Algebra for JEE (Advanced), 3rd edition 9353503779, 9789353503772

Citation preview

,.....-

Theory of Equations

~FUNCTION

uA

2

LCtf(X)"' ax f(.r)

~

l .

+ where a, b, c, C

+[JX ;

C]

E

Rand a 'F- 0. We have,

.

2.27

5. a < 0 and D = 0, so f (x) ::, 0, \::/ x E R, i.e .J(x) is negative for all values of x except at vertex where f(x) is 0.

r1 +-x+a a

a ,

.2

~ 0 [ ·'

y'

+!!__x+lL+~-£] a ·· 4a a 4a

··[(,+ :J

2

2

6. a < 0 and D > 0. Let/(x) = 0 have two roots a and /3 (a < /3) . Then f(x) < 0, V x E (- oo, a) U (/3, 00 ) andf(x) > 0, V x E (a, /3) .

4•::/] ··[(,+ :. )'-4~'] ~ l· 0, the parn o ~ o~ens !fill)_..Jc d for a < 0, the parabola opens downwards. This gives UJJ"'lll"" an the following cases: I. a>0andD O, V x ER, i.e- .f (x) is positive for all values

(v)

y

y = x 2 - 4x + 6

= (x-2)2 + 2

V I

-= o l - - - ~ - - ~x

Now

(x - 2)2 ~ 0 for all real x.

Then

(x - 2) 2 + 2 ~ 2 for all real x.

Hence, the minimum value of y ( or x 2 - 4x + 6) is 2, which is the minimum height of the graph above the x-axis.

(i) (ILL UST~TION:

2. a < 0 and D < 0 so f (x) < 0, 'v r E R, i.e.,f(x) is negative for all \'alues ofx. Range of function is (---=, -D/(4a)] . r = -b/(2a) is a point of maxima.

What is the maximum height of any point on the curve = -x2+ 6x - 5 above the x-axis? ·

y

, y = -x2 + 6x- 5

y' 3. a>OandD=O, sof(x)~O, V xE R, i.e.,f(r) is positi ve for all values of x except at vertex where f (x) is O.

(ii)

ILLUSTRATION

I

~

2.~9

Find the largest natural number a for which the ma,ximum value of f(x) = a -1 + 2x - x2 is smaller than the minimum value of g(x) = x2 - 2ax + 10 - 2a. A

I

=4-(x-3)2 Now (x - 3)2 ~ 0 for all real x , Hence, the maximum value of y ( or - x 2 + 6x- 5) is 4, \Vhich is the maximum height of the graph above x-axis.

y

(iii)

4 · a> 0 and D > 0. Let /(x) = 0 have lworeal roots a and {3. If a < {3, then flx) > 0, Vr E (--00, a) u (/3, co) and f(r) < 0, V x E (a, {3).

2~9B,

y

Sol.

.f(x) = a - 1 + 2x - x2 = a - (x2 - 2x + I)

= a - (x -

1)2

Hence, the maximum value off (x) is a when (x - I )2 = 0 or x = 1 g(x) = x2 - 2ax+ I0 - 2a = (x - a) 2 + 10 - 2a - a 2 Hence, the minimum value of g(x) is l 0 - 2a - a2 when (x- a)2 = O or x = a. (iv)

Now given that maximum off (x) is smaller than the minimum of g(x). Thus,

2.28

or

Algebra

a 0 and 2ax2 + 3bx + Sc = 0 does not have real roots. Let

⇒

r? + 2n -

24 > 0 and n > 2 (n + 6) (n - 4) > 0 and n > 2

But

f(x) > 0, V x E R, if a> 0 orf(x) < 0 \;/ x Sc= f(0) > O

⇒

f(x) > 0, V x

⇒

n > 4 as n E N and n > 2 ⇒ n~5 Hence, the least value of n is 5.

⇒

E

E

R, if a< O

R

2

2ax + 3bx + 5c > 0, V x 2a - 3b +Sc > O

E

R (for x =-ll

ILLUSTRA TION 2.1 05

ILLUSTRA TION 2.1 02

Given that a, b, care distinct real numbers such that expressions ax2 + bx + c, bx2 + ex + a and cx2 + ax + b are always nonnegative. Prove that the quantity (a2 + b 2 + c2)/(ah + be+ ca) can never lie in (-oo, 1] u [4, 00). Given,

2 If ax +bx + 6 ==O does 11ot h · ave d'1stmct real roots then find the least value of 3 0 + b. '

01 .~ ' Given, equation ax~+ bx + 6 == 0 does not have distinct 1eal roots. Hence, or But

a-:;:. b-:;:. c, a, b, c E R. Now, ⇒

Ix - a+ 31 + Ix - 3al = l(x - a+ 3) + (x - 3a)I (x - a+ 3) (x - 3a) ~ 0 \;/ x E R ⇒ x 2 + x(3 - 4a) + 3a(a - 3) ~ 0 , \;/ x E R For the above inequality,

f(x) = '2a.x2 + 3bx + Sc

(n - 2)x2 + 8x + n + 4 > o; V x E R D = 64 - 4(n - 2) (n + 4) < 0 and n - 2 > 0 16 - (n 2 + 2n - 8) < 0 and n > 2

ax2 + bx+ c~ 0 b2 - 4ac ~ 0 and a > 0 bx 2 +ex+ a~ 0

R

If c is positive and 2ax2 + 3bx + Sc= 0 does not have any real roots, then prove that 2a - 3b + Sc> 0. ·

3

⇒

Sol.

03

lx-a+31+1 x-3a1=12 x-4a+31

-l'LLUSTR ATION

4 ? -x--8x+ 10

Find the least value of n such that (n - 2) x 2 + 8x + n + 4 > O, \;/ x E ~ where n E N. ⇒

(a2 + b 2 + c2 )/(ab + be + ca) can never lie in

=> =>

=>

ILLUSTR ATION 2.101

Sol

Sol.

⇒ ⇒

3 ⇒

(S)

D~O

2

a= -

a2+b2+c2 ---->l ab+bc+ca

For a E R, if Ix - a+ 31 + Ix - 3al = 12.x- 4a + 31 is true\;/ x E then find the values of a. I

[·: c=.f(O)= 10]

3b - =-12

⇒

a2+b2+c2 - - ~- < 4 (4) ab+bc+ca Now, (a - b) 2 t (b - c) 2 + (c - a)2 > 0 (': a, b, care distinct)

(2)

b

b=-8

(3)

or

ILLUSTRA TION 2.1

- - X -=-12 la 2

or

2

(1)

(b 2 - 4ac) - - - - = -2 4a From (2). we get

or

(2)

(-oo; l] U [4, oo),

b

- - =3 2a

b

a - 4bc ~ 0 and c > 0

From (4) and (5),

and

or

⇒

⇒

:a ,-~)

b:: - + 10=-2 4a

c 2 - 4ab ~ 0 and b > 0 cx 2 +ax+b~0

Equality cannot hold simultaneou sly in (1), (2), and (3). a2 + b2 + c 2 - 4ab - 4bc - 4ac < 0

ILLUSTRA TION 2.1 00

Sol.

⇒

⇒

(I)

=> =>

f(x) == ax2 + hx + 6 ~ 0, V x E R if a < 0 f(x) == ax2 +bx+ 6 ~ 0, V x E R , 'f > 0 1 f(O) == 6 > o ' a J(x) == ax2 + bx+ 6 > 0 , \-I R v XE ./(3) == 9a + 3b + 6 ~ o 3a+b~-2

Therefore, the least value of 3 + b . a 1S-2.

I ~Nz.1 □ 6

I

11-1,IJ

2 . · tnn omial P(x) = ax + bx + c 1s su ch tha t the uad fatl ) _ x · q has . no rea l roo ts. Pr ov e A p(x - , tha t m tlu.s ca se uatiotl eq · nP ( .x == x ba s no rea l roo ts eit he r. uatt 0 . cq . the equ ati on ax 2 +bx + c = x ha s no rea l roo s,n ce · sc,I, . p(x) _ x = ax2 ts, th e b l) . , + ( . , + c ass um es, va lue s of on e . -pre-ss 10 11 R ay P (x) - x > 0. Th en l ;\ 'q \' E •S

·0

p.(

· ))

sif ' .PlPlxo)) - P(xo) > 0 , """ Xo- i.e .. P(.Y0) > :'"o an d he nc e P ( P (.~0)) > x fl,r at1) · b a root of the 0 . Th ere for e, tomt h de gre e eq ua tlo . ca11not e n P (P (x) ) = x .

,\ (1

,~~

Theory of Equations

!S-rfl.A

2. 1 □ 7

u sT RA T IO N

2 _ e inequality (m.i + 3x + 4)/ (x + 2x + 2) < S is sat isf ied for Jt th R then find tl1e va lue s of m. 31\ XE •

(x - a)( x- c ) . Le t Y = ...: ___~_....:... ( x (x - b)

Sol. or

2.2 9

* b)

2

x + (- a - c - y )x + ac + by = 0 N ow , x is rea l. He nc e,

= (a + c + y)2 - 4(a c + hy ) ~ 0 , Vy y2 + 2(a + c - 2/J)y + (a - 2 D

or

c ) ~ 0 , Vy

D~O

E

R

E

R

4(a + c - 2h)2 - 4(a c)2 :S 0 (a + c - 2b + a - c) (a + c - 2h - a + c ) :S O (a - b) (c - h) ~ 0 , wh ich is tru e if a < b --: c or a --" b > c

or or

Sil- We bave. x~ + 2 Y + 2 = (x + 1)2 + 1 > O. V x ,n.·/ +3 x +4 fore 0 (c) a > 0 3. If ax 2 + bx + c = 0 ha s

+2

2 ,tm - 5)x - 7x - 6 < 0 , V x E R m - 5 < 0 an d D < 0 m < 5 and 49 + 24 (m - 5) < 0 71

or ::=,

m< -

24

l U. U ST RA T I O N

2 . 1 OB 2

Find the values of k for . lx wh ich \ 2 +k x+ l \ < 2, V x ER x +x +l

Sol

_r

We have ,

\xi < a ⇒ -a < x

O and x2 + (2 - k)x + 1> 0 For abo ve in eq ua lit ies . to b e tru e fo r a ll va lue s of x , th e 1r discriminants must be ne ga ti ve. He nc e, (2 + k)2 - 36 < 0 an d (2 - k)2 - 4 < 0 (2) ~ (k + 8) (k - 4 ) < 0a nd k(k - 4) < 0 (3) ~ -8 < k < 4 an d O < k < 4 Therefore, O < k < 4 . IL LU ST RA TI ON

im agi nar y roo ts an d a~ c < b, the n pro ve tha t 4a +c < 2b . 4. Le t x, y, z E R suc h tha t x + y + z = 6 an d xy + y z + zx = 1. Th en fin d the ran ge of va lues of x , y . and z. 5. If x is rea l an d (x 2 + 2x + c)/(:< + 4x + 3c) can tak e all rea l va lue s, the n sho w tha t O ~ c ~ 1. 6. Prove tha t for all rea l va lue s of x and _v. + 2xy - 3y2 - 6x . - 2y ~ - 11. 7. Fin d the co mp lete set of values of a suc h tha t (x 2 - x)/ ( 1 - ax) att ain s all rea l val ue s. 8. If the qu ad rat ic eq ua tio n + bx + 6 = 0 do es no t ha, e real roo ts an d b E R+. the n pro ve tha t

a>

' Jb2 max\ .. ~4 . b - 6lj .

9. If x is real an d the roo ts of the equa tion ,n : --r bx - ,. = 0 are im ag ina ry, the n pro ve tha t a:x : + abx + ,1c i~ ah\-ay s po siti ve . tO Le t a , b, c be rea l. l f m·: + bx + c = 0 h:.\s C\\ v rea l roo ts a an d /3, wh ere a < - I and /J --,, I .. then ~h1..,\\ tha t I + + < 0.

~ \!!.\ Cl

ll

ti. lf ./+ (o h) x t t i ,, t,) = O, whe1'\.' 0J(0) < 0 and when a < 0, then/ (0) > 0 af(0) < 0. ⇒ C This is equivalent to oc < 0 or - < 0 . a

4. a, /3 > k Graphically: Graphs for a > 0

Graphically: Graphs for a > 0

,

Graphs for a < 0

Y\~ x

Graphs for a < 0

0

n

-b/2a

x' +------,----.

X

X

k

- b/2a

r \X ' J !

i

Conditions: Conditions: (a) ef(O) > 0 (' : when a > 0,.1{0) > 0 and when a < 0,.1{0) < 0) (b) -b/2a > 0 (c) D ~ 0 2.

(": when a > 0,.l{k) > 0 and when a < 0,.l{k) < 0)

(a) aj(k) > 0 (b) -b/2a > k (c) Dc:O

5.

a, /3 0 (c) D ~ 0 GraphicaUy:

Graphs for a > 0

- b/2a

Graphs for a > 0

\

hi \

- b/2a

- ~

./---T~-/~~ 0 X

J.

k

Graphs for a < 0

x'--- '._.,'- ---;.~ X

x

k

x'~

x

Graphs for a < 0

X

x'-. '

- b/2a

O

'T \'

Conditions:

Conditions: (a) a.f(k) > 0 (': when a > O,f(k) > 0 and when a < O,,f(k) < 0) (b) - b/(2a) < k

(c) D "2'. 0 < k < f3 a 6.

(a) a/(0) > 0

(one root is sma lier than k and other root is I grentt1r than k)

(b) -b/2a < 0

(c) D "?. 0

e sign) 3_ a< 0 < /3 (roots of opposit Product of roots, a/3 < 0 C

- < O, or ac < 0 a D= b2 -4ac > 0

WhCll'a > O,j(k) < 0 and when a < O,f(k) > 0 aj(k) < 0. In this case, there is no need to check discriminant. ⇒

J

_ __

~

~ r~ot lying in (k,, k2) xactlY one •~-

7"xfoc~,\\

1, 6,ac \

k1

,'i7 ~' \ I -,, Graphs for a < 0

~

x

(xii) (xiii ) (xiv) (xv)

• x

'C l/ k2

1~

·'

~

•x

2).

Grap hs for a > 0

,~ .. h~~

X

k1

k2

;>;.'.\

t

k,

-

-b/2 a

or or ⇒

(ii) Both the roots are equa l. So, =

I

D 0 and af(k ) > 0 2 (b) k1 < -b/(2 a) < k 2 ~

D= 0 m = 9 orm

(iii) Both the roots are imag inary . So,

;

(c) D

(m - 3) 2 - 4m > 0 m2 - 10m + 9 > 0 (m - l)(m - 9) > 0 m E (-oo, l) U (9, oo)

⇒

⇒

k2

I

2

Grap hs for a < 0 k~,

2.31

Let./{x) = x - (m - 3) x + m = O (i) Both the roots are real and di stinc t. So , D >0

fron1 the ecrraphs. v,1e can see that . f(k 1) and f(k 2) have · ,, ·te sign opP 0 •1 ~ . Hen ce,f( k 1)f(k2) < 0. ·,. case also, there is no need to chec k discr imin ant. ln th k • ~- Both the roots lying in the inter val (k , k 1

Theory of Equations

exac tly one root lies in the inter val (I, 2) both the roots lie in the inter val (I, 2) at least one root lies in the inter val (I , 2) one root is grea ter than 2 and the othe r root is smal ler than

Sol.

/ f ' - k2

x'•

_ _ _ _ _ _ ___

(x) both the roots are grea ter than 2 (xi) both the roots are smal ler than 2

~

k,

__

0

m 0, 111 > 0 and 111 (- = . I] u (Q, .,._,)

ILLU STR ATIO N 2 . 1 1 0

\ I I

I \

Letx2 -(m - 3)x + m = O (m E R) be a quad ratic equa tion . Find the value s of m for whic h the roots are (i) real and disti nct (ii) equal (iii) not real (iv) opposite in sign (~) equal in mag nitud e but oppo site in sign (Vi) Positive (Vii) · negative 10

./i

(x)

a 2 = aa 2 +ba+ c--a 2

a 2 =--a 2

Both the roots are greater than 2. So, b f (2) > 0. D?:. 0. - - > 2

and /(/3)

2a

=

m < IO and m

⇒

m E [9, 10)

E (-

Now,f(a)f(/3)

I]

y

= -3 a 2 a 2 /3 2 < 0

4

ILLUSTRATION

2a

2.

1 1

2

The equation ax2 +bx+ c = 0 has real and positive roots.Prove that the roots of the equation a 2x2 + a(3b - 2c).~ + (2b - c:) x (b - c) + ac = 0 are real and positive.

y

J.

(": f3 is a root of-ax:1 - bx - c == U

Hence,f(x) = 0 has one real root between a and /3.

b / (2)> 0, D?:.0, -- 0 or h/a < 0 i.e., a and b have opposite signs. Product of root is ('/(I > 0

(l(J , =)

(xiii) Both the rool.b fie in the interv 0 andf(2p - (J

b I < -- < 2

i.u., a and c ha ve sa me sign .

⇒

J(J ,, m

(2)

Now, for cq11111inn we have /)

⇒

5 .-- m < 7

(3)

2a Thus, no such m exists. (xiv) Case I: Exactly one root lies in (l , 2) . So, f(J).1(2) < 0 ⇒ m > 10 Case II: Both the roots lie in (l , 2) . So, from (xiii), m E ¢. Hence, m E (l 0, 00).

o\J b

i/\' 2

~c ·)r + (~b - dlb - t') + ac = 0.

1 a(.l/i

2C') ~ 4o 2l(2h 2

c) (b - c)

+ ac]

2

,/j >h i I2hO

:: l::-_u_~-3-)+ I

=O

(a - 5)

[Using (2) and (3)] ots are positive. the rO 1w1ce, 11. .1... LJS

t real values of a do the roots of the equation 11a forW . b _2 _ . -ca2 _ l) = 0 he etween t h e roots of the equation 21 2(a+l)x +a(a- l)=O. · r(x)==x 2-2x - (a 2 -l)=O SO·1 .II .fi(x) =x2- 2(a + l)x + a(a - 1) = O (1) we have froll1 ' ,----2±.J4+ 4(a2 -1) x=- ~--- =l±a . 2

b

(a - 5)

ae(5, (l)

or

1 ---

,RATIO N 2. 1 l 3

or

2.33

Hence, (t+ 1/ - (a - 3)1(1+ l)+(a - 4)t2 = 0 or t2 + 21 + 1 - (a - 3) (t2 + 1) + (a - 4)r = 0

a t

_ _ _ _ Theory ?.!_Equations

Letf(t) = t2 - kt - 3, where t = sin x. Since equation has exactly two di.stinct real roots in [O, rr),f(r) = 0 must have exactly one root in (0, 1). Now, f(O) = -3. So, we must have f (I) = -k - 2 > 0 or k I 0

⇒

/a/> JO

""

/2,... , ,-12

Since th e rwdut.;I CJf r1A-1I 11, ~ l,1)1I, u,r,1 •. which mean.b there J!> 1_,11ly 1m1: ,wt"' (

j(- .Ji)j( ,/5_) /

~

0

W..:. ha\ c / (I)

U JJ,,.t

3 or a

EXAMPLE 2. 1 4

J2 Pulimf!, ttm.r ..

~ -

4 Prove th at for all u, b e R, the fun~tion / l r) = 3.r - -i./..,. ('.t: 1 has exac tly ono point wlwrt· lkri" ,Hi""' b1:.~om-.·s ~ro.

4 )

(I

I

1 \ ,I

121( \ !

\

I

i/1

,( I u 11 , 11 f) f

I I () \ .' I

a 1111 ly1101111al 11l\kgn:c J.

S1J , ll lia:, ~xuc1l y u 11c real root. Thus. derivative becomes zero 31 cx,11.:1l y 011..:. point.

A

Y s ltJ~ - - -- Exe -rcise~ y2 -mi!MIN®WS" ·~ 111 ......

1.

The value of the expressi. on x 4 - 8x 3 + 18x 2 - 8x + 2, when ,;; . x== 2 + \13, IS (1) 2 (2) 1 (3) 0 (4) 3

z.Ifx== l+ 3+

2+

(I)

(2)

J%

, then value of x is

1

(3)

~

(4)

JI

2 3. The swn of the non-rea l roots of (x + x - 2) (x 2 + x - 3) = 12 is (1) -1 (2) 1 (3) -6 (4) 6 4. The number of irration al roots of the equatio n

4x 5x 3 ---+----=-is 2 x2 +x+3 x -5x+3 2 (2) 0 (1) 4 (3) 1

(4) 2 5. The curve y =(A+ l)x 2 + 2 intersec ts the curve y =AX+ 3 in exactly one point, if A e_quals (1) {-2, 2} (2) {1} (3) {-2} (4) {2} 2 6. If the express ion x + 2(a + b + c)x + 3(bc + ca + ab) is a perfect square, then (1) a= b = c (2) a=± b = ± c (3) a= b

*c

(4) none of these

7. If (ax2 + c)y + (a'x2 + c') = 0 and xis a rational function of y and ac is negativ e, then (1) ac' + a'c=O (2) a la'=c!c ' (3) cl-+c2 =a' 2 +c' 2 (4) aa' +cc'= 1 8. If a, b, c are three distinct positive real number s, then the number of real roots of ax2 + 2b\x\ - c = 0 is (1) 0 (2) 4 (3) 2 ( 4) none of these 9. Leta b andcbe realnum berssuc hthat4a +2b+c= Oandab > O. The~ the equatio n a:l- +bx+ c = 0 has (1) comple x roots (2) exactly one root (3) real roots (4) none of these 10. If a e (-1, 1), then roots of the quadrat ic equation (a - 1) x2 + ax + (1) real (3) both equal

P

= 0 are (2) imagina ry

mx2 + (2m _ 1)x + (m - 2) = o are rational are given by the express ion [where n is integer] (1) n2 (2) n(n+2)

(4) none of these

2

+ ni2)

m2,

a 2)

(2) ( cl

-

m2,

cl)

( 4)

none of these 14. lfx is real, then xl(x 2 - 5x + 9) lies between (1) - land - I/I J (2) l and -- 1/11 (3) I and 1/ 1 l (4) none of these 15. Jfx 2 + ax - 3x - (a + 2) = 0 has real and distinct ronts. then the minimu m value of (ci2 + I )!(cl+ 2) is l I (I) l (2) 0 (4) (3) --

2

4

16. If a, b, c, de R, then the equation (x2 + ax - 3b) (x2- ex+ h) (x 2 - dx + 2b) = 0 has ( l) 6 real roots (2) at least 2 real roots (3) 4 real roots (4) 3 real roots 17. Ifthero otsofeq uation( a-1 )(x2 +x+ 1)2=(a + l )(x4 + x.2 + 1) are real and distinct, then the value of a e (1) (-oo, 3] (2) (- 00 ,-2) U (2, OOJ (3) [-2, 2] (4) [-3, oo) 18. If b 1b 2 = 2(c 1 + c 2), then at least one of the equatio ns x 2 + b 1x + c 1 = 0 and x 2 + b 2x + c = 0 has 2 (1) imagina ry roots (2) real roots (3 ). purely imagina ry roots (4) none of these 19. Suppose A, B, Care defined as A = a 2 b + ab 2 - a 2c - ac 2 • B = b2 c + bc 2 - a 2 b - ab 2 , and C = a 2 c + ac 2 - b 2c - be=. where a > b > c > 0 and the equation A..:c2 + Bx + C = 0 has equal roots, then a, b, c are in (1) A.P. (2) G.P. (3) H .P. (4) A .G .P. 20. lf a, f3are the roots ofx 2 - px+q = O,md a '. {3' a re the roots of x 2 - p'x+ q ' = O, then thevalu eof( a - a.')2 + ({3- a.'f +{ a -/3"1 2 + ({3 - {3')2 is (1) 2{p2 - 2q + p' 2 - 2q' - pp') (2) 2{p 2 - 2q + p':. - 2q' + qq'} (3) 2lp2 - 2q - p' 2 - 2q' + pp ' } (4) 2{p 2 - 2q - p' 2 - 2q' - 1/1/'} 2·1. 1fa, f3 are the roots of thci equntil,n ,1x~ + b.\· + c = 0. then the value of (c,a.:. + C') /((la+ h) + (11/3 ~ + d 1 l,1/3 + h) is (I)

( 4) none of the8e

11. The integral values of m for which the roots of t~e equatio n

(3) n(n + l)

13. If the roots of the equation x.2 + 2ux + h = 0 arc real and distinct and they differ by at most 2m, then h li es in the interval

(3) [a

1 3+ - 2 ...oo

J%

12. x 2 - xy + 4x - 4y 1- 16 "· 0 re present:,; (l) apoinl (2) acirclc (3) a pair ofslraig ht lines (4) none of these

( I ) ( a2, a 2

1

__ I

(3)

h(h:. - 2a - 1 2 (4) a < 2 and f3 > -1 (2) a < - and

38. The quadratic equations x2 - 6x + a = 0 and x2 - ex+ 6 = 0 have -one root a in common. The other roots of the firn and second equations are integers in the ratio 4 : 3. Then . 2 equation x - 3x - 4 = 0 has (1) both roots more than a (2) both roots less than a (3) one root more than a and other less than a (4) Can't say anything 39. If a and /3, a and y, a and Dare the roots of the equarioill 2 ax + 2bx + c = 0, 2bx·? +ex+ a= 0 and ex·? + ax + 2b = 0. respectively, where a, b, and c are positive real numbers. then a + a2 =

If one

(I) abc

(]J

(3) - I

root of x2 - x - k = 0 is square of the other, then k = 2 ±-Js (2) 2 ±,!3 (3) 3 ±-./2 (4) 5±-./2

31. lf a and f3 be th e roots of the equation x2 + px - 1/(2/) = 0, where p ER. Then the minimum value of ci + /1' is

( l J 2-./2

(2) 2 -

-.f2

(3) 2

(4) 2 + -.f2

32. If a, f3 are the roots ofx2 + px + q = 0 and y, oare the roots of z O .h (a - y )( a - 8) x + px + r= , t en---'--- - -

-

(/3 - r>UJ - o)

(I) 1

(2) q

(3) r

(4) q +r

33. The value of m for which one of the roots of x2 - 3x + 2m = 0 is double of one of the roots of x2 - x + m = 0 is (1) -2 (2) I (3) 2

(4) none of these

/3 < 0

(2) a+ 2b + c

(4) 0 2

40. If the equations ax + bx+ c = 0 and x3 + 3.r~ + 3.r + 2 = 0 have two common roots, then (I) a = b = c (2) a = b :i= c (3) a = - b = c (4) none of these 41. The number of values or a for which equations x 3 +ax+ I = 0 nml x" + ax~ + I = 0 have a common root is

(I) 0

(2) I

(3) 2

(4) infinite

42. The number of values of k for which [x2 - (k- 2)x +!(]'I? [ X- + kx + (2k - I)] is a perfect square is (I) 2 (2) 1 (3) 0

(4) none of these

.

~ of x satisfying the equation · 0 2 r,-;;15)"2_3 . 'fhC Sl1111r.7 ,2- 3 + I == (3 + 8."I) . IS J3· -r g-..;l S) (2) 0 (3 I (I) 3 (4) noneofthese ~

~

~

S6. If roots of an cquatil,n x" - I ;:: 0 arc I. 111• " > .. .. o,,

the value of(I - o 1) (I - o 2 ) (I (\)

2

tion cx2 + x + l )" + I = (x + x + l) (x 2 - x - 5) for 'fhC equa) ,1·11 have number of solutions JJ, -2. 3 V. , sE \ (2) 2 ( 3) 3 (4) zero (.1)

re the roots of x2 + px + q

lI)

=?nnd x2" + p"x" + q" = 0

45 . \fO'd·t~(X}/J). (WO'.) are the roots ol x"

1 \

1-

(x + 1)"

= O,

nn ,I\ 11(E /v I thC 11 (2) may be any integer n1st l,~, 'an· odd integer . (I )

11

_ be

') 11n1~1

an even mteger (4) cannot say anything

. )\)inotnial with integral coefficients such that for ' r) 1s n Pl P I P( a) =P(b) = P(c) = P(d) = 3 46, ,. i··tinctintegersa, b,c,c, · ) . t()\lf (I~ ) - 5 (e is an mteger , t11en lf P(c _- (2) e = 3 I) e - \ l _ (4) norealvalueof e (3) e- 4 r:l ·) == y2 + bx + c. where b, c E R. lff(x) is a factor of 47· Ll.'t ·1 ·'.~ + ·6./ + 25 and ,,-'x4 + 4x-~ +28x +5, then the least botl1 -' . ra\ue off(x) is (I) 2 (2) 3 (3) 5/2 (4) 4 (.1

2

48

C0nsider the equation x + 2, - 11 = 0, where n EN and · n E ["- , JOO] . The total number of different values of n so that the given equation has integral roots is (1) 8 (2) 3 (3) 6 (4) 4

2 49 . The total number of integral v~lues of a so that x - (a+ 1)x +a - 1= Ohas integral roots 1s equal to

(I) 1

(2) 2 (4) none of these

(3) 4

l2) O

(3) 1

(4) 2

51. The number of real solutions of the equation (911 ot = -3 - x - _; is (1) 2

(2) 0

(3) I

(4) none of these

52. The number of real solutions of •x is /are (IJ 6 (2) i (3) 0

T

2 .J 5 - 4x - x

2

= 16

(4) 4

53. Let p(x) = ()be a polynomial equation of the least possible degree, \.\irlrn:rional coefficients. having ifi + if49 as one of its roots. Ther, the product of ~II th e roots of p(x) == 0 i:-; 0) 56 ( 2 ) r,3 (3) 7 (4) 49 54 1 · If u., /3, ·1, v P-re ti-~ rry:-;;t~of the equation x4 + 4X - C,l- + 7[ 9 ~ '= 0, thtt.1.c ', ?-.foe r.,f () - rlJ ( I I I I I a ) IS

1/

(J)

11

(I -

11 11

1)

{..\)

0

11

57. If tun 0 1, tan O~. tan A_1 arc the real roots of thl· .,·' + (h - a) .,· - /, == O. where (} 1 + (}~ 1 O, E ( ll. + 02 + O_, is cqunl tn (I) ,r/2

( 2) ,r/4

(3) 3 ,r/ 4

1• thl.'n will bl'.

l" 1 I) / ,r}. t hl' II f),

(4) ,r

58. If a, {3, yare the roots of x-' - x2 · I - 0 then th e vn luc or ( I + a)/( I - a) + ( I + µ) /( I -· /J) I ( I I y)/( I - y) is cqun I to (\) --5 (2) - (1 ('.\) 7 (4) 2 . " , . , t- K ~ 59. If a, {3, y, 8 arc the roots of the cqu allon x I\. X r + Lx + M = O. where K, L, and 1'v/ arc rea l numbers. !hen thi; 2 minimum value of a 2 + {3 2 + y2 + 8 is (I) 0

(2) - I

(3) 1

(4) 2

60. Set of all real values of a such that

r(x) · · -

(2a -1)x 2 + 2(a + l)x

+ (2a -

I)

_:____...:._----=-........:..-- ' - - - - -

x 2 - 2x + 40

negative is (l) (-oo, 0) (2) (0, oo)

is a Iway~

(3) ( oo, 1/2)

61. lfa, b ER, a :t Oand the quadratic cqualion has imaginary roots, then (a + h + I) is

(4) No ne

ax2 -

I

h,r '

0

(1) positive

(2) negative

(3) zero

(4) dependent on the sign of h

62. If the expression [mx - l + (1 /x)] is non-negati ve for ,111 positive real x, then the minimum value of m must be (1) - 1/2

(2) 0

(3) 1/4

(4) l/2

63. Suppose that f (x) is a quadratic expression positive fo r all real x. If g(x) = f (x) + f'(x) + f"( x ), th en for any n.:al

50. Toe number of integral values of a for which the quadratic equation (x - a) (x - 1991 ) + 1 = 0 has integral roots are (1) 3

(2)

11

o,) ...

rh ( 1) (

x (wheref'(x) andf"(x) represenl Isl and 2nd d~ri vati vt:,

respectively) (1) g(x) < 0

(2) g(x) > 0

(3) g(x) = 0

(4) g(x) ~ 0

· 64. Let a, b, c E R with a > 0 such th at the equ ation ax2 + bcx + b3 + c3- 4abc = 0 has non-real roo ts. lf P(x) = ax 2 + bx + c and Q(x) = a_t2 + ex -L h. th en (I) P(x) > 0 for all x E R and Q(x ) < 0 for all x '= R. (2) P(x) < 0 for all x E Rand Q(x) __.. 0 for all x '= R. (3) neither P(x) > 0 for all x E R nor Q(:c ) __.. 0 for all -c "' R. (4) exactly one of P(x) or Q(x ) is positive fo r a ll ri:al r. 65. Let/(x) = ax2 - bx .,.. c2 , b :t O and } l:c ) :t O for all Then (I) a + c2 < h (2) ➔ LI • ( ~ • '2h 2 (3) 9a - 3b .i. c ,,, 0 (.:I) non.: l•fth.:,.:

I

c

R

,,. Lct/(x) == a/- + hx - c, u, h. c £ f{ I t'/ I \ ) tJkc-- r.:al \ aluc, 6 .or rca I valtics , , of ·r and non -1..:,11 \ ,tl ui:-, for non-rcJI \aluc ,

I of x, then

(IJ o IJ (2) /J 0 (1)

l'

()

(4 J f1(Jlhin g can he -.ard about a, h. ,:.

..:.. 2'...:.. 4 =2 _

A __:l=..: ge:.::.. br:.::.. a _ _ _ _ _ _ __

_

_

_

--7

_ _ _ _ _ _ __ _ - - ~ - _ _ _ _ _ _ _ _ _

67. lfboth roots of the equation ax +x + c - a = 0 are imaginary and c > - \. then (I) 3a > '.! + 4c (2) 3a < 2 + 4c (3) c < a (4) none of these

2 76. If roots -. of x - (a - 3)x +a = 0 arc such that at-le as t one them 1s greater than 2, then or (I) aE[7,9] (2) aE[7,oo) (3) aE[9, 00 ) (4) aE[7,9)

61\. lf (b: - 4ad ( I + 4a: ) < Ma 2• a, < 0. then ~he maximum value. of quadratic expression ax· + b.r + c 1s always less

77. All the value~ of m for which both the roots of the equatio x2 - 2mx + m - I = 0 are greater than - 2 but less than 4 /

2

(2) 2

69 _ lf the equation

(4) - 2

(3) - I

1.,) + b.r + cl = k has four real roots, then 4c-b 4

2

4c -b 2 0 0,

then the set of points (x, y ) satisfying the equation

/a( x2 + ~ ) + (b + l)x + c/ = lax2+ bx+ cl+Ix+ YI consists of the region in the xy-plane which is ( l) on or above the bisector of J and Ill quadrant (2) on or above the bisector of JI and IV quadrant (3) on or below the bisector of I a·nd IJI quadrant (4) on or below the bisector of II and IV quadrant x2

73. Gi ven x,y ER,x2+/> 0. Thentberangeof ~

(!)

(3)

+

2

y

2

x + zy +~

( 10-4✓5 ' 10+4✓5] I

30

30

(5 -4✓5 5+4✓5] 15

'

15

2

(10-4✓5 10+. 4✓5)

(2

)

(4)

15

'

15

(20-4✓5 20t4✓5) 15

'

15

74. x 1 andx 2 are the roots of ax2 +bx+ c = 0 and x 1x2 < 0. Roots of x 1(x - xi+ x 2 (x - x 1)2 == 0 are (I) real and ofopposite sign (2) negative (3) positive (4) nonreal 75. If a, b, c, d are four consecutive terms of an increasing

A.P., then the roots of the equation (x- a) (x- c) + 2(x- b) (x- d) == 0 are (l) non-real complex (2) real and equal (3) integers (4) real and distinct

~

(2) m > 3 (4) I < m < 4

79. The interval of a for which the equation tan 2 x - (a - 4) tan x + 4 - 2a = 0 bas at least one solution \:f x E [0, 7u'4] (I) aE(2,3) (2) aE[2,3] (3) aE(l,4) (4) ae[I,4] 80. The range of a for which the equation x2 + ax - 4 == Oha. its smaller root in the interval (-1, 2) is (l) (-oo,-3) (2) (0,3) (3) (0, oo) (4) (-oo, -3) U (0, 00

(4) none of these

71. If the equation ax 2 + bx + c = x has no real roots, then the equation a(ax 2 +bx+ c) 2 + b(ax 2 + bx+ c) + c = x will have (1) four real roots (2) no real root (3)

(I) - 2 0) has sec 2 0 and cosec 2 0as its roots, then which of the followin g must hold good? (I) b + c = 0 (2) b2 - 4ac ~ 0 (3) c ~ 4a (4) 4a + b ~ 0

89 _ 1fa0.a 1.ac.a3 areall positive , then4o 0 _,J + 301 x2+ 2o x +o 1 = O 2 has at least one root in (- I. 0) if ·

2o 2 = 3o 1 + o 3 and o0 + o < 2

= 0 have

q-q p'- p

(2)

p' - p

q- q

2.43

I

pq' - p'q q - qI

(I)

(4) OE(0.1 /3)

(I) a0 + a 2 = a, + o 3 and 4o + 2o > 0 2 (2) 4a0 + 2a 2 < 3o 1 + o 3

Theory of Equations

---

6. If the equation s x2 + px + q = 0 and x2 + p'x + q' a common root, then it must be equal to

(2) a < 0

(3) aE(- oo.1 /3)

(3) 4a 0 +

-

-

- ----;;;equation 22'" + (a - I )2'·+t + a = 0 has roots of o;positc 87• . signs. then ex.I1austwe set of values of a is

-

2

a,

(2)

(I =

b =0

(4) a,harero otsofx 2 +x+2=0

15. I f/(x) is a polynom ial of dcB_rec 4 with rational coefficie nts

and touches x-axis at (✓2, 0), then for the equation / (x) = 0,

(I) sum of roots is 4~ (3) product of roots is -4

(2) sum of roots is 0 (4) product of roots is 4.

1

I

2.44 Algebra

- J+f s

of the foIIo"·· 24. Let a, b, c E Q'· satisfying a > b > c. Which 1 d f polyn 0" . statement(s) hold true for the qua ra ic 2 a - 2b)? tlii; .f(x) = (a + b - 2c)x + (b + c - 2a) x + (c + s upwards ( I) The mouth of the parabola Y =f (x) open r~tionaI (2) Both roots of the equation.f(x) = 0 are h '.s positive (3) The x-coordinate of vertex of the grap tive (4) The product of the roots is always nega

2

ax2 +bx + c. Consider the following diagra ~ 25• Let f (x) =

16 · The real .\" satisfying the equation

+x x' +~ .\ (I)

(3)

2

)- 7= Ois/arc -6( x+~ +~ X X

3+ ✓5

(2)

2

3 - ✓5

(4)

2

-3 - ✓S

2

2 17. I f x.y ER and 2\' + 6xy + sy2 = I, then

(I)

\x\$ ~

(2)

Then

/x/~\15

y= ax 2 + bx

(4) y2 s4

(3) / $ 2

E R, then which of 18. lf.flx) = ax2 + bx + c, where a -:t- O, b, c real roots ? t11e following conditions implies that .f(x) has (1 ) a+ b +c= 0 (2) a and c are of oppo site signs

2

(3) 4ac - b < 0 (4) a and b are of opposite signs .

r 2 +5 one real root, 19. If -· - - = x - 2 cos (111 + 11x) has at least 2 then (1 ) number of possible values of x is two (2) number of possible values of x is one

(3) the value of 111 + 11 is (211 + l )rt (4) the value of m + 11 is 2111t 2 +c= 0, bx2~ 2cx+a=0 20. Letthreequadraticequationsax -2bx roots . Then and cx2 - 2ax + b = 0, all have only positive which of these are always true? 2 (1 ) b = ac 2 (2) c = ab

common (3) each pair of equations has exactly one root on (4) each pair of equations has two roots comm 1)x + 9a - 5 = 0, 21. for the quadratic equation x2 + 2(a + which of the following is/are true? . ( l) If 2 < a < 5, then roots are of opposite sign. (2) If a < 0, then roots are of opposite sign. (3) If a > 7, then both roots are negative. (4) If 2 5 a 5 5, then roots are unreal.

bcx2 +2(b +c - a) 22. Ifa, b, c E Rand abc .O (3)

C
0 (l) c < o (4) abc < 0 (3) a+ b-c >0 2 figure. ~f PQ~ 26. Graph ofy= ax +bx+ c is as shown in the followmg is/ the of h whic OR = 5 and OB = 2.5, then

true? (l) AB= 3 (2) y(-1) < 0 (3) y ~ 7 for all x ~ 3 Q (4) ax2 + bx + c = mx has real roots for all real m 2 has two real r 27. If the equation ax +bx + c = 0 (a > 0) which of a and f3 such that a < -2 and /3 > 2, then following statements is/are true? 2 (2) c < 0, b - 4ac > 0 (1) a- /bl+ c < 0 (4) 9a- 3/b /+c 0 (4) none of these (3) c(4a -2b + c) > 0 2 29. Ifco sx- y2 -Jy -x -l~0 , then

(1) y 2:'. I

(3) y = 1

(2) x ER (4) X = 0

' 2 ual real roots for a 30. If ax + (b- c)x +a- b- c =0 has uneq c ER, then (2) a < O< b (1) b < 0 < a (4) b > a > O (3) b , cos a) and

the other in (sin a, 00)

3 34 If the roots of the equation x + px 2 + qx - 1 == o fonn au · increasing G.P., where p and q are real , then (1) p+ q= 0 (2) pE (-3 , 00) (3) one of the roots is unity (4) one root is smaller than l and one root is greater than l 35, Ifroots ofax2 +b x+ c = 0 are a and j3 and 4a + 2b + c > 0, 4a - 2b + c > 0, and c < 0, then pos sible value/values of [a] + (/31 is/are (where [·1 represe nts greatest integer function)

(1) -2

(2) -1

(3) 0

2.45

2. lf R(x) == px2 + (q - I )x + 6 has no ~istinct real roots and p > O, then the least value of 3p + q ts (\) - 2 (2) 2/3 (3) _ 113 (4) none of these 3. Range off (x) = [R(x)] /(x 2 - 3x 2 + ) is

(2)

(4) If

Theory of Equations

(4) none of these For Problems 4-6 Let f (x) == x2 + bix +Ct, g(x) = x2 + b x + c2. Let the real roo ts:~ f(x) = 0 be a, f3 and real roots of g(x) 2 == 0 be a+ h, f3 + h. The lea value off (x) is -1/4. The least valu e of g(x) occurs at x = 712. 4. The least value of g(x) is 1 l 1 (1) -4 (2) -1 (3) (4) 3 2 5. The value of b2 is (1) -5 (2) 9 (3) -8 (4) -7 6. The roots off (x) == 0 are (1) 3, -4 (2) -3, 4 (3) 3, 4 (4) -3, -4 For Problems 7-9 In the given figure, vertices of MB C lie ony = f(x) = d- +b x+ c. The MB C is right-angled isosceles triangle whose hypotenuse AC== 4 -f2: units.

(4) 1

y

~Y

36. The equation ( + (_x_y = a(a - 1) has ~x +l) ~x -1 ) (1) four real roots if a > 2 (2) four real roots if a < - l (3) two real roots if 1 < a < 2 (4) no real root if a < - 1 37. If the quadratic equations, x2 + bx+ c = 0 and bx2 +ex + 1 =0 have a common root then (1) b + c + 1 = 0 (2) b2 + c 2 - 1 == be - b - c (3) b + c- 1 == O (4) b2 + c2 + 1 =be + b + c 38. If the inequality cot2x + (k + l )cot x - (k- 3) < 0 is true for at least one x E (0, n/2), then k E (1) (-o,3-2"✓5) (2) (3, 00 ) (3) (-1, oo) (4) (-00, 3)

linked Comprehen~ion Typ~··...

/,., ,.,' · . . . . .

Ill

For Problems 1-3 Consider an unknown polynomial whi ch when divided by_(x - 3) . ers and (x - 4) leaves remamd d tively.x Let R(x) be 2 a~ .1 _ x _ ). the remainder when this polynomial , r~s_pec 1s dtvtded -by ( 3) ( 4 1. If equation R(x) = x2 +a x+ l has t distinct real roots, wo then exhaustive values of a are (1) (-2 ,2) (2) (-"' °,-2 )u( 2,o o) (3) (-2, oo) (4) all real numbers

y

=f

(x)

= a:x2 + bx + c

B

1. y = f(x) is given by (1) y

=x 2 -

2✓ 2

(2) y= x2- 12

x2

(3) y= --2

2

(4) y =

8. Minimum value of y = f(x) is (I) -4

x2

~-2 ✓2 2"2

(2) - 2

(3) - 2"2

(4) none of these 9. Number of integral values of k for which one root off (x) = 0 is more than k and other less than k ( I) 6

(2) 4

(3) 5

(4)

(2) 2

(3) 3

(4) O

7 For Problems 10-12 Let f (x) = 4x 2 - 4ax + a2 - 2a + 2 such that minimum value of /(x ) for x E [0, 2] is equal to 3. 10. Number of values of a for whi ch global minimum value that is equal lo 3 for x E [O, 2], occurs at the end point of interval (0, 2] is (I) I

2.46

t

Algebra

11. Number of values of a for which global minimum value, that is equal to 3 for x E [0, 2), occurs for the value of

23. The given inequality has at least one positive solution rot aE

x lying in (0, 2) is

ll) I

(2) 2

(3) 3

(4) 0

12. Values of a for whichf (.Y) is monotonic for x E [0, 2] arc given by ( l) a ~ 0 or a ~ 4 l3) o > O

For Problems B-15 Consider the equation 2 + lx2 + 4x + 31 = 111, m E R B. Set of all real values of m so that the given equation has three solutions is ( l) {3} (2) {2}

(3) {I}

(4) {O}

15. Set of all real values of m so that the given equation have two solutions is (1) (3, oo) (2) (2, 00) (3) {2} U (3. oo) (4) None of these

For Problems 16-18 Consider the quadratic equation ax 2 - bx + c = 0, a, b, c EN, which has two distinct real roots belonging to the interval (I, 2). 16. The least value of a is (] ) 4 (2) 6

(3) 7

(4) 5

17. The least value of bis (I) IO (2) 11

(3) 13

(4) 15

18. The least value of c is (1) 4 (2) 6

(3) 7 For Problems 19-21 Consider the inequation x2 + x + a - 9 < O.

(4) 5

(I) (- oo, 3)

(2) [2, 00 )

(3) (3, oo)

(4) [- 2, 00)

For Problems 25-27

21. !he va!ue~ of the real parameter a 1nequat1on 11.1 true '.;/ x E (- 1, 3):

80

(I) (-oo, - 3]

(2) ( 3, "" )

( 3 ) [ 9, '-""')

(4) ( -

that lhc , · given

3714)

For Problems 22-24 -1-

27. If (a+ b + c)c < 0 < a(a + b + c), then ( 1) one root is less than 0, the other is greater than 1 (2) one root lies in (--oo, 0) and other in (0, 1) (3) both the roots lie in (0, 1) (4) one root lies in (0, 1) and other in (I , oo) For Problems 28 and 29

lpx2 + qx + rl ~ 1Px 2 + Qx + RI Vx E

R and

3 ~ 0 1 .. . ' w icrc a rs a real

22. The given inequality has at least one negative , I . aE so ut1on for (3) (-2,oo)

28. Which of the following must be tme? (l) (3)

IPI ~ IPI lpl=IPI

(2)

IPI s; IPI

(4) Allofthese

29. Which of th e fol lowing must be true? ( l) lal s; IDI (2) lctl ~ IDI

lal = IDI

(4) None of these

For Problems 30-32

(4) none of these

Consider the inequality ~ __aJ' _ a parameter.

26. If a(a + b + c) < 0 _]

w,

25. If !bl > la+ cl, then (I) one root ofj(x) = 0 is positive, the other is negative (2) exactly one of the roots off (x) == 0 lies in (- 1, I) (3) l lies between the roots off (x) == 0 (4) both the roots ofj(x) == 0 are less than I

d=q2-4pr > 0 andD = Q - 4PR > O

(4) (---oo, 9)

(2)

(I) (- 00 ,2) (2) (3,oo)

)

2

2-0. The values of the real parameter a so that the given . . . mequation has at least one negative solution :

(-w,347 )

00

(a.I{µ) < O) is the necessary and sufficient condition for~ particular real numberµ to lie between the roots of a quadratic equation f(x) = o, where.l{x) == ax2 +bx+ c. Again iff(µr)f(~) < 0, then exactly one of the roots will lie between µ and µz.

Given

19. :°e va!ues of the real parameter a so that the given mequat1on has at least one positive solution: (IJ (--=, 37/4) (2) (--oo, oo)

(3)

(3) (2, oo)

1

14. Set of all real values of 111 so that given equation have four distinct solutions. is (l) (0. 1) (2) (1. 2) (3) (1 , 3) (4) (2, 3)

(3) (3 , =)

(2) [3,oo) (4) [- 2,

24. The given inequality has at least one real solution for a ":

l2) 0 ~a~ 4

l4) none of these

(I) (- oo, - 2)

(4) (2,3)

Consider the equation x 4 + 2 .J + .2 m ·' + 2ar + l = 0, where a eR. 30. _r~tqu~t!on has at least two distinct positive real roots then a possible values of a are ' { I) (- oo, - 1/4) (2) (5/4,oo) (3) (- oo, - 3/4) If . (4) none of these 3 I. eq uati on has nt least two di st111 , . . all possible v·tlLtc ·· , ct negative real roots, then ' so 1a are (I) (3/4,oo) (3) {- oo, 1/4) {2) (-5/4, co) 4 32. lfexactl t ( ) none of these ' y wo roots are positive then the number of . and two roots are negative, (I) 2 mtegral values of a is (2) 1 (3) 0

(4) 3

Theory of Equations

,~

problems 33 and 34

.

.

fbe real numbers x 1, x 2 , x 3 sattsfymg the equation x3 _ x2 + {3.x + r,:c Oare in A.P. 33 . All possible values of f3 are (I)

(-oo, ½)

(3) (

½,

00

(2)

(4)

)

(-oo, -±)

b. If x 2'

(3) (

¾-oo)

(2)

(-± ,oo)

¾,+ 00)

(

- _I

27 '

+

If x 2 +ax+ b =- 0 has ro~s a, f3 anci x 2 + px + q = 0 has roots - 2/ a, r, then - d. . If x' + ax + b = 0 has roots a, f3 and x 2 + px + q = 0 ha s roots - l/(2a), y; then

00 )

-

U

A.x + 9 = o.

(2) (0, oo) (4) (-oo,-6)

36. If the equation has no real root, then A lies in the interval (1) (-00, 0) (2) (-oo, 6) (3) (6, oo) (4) (0, oo)

37. If the equation has only two real roots , then the set of values of A is

( 1) (-oo, -6)

(2) (-6, 6) (4) ¢

(3) {6}

1. Match the following for the equation

Listi

/ b. Two real roots

,c.

Three real roots

/

(I) (2) (3)

(4)

p q r q

b q r p

C

s

p

r r s

d s

p q r

er

Then match the following lists :

r. a = - 2

List I

s. a?: 0

The number of elements in set Sis -----b. The sum of all possible values of(a + /3) of the pair {a, /3) in set Sis a.

ListU

c.

a. one root is positive and the other is negative p. 0 for the equation (m - 2)x2- (8 - 2m)x - (8 - 3m) = 0 -- - q. infinite b. exactly one root of equation. . x2 _ m(2x _ S) _ 15 = o Jies ·m mterval (0, l)

both roots lying on either sides of 1

I

p.

bx + C = 0 where (I -:;:. 0 is satisfied by a, /3, and /J2, where a/3 -:t=. 0. Let set S be the set of all possible unordered pairs ( a, /J) .

the equation x2 + 2(m + l)x + 9m - 5 - O has both roots negative the equation x2 + 2 (m - 1_)x. + m + 5 - 0 has

List II

Listi All four real and distinct roots.

s. If ax +

p. a < -2

List I (Number of positive integers for which)

d.

s.

2

2. Match the following lists:

c.

-r

I-

Codes

where a is a parameter.

a. No real roots

--r. -(l -,_::-4bq )2 = (a+ 2bp)( - 2p - 4aq)

a.

a

I

-

Match the values of 'a ' in List II for the types of roots in List I.

2

(6, oo)

q. (4 - hq )2 = (4a + 2pb)(- 2p - aq)

4. Consider equation (x 2 + x)2 + a(x 2 + x) + 4 = 0

35. If the ~quation has four real and distinct roots, then)., lies in the mterval

(I) (-00, -6) (3) (6, oo)

-

- List II p. (l - bq}2 = (a - pb)(p - aq)

+~,~+h = 0 has roots

c.

(4) none of these

For Problems 35-37 4 Consider the equation x

--

a, f3 and x 2 + px + q = (J has roots I/ a , y, then

34. All possible values of yare (l) (-

3. Match the following lists : ---- List I a. If x 2 + ax +I J = 0 has roots a, f3 and x 2 + px + q = 0 has roots - a, y, then

2.4 7

r. s.

-I 2

d,

List TI p.

2

q . .)

The sum of nil possible va lues of af3 of the pair (a. /3) in set Sis

r.

I

-4

The sum of nil possibk vuhlt'S l)r if+ f oftl1e ~ pair ( O'., /3) in se t S is. where a, f3 e R is -- - ·----· -----~

Codes a ( 1) q (2) r (3) q (4) r

b s s s

C

d

s q r

p p

s

p

q

r

r

2.48

Algebra

6. Consider equation x 4 - 6x 3 + 8x 2 + 4ax - 4a 2 Then match the following lists: List l a.

lf equation has four

p.

= 0, a

R.

E

List -n -----

- - ----0 E

~I

distinct roots then b.

c. d.

-- - - If equation has exactly two q. O E (- 1/2,2) distinct roots th en -----If equation has no rca I OE (- oo, - J/2)u(2,oo) roots then

,·.

If equation has four distinct positive roots then

Codes a (l) q (2) r (3) q (4)

q

b s

C

s

q

s

r

r

p

s.

(-oo,2)

OE

p p p

Ill 1. Ifx = 2 + 2

213

+2

113

then the value of x

J.JTx = ~ ~ ~3x

4

?f

odd_degree. When 15. Polynomial P(x) contains only terms ' · IC d by (x _ 3) , the remamder · d !VIC . 1s 6. If P(x) 1,i P(X ) IS . 'd' e-d b y (X 2 - 9) ' then the remainder is g(x). Then th e d IV) value of g(2) is _ _ __ . J 16. fa and /3 are the roots of the equation x2 - 6x + 12 = 0 and 8

(/3 ~

the value of (a - 2)24

d r

s

13. The quadratic equation x 2 + mx + n = 0 has roots which a · twice those of x 2 + px + m = 0 ancI m, n an d P * O.Then th,re value of nip is _ _ __ 14. Suppose a, b, c are the roots of the cubic x3 - x2 - 2 "'0 · · Then the value of a 3 + b 3 + c 3 1s _ _ _ _.

3

-

6x2 + 6x is_ _ . x4

8

6) + I is 4a, then the value

of a is . Leta and b be the roots of the equation x2 - I Ocx - 11 d=Oand 17. t h oseo f x 2 - JO . ax- llb=Oarec ' dthenthevalueofa+b+c+d• when a* b * c * d, is____ . Leta,bE Rand ab:;,. I. Tf6a2+20a + 15=0and l5b2+20b 18. 4030b 3 . + 6 = O then the value of l)3 is _ _ . ( 2 ab -9 ab+ If there exists at least one real x which satisfies both t~e 19. equations x2 + 2x sin y + I = 0, wher_e y : (0, n /2), and ax 2 + x + I = O, then the value of a + sm y is_ _ __

is

20. Jfthe equation x2 + 2(A. + l)x + A.2 +A.+ 7 = 0 bas only negative roots, then the least value of A. equals_ _ __

3. Sum of the values of x satisfying the equation

21. All the values of k for wruch the quadratic polynomial J(x) = - 2x2 + kx + k2 + 5 has two _di~tinct ~eras and only one of them satisfying O < x < 2, he m the mterval (a, b). The value of(a + I Ob) is _ _ __ 2 22. If set of values of a for whicb/(x) = ax - (3 + 2a)x + 6, a* O is positive for exactly three distinct negative integral values ofx is (c, d] , then the value of dis _ _ __ 23. a b and c are all different and non-zero real numbers in a;ithmetic progression. If the roots of quadratic equation I 1 2 ax + bx + c = 0 are a and /3 such that a + /3 , a+ /3, and

2. If

+4

then the value of

.J2x+ ✓2x+4 = 4 is_ _ . . 4. lfa2-4a+l =4, tbentbevalueof is equal to _ _ __

5.

a3

-

a2 + a - I

a 2 -I

(a2:;,.I)

and b are positive numbers and each of the equations real roots, then the smallest possible value of (a+ b) is _ _ __ If a

x2 + ax + 2b = O and x2 + 2bx + a = 0 has 6. Given that x2

- 3x + 1 = 0, then the value of the expressiony = x 9 + x 7 + x-9 + x- 7 is _ _ __

7. If sin 2 a , cos 2 a and -cosec 2 a are the zeros of P(x) = x 3 + x2 + ax + b (a, b E R), then P(2) equals_ _ _ .

8. If the equation x2 - 4x - (3k - 1) /x - 2/ - -2k + 8 = 0, k E R, has exactly three distinct solutions, then k is equal to _ _ .

9. If cos 2 !:.. is a root of the equation

a, b, E + a is

x2 + ax + b = 0 where 8 Q, then the least value of the expression x 2 + 8bx

JO. Given u and /3 are the roots of the quadratic equation .x2-4x+k =O(k:;t0). Jfa/3, a/32 + «/3, c?+ /33 are in geometric progression, then the value of 7k eq uals _ _ __

11. Let a1, /3 be the roots of .x2 - 6x + p = 0 and £Xi, /32 be the roots of} - 54x + q = 0. If a1, /31, O,z, /32 form an increasing G.P, then the value of(q - p) is _ __ _

12. Let o: and

/3

be the solutions of the quadratic equation I

I

x2- 1154x + 1 = 0, then the value of a 4 +/3 4 is equal to

·a?- + [32 are

in geometric progression, then the value of :!. will be____ c 5

3 - 6x -

24. Let P(x) =

7

9x- and Q(y) = - 4y 7 + 4y +

13

2

. If

there exists unique pair ofreal numbers (x,y) such thatP(x) Q(y) = 20, then the value of(6x + I0y) is _ _ __ 4

25. If equation x - (3m + 2)x 2 + m 2 = O (m > 0) has four real solutions which are in A .P. , then the value of m is

26. If the equation 2x 2 + 4xy + 7y 2 - I 2x - 2y + t = O, where t is a parameter has exactly one real solution of the form (x, y), then lhe sum of (x + y ) is equal to

----

27. Let P(x) = x 3 - 8x 2 + ex - d be a polynomial with real coefficients and with all its roots being distinct positive integers. Then the sum ofall possible values of c is _ _ __ 4

3

.

2

28. Let P(x) =x + ax + bx +ex+ dbe a polynomial such that P(l) = 1, P(2) = 8, P(3) = 27, P(4) = 64, then the value of

P(5)is _ __

Theory of Equations 2, 49

11n,l ,. .• o, thon the v11lt11.: ol'[ /3. If a10 - 2a8 . 1s I , then the value of ___:_::__...;;...

/3

11

for n ~

- 6x - 2 = 0, with 2~

II

(I) I

(2) 2

(4) 4

(3) 3

(HT-JEE, 2011) 4. The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation p(p(x)) = 0 has ( 1) only purely imaginary roots (2) all real roots (3) two real and two purely imaginary roots (4) neither real nor purely imaginary roots (JEE Advanced 2014) ,r

(3) . ( 0,

}s, 0) (}s,f)

(2) (-

}s)

(4)

(4) -,,[3 (HT-JEE, 2011)

3. Let a and f3 be the roots of x2

(-_!_ __1 ) 2 ' Js

(I)

2 2. A value of bforwhichtheequationsl+bx- 1 =0,x +x+b=0

have one root in common is (3) ~ (2) -;-,,{3 (I) -'5.

(2) 2 sec 0 (4) 0

,r

5. Let- - < 0 < - - . Suppose a 1 and /31 are the roots of the 12 6 2 . equation x - 2:t sec 0 + I= 0 and ai and {32 are the roots of 2 the equation x + 2x tan 0 - 1 = 0. If a 1 > /3 1 and exi > {32, then a 1 + /32 equals

(JEE Advanced 2015)

linked Comprehension Type For Problems 1 and 2 Let p , q be integers and let a, /3 be the roots of the equation, x2 -x - I = 0 where a :;t: /3. For n = 0, 1, 2, ... , let 0 11 =pa"+ q/J'. Fact: If a and b are rational numbers and a + b ✓5 = 0, then a = 0 =b. J.

0 12

=

( I) a 11 (3)

-

(2) a 11 +al0

a 10

(4) a 11 + 2al0

2011 + 0!0

2. If a 4 = 28 , thenp + 2q = (4) 12

(3) 7

(2) 14

( 1) 21

(JEE Advanced 2017) Numerical Value Type 1. The number of distinct real roots of x 4 - 4x 3 + 12x2 + x - I

(HT-JEE, 2011)

= 0 is___

Answers Key

\

EXERCISES Single Correct Answer Type

1. (2)

2. (4)

3. (1) 8. (3) 13. (3) 18. (2) 23. (2) 28. (3) 33. (1) 38. (3) 43. (2) 48. (1)

6. (1)

7. (2)

11. (3) 16. (2) 21. (3)

12. (1) 17. (2) 22. (1)

26. (2) 31. (4) 36. (2) 41. (2) 46. (4)

27. (1) 32. (1) 37. (]) 42. (2) 47. (4) 52. (3) 57. (2)

58. (1)

62. (3) 67. (2)

63. (2) 68: (2)

51.. (2) 56. (1) 61. (J) 66. (I)

53. (I)

4. 9. 14. 19.

(4) (3) (2) (3) 24. (2) 29. (3) 34. (3) 39. (3) 44. (4) 49. (1) 54. (3) 59. (2) 64. (4) 69. (1)

5. 10. 15. 20. 25. 30. 35.

40. 45. 50. 55. ,60. 65. 70.

(3) (1) (3) (1) (2) (1) (3) (1) (3) (4) (2) (1) (2) (3)

71. 76. 81. 86.

(2) (3) (2) (1)

J 72. (2)

77. (3) 82. (4) 87. (3)

73. 78. 83. 88.

(2) (2) (1) (2)

Multiple Correct Answers Type 1. (1 ), (2) 2. 3. (1), (4) 4. 5. (1), (3) 6. 7. (1 ), (2), (3) 8. 9. (1 ), (2) 10. 11. (1), (3) 12. 13. (2), (3) 14.

15. (2), (4) 17. (I), (3) 19. (2), (3)

16. 18. 20.

21. (2), (3), (4)

22.

74. 79. 84. 89.

(1) (2) (4) (1)

(1), (2), (4) (I), (3) (1), (2) (1), (4) (2), (4) (1), (4) (2), (3), (4) (1), (2), (3), (4) (1), (2), (3) (1 ), (2), (4) (3), (4)

75. (4) 80. (1) 85. (4)

~---------;~--~~m~~~----- -----~i (1), (2).(3) ;~: (l).(2). (3).(4)

24. (l),(2),(3) 26. (1).(3),(4)

•. ,) ,~) 27- (1U-- . ,,o. (.i).(4) ;1, (\), (4)

28 • 30. 32. 34.

.Ii.(.~).(➔) _

-i~- (1). (2). (3)

·

1) (4)

9. (- 1.25) 13. (8)

(\) '2) ' ' . (3) (3),(4) (1), (2), (3) (1), (3),(4) 36. (I), (2), (3) 38. (1), (2)

37. ( . . ~-ed comprehension Type unt. (➔) 2. (3) 3. 6. (3) 7. (4) 8. tl. (4) 12. (1) 13. ~. (4) f7. (2) 18. 1 (!) Jl. 22. (4) 23. ~6. (1) 27. (2) 28. 31. (1) 32. (3) 33. J6. (2) 37. (4)

(3)

4. (I)

(3)

9. (3)

(1)

14. (4) 19. (4)

(2) (3)

24. (2)

(2) (1)

29. (1) 34. (2)

17. 21. 25. 29.

(6) 33. (5)

37. (3) 41. (5)

18. 22. 26. 30.

(12) (- 0.75) (3)

(9) 34. (3)

38. (4) 42. (3)

19. 23. 27. 31.

( 1.5) (3) (36) (- 1) 35. (-9)

39. (315) 43. (8)

12. (6)

16. ( 12) 20. (6) 24. (3) 28. (149) 32. (3) 36. (- 1) 40. (3)

5 (

ARCHIVES

15. (3)

JEE MAIN

20. (3) 25. (2) 30. (3)

Single Correct Answer Type 1. (3) 2. (2) 3. (1) 6. (1) 7. (4)

35. (3)

JEE ADVANCED Single Correct Answer Type 1. (2) 2. (2) 3. (3)

1. a~ s: b ➔ r; c ➔ q; d ➔ p. 2. a~ r; b ➔ r; c ➔ q; d ➔ p. 3. a~ s; b ➔ p; c ➔ q; d ➔ r. 4. (I) 5. (1) 6. (4)

Multiple Correct Answers Type l. (1), (4)' Linked Comprehension Type

1. (2) 3. (6) 7. (9)

-JO. ( 16) 11. (540) 14. (7) 15 (4) - •

lO: (~~

Matrix Match Type

Numerical Value Type 1. (2) 2. (4) 5. (6) 6. (6621)

(1210) (7) (6)

Theory of Equations 2.51

4. (4) 8. (2)

2. (4)

Numerical Value Type 1. (2)

4. (4)

5. (4)

4. (4)

5. (3)

.In eq u al it ie s In vo lv in g Means

6

E MEANS OF TW O POSITIVE

~ Nuu M ,. ~BE . Rs

~

Let A, ~ ~n

. met1.c. g eometTiC and hann omc. mea ns of. d H be anth

wnbers a and b. rwo positive n 2ab a+b G= J;b and H = - - b , A Then , 2 a+ These tlu·ee means

are relat ed as A

;m" .

.

⇒

-Fob and H = ·

~ (✓a

~a+ b = • -G= - - -"\Jab 2

(1)

-2ab a+ b - 2

= J;;,; ⇒

1

c✓a- -fh)2?. o

(3 )

a+b

a

1

-+ - , whe re a, b, c > O. b c

1

Similarly, ax + by > 2 ✓abxy Multiplying (1 ) and (2), we get

1

1

~;--c:-

_·

ur"7....~~~~~

ti TRA I • L.LL ,, - ,J'S' ,., -'=· -'•"·. iip~ • .• . 6~-~ -

1

If a b and c are distinct posi tive real num bers such that ' ' (1 + a)(l + b)(l + c) > a+ b + c = 1, then prov e that (l - a)(l 8· - b)(l - c)

[ S~h ] a+b + c = l a= 1 - b -c l +a= (l-b) +( l -c) > 2 ✓( 1 - b)(l - c) . ==> --- --- :(.Using A.tvl. '?. G .M.) Similarly, l + b > 2 ✓0 - c)( l - a) or

(I)

(2)

and

(ab+ xy) (ax+ by) > 4abx y

1 + c > 2✓(1 - a)( I - b)

Mul tiply ing, we get . ( l + a)( l + b)(I

6.2

2

Prove that b c?-+c?-cl-+ cl-b 2 > abc(a + b + c),

1

- +-

2 - + - + - >2- - + 2- - + a b c a+b b+ c a+ c

(2)

6. 1

ab+ xy Ci;= --> -va uxy 2 ax + xy > 2 ✓abxy

1

2

Add ing, we get

Sol. Using A.M . ?. G.M . , we have

JID&

(2 )

l l 2 - +7 .9___Q_ > - - .Q__f_ > - - and .f__g _ > - -2 a+b ' 2 b+ c 2 c+ a

Prove that (ab+ xy) (ax+ by) > 4abxy; whe re a, b, x, Y > 0.

ILL UST RAT ION

c+a

1

- +-

From (1) and (2), we get A?. G?. H Oearly, A = G = H if a= b.

⇒

. · b+c

-J;;E}

G?. H

ILLU STR ATI ON

2 2 1 1 Prov e th at --+ 2--+ --< -+

. b a+

a+b

2

rsoe) Usin g A.M. > H.M ., we get

a+ b

=--raE {

(I)

ll_LL._U,s;:;; -~• ~ -6.3 1

.Jb)2

2 .

A ?. G G-H=-✓ ab-

2

lic ·+ c 2a 2 > 2abc 2 2 2

2ab

-

==>

c a + a b 2 > 2a 2bc 2 2 2 a b + b c 2 > 2ab2c Add ing (1), (2) and (3), we get 2 2 2 2b c + 2c a 2 + 2a 2b2 > 2abc 2 + 2bcc l + 2cab2 22 2 2 ==> b c + c a + a_2b2 > abc(a + b + c)

- -.-b a+

n

r:, > v(b2c2)(c2a2)

Sim ilarl y,

proof: a+b A =- :,- , G =

b2c2 +c2a2 2

?. G ?. H.

We have

~ G.M., we have

[ Sol. l Using A.M.

whe re a, b, c > 0.

or

+ c)

> S( _ a)( 1 - b)( l - c) 1

(1 + a )(1 + b)(I + c) > 8 (l - a)(l - b)(l - c)

6 ,2

Algebra

_ILLUSTRATION.

6.S

Find the minimum value of

soi:

4sin2 X

+ 4cos2 .\'

4 x + -16 + 2y + -18 + 9 z + -9 ~46

forxE R.

)I

X

UsingA .M. 2 G.M. , we have

Z

So, equation (I) holds only if 4x +

.,

9

I.Q. = 16

2y + 18

y:::: 12

.,

X

where a , b, c >

c

b

,

On multiplying both sides by abc, we have to prove thar 8/27 > (I - a) (1 - b) (1 - c) > Babe. Now, (1-a) +(l-b) +(J- c) > [(1 -a) (1- b) (l -c)] l/3 .

3

;t f(a;)

⇒

2'. - -

0.

l§«,I.

0. Therefore , graph of function

11

-

1

{_!.-1}{_!.-1}{!-1} 8;

_B_ > a 27abc

=>

r

1

c. n - · - · - · ... · -

If a+ b + c = I , then prove that

f(xl is concave downward.

So, we have f

a,,

a3

a2

I

1

ILL USTRA TION

Alternat ive Method:

We have a lready proved A.M.-G.M. inequality for n positive rea l numbers a 1, a 2 , a 3 , ... , a,,. Let us see another proof. Consider the fun ction f(x) = log x V x > 0.

= --1,
(ab) , (c + b)/2 > (cb), and (a + c)/2 >(ad·. Multiply ing these three inequaliti es·, we have a+bb+ c a+ c - - -- -- >obc

I,log(a; ) - --n 1= 1

2

or or

2

2

(a+ b)(b + c)(c +a)> 8abc (l - a)( l - b)(l - c ) > 8abc

n

L, log(a;) N ow,

i= l

= log a 1 +Joga 2 + ... +Joga,, n

n

= log (a 1a 2 a 3 . .. a

lfyz

th + z x + xy = 12, where x , y , z are positive values, find e

greatest value of xyz .

11 )

n

~ log((a aa .a) J 1 2 3

ILLUST RATION 6. 1 2

Sol. ' Using A.M. ~ G.M., we have 2 2 2 1/ 3 xy + yz + zx ~ ( xyz ) 3

r

~ ~ (x2 y2z2 )113

3 (XJZ)max

12)

Weh ave

~

312

=(3

A.M.~H.M.

s

s +· ·· +- s S-a

--+

S-a ~ --:.._,- - -a2

usTR. ATION 6. 1 3

---__: .:.!.!.. n ~

. .

11-1-

If a, band c are pos1hve then prove that 3 C b a +-+--~-;;; c+a a+b 2

So,I We have 3 c b a -+--+-->2 b+C c+a a+b 3 c b a -+l+ --+ l+-- +I >-+3 2 a+b c+a b+c a +b +c a+ b +c a +b +c - - > -9 - - - + - - - + -a+b - 2 c +a b +c

_I_ +-1- +-1- > b +c

9 a+b - 2(a+b +c)

c +a

Now, using A.M. ~ H.M., we have I

I

I

- - + -- +- 3 b +c c +a a+b :--- ------- ---> -;-- -:-:+(b+c )+(c+ a)

n

n S-a 1 S-a 2

S-an

- - +- - +··· +s -

s

s

or

or or

S

S

-a,

S-a 2

S

n2

-->~ + - - + ···+ S-an - n-1

ILL~SJ.RAT ION 6.1 6

If 0 1 + 0 2 + a3 + a4 + ··· +an= I\;/ a;> 0, i = 1, 2, 3 ... , n, then find the maximum ~alue of a/ a2 a3 a4 a5... a,,.

Using A.M. ;?: G.M. a1 a1 a +a +a +· ·· +a n -+-+ 2 · 2 3 4 2 n+I

- (a+b)

3

_I_ +-1- +-1- > _ _9__ b +c c +a a+b - 2(a+b +c) ILLUS TRATI ON 6. 1 4

Prove that 2n > I + n .J2n-l , V n > 2 where n is a positive integer.

SoL 2n > l + n .J2n-l 2n -I

-- > n

⇒

Hence, required maximum value is

(n + 1/+l

x i(n- 1) /2

2-1

Now, (2n - 1)/(2 - I) is the sum of a G.P. whose first tenn is I and common ratio is 2. We have to prove that I + 2 + 22 + i3 +... + 2n-l > n x i O; z. = 1' 2 ' ... ' n an hted , _., , ' , , me~ ns are defi ned as follo ws:

Wei ghte d arith met ic mea n, Aw =

(z+z +z+ •·-ztime-.,

x+y +z

xx

or

1. Prov e that (

rx+ y +z]x+y+z 3 (x,Y,Z >O)

>L

+ y2 + z2

x2

4 sec 4 a {3 - - - + -sec -->8 2

tan

an

(x+x +x+ ... xtim es)+ (y + y + y + · · · y time s) +

X X )I V -+· -+ -+ -+ -+ -+ -+ ~~ s

tan 2

X y Z >X y z

lx x + y +z

x ,- )' 2 \ X X y y 8, > - · - · - · -l · ·-·-·( y x xyyyxx _ }

wn

+ ... + -

la LLI..JSTRATl □ N 6 • l g Prov e that

8

)I -

a2

Jf w1 = w2 ~ w3 -_ .. • =. w n• then thes e mea ns are sani ea1 corr espo ndin g simp le mea ns .

yyyxx

x-

Wz

+-

For thes e mea ns also , we have Aw > G ?:.H w w

} \ X = -+ -+ -+ -+ 2- +2 y-

,

w, a,

4 scc 4 a f3 (x+ 1) 2 (y + 1)2 - -- + -scc- = - - - + - -Ian ~ {3 tnn 2 a y x

,

W1 +w2 + .. . +wn

Weighted harm onic mea n , I-fw =

w,a, + w2a 2 + ... + w,,a,, w, + w + ...+ w 2

11

or

x 1• : X )" Z

>(

x+y +z ) 3

x +v+:

.

J

~rln ') Me::, n', l ncq 11allt1P ,, l nvo

fro1J1

fW ·

(2) , we ge t ( 1) an·d ] ,·+ ,·+ -

.2

+ v2 + ;:2

.

>

.:_:- .--. :-_-[ .r+ .1 +-

. -

,.

-

.

X y ·'

z-

.

.

. -

> [ ·' + .l' + z1•+.1 + _ J

of .r2y1 is 1 So , lh e grc atc sl va lue 16

.2 0 s-rRATl □ N 6

)- -211(11+1)

11-1.-LJ

pro ve

22 X 33 X

I that 1 X

... X

+I -211 3

II

n
5

wh en

Si nc e -2 < x < 7 > O 4 x + 2 > 0 and 7 - x p q5 w her e (7 - x )4 (2 + x>5 or of lue va um xim ma We ha ve to find p + q = 9.

]2 == 2+ 2 +3+J ns Using we igh ted m ea

32== 3

1+ (2 +2 )+ (3 +3 +

3) + ... + (n +n + ...

·(1]::(tJ ~[W'(tJ']¾

n tim es )

1+ 2+ 3+ ... +n ~

~1 , 1~ -,- +. ., + ... +n , n( n + 1)

(1

1

·

2

2

.. +n · ... • n" ) 1+2 +3+ .

2

1 ~ (1

1

--

.,

-2 - · ... · 11 11)11(11+1)

2 n(n + 1)(211 + 1)

_ __.::.6_ _ _ ~

2

(11

.2 2 · ... ·11 ")1 1 ( 11+!)

n( n + 1) 2

> 2n +l --

l (l 1 . 2 2 . ... · 11 ")n (n+

:s:; ( 2,~+ 1 11 . 22 . 33 ... . . n"

ON

[(:J(f)']¾$p;q

⇒

(:J(fJ

~I

p4q5 ~ 445 5 (2 + m va lue of (7 - x)4 Th ere fo re, ma xi mu

⇒

x/

4 5

is 4 5

)- 2-

6 .2 1

v

2

+

1

') zvX + - = .I + ev Sol. W e ha 27 4 I a ns, we ha ve Us ing we igh ted me

3x +4 y =5

~ G.M. for we ig ht ed m ea

2:3

n - + -- - + -27 4 va lu e of ,,y :- w he m I mu xi ma the Fi nd wh ere x, y, z > 0.

lie in the first .x2y 3 wh er e x an d y of lue va t tes ea gr Find the + 4y = 5. quadrant on the lin e 3x (1 ) 5 = 4y + 3x t tha Sol Gi ven er 3 sion .x2y , we co ns id Since we ha ve expres

2(~x ] 3(; )=

.

6 .2 3

TI O N

X

)

n(n + l )

ILL U ST RA TI

⇒

IL LU ST RA

2

Using A.M.

x lie s

Sol.

1 sol, 1~ ==

3

6. 7

ns , we ge t

II

⇒

.-1yz

~ (,61 )" .ifi. . V9 II

6 .~ .~. e of xy z is ( 6 ) So , m ax im um va lu 11

=1 :

~6.!_8~ ~~ - - - - -~ -- -- -- -_ -- -~ -~-=-_-_-_:_-_-, .-:,;-, ,-...,-. ,.- ----=-N-:-o-w-, J~-o~in~M;-;2- :a::n:d~A~ l ~x3~,.fl/(:x3~)))..- - - -._ _=_

l _· -2 ----------

. tM MA be such that M2M3 · M3A3 · · Let pom 3 011 2 3 . /( )

r

So, coordinates of M3 are (

l. Prove that

(a:::' >a'b" >{a;f'; where a,b>0 2. Provetha td;bq>

b )p+q ap+ p ;wherea , b,p, q >O. ( p+q

3. Prove that pxq -r + qx"-P+ ,._,p-q > p + q + r, where p, q, r arc distinct numbers and x > 0, x :f. 1. 4. Given are positive rational numbers a, b, c such that a+ b + c = 1, then prove that a0 bbcc + ibcca + acbacb ~I. 5. If a and bare positive numbers such that a2 + b2 = 4, then find the maximum value of a2b. 6. Find the greatest value of x2y3z4 if x2 + y2 + l- = I, where x, y, z are positive.

Further on M3A4 , consider point M4 such that M3M4 : M4A4:::: I :) So coordinates of M4 are

(;' +x,

:x

1 +x4 ,

25

X

f(x,)+f (x,): f(x,)+ f(x,)).

Similarly, we get point M11 whose coordinates are 11

Ix;

II

lf(x;)

i=L__.:.:i-:.:,_1_ _ 11

11

All the points M , M3, 2

.. . ,

M,, Iie in the region which is above the

graph of the function y =f(x). y

ANSWE RS

6.

x1 +x2 +x3 .f(x1)+ . X2 +/(x3)] ' 3 , 3

y= f(x)

rl5/2

MEANS OF mth POWER Consider n positive quantities a 1, a 2, a3, ... , a,,. Then arithmetic . al m + az"' + ... + a m 11 mean of mth power 1s , where m is any real n number except zero. We have the following inequalities for this mean : al +a2 + ... + a11 )m ()') at +a2 m + ...+ a/' > ( -'-----==-------"- ; If m < 0 or m > 1

n

.. ) (u

A - - + - - - - - - -LL---- ---+X B 0

In the above figure, from point M,,, draw vertical line, whicb intersects the graph of y =f(x) at A and x-axis at B.

n

at +a2 m+ ... +a,,m n


(+ n

I ILL UST RA! ION 6.2 6

, sec A + sec B Prov e that in acute angled triangle ABC + sec c~ 6.

l

func tion is conc ave dow nwar .d ' point M ,, h of the . . the graP grap h of the function y =fi(x) · w11en 1-011 whic h 1s belo . w the .0 the reg lies I

111

[ ,,

that case,

f,J(X;)

X;

~

Triangle ABC is acute angled . 0 O such that a + b ~ 2, then show

[SoLJ We have (a3 )1/J +( b3 )1/3 2


0.

Sol,

h+ c

c +a

1_j__

l_j_

(_ , _

_

2

a+b

2

a +!? +-- > a+ b + c ; wher e a+b

N

+ h + -;

ow,

_

1_\_2_ _ _ _ _ _

+ c + ~; )

~l

\

3

~[(a+{)+:~Hc+~)I

2

2 1>2 + c2 I // + c- > (b+ c) or -- >- (b+ c) 2 2

b+ c

_( _

2

2

Simi larly .

a: + b: l c2 + a 2 I - - - >-( a+b ) and - - > -(c +a) a+b 2 c +a 2

~l6:~ j"¥

7 (a+ ±J +(h +~f +( c +± J ?:: :

Addi ng. we get

b2 + c2 c2 + a2 a2 + b2 - --+ - - - + - - - > a +b+ c b+c c +a a +b 1. Prov e that a 4 + b4 + c 4 > abc(a + b + c), wher e a, h, c>o.

ILL UST RAT ION 6.2 8

2. If C,. =

a-8 + b8 + cs l l 1 Prov e that , 3 , >- +-+ -. a~b c a b c

Soi.-

8

or

as + b; + c > ( a+~ + c

J J J

2

2

b

2

+!a b + 2bc

So,

a 8 +b 8 +c 8 2 2 2 (3 ab+ 3bc +3ca) - - - - >ab c

or or

3

2ca

(': A.M . > G.M .)

cl-+ b 2 + c 2 > ab + be + ca 9

a 8 + b8 + c8 > a 8 +b 8 + c 8

c?b3c3

cl-b 2 c2(ab + be+ ca)

It2 _·__s_o_lv_e_d_E_x_am_ p_le_s__ _- □I ]

Let x 1, x 2 . ... , x,, be posit iye real num bers and we define S = x 1 + x 2 + · · · + x 11 • Prove that (l +x 1)(l +x2)

Sol.

ab+ bc ca > -- -+abc

Sol.

2

J) 75 a+±) + b+-;l ) + ( c+~ ?::4 .

l

a

So,

or

or

I

b 3

3

c?:

a+b +c

1 l l 3 -+- +-? :a b c 2

21 31

.

sn

+ --- +-;;T

We have to prove that

11

Sinc e A.M, ?:: H.M ,, we have I

s 3

+

( I+ x 1) + (l + x 2 ) + ·.. +(I+ x,,) ~ [( l + x,)(l + _y

2

(

- +- +-

s2

s'

-+ -,,

, .

A.M.?::_G.M.

If each of a, ban d c is posi tive and a + b + c = 6, then show (

(l +x11)5: 1 +S+

Prod uct on left-h and side suggests that we must consider G.M. of ( I + x 1) , ( I + x ), ... , ( I + x ). Also, ( 1 + x 1), ( l + x2) ... ., (I +J,l 2 11 are positive. Now ,

ILL UST RAT ION 6.2 9

2

...

s2 s3 ( I +x 1)( 1 +x)) .. . (1 +x )5: l + S + + - + .. 11 2! 31 •

or

that

✓3 . 2

+j

But

+(b +}J ?:'. 2; .

4. ln MBC , prov e that sin A + sin B + 3 sin C 5:

(a +~+ c

J~a+ +c

> [ (abc )'"

, then prov e that

3. Jfa+ b= l ,a> O, b> O,pro ve that ( a+± Y

+b: +cs > ( a+~ + c 8

·

r!(n - r)!

F, +Jc ;+ · · · + Jc: < ✓ n (2" - I) .

We have to prov e that as+ b8 +cs > a2 b2 c2(bc +ca + ab). Now , a

n1

3 I =- =-

6

2

Now

2

) .. -(I +x,,)l'

11 Inequali tle~ Involvln9 Mea ns 6, -

x4

2'. () 2 ,1 x - 4x + 8x + 16 " f(x ) ~ 0 and the least value is 0. Hence, lhe greatest value is I/ 12

l 1- _NoW,

,R + 2 6

__!J_ 4 2 - . 4

-

X

X

I l s:: j(x) ⇒ 12 /(x ) ~

12

e Again using A.M. 2'. G.M. , we hav

or

2x6 + 8x2 - 4x4 2'. 4x4 2'. O 4 8 2 6 )6 > Q X + X - 4x + 8x2 +

Also,

x4 ~

0

$

O

J2 = O

I

2

' / EX AM PL E 6, 6 . 5 should b d. , In how many parts an integer N 2'.'ized e issected so that . the parts is maxim

the product of S01:

Using AM - G •M., we get · ·· >

x1 +x2 + .. . + ';\:II

2 6 2x +8x 4 - -2- > 4 - X

or

+( y+½J2

( x+}J.2 +( y +½

2

+

Now, A.M. 2'. G.M. x

4I +y2+ y + 4 $0 2

4

z-1

? +x +y +y $ -2

es of.the function Calculate the greatest and least valu 4

= 1, then

_!. 2'. z2(x 1 +y+x+ y')

2

X

+ I6

---2

- a,a2a3)

b3

6

x+y '

Y

2

16x +y

213

-2 '.4 or 216 113 b 2'. 6 X 4 or Adding Eqs. (1) and (2 ), we get 113 113 a + b 2'.6 (2 + 4 )

8

'+ 16x

[s.ol3J Using A.M. 2'. G.M. , we get . 2

3

(': bis G.M. of a and c)

2 ~3 c + 2ab 2 - - - > (c aca ) 3 2 2ab > 3ac + c

If x and y are real numbers such that find the values of x and y.

+a2a3+a1a3 > (

0 1a2

2 1/3 c2 +ab+ab > (c 2b2c,) 3

f-e:~~MP~:E·s.~

J

a3

>? 216 - -

itive real numbers, so Since a, b, c arc three distinct pos 2 ab, we get applying A.M. > G.M . in c , ab,

i Sol.

or

a, +a2 +a3 >- ( a a?a, )1/3

or

e real numbers in 0.P ., then If a; b, car e three distinct positiv 2 prove that c + 2ab > Jae.

11

or

·

,

2:: (x I.\' 2

.. -.\

. )1 1,, 1

~- )" - ( X1 + X?- + ··· + ,,,, x 1x2 ......\' ,, < 11

~~ - - - - .- :- ~-: -- :~ ~- -~ - ~ 2 )2 x, ~x, :,:•::s ⇒ (x+~) + +z+; J ~

Algebra

. f Therefore, max:1mwn value o x 1 .,.2 -'.3 · · · -x" is obtained when :,x! :, ; ;,:" ;:e; are all equal. Now,

x, + X2 + ... + x,, )" I . I1 . Now, function to be maximized is _!.._ .--'=- ----w 11c is ( . . of a discre /1 te fun ct1011 n. In order to arrive at some possible neighborhood, we make it continuous first. Thus, changing the variable n to x , we have /(x)

N)x

-I

or

EXA MPL E

6.7

(

100

1

>

three posit ive real numb ers, then 2

~

find,~ . ,,

Using A.M . 2:: G .M ., we get a 2 + 1 2:: 2a

a2 + 1 - - > -2a-

b+

C

b+

-

C

b2 + 1 2b Simil arly, - - 2:: - -

a+ c

and

If x + y + z = 1 and x, y , z are positive, then show that

a+c

c2 + 1 2c -->a+b - a+b

(J1

Addi ng (1), (2) and (3), we get

(x+H+ +:J++H >l~O. SoL

1)

a2 + 1 b2 + 1 c + 1 minimum value of ~ + ~ + ~ ·

⇒

f'(x) = 0 for x = Nie Hence. the neare st intege r is [Nie] or [Nie]+ 1, where [x] denot es the greate st intege r less than or equal to x .

(

If a, b an d c are

Sol.

1n(:~)-1)

2

EXA MPL E 6.B

= 0. i.e ..

f'(x) =f(x ) (

+z I > 9

I

y

y+ y

= (~

For maxim a.f'(x )

+

x

a 2 + l b2 + 1 c 2 + 1 2b --+--+ - - : 2 : :2a - - + - - + -2cb +c

A .M. of 2nd powe r > 2nd powe r of A.M.

Now,

a+ c

2a

2b

b +c

a+ c

a+ b

b +c

a+ c

a +b

2c --+--+-

⇒

= (~ + b+ c

a+ b

2) + (--3!! .._ + 2) + (a+b ~ + 2) -6 a+c

a+b +c a+b +c) -_ 2 (a+b +c + - - - + - - - -6 b+c a+c

3

a+b

Now,

or

IT

a+b +c a+b +c a+b +c +- - - + - - b+ c a+c a+b

(x+ ~)' + +~)' +(,+;)'

3

3

I -1 +-1 ) > -I ( x+ y +z+ -+ 9 X y Z

(x+H+ +:;J+( z+;J' /!_(I+ _I_+ _I_+ !_)2 3

9l

x

3 ~-b+---- -a+ r c -- a+c

2

z [·: x +y +z == I]

- -- + - - - +- a + b + c a+b +c a+b + ~-3

Hence, minimum value is _ 3

- 3

6.13

Inequalities Involving Means

1□ 2

I I. For x - (a+ 3)/x/ + 4 of a is

J. The mini mum valu e of num bers x, y, z is 2 (2) 2'✓ (I) -fi

x4

--- =- -xyz

(3)

for posit ive real

4'12.

(4) 8'12.

dinat e axes. If 2. A rod of fixed lengt h k slide s alon g the coor the mini mum it meet s the axes at A(a, 0) and B(O, b), then

va lue of (a+ ~

J2 + ( b + ¾J2 is (2) 8

(!) 0 (3)

k2 -4+_i_

k2

(4)

k2+4+ _i_

2

k2

(3) (III) is corre ct (4) none of the abov e are corre ct 3 4 5 2 3 4 3 b, c > 0), then the 4. If ab 2 c , a b c , a b c are in A.P. (a, mini mum value of a + b + c is (4) 9 (3) 5 (2) 3 (I) 1 1 is 5. lfy = y - 1 + r x- , then the least value ofy (4) 3/2 (3) 2/3 (2) 6 (I) 2 l b+(a +b) l c (for 6. Mini mum value of (b+c ) l a+(c +a) real posit ive num bers a, b, c) is (4) 6 (3) 4 (2) 2 (I) 1 then their sum is 7. If the prod uct of n posit ive numb ers is n", (2) divis ible by n (I) a posit ive integ er 2 4) neve r less than n ( (3) equal ton+ J/n when .xyz = abc, 8. The mini mum value of P = bcx +cay + abz, IS

(4) 4abc (3) abc of th e equa tion 9. Jf /, m, n are the three posit ive roots value of (I ll) mum mini the x3 _ ax2 + bx _ 48 = 0, then + (2 /m) + (3 /n) equa ls (4) 5/2 (3) 3/2 (2) 2 (I) I .P., then equa tion JO. Jf positive num bers a, b, c are in H has R) E (k x2 ~ kx + b101 _ 101 _ c10 1 == o (2) 6abc

0 2 (I) both roots positive

(2) both roots nega tive (3) one posit ive and one nega tive roof (4) both roots imag inary

(2) (- 3, oo) (4) [I, oo)

the mini mum value 12. If a, b, c are the sides of a triangle, then c h a 1.s equa I to of - - - + - - - + b+c -a c+a -h a+h -c (4) 12 (3) 9 (2) 6 (1) 3 mini mum value of 13. If a, b, c, d E R+ - {I }, then the logd a + Jogb d + log0 c + loge b is (2) 2 (1) 4 none of these (4) (3) 1 ab . 1 ays ac be + - - + - - 1saw 14. Ifa, b,c e R ,th e n - - + b+c a+c a+b

2

18 is 3. The least value 6f 6 tan + 54 cot + tan 2 (/J, 54 cot2 , (I) 54 when A.M . ~ G.M . is applicable for 6 18 6 tan 2 (/J, 54 cot2 , (II) 54 when A.M. ~ G.M. is applicable for and 18 is adde d furth er 2 2 (III) 78 when tan = cot (I) (I) is correct, (II) is false (2) (I) and (II) are corre ct

(I) 3abc

(1) (-oo, - 7]u[ l,oo ) (3) (-oo, - 7]

+ y4 + 2 2

= 0 to have real solut ions, the rang e

(I)

$

(3)

$

1 -(a+ b+c ) 2 1 -(a+ b+c )

(2)

~}__Mc

(4)

~ !._f;bc

3

2

3

+ 15. Ifa, b,ce R , tben (a+ b+c )

(

I ys 1 1) is. awa

J +-;; -;;+b

(2) ~ 9 (4) none of these

(I) ~ 12 (3) $ 12

of 16. If a, b, c E R+, then the mini2mum2 value 2 2 2 l to equa 2 is ) b + c(a + ) a + a(b + c ) + b(c (4) 6abc 3abc (3) 2abc (2) (l) abc H.P. , then 17. If a, b, c, d E R+ and a, b, c, dare in >c+ d a+b (2) b+c (1) a+d > these of none (4) d + b > (3) a+ c then the max imum 18. If a, b, c E R+ such that a+ b + c = 18, 2 3 4 value of a b c is equa l to 3 2 8 (2) 2 18 X 3 (1) i X 3 (3)

i

9

x3

2

3 19 (4) 2 x3

. . . (x -2)( x-1) mum valu e off(x ) 1s mm1 The 3. > x ,V ----'= f(x) 19. · (x-3 ) equa l to (2) 3 + 2"YJ (I) 3 + 2-..fi 2-2 (4) 3'✓ (3) JV2 + 2 2 3 a + I)~ is 20. ff a > 0, then least value of (a + a + 4 2 (2) 16a (I) 64a 3 (4) none of these (3) l6a

Ill

sid ~, of a rr· . iang 1e, J. If A is lhe nrcn and 2s the sum of the , t:~ then ?

(I) A ::;~ 4

,,-

'

(2) .-l :5 ~ 3 3✓

1

(3) A < -· -

Jj

(4) none of these

6 -14

Algebra

2·

lfx. -"· =are posit ive numb ers in A.P., then 2 (2) -~Y + yz ~ 2xz x+v v+z (3) ---+ -·- - ~ 4 (4) none of these 2_,·-x 2y-=

For Problems 4-6 · . Equation x 4 + ax,3 + bx2 + ex + I = 0 has real roots (a, b c ' are non-negative). 4. Mini mum non-n egati ve real value of a is

(I) y ~ xz

3 • For posit ive real numb ers a. b, c such that a + b + c = /J, whic h one holds? ( 1) ( p -a)( p - b)(p - c)-S. 8 1 µ-

27

(2) (p - a) (p - b) (p - c) '?:. Sabe be ca ab ( ') ~ -++-' 5.p a b c ( 4) none of these 4. If first and (211 - l) th terms of an A.P., G.P., and H.P. are equa l and their 11 th tenns are a, b, c, respective ly, then (1) a=b =c (2) a+c =b (3) a > b >c (4) ac-b 2 =0 5. If a > 0, b > 0, c > 0 and 2a + b + 3c = 1, then 4

,

, .

4

,

, .

1

(1) a b-c 1s greatest then a= 4

1

(2) a b-c- 1s great est then b = -

4

4 ,

, .

(3) a b-c 1s great est then c

= -1

12

(4) great est value of a 4 b 2 c 2 is _l_ 9-4 8

IH For Prob lems 1-3 If roots of the equation.f{x) = x 6 - 12x 5 + bx4 + cx 3 + dx 2 +ex+ 64 = 0 are positive, then 1. whic h has the great est abso lute value ? (1) b (2) c (3) d (4) e 2. whic h has the least abso lute value ? (1) b (2) C (3) d (4) e 3. rema inde r when.f{x) is divid ed by x - I is (1) 2 (2) 1 (3) 3 (4) 10

(I) JO (2) 9 (3) 6 5. Mini mum non-n egative rea l value of b is (I) 12 (2) JS (3) 6 6. Mini mum non-n egati ve real value of c is (l) 10 (2) 9 (3) 6

ADV ANC ED

f(x) =

1. The least value of a X

E

R for whic h 4 a x2 + _!_ 2 I, for al I X

> 0, is

(1)

I

64

(2 ) 32

( 3) 27

l (4 ) 25

(JEE Advanced 2016)

(4) 4

2. Let x 2

4x 2 +8x +13 . 6(l+ x) ' is _ _ .

3x + p = 0(~as t;v)o .posi tive roots a and b, then mini mum value of - +- 1s _ _ _ . a b .. 12- yz 3. If x, y, and z are posit ive real numb ers and x = - .The maxi mum value of (xyz) equa ls _ _ _ Y+z 4. If a, b, and c are posit ive and 9a + 3b + c = 90, then the maxi mum value of (log a+ log b + log c) is (base of the logar ithm is l 0) _ __ _

1 \

1 \

\

5. Give n that x, y, z are posit ive reals such that xyz = 32. Toe mini mum value of x 2 + 4.xy + 4y2 + 2z 2 is _ -__. 6. If x, y E R+ satis fying x + y = 3, then the maximum value of x 2y is _ _ _ .

7. For any x, y E R, xy > 0 . Then the mini mum value of 2x x 3 y 4y 2 .

-+-+--

y3

3

9x 4

IS ---

8. Let a, b, c, d and e be posit ive real numb ers such that a+ b + c + d + e = 15 and ab 2 c 3cfe 5 = (120)3 x 50. Th~n 2 the value of a + b 2 + c 2 + cf + e 2 is -_ _ - _.

x xJ xf xJ

9. Cons ider the syste m of equa tions + + + +xJ,= 5 and x 1 + 2x2 + 3x + 4x + 5x = 151 wher e x , x , x,, X.i,X: 3 4 5 are posit ive real numb ers. Then num ber of 1 2 (x 1,x2,x_\,X-1-,x:l is - -

Num erica l Value Type

Single Corr ect Ans wer Type

(4) 10

1. For x ~ 0 , the smal lest value of the funct ion

g__. -- -- -- -- -- -- -- -- -- .-Ar--ch--ive- -s-·- - - - - - - - - - - --

JEE

(4) 4

--

- -I

_,, a..l, 1. The mini mum value of the sum of real numb ers a ' 3a- 3 , 1, a 8 , and a 10 with a > 0 is _ _ .

(IIT-~E 2Qttl

p lnequalities Involving Means

6.15

Answers Key - - -- - -----__,) ~ - - - -------:--__::~ ~ ~ ~~ Numerical Value Type

~ Single correct Answer Type 1, (2)

. (4) 6 11. (4) 16. (4)

2. 7. 12. 17.

(4) (4) (l) (1)

3. 8. 13. 18.

(2) (l) ( 1) (4)

4. 9. 14. 19.

(2) (3) (l) ( l)

Multiple correct Answers Type 1, (1), (2)

2. (1), (3) 5. (1), (2), (3), (4)

4. (3). (4)

Linked comprehension Type

t. (3) 6. (4)

2. (1)

3. (2)

4. (4)

5. (3)

10. (3) 15. (2) 20. (3)

1. (2) 6. (4)

2. (3) 7. (2)

3. (8)

4. (3)

8. (55)

9. ( \ )

ARCHIVES

JEE ADVANCED Single Correct Answer Type

3. (l ), (2)

1. (3) Numerical Value Type

1. (8)

5. (3)

5. (96)