Download JEE Advanced 2025 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 $x_0$ be the real number such that $e^{x_0}$ + $x_0$ = 0 . For a given real number $\alpha$, define
   $g(x)$=$\frac{3xe^x+3x-\alpha e^x-\alpha x}{3(e^x+1)}$
   for all real numbers $x$. Then which one of the following statements is TRUE?
    
   1. For $\alpha$=2, $\lim \limits_{x \to x_0} \left|\frac{g(x)+e^{x_0}}{x-x_0}\right|$=0
   2. For $\alpha$=2, $\lim \limits_{x \to x_0} \left|\frac{g(x)+e^{x_0}}{x-x_0}\right|$=1
   3. For $\alpha$=3, $\lim \limits_{x \to x_0} \left|\frac{g(x)+e^{x_0}}{x-x_0}\right|$=0
   4. For $\alpha$=3, $\lim \limits_{x \to x_0} \left|\frac{g(x)+e^{x_0}}{x-x_0}\right|$=$\frac{2}{3}$
2. Let $ℝ$ denote the set of all real numbers. Then the area of the region
   $\left\{(x, y) ∈ ℝ × ℝ ∶ x > 0, y > \frac{1}{x}\right.$,$\left. 5x − 4y − 1 > 0, 4x + 4y − 17 < 0 \right\}$
   is
    
   1. $\frac{17}{16}-\log_e4$
   2. $\frac{33}{8}-\log_e4$
   3. $\frac{57}{8}-\log_e4$
   4. $\frac{17}{2}-\log_e4$
3. The total number of real solutions of the equation
   $\theta$=$\tan^{-1}(2\tan \theta)$$-\frac{1}{2}\sin{-1}\left(\frac{6\tan \theta}{9+\tan^2\theta}\right)$
   is
   (Here, the inverse trigonometric functions $sin^{−1} x$ and $tan^{−1} x$ assume values in $\left[− \frac{\pi}{2}, \frac{\pi}{2}\right]$, and $\left(− \frac{\pi}{2}, \frac{\pi}{2}\right)$, respectively.)
    
   1. 1
   2. 2
   3. 3
   4. 5
4. Let $S$ denote the locus of the point of intersection of the pair of lines
   $4x − 3y$ = $12\alpha$, $4\alpha x + 3 \alpha y$ = 12 , where $\alpha$ varies over the set of non-zero real numbers. Let $T$ be the tangent to $S$ passing through the points $(p, 0)$ and $(0, )$, $q > 0$, and parallel to the line $4 x − \frac{3}{√2} y$ = 0.Then the value of $pq$ is
    
   1. $-6\sqrt{2}$
   2. $-3\sqrt{2}$
   3. $-9\sqrt{2}$
   4. $-12\sqrt{2}$

SECTION 2 (Maximum Marks:16)

• This section contains FOUR (04) 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 $I$=$\begin{equation*} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \end{equation*}$ and $P$=$\begin{equation*} \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \end{equation*}$. Let $Q$=$\begin{equation*} \begin{pmatrix} x & y \\ z & 4 \end{pmatrix} \end{equation*}$ for some non-zero real numbers $x$, $y$, and $z$, for which there is a 2 × 2 matrix $R$ with all entries being non-zero real numbers, such that
   $QR$ = $RP$.
   Then which of the following statements is (are) TRUE?
    
   1. The determinant of $Q − 2I$ is zero
   2. The determinant of $Q − 6I$ is 12
   3. The determinant of $Q − 3I$ is 15
   4. $yz$=2
6. Let $S$ denote the locus of the mid-points of those chords of the parabola $y^2$ = $x$, such that the area of the region enclosed between the parabola and the chord is $\frac{4}{3}$. Let $ℛ$ denote the region lying in the first quadrant, enclosed by the parabola $y^2$ = $x$, the curve $S$, and the lines $x$= 1 and $x$= 4 . Then which of the following statements is (are) TRUE?
    
   1. $(4, \sqrt{3}) \in S$
   2. $(5, \sqrt{2}) \in S$
   3. Area of $R$ is $\frac{14}{3}-2\sqrt{3}$
   4. Area of $R$ is $\frac{14}{3}-\sqrt{3}$
7. Let $P(x_1, y_1)$ and $Q(x_2, y_2)$ be two distinct points on the ellipse
   $\frac{x^2}{9}$+$\frac{y^2}{4}$=1
   such that $y_1$ > 0, and $y_2$ > 0 . Let $C$ denote the circle $x^2 + y^2$ = 9 , and $M$ be the point (3, 0).Suppose the line $x = x_1$ intersects $C$ at $R$, and the line $x = x_2$ intersects $C$ at $S$, such that the$y-$coordinates of $R$ and $S$ are positive. Let $\angle{ROM}$ = $\frac{\pi}{6}$ and $\angle{SOM}$ = $\frac{\pi}{3}$, where $O$ denotes the origin (0, 0). Let $|XY|$ denote the length of the line segment $XY$.
   Then which of the following statements is (are) TRUE?
    
   1. The equation of the line joining $P$ and $Q$ is $2x + 3y$ = $3 (1 + \sqrt{3} )$
   2. The equation of the line joining $P$ and $Q$ is $2x + y$ = $3 (1 + \sqrt{3} )$
   3. If $N_2$ = $(x_2, 0)$, then $3 |N_2Q|$ = $2 |N_2S|$
   4. If $N_1$ = $(x_1, 0)$, then $9 |N_1P|$ = $4 |N_1R|$
8. Let $ℝ$ denote the set of all real numbers. Let $f: ℝ → ℝ$ be defined by
   $f(x)$=$\left\{\begin{array}{cc} \frac{6x+\sin x}{2x+\sin x} & \text { if } x \neq 0, \\ \frac{7}{3} & \text { if } x=0 \end{array}\right.$
   Then which of the following statements is (are) TRUE?
    
   1. The point $x = 0$ is a point of local maxima of $f$
   2. The point $x = 0$ is a point of local minima of $f$
   3. Number of points of local maxima of $f$ in the interval $[\pi, 6\pi]$ is 3
   4. Number of points of local minima of $f$ in the interval $[2\pi, 4\pi]$ is 1

SECTION 3 (Maximum Marks:32)

• This section contains EIGHT (08) questions.
• The answer to each question is a NUMERICAL VALUE.
• 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.
• 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: +4 If ONLY the correct integer is entered;
  • Zero Marks: 0 In all other cases.

9. Let $y(x)$ be the solution of the differential equation
   $x^2\frac{dy}{dx}$+$xy$=$x^2$+$y^2$, $x > \frac{1}{e}$,
   satisfying $y(1)$ = 0 . Then the value of $2\frac{(y(e))^2}{y(e^2)}$ is..............
10. Let $a_0$, $a_1$, … , $a_{23}$ be real numbers such that
    $\left(1+\frac{2}{5}x\right)^{23}$=$\sum \limits_{i=0}^{23}a_ix^i$
    for every real number $x$. Let $a_r$ be the largest among the numbers $a_j$ for $0 ≤ j ≤ 23$.Then the value of $r$ is ___________.
11. A factory has a total of three manufacturing units, $M_1$, $M_2$, and $M_3$, which produce bulbs independent of each other. The units $M_1$, $M_2$, and $M_3$ produce bulbs in the proportions of 2: 2: 1, respectively. It is known that 20% of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by $M_1$, 15% are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by $M_2$ is $\frac{2}{5}$.
    If a bulb is chosen randomly from the bulbs produced by $M_3$, then the probability that it is defective is ___________.
12. Consider the vectors $\vec{x}$=$\hat{i}+2\hat{j}+3\hat{k}$, $\vec{y}$=$2\hat{i}+3\hat{j}+\hat{k}$, and $\vec{z}$=$3\hat{i}+\hat{j}+2\hat{k}$.
    For two distinct positive real numbers $\alpha$ and $\beta$, define
    $\vec{X}$=$\alpha \vec{x}+\beta \vec{y}-\vec{z}$, $\vec{Y}$=$\alpha \vec{y}+\beta \vec{z}-\vec{x}$ and $\vec{Z}$=$\alpha \vec{z}+\beta \vec{x}-\vec{y}$.
    If the vectors $\vec{X}$, $\vec{Y}$, and $\vec{Z}$ lie in a plane, then the value of $\alpha + \beta − 3$ is_____________
13. For a non-zero complex number $z$, let $arg(z)$ denote the principal argument of , with $−\pi < arg(z) ≤ \pi$. Let $\omega$ be the cube root of unity for which $0 < arg(\omega) < π$. Let
    $\alpha$=$arg \left( \sum \limits_{n=1}^{2025}(-\omega)^n\right)$.
    Then the value of $\frac{3\alpha}{\pi}$ is ________________.
14. Let $ℝ$ denote the set of all real numbers. Let $f: ℝ → ℝ$ and $g: ℝ → (0, 4)$ be functions defined by
    $f(x)$ = $log_e(x^2 + 2x + 4)$, and $g(x)$ = $\frac{4}{1 + e^{−2x}}$.
    Define the composite function $f ∘ g^{−1}$ by $(f ∘ g^{−1}) (x)$ = $f(g^{−1}(x))$, where $g^{−1}$ is the inverse of the function $g$. Then the value of the derivative of the composite function $f ∘ g^{−1}$ at $x$ = 2 is ______________.
15. Let
    $\alpha$=$\frac{1}{\sin 60° \sin 61°}$+$\frac{1}{\sin 62° \sin 63°}$+...+$\frac{1}{\sin 118° \sin 119°}$. Then the value of
    $\left(\frac{cosec 1°}{\alpha}\right)^2$
    is..........
16. If
    $\alpha$=$\int \limits_{\frac{1}{2}}^{2}\frac{\tan^{-1}x}{2x^2-3x+2}dx$,
    then the value of $\sqrt{7} \tan \left(\frac{2\alpha}{\sqrt{7}}{\pi}\right)$ is............
    (Here, the inverse trigonometric function $tan{−1} x$ assumes values in $\left(− \frac{\pi}{2}, \frac{\pi}{2}\right)$ .)

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