I Yr JEE Mains Maths (Functions) 17-09-2023

DELTA JUNIOR COLLEGE
I Yr MAINS MATHEMATICS
TOPIC: Functions
Date: 17 - 09 - 2023 Time : Hr.

Max. Marks :
SECTION – I
20 x 4 = 80 M
Instructions:
• This section Contains 20 Multiple choice Questions (MCQs). Candidate need to
attempt ALL the Questions.
• Candidates will get 4 marks for each correct answer, there will be a deduction of one
mark for each wrong
answer.
1. A certain polynomial P (x ), x  R when divided by x – a, x – b, x – c leaves

remainders a, b, c respectively. The remainder when P (x ) is divided by (x – a)(x –
b)(x – c) is (a, b, c are distinct)
(A)0 (B)x (C)ax + b - c (D) ax 2 + bx + c
2. If f : R → R, g : R → R be two given functions then f (x ) = 2 min {f(x) - g(x), 0} equals
(A) f(x) + g(x) – |g(x)- f(x)| (B) f(x) + g(x) + |g(x)- f(x)|
(C) f(x) - g(x) + |g(x)- f(x)| (D) f(x) - g(x) - |g(x) - f(x)|
3. If f (x ) = 3|x | −x − 2 and g(x)=sin x, then domain of definition of fog(x) is
   7 11   7 
(A) 2n  +  (B)  2n + , 2n +  (C) 2n  + 
 2 n 1 n 1  6 6   6 n 1
  7 11 
(D) {(4m + 1) :m I }  2n + , 2n + 
2 n 1  6 6 
(1 + sin  x )t − 1
4. If f (x ) = lim , then the range f(x) is
t → (1 + sin  x )t + 1
(A){-1, 1} (B) {0, 1} (C) {-1, 1} (D) {-1, 0, 1}
n
5. If f : (0,  ) → R , defined by f (x ) =  [1 + sin kx ] , where [x ] denotes the integral part
k =1
of x, then the range of f(x) is

(A) {n − 1,n + 1} (B) {n − 1, n, n + 1} (C) (n, n + 1) (D) none of these
6. Let f (x ) = (x + 1)2 − 1, x  −1, then the set S = { x : f (x ) = f −1(x )} is

 −3 + i 3 −3 − i 3 

(A) 0, − 1, ,  (B) {0, 1, -1} (C) {0, -1}

 2 2 

(D) empty
 1 
7. Range of the function f defined by f (x ) =   (where [.] and {.} respectively
 sin{ x } 
denote the greatest integer and the fractional part functions) is
(A)I, the set of integers (B)N, the set of natural numbers
(C)W, the set of whole numbers (D) {2, 3, 4,….}
8. A function F(x) satisfies the functional equation x 2F (x ) + F (1 − x ) = 2x − x 4 for all real
x. Then F(X) must be
(A) x 2 (B) 1 − x 2 (C) 1 + x 2 (D) x 2 + x + 1
1 − x 
9. If 2 f (x − 1) − f   = x , then f(x) is
 x 
1 1  1− x 1
(A) 2(1 + x ) + (B) 2( x − 1) − (C) x 2 + +3 (D)none of these
3  1 + x  x x2
10. If f : R → R, g : R → R be two one-one and onto functions such that they are the
mirror images of each other about the line y = a. If h(x) = f(x) + g(x), then h(x) is
(A)one-one onto (B) one-one into (C)many-one onto (D)many-one into
11. If a, b be two fixed positive integers such that
1
f (a + x ) = b + [b 3 + 1 − 3b 2 f (x ) + 3b{ f (x )} 2 + { f (x )} 3 ] 3
for all real x, then f(x)is a
periodic function with period
(A) a (B)2a (C)b (D)2b
12. If [2sin x]+[cos x]=-3, then range of the function f (x ) = sin x + 3 cos x in [0,2 ] is
(where [.] denoted the greatest integer function)
 1
(A) [-2, -1) (B)(-2, -1) (C)  −1, −  (D)none of these
 2
3
−3x +2
13. Function f : (−, −1) → (0, e 5 ] defined by f (x ) = e x is
(A) Many one and onto (B) Many one and into
(C) one one and onto (D) one one and into
14. If f : R → [0, ) is a function such that f (x − 1) + f (x + 1) = 3 f (x ) , then period of f(x)
is
(A) 2 (B) 6 (C)12 (D)None of these
15. The maximum value of x 2y , subject to constraints x + y + 2x 2 + 2xy + 3y 2 = k

(constant) x , y  0 is
k2 4k 3 4k 3 + k 2
(A) (B) (C) (D) None of these
( ) ( ) ( )
2 3 3
2 + 15 3 + 15 3 + 15
 y y
16. If f  2x + ,2x −  = xy , then f(m, n) + f(n, m) = 0
 8 8
(A) only when m = n (B) only when m  n

(C) only when m = −n (D) for all m and n
1 − x    
17. Let f (x ) = ln   . The set of values of '  ' for which f ( ) + f ( ) = f  2  is
2
1 + x    −  +1
satisfied are
(A) (−, −1)  (1, ) (B)(-1, 1) (C)(0, 1) (D)None of these
18. The period of the function f (x ) = cos 2 {2x } + sin2 {2x } is (where {.} denotes the
fractional part of x)
 1
(A)1 (B) (C) (D) 
2 2
19. Consider a real valued function f(x) satisfying 2 f (xy ) = ( f (x ))y + ( f (y ))x x , y  R and
n
f(1) = a where a  1 , then (a − 1) f (i ) equals
i =1
(A) a n (B) an +1 (C) a n −1 + a (D) a n +1 − a
20. If f(2x + 3y, 2x - 7y) = 20x, then f(x, y) equals
(A)7x - 3y (B) 7x + 3y (C) 3x - 7y (D) x - y
SECTION – II
5 x 4 = 20 M
Instructions:
• This section Contains 10 Numerical Questions. Candidates need to attempt Any 5
questions out of 10.
• Candidates will get 4 marks for each correct answer, there is no negative marking
21. What is the number of pairs (x, y ), x, y  R , satisfying 4x 2 = 4x + 2 = sin2 y and

x 2 + y2  3 ?
2F (n ) + 1
22. If F (n + 1) = n = 1, 2,... and F(1) = 2 then what is the value of F(101)?
2
23. Let f : R → R be a periodic function such that

1
f (T + x ) = 1 + [1 − 3 f (x ) + 3 f (x ) + 3( f (x ))2 − ( f (x ))3 ] 3
where T is a fixed positive
number, then period of f(x) is KT. Find K
24. If f (x + y ) = f (x ) + f (y ) − xy − 1  x, y  R and f(1)=1, then what is the number of

solutions of f (n ) = n,n  N ?
1
25. If for x  0, f (x ) = (a − x n ) n , g(x ) = x 2 + px + q, p, q  R and the equation g(x) – x = 0
has imaginary roots, then what is the number of real roots of the equation
g(g(x)) - f(f(x)) = 0 ?
26. If x and y satisfy the equations y = 2[x] + 3 and y = 3[x - 2] simultaneously, then
[x + y] is equal to?
27. Let f(x) be a polynomial of degree n, an odd positive integer, and is monotonic,
then what is the number of real roots of the equation
1
f (x ) + f (2x ) + f (3x ) + ...... + f (nx ) = n (n + 1) ?
2
1
28. If f(x) is an even function and satisfies the relation x 2 f (x ) − 2 f   = g (x ) , where
x
g(x) is an odd function, then what is f(5)?
29. The function f(x) is defined for all real x. If f (a + b ) = f (ab )  a and b and
 1 7
f  −  = , then what is f(2022)?
 2 2
30. If f : R → R is a function satisfying the property f (2x + 3) + f (2x + 7) = 2, x  R ,

then what is the period of f(x)?

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DELTA JUNIOR COLLEGE

Date: 17 - 09 - 2023 Time : Hr.

1. A certain polynomial P (x ), x  R when divided by x – a, x – b, x – c leaves

of x, then the range of f(x) is

(A) 2 (B) 6 (C)12 (D)None of these

15. The maximum value of x 2y , subject to constraints x + y + 2x 2 + 2xy + 3y 2 = k

(A) only when m = n (B) only when m  n

(A) (−, −1)  (1, ) (B)(-1, 1) (C)(0, 1) (D)None of these

(A) a n (B) an +1 (C) a n −1 + a (D) a n +1 − a

20. If f(2x + 3y, 2x - 7y) = 20x, then f(x, y) equals

(A)7x - 3y (B) 7x + 3y (C) 3x - 7y (D) x - y

21. What is the number of pairs (x, y ), x, y  R , satisfying 4x 2 = 4x + 2 = sin2 y and

23. Let f : R → R be a periodic function such that

24. If f (x + y ) = f (x ) + f (y ) − xy − 1  x, y  R and f(1)=1, then what is the number of

30. If f : R → R is a function satisfying the property f (2x + 3) + f (2x + 7) = 2, x  R ,

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