Download JEE Advanced 2023 Mathematics Question Paper - 2

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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. Let $f:[1, \infty) \to R$ be a differentiable function such that $f(1)$=$\frac{1}{3}$ and $3\int \limits_1^xf(t)dt$=$xf(x)-\frac{x^2}{3}$, $x \in [1, \infty)$. Let $e$ denote the base of the natural logarithm. Then the value of $f(e)$ is
   1. $\frac{e^2+4}{3}$
   2. $\frac{log_e4+e}{3}$
   3. $\frac{4e^2}{3}$
   4. $\frac{e^2-4}{3}$
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 $\frac{1}{3}$, then the probability that the experiment stops with head is
   1. $\frac{1}{3}$
   2. $\frac{5}{21}$
   3. $\frac{4}{21}$
   4. $\frac{2}{7}$
3. For any $y \in R$, let $\cot^{-1}(y) \in (0, \pi)$ and $\tan^{-1}(y) \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then the sum of all the solutions of the equation $\tan^{-1}\left(\frac{6y}{9-y^2}\right)$+$\cot^{-1}\left(\frac{9-y^2}{6y}\right)$=$\frac{2\pi}{3}$ for $0 < |y| < 3$, is equal to
   1. $2\sqrt{3}-3$
   2. $3-2\sqrt{3}$
   3. $4\sqrt{3}-6$
   4. $6-4\sqrt{3}$
4. Let the position vectors of the points $P$, $Q$, $R$ and $S$ be $\vec{a}$=$\hat{i}$+$2\hat{j}-$$5\hat{k}$, $\vec{b}$=$3\hat{i}$+$6\hat{j}$+$3\hat{k}$, $\vec{c}$=$\frac{17}{5} \hat{i}$+$\frac{16}{5} \hat{j}$+$7\hat{k}$ and $\vec{d}$=$2\hat{i}$+$\hat{j}$+$\hat{k}$, respectively. Then which of the following statements is true?
   1. The points $P$, $Q$, $R$ and $S$ are NOT coplanar
   2. $\frac{\vec{b}+2\vec{d}}{3}$ is the position vector of a point which divides $PR$ internally in the ratio 5:4
   3. $\frac{\vec{b}+2\vec{d}}{3}$ is the position vector of a point which divides $PR$ externally in the ratio 5:4
   4. The square of the magnitude of the vector $\vec{b}×\vec{d}$ is 95

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 none of the options is chosen (i.e. the question is 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 +2 marks;
  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 (i.e. the question is unanswered) will get 0 marks; and choosing any other combination of options will get -2 marks.

5. Let $M=(a_{ij})$, $i, j \in \text{{1, 2, 3}}$, be the 3×3 matrix such that $a_{ij}$=1 if $j+1$ is divisible by $i$, otherwise $a_{ij}$=0 . Then which of the following statements is(are) true?
   1. $M$ is invertible
   2. There exists a non-zero column matrix $\begin{equation*} \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} \end{equation*}$ such that $\begin{equation*}M \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} \end{equation*}$=$\begin{equation*} \begin{pmatrix} -a_1 \\ -a_2 \\ -a_3 \end{pmatrix} \end{equation*}$
   3. The set {$X \in R^3$ : $MX=0$} $\neq$ {0} where 0 =$\begin{equation*} \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \end{equation*}$
   4. The matrix $( M-2I)$ is invertible, where $I$ is the 3 × 3 identity matrix
6. Let $f:(0,1) \to R$ be the function defined as $f(x)$=$[4x]$$\left(x-\frac{1}{4}\right)^2$$\left(x-\frac{1}{2}\right)$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements is(are) true?
   1. The function $f$ is discontinuous exactly at one point in (0,1)
   2. There is exactly one point in (0,1) at which the function $f$ is continuous but NOT differentiable
   3. The function $f$ is NOT differentiable at more than three points in (0,1)
   4. The minimum value of the function $f$ is $-\frac{1}{512}$
7. Let $S$ be the set of all twice differentiable functions $f$ from $R$ to $R$ such that $\frac{d^2f}{dx^2}(x)$ > 0 for all $x \in (-1, 1)$. For $f \in S$, let $X_f$ be the number of points $x \in (-1, 1)$ for which $f(x)=x$. Then which of the following statements is(are) true?
   1. There exists a function $f \in S$ such that $X_f$= 0
   2. For every function $f \in S$ , we have $X_f \leq 2$
   3. There exists a function $f \in S$ such that $X_f$= 2
   4. There does NOT exist any function $f$ in $S$ such that $X_f$=1

SECTION 3 (Maximum Marks:24)

• This section contains SIX (06) questions.
• The answer to each question is a NON-NEGATIVE INTEGER.
• For each question, enter the correct integer corresponding to the answer using the mouse and the on- screen virtual numeric keypad in the place designated to enter the answer.
• Answer to each question will be evaluated according to the following marking scheme:
  
  • Full Marks: +4 If ONLY the correct integer is entered;
  • Zero Marks: 0 In all other cases.

8. For $x \in R$, let $\tan^{-1}(x) \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then the minimum value of the function $f: R \to R$ defined by $f(x)$=$\int \limits_0^{x \tan ^{-1}x}\frac{e^{(t-\cos t)}}{1+t^{2023}}dt$ is
9. For $x \in R$, let $y(x)$ be a solution of the differential equation $(x^2-5)\frac{dy}{dx}-$$2xy$=$-2x(x^2-5)^2$ such that $y(2)$=7. Then the maximum value of the function $y(x)$ is
10. Let $X$ be the set of all five digit numbers formed using 1,2,2,2,4,4,0. For example, 22240 is in $X$ while 02244 and 44422 are not in $X$ . Suppose that each element of $X$ has an equal chance of being chosen. Let $p$ be the conditional probability that an element chosen at random is a multiple of 20 given that it is a multiple of 5. Then the value of 38$p$ is equal to
11. Let $A_1$, $A_2$, $A_3$, ..., $A_8$ be the vertices of a regular octagon that lie on a circle of radius 2. Let $P$ be a point on the circle and let $PA_i$ denote the distance between the points $P$ and $A_i$ for $i$=1, 2, ..., 8. If $P$ varies over the circle, then the maximum value of the product $PA_1$•$PA_2$...$PA_8$, is
12. Let $R$ =$\begin{equation*} \left\{ \begin{pmatrix} a & 3 & b \\ c & 2 & d \\ 0 & 5 & 0 \end{pmatrix} \right. \end{equation*}$ : a, b, c, d$\left. \in \text{{0, 3, 5, 7, 11, 13, 17, 19}} \right\}$. Then the number of invertible matrices in $R$ is
13. Let $C_1$ be the circle of radius 1 with center at the origin. Let $C_2$ be the circle of radius $r$ with center at the point $A$ = (4,1), where 1 < $r$ < 3. Two distinct common tangents $PQ$ and $ST$ of $C_1$ and $C_2$ are drawn. The tangent $PQ$ touches $C_1$ at $P$ and $C_2$ at $Q$. The tangent $ST$ touches $C_1$ at $S$ and $C_2$ at $T$. Mid points of the line segments $PQ$ and $ST$ are joined to form a line which meets the $x$ -axis at a point $B$ . If $AB =\sqrt{5}$ , then the value of $r^2$ is

SECTION 4 (Maximum Marks:12)

• This section contains TWO (02) paragraphs.
• Based on each paragraph, there are TWO (02) questions.
• The answer to each question is a NUMERICAL VALUE.
• For each question, enter the correct numerical value of the answer using the mouse and the on- screen virtual numeric keypad in the place designated to enter the answer.
• If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.
• Answer to each question will be evaluated according to the following marking scheme:
  
  • Full Marks: +3 If ONLY the correct numerical value is entered in the designated place;
  • Zero Marks: 0 In all other cases.

PARAGRAPH "I"

Consider an obtuse angled triangle $ABC$ in which the difference between the largest and the smallest angle is $\frac{\pi}{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.

14. Let $a$ be the area of the triangle $ABC$. Then the value of $(64a)^2$ is
15. Then the inradius of the triangle $ABC$ is

PARAGRAPH "II"

Consider the 6 × 6 square in the figure. Let $A_1$, $A_2$, ..., $A_{49}$ be the points of intersections (dots in the picture) in some order. We say that $A_i$ and $A_j$ are friends if they are adjacent along a row or along a column. Assume that each point $A_i$ has an equal chance of being chosen.

16. Let $p_i$ be the probability that a randomly chosen point has $i$ many friends, $i$ = 0,1,2,3,4 . Let $X$ be a random variable such that for $i$ = 0,1,2,3,4, the probability $P(X = i)=p_i$ . Then the value of $7E (X)$ is
17. Two distinct points are chosen randomly out of the points $A_1$, $A_2$, ..., $A_{49}$. Let $p$ be the probability that they are friends. Then the value of $7 p$ is

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