Q: Range of the function f(x)=log2(2−log2(16sin2x+1)) is :

f(x)=log2(2−2log2(16sin2x+1)) 0≤log2(16sin2x+1)≤log217⇒2−2log217≤2−2log2(16sin2x+1)≤2 ⇒0<2−2log2(16sin2x+1)≤2⇒f(x)≤1

Q: The value of a and b for which ∣∣e∣x−b∣−a∣∣=2, has four distinct solutions, are :

For any b∈Re∣x−b∣ is∣∣e∣x−b∣−a∣∣ has four distinct solutions a>3 so a∈(3,∞)

Q: Find the number of real ordered pair(s) (x,y) for which : 16x2+y+16x+y2=1

(22x2+2y−22x+2y2)2=1−22x2+2y2+2x+2y+1≥0 ⇒(x+21)2+(y+21)2≤0

Q: The maximum value of sec−1(2(x2+2)7−5(x2+3)) is :

sec−1(−25+2(x2+2)2)=sec−1(−25+(x2+2)1)≤sec−1(−2)=π−sec−1(2)=32πx2+2≥2(x2+21≤21)(−25+x2+21≤−25+21=−2)

Q: Let A be the greatest value of the function f(x)=logx[x], (where [.] denotes greatest integer function) and B be the least value of the function g(x)=∣sinx∣+∣cosx∣, then :

f(x)=logx[x]⇒f(x)∈[0,1] g(x)=∣sinx∣+∣cosx∣ ⇒g(x)∈[1,2]

Q: Solution of the inequation {x}({x}−1)({x}+2)≥0 (where {⋅} denotes fractional part function) is :

0≤{x}<1 {x}({x}−1)({x}+2)≥0 ⇒{x}=0⇒x∈z

Q: - Let f(x),g(x) be two real valued functions then the function h(x)=2max{f(x)−g(x),0} is equal to :

(c) f(x)−g(x)+∣g(x)−f(x)∣

Question 1

Range of the function f(x)=log2​(2−log2​​(16sin2x+1)) is :

Accepted Answer

f(x)=log2​(2−2log2​​(16sin2x+1))
0≤log2​​(16sin2x+1)≤log2​17⇒2−2log2​17≤2−2log2​​(16sin2x+1)≤2
⇒0<2−2log2​​(16sin2x+1)≤2⇒f(x)≤1

Question 2

The value of a and b for which ∣∣​e∣x−b∣−a∣∣​=2, has four distinct solutions, are :

Accepted Answer

For any b∈Re∣x−b∣ is∣∣​e∣x−b∣−a∣∣​ has four distinct solutions a>3 so a∈(3,∞)

Question 3

The range of the function : f(x)=tan−1x+21​sin−1x

Accepted Answer

Domain =[−1,1] and both are increasing functions.
∴x=−1, we get minimum value &x=1, we get maximum value.
[−4π​−4π​,4π​+4π​]=[−2π​,2π​]

Question 4

Find the number of real ordered pair(s) (x,y) for which : 16x2+y+16x+y2=1

Accepted Answer

(22x2+2y−22x+2y2)2=1−22x2+2y2+2x+2y+1≥0
⇒(x+21​)2+(y+21​)2≤0

Question 5

For a real number x, let [x] denotes the greatest integer less than or equal to x. Let f:R→R be defined by f(x)=2x+[x]+sinxcosx. Then f is :

Accepted Answer

(a) One-one but not onto

Question 6

The maximum value of sec−1(2(x2+2)7−5(x2+3)​) is :

Accepted Answer

sec−1(−25​+2(x2+2)2​)=sec−1(−25​+(x2+2)1​)≤sec−1(−2)=π−sec−1(2)=32π​​x2+2≥2(x2+21​≤21​)(−25​+x2+21​≤−25​+21​=−2)​

Question 7

Number of ordered pair (a,b) from the set A={1,2,3,4,5} so that the function f(x)=3x3​+2a​x2+bx+10 is an injective mapping ∀x∈R:

Accepted Answer

f′(x)=x2+ax+b is injective if D≤0
a2−4b≤0
 If a=1,b=1,2,3,4,5a=2,b=1,2,3,4,5a=3,b=3,4,5a=4,b=4,5a=5​ Number of pair =5 Number of pair =5 Number of pair =3 Number of pair =2 b has no value ​

Question 8

Let A be the greatest value of the function f(x)=logx​[x], (where [.] denotes greatest integer function) and B be the least value of the function g(x)=∣sinx∣+∣cosx∣, then :

Accepted Answer

f(x)=logx​[x]⇒f(x)∈[0,1]
g(x)=∣sinx∣+∣cosx∣
⇒g(x)∈[1,2​]

Question 9

Let A=[a,∞) denotes domain, then f:[a,∞)→B,f(x)=2x3−3x2+6 will have an inverse for the smallest real value of a, if :

Accepted Answer

f(x)=2x3−3x2+6
f′(x)=6x2−6x≥0.⇒x∈[1,∞)
and f(x)∈[5,∞)

Question 10

Solution of the inequation {x}({x}−1)({x}+2)≥0 (where {⋅} denotes fractional part function) is :

Accepted Answer

0≤{x}<1
{x}({x}−1)({x}+2)≥0
⇒{x}=0⇒x∈z

Question 11

- Let f(x),g(x) be two real valued functions then the function h(x)=2max{f(x)−g(x),0} is equal to :

Accepted Answer

(c) f(x)−g(x)+∣g(x)−f(x)∣

Question 12

Let R={(1,3),(4,2),(2,4),(2,3),(3,1)} be a relation on the set A={1,2,3,4}. The relation R is :

Accepted Answer

(c) not symmetric

Question 13

The true set of values of ' K ' for which sin−1(1+sin2x1​)=6Kπ​ may have a solution is :

Accepted Answer

1+sin2x∈[1,2]
1+sin2x1​∈[21​,1]
sin−1(1+sin2x1​)∈[6π​,2π​]

Question 14

A real valued function f(x) satisfies the functional equation f(x−y)=f(x)f(y)−f(a−x)f(a+y) where ' a ' is a given constant and f(0)=1,f(2a−x) is equal to :

Accepted Answer

f(x−y)=f(x)f(y)−f(a−x)f(a+y)
Put x=y=0
f(0)=[f(0)]2−f(a)f(a)
⇒f(a)=0[∵f(0)=1]
Put x=a and y=xf(a−x)=f(a)f(x)−f(0)f(a+x)
⇒f(a−x)​=f(a+x)⇒f(2a−x)=−f(x)​

Question 15

Let g:R→R be given by g(x)=3+4x if gn(x)=g∘g∘go……og(x)n times. Then inverse of gn(x) is equal to :

Accepted Answer

(a) (x+1−4n)⋅4−n

Question 16

Let f:D→R be defined as :f(x)=x2+4x+3ax2+2x+a​ where D and R denote the domain of f and the set of all real numbers respectively. If f is surjective mapping, then the complete range of a is :

Accepted Answer

(d) 0<a<1

Question 17

If f:(−∞,2]⟶(−∞,4], where f(x)=x(4−x), then f−1(x) is given by :

Accepted Answer

f(x)=4x−x2=yx2−4x+y=0f−1(x)=2−4−x​

Question 18

If [5sinx]+[cosx]+6=0, then range of f(x)=3​cosx+sinx corresponding to solution set of the given equation is : (where [.] denotes greatest integer function)

Accepted Answer

[5sinx]+[cosx]=−6
⇒−1≤cosx<0 and −5≤5sinx<−4
−1≤sinx<−54​

Question 19

If f:R→R,f(x)=ax+cosx is an invertible function, then complete set of values of a is :

Accepted Answer

f(x)=ax+cosx
f′(x)=a−sinx
if f(x) is invertible, then
f′(x)≥0 or f′(x)≤0
⇒a≥1 or a≤−1

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Solutions for Advanced Problems in Mathematics for JEE (Main & Advanced) by Vikas Gupta

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