E v h
log|a|b 1 has T -J E c ti K .S
(a) one root in (,IaI) (b) oene root in (b,L). Following questions are assertion and reasoning type
j . questions. Each of these questions contains two
b r
(c) one root in (a , bO) (d) no root in E(a , b) statements , Statement 1 (Assertion) and Statement 2
(Reason). Each of these questions has four alternative
 answers , only one of them is the correct answer. Select
32. Let p,qQ and cos2 be a root of the equation
8 the correct answer from the given options :
x2 + px + q2 = 0 , then : (a) Both Statement 1 and Statement 2 are true and
Statement 2 is the correct explanation of Statement 1.
(a) |sin||cos|p0 for all R , where
(b) Both Statement 1 and Statement 2 are true
[.] represents the greatest integer function.
but Statement 2 is not the correct explanation of
3 Statement 1.
(b) Value of log |q|
2 2 (c) Statement 1 is true but Statement 2 is false.
(c) 8q24p0 (d) Statement 1 is false but Statement 2 is true.
(d) |sin||cos|2p0 for all R, where
36. Let a ,b,cR,a0, f(x)ax2bxc ,where
[.] represents the greatest integer function.
b24ac. If f (x) = 0 has , as two real and
33. Let S :2560,R and a,bS.
distinct roots and f(xk)f(x)0,,kR,
If the equation x274x3sin(axb) is satisfied has exactly one real root between  and  , then
for at least one real value of x , then
Statement 1 : 0|ak| 
(a) minimum possible value of 2a + b is /2
(b) maximum possible value of 2a + b is 7/2 because
(c) minimum possible value of 2a + b is /2
Statement 2 : the values of 'k' don't depend upon the
(d) maximum possible value of 2a + b is 11/2 values of '' .
[ 3 ] Mathematics for JEE-2013
e-mail: [email protected]
Author - Er. L.K.Sharma
www.mathematicsgyan.weebly.com

37. Statement 1 : If a,b,cR , then at least one of the Statement 2 : sin1(x)cos1(x) 0 for all
2
following equations ..... (1) , (2) , (3) has a real solution
x[1,1].
x2 + (a – b) x + (b – c) = 0 ........ (1)
x2 + (b – c) x + (c – a) = 0 ........ (2)
39. Statement 1 : If equation x2(1)x10 is
x2 + (c – a) x + (a – b) = 0 ........ (3)
having integral roots , then there exists only one
because integral value of ''
Statement 2 : The necessary and sufficient condition because
for at least one of the three quadratic equations , with
Statement 2 : x = 2 is the only integral solution of the
discriminant  , , , to have real roots is
1 2 3 equation x2(1)x10, if I.
   0.
1 2 3
40. Let f(x)ax2bxc,a,b,cR and a0.
38. Statement 1 : If the equation
 Statement 1 : If f(x)0 has distinct real roots , then
x2x sin1(x26x10)cos1(x26x10)0
2
the equation f '(x)2 f(x).f"(x)0 can never
is having real solution , then value of '' must
have real roots
be 2log 8 s
1 because c
2 i
t
because Statement 2 : Ifa f(x)0 has non-real roots , then
m
they occur in conjugate pairs.
e
h
t
a
a
M m
r
e a
E v h
E
T -J c ti K .S
II e L .
j .
b r
E
O
[ 4 ] Mathematics for JEE-2013
e-mail: [email protected]
Author - Er. L.K.Sharma
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5. If ,,,R and , then :
(a) f '(x)0  xR{,}.
Comprehension passage (1) (b) f (x) has local maxima in (,) and local minima
( Questions No. 1-3 )
in (,).
Let a ,bR{0} and ,, be the roots of the (c) f (x) has local minima in (,) and local maxima
2 1 1 in (,).
equation x3ax2bxb0. If   , then
  
(d) f '(x)0  xR{,}
answer the following questions.
6. If ,,, are the non-real values and f (x) is
1. The value of 2b + 9a + 30 is equal to :
defined  xR , then :
(a) 2 (b) – 5 (a) f ' (x) = 0 has real and distinct roots.
(c) 3 (d) –2 (b) f ' (x) = 0 has real and equal roots.
(c) f ' (x) = 0 has imaginasry roots.
()2()2()2 c
2. The minimum value of is equal (d) nothing can bet cioncluded in general for f ' (x).
()2 a
m Comprehension passage (3)
to :
( Questions No. 7-9 )
e
1 1 h
(a) (b) Consider the function
2 9 t
a f (x) = (1 + m) x2 – 2(3m + 1)x + (8m + 1) ,
a
1 1 M wherme mR{1}
(c) 8 (d) 3 e a r
E v h 7. If f (x) > 0 holds true  xR , then set of values of
E
aT -bJ c ti K .S 'm' is :
3. The minimum valueI oIf ies equal to : L . (a) (0 , 3) (b) (2 , 3)
bj .
b r
E (c) (–1 , 3) (d) (–1 , 0)
2 O 3
(a) (b)
3 4 8. The set of values of 'm' for which f (x) = 0 has at least
one negative root is :
1 3
(c) (d)
3 8  1 
(a) (,1) (b)  ,
 8 
Comprehension passage (2)
 1   1 
( Questions No. 4-6 ) (c) 1,  (d)  ,3
 8   8 
Let , be the roots of equation x2axb0 ,
9. The number of real values of 'm' such that f (x) = 0
and ,be the roots of equation x2axb 0.If has roots which are in the ratio 2 : 3 is /are :
1 1
S x:x2a xb 0, xR and f :RSR (a) 0 (b) 2
1 1
(c) 4 (d) 1
x2axb
is a function which is defined as f(x) ,
x2a xb
1 1
then answer the following question. 10. Let , be the roots of the quadratic equation
4. If ,,,R and , then m2(x2x)2mx30,where m0 & m , m are
1 2
(a) f(x) is increasing in (,)   4
two values of m for which    is equal to .
(b) f (x) is increasing in (,)   3
(c) f (x) is decreasing in (,) m2 m2  3P
IfP 1  2 ,then value of   is equal to ....
(d) f (x) is increasing in (,) m m  17
2 1
[ 5 ] Mathematics for JEE-2013
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Author - Er. L.K.Sharma
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11. Let a , b , c , d be distinct real numbers , where 13. If the equation x4 – (a + 1) x3 + x2 + (a + 1) x – 2 = 0
the roots of x2 – 10 cx – 11d = 0 are a and b. If the is having at least two distinct positive real roots ,
roots of x2 – 10ax – 11b = 0 are c and d , then value then the minimum integral value of parameter 'a' is
equal to ..........
1
of (abcd) is ..........
605
14. If the equations ax3 + 2bx2 + 3cx + 4d = 0 and
ax2 + bx + c = 0 have a non-zero common root , then
12. If a , b are complex numbers and one of the roots of
the minimum value of ( c2 – 2bd )( b2 – 2ac ) is equal
the equation x2 + ax + b = 0 is purely real where as the
to ..........
a2(a)2
other is purely imaginary , then value of   15. If nI and the roots of quadratic equation
 2b 
x22nx19n920 are rational in nature , then
is equal to ..........
minimum possible value of |n| is equal to ..........
s
c
16. Match the following columns (I) and (II) i
t
a
Column (I) Colummn (II)
(a) If roots of x2 – bx + c = 0 are two consecutive (pe) –2
h
integers , then (b2 – 4c) is
t
a
a
(b) If x2,4, then least value of the expressionM (qm) 0
(x2 – 6x + 7) is : e a r
E v h
E
(c) Number of solutionsT o-f Jequatciont i|x21|3 K4. iSs /are (r) 2
II e L .
(d) Minimum value of fb (jx)|2x4|r|6.4x| is : (s) 1
E
O
17. Match the following columns (I) and (II)
Column (I) Column (II)
(a) If (22)x2(2)x1  xR, then  (p) (0 , 4)
belongs to the interval
 2
(b) If sum and product of the quadratic equation (q) 2, 
 5
x2(255)x(2234)0 are both
 5
less than one , then set of possible values of  is (r) 1, 
 2
(c) If 5x(2 3)2x 169 is always positive then set of x is
(d) If roots of equation 2x2(a28a1)xa24a0 (s) (2,)
are opposite in sign , then set of values of a is
[ 6 ] Mathematics for JEE-2013
e-mail: [email protected]
Author - Er. L.K.Sharma
www.mathematicsgyan.weebly.com