Exam (elaborations) jee

JEE (Advanced) 2023 Paper 2
Mathematics
SECTION 1 (Maximum Marks: 12)

 This section contains FOUR (04) questions.
 Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is the
correct answer.
 For each question, choose the option corresponding to the correct answer.
 Answer to each question will be evaluated according to the following marking scheme:
Full Marks : 3 If ONLY the correct option is chosen;
Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered);
Negative Marks : 1 In all other cases.


Q.1 1
Let f :[1,  )   be a differentiable function such that f (1)  and
3
x x3
3  f (t ) dt  x f ( x)  , x  [1, ) . Let e denote the base of the natural logarithm. Then the
1 3
value of f (e) is

e2  4 log e 4  e 4e 2 e2  4
(A) (B) (C) (D)
3 3 3 3



Q.2 Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses
1
are same. If the probability of a random toss resulting in head is , then the probability that the
3
experiment stops with head is


1 5 4 2
(A) (B) (C) (D)
3 21 21 7




Q.3  
(
For any y   , let cot 1 ( y )  (0,  ) and tan 1 ( y )   , ). Then the sum of all the solutions
2 2
1 9  y
)  2 for 0  | y |  3, is equal to
2
6y
of the equation tan 1 ( )  cot (
9 y 2
6y 3

(A) 2 3  3 (B) 3  2 3 (C) 4 3  6 (D) 6  4 3




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
Q.4 Let the position vectors of the points P, Q, R and S be a  iˆ  2 ˆj  5kˆ , b  3iˆ  6 ˆj  3kˆ ,
 17 16 ˆ 
c  iˆ  j  7kˆ and d  2iˆ  ˆj  kˆ , respectively. Then which of the following statements is
5 5
true?


(A) The points P, Q, R and S are NOT coplanar
 
b  2d
(B) is the position vector of a point which divides PR internally in the ratio 5 : 4
 3 
b  2d
(C) is the position vector of a point which divides PR externally in the ratio 5 : 4
3
 
(D) The square of the magnitude of the vector b  d is 95




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,JEE (Advanced) 2023 Paper 2


SECTION 2 (Maximum Marks: 12)

 This section contains THREE (03) questions.
 Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is
(are) correct answer(s).
 For each question, choose the option(s) corresponding to (all) the correct answer(s).
 Answer to each question will be evaluated according to the following marking scheme:
Full Marks : 4 ONLY if (all) the correct option(s) is(are) chosen;
Partial Marks : 3 If all the four options are correct but ONLY three options are chosen;
Partial Marks : 2 If three or more options are correct but ONLY two options are chosen, both of
which are correct;
Partial Marks : 1 If two or more options are correct but ONLY one option is chosen and it is a correct
option;
Zero Marks : 0 If unanswered;
Negative Marks : 2 In all other cases.
 For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct
answers, then
choosing ONLY (A), (B) and (D) will get 4 marks;
choosing ONLY (A) and (B) will get 2 marks;
choosing ONLY (A) and (D) will get 2marks;
choosing ONLY (B) and (D) will get 2 marks;
choosing ONLY (A) will get 1 mark;
choosing ONLY (B) will get 1 mark;
choosing ONLY (D) will get 1 mark;
choosing no option(s) (i.e. the question is unanswered) will get 0 marks and
choosing any other option(s) will get 2 marks.




Q.5 Let M  (aij ), i, j  {1, 2,3}, be the 3  3 matrix such that aij  1 if j  1 is divisible by i ,
otherwise aij  0 . Then which of the following statements is(are) true?


(A) M is invertible
 a1   a1    a1 
    
(B) There exists a nonzero column matrix  a2  such that M  a2  =   a2 

a   a   a 
 3  3  3
0
(C) The set { X   : MX  0}  {0} , where 0 =  0 
3  
0
 
(D) The matrix ( M  2 I ) is invertible, where I is the 3  3 identity matrix




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Q.6 2
 1  1
Let f : (0,1)   be the function defined as f ( x)  [4 x]  x    x   , where [ x] denotes
 4  2
the greatest integer less than or equal to x . Then which of the following statements is(are) true?


(A) The function f is discontinuous exactly at one point in (0,1)

(B) There is exactly one point in (0,1) at which the function f is continuous but NOT
differentiable

(C) The function f is NOT differentiable at more than three points in (0,1)

1
(D) The minimum value of the function f is 
512




Q.7 d2 f
Let S be the set of all twice differentiable functions f from  to  such that ( x)  0 for
dx 2
all x  (1,1). For f  S , let X f be the number of points x  (1,1) for which f ( x)  x. Then
which of the following statements is(are) true?


(A) There exists a function f  S such that X f  0
(B) For every function f  S , we have X f  2
(C) There exists a function f  S such that X f  2
(D) There does NOT exist any function f in S such that X f  1




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