mathematics paper 2

JEE(ADVANCED)-2023-Paper-2-MPC-1




FIITJEE
SOLUTIONS TO 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.


1 x x3


Q.1. Let f : [1, )  be a differentiable function such that f 1 3 and 3 1 f t dt xf x 3,x




[1, ). Let e denote the base of the natural logarithm. Then the value of f (e) is e2
4 log 4 e
e
(A) (B)
3 3
2 2
4e e 4
(C) (D)
3 3

Sol. C
x 3
x

3 f x dt xf x 3


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,JEE(ADVANCED)-2023-Paper-2-MPC-2



1


3f x f x xf x x2 (using Newton’s Leibniz theorem)
dy 2
y x

dx x
2
I.F. = e xdx e 2logx 1

x2
solution is
y
2 loge x c


x
y x2 loge x cx2




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But f 1

c

y = x1 logex + x2

1 4
Now y(e) = e2 e2 e2
3 3

Q.2. Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the

probability of a random toss resulting in head is , then the probability that the experiment stops with head

is

(A) (B)


1


9


y




3
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, JEE(ADVANCED)-2023-Paper-2-MPC-3


(C) (D)


Sol. B
1 2
P H ,P T



3 3
P = P(H H) + {P(T H H) + P(T H T H H) + P(T H T H T H H) + … )
+ {P(H T H H) + P(H T H T H H) + P(H T H T H T H H) + .. )
1 2 1 2 1 2 1
9 3 9 3 3 3 9 1 2 1 1 2 1 2 1
2 1 2 1
1 27 3 3 9
2 2 ..... +
9
1 1
9 9 ....
3 3 9 3 3 3 3 9




.



Q.3. For any y , let cot 1 y 0, and tan 1 y , . Then the sum of all the solutions of the


2 2
1 1
6y 9 y2 2
equation tan 2 cot for 0 < |y| < 3, is equal to



9 y 6y 3

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

Sol. C
If 0 < y < 3 then given equation can be written as
1 6y 1 6y 2
tan 2 tan 2




9 y 9 y 3



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, JEE(ADVANCED)-2023-Paper-2-MPC-4


1 6y
tan
1 6y
tan tan 2
tan
9 y 3
6y 2
2 3 3y 6y 9 3 0
9 y
6 36 108 6
y 3
2 3 2 3
If – 3 < y < 0 then given equation can be written as

1 6y 1 6y 2
tan 2 tan 2




9 y 9 y 3


1 6y
tan 2 9 y 6




ˆ ˆ
Q.4. Let the position vectors of the points P, Q, R and S be a iˆ 2ˆj 5k , b 3iˆ 6ˆj 3k ,

16 ˆ ˆ
c 17 iˆ ˆj 7k and d 2iˆ ˆj k , respectively. Then which of the following statements is true?
5 5
(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

Sol. B



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www.fiitjee.com.