Download JEE Advanced 2026 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 $\vec{a}$, $\vec{b}$ be two vectors, and let $P$, $Q$ and $R$ be the points with position vectors $\vec{a}$, $\vec{b}$ and $\vec{a}+\vec{b}$, respectively, with respect to the origin $O$. If $|\vec{a}+\vec{b}|$=$\sqrt{21}$, $|\vec{a}-\vec{b}|$=3, and $\vec{a}$ and $(\vec{a}-\vec{b})$ are perpendicular to each other, then the area of the triangle $𝑂𝑃𝑅$ is
    
   1. $\sqrt{3}$
   2.   $\frac{\sqrt{3}}{2}$
   3.   $\frac{3\sqrt{3}}{2}$
   4.  $\frac{3}{2}$
2. Let $𝑇$ be the tangent to the parabola $𝑦^2 = 16𝑥$ at the point (64, 32). Let $𝐿$ be the tangent to the same parabola at another point $(𝑥_1 , 𝑦_1)$ on the parabola. If $𝐿$ and $𝑇$ are perpendicular to each other, then the distance between the point $(𝑥_1 , 𝑦_1)$ and the focus of the parabola, is
    
   1. $\frac{15}{4}$
   2.   4
   3.   $\frac{17}{4}$
   4.   5
3. Let $y: (-\infty, \infty) \to (0, \infty)$ be the solution of the differential equation
   $\frac{dy}{dx}=\frac{e^{5x}y^3+y^3}{e^x+e^xy^4}$,
   satisfying $y(0)=\frac{1}{\sqrt{2}}$. Then the value of $y(\log_e2)$ is
    
   1. $\sqrt{\frac{5+\sqrt{35}}{2}}$
   2.   $\sqrt{\frac{7+\sqrt{53}}{2}}$
   3.   $\frac{7+\sqrt{53}}{2}$
   4.   $\frac{5+\sqrt{35}}{2}$
4. The value of the definite integral
   $\int \limits_{0}^{2}\frac{1}{3^x+3}dx$
   is
    
   1. $\frac{1}{2}$
   2.   $\frac{1}{3}$
   3.  $\frac{\log_e3}{3}$
   4.  $\frac{\log_e3}{2}$

SECTION 2 (Maximum Marks:20)

• This section contains FIVE (05) 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: -1 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 -1 marks.

5. Let $ℝ$ denote the set of all real numbers. Consider the polynomial function $𝑓: ℝ → ℝ$ defined by
   $f(x)=\frac{d^{10}}{dx^{10}}((x^2-1)^{10})$, for all $x \in R$.
   Here $\frac{d^{10}}{dx^{10}}((x^2-1)^{10})$ is the $10^{th}$ order derivative of the function $(x^2-1)^{10}$.
   Then which of the following statements is (are) TRUE ?
   1. The coefficient of $x^8$ in the polynomial $f(x)$ is $(-10)\left(\frac{18!}{8!}\right)$.
   2.   The value of $𝑓(1) + 𝑓(−1)$ is equal to $10! 2^{11}$
   3.  The degree of the polynomial $𝑓(𝑥)$ is 10
   4.   The constant term of the polynomial $f(x)$ is $- \left(\frac{10!}{5!}\right)$
6. Let $𝑎$, $𝑏$, $𝑐$ be positive integers in arithmetic progression such that the equation $𝑎𝑥^2$ + $𝑏𝑥$ + $𝑐$ = 0 has only integer solutions.
   Then which of the following statements is (are) TRUE ?
    
   1. $𝑐 − 𝑏$ is an integer multiple of $𝑎$
   2.   Both the roots of the equation $𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0$ are odd integers
   3.   If $𝑐 = 15$, then $𝑎𝑏 = 8$
   4.   If $𝑏 = 8$, then $𝑥 = 3$ is a root of the equation $𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0$
7. Let $𝐿$ be the straight line joining the points $𝑃(1, 2, −1)$ and $𝑄(2, 3, 1)$. Let $𝑆$ be the foot of the perpendicular drawn from the point $𝑅(4, −1, 5)$ to the line $𝐿$. Another line passing through $𝑅$ intersects $𝐿$ at a point $𝑇$ such that the point $𝑆$ divides the line segment $𝑃𝑇$ internally in the ratio $|𝑃𝑆| ∶ |𝑆𝑇$| = 1 ∶ 2 , where $|𝑃𝑆|$ and $|𝑆𝑇|$ are the lengths of the line segments $𝑃𝑆$ and $𝑆𝑇$, respectively. Then which of the following statements is (are) TRUE ?
    
   1. The orthocentre of the triangle $𝑃𝑅𝑇$ is $\left(\frac{23}{5}, -4, \frac{31}{5}\right)$
   2.  The orthocentre of the triangle $𝑃𝑅𝑇$ is (4, 3, 5)
   3.  The area of the triangle $𝑃𝑅𝑇$ is $6\sqrt{5}$
   4.  The area of the triangle $𝑃𝑅𝑇$ is $18\sqrt{5}$
8. Let $𝑦 = 𝑓(𝑥)$ be the real valued function defined on the interval $(0, ∞)$, satisfying $𝑦(1) = 0$ and the differential equation
   $x\frac{dy}{dx}=y-x^3$.
   Then which of the following statements is (are) TRUE ?
    
   1. The function $𝑓$ has a local minimum at $𝑥 = \frac{1}{\sqrt{3}}$
   2.   The function $𝑓$ has a local maximum at $𝑥 =\frac{1}{\sqrt{3}}$
   3.   The function $𝑓$ is increasing in the interval (1, 2)
   4.  If $g(x)=4x^3-5x^2+\frac{3}{2}x$ for $x > 0$, then the number of elements in the set {$𝑥 ∈ (0, ∞) ∶ 𝑓(𝑥) = 𝑔(𝑥)$} is 2.
9. Let $ℝ$ denote the set of all real numbers and let $𝑖 = \sqrt{−1}$. Consider the matrices
   $S=\begin{equation*}\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \end{equation*}$ and $T=\begin{equation*} \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \end{equation*}$. Let $a$, $b$, $c$, $d$ be the real numbers such that $ST=\begin{equation*} \begin{bmatrix} a & b \\ c & d \end{bmatrix} \end{equation*}$. Let $H$={$x+iy: x, y \in R$ and $y > 0$}.
   Then which of the following statements is (are) TRUE ?
    
   1. $\frac{b+ia}{d+ic}=i$
   2.   If $\omega=\frac{-1+i\sqrt{3}}{2}$, then $\frac{a \omega +b}{c \omega + d}$=$\omega$
   3.  If $𝑚$ is an integer greater than 2 such that $(𝑆𝑇)^2$ = $(𝑆𝑇)^𝑚$, then $𝑚$ is an integer multiple of 8
   4.  If $z \in H$, then $\frac{az+b}{cz+d} \in H$

SECTION 3 (Maximum Marks:20)

• This section contains FIVE (05) 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.
• Answer to each question will be evaluated according to the following marking scheme:
  
  • Full Marks: +4 If ONLY the correct numerical value is entered in the designated place;
  • Zero Marks: 0 In all other cases.

10. Let $ℕ$ denote the set of all positive integers. Consider the sets $𝐴$ = {1, 2, 3, 4, 5} and $𝐵$ = {1, 2, 3, 4, 5, 6, 7} . Let $𝑆$ be the set of all functions $𝑓: 𝐴 → 𝐵$ such that $𝑓(2) ≠ 2$ and $𝑓(4) ≠ 4$. Consider the set $𝑇$= {$𝑓 ∈ 𝑆$ ∶ there exists a function $𝑔 ∶ 𝐵 → ℕ$ such that $𝑔(𝑓(𝑥)) = 2^𝑥$ for all $𝑥 ∈ 𝐴$} . Then the number of elements in the set $𝑇$ is ___________.
11. A bookshelf contains 6 distinct books of Mathematics and 5 distinct books of Physics. From these 11 books, 6 books are chosen at random. Let $𝑋$ be the absolute value of the difference between the number of Mathematics books chosen and the number of Physics books chosen. If $\alpha$ is the mean of the random variable $𝑋$, then the value of $77\alpha$ is ___________.
12. Consider a data consisting of 10 observations $𝑥_1$, $𝑥_2$, … , $𝑥_{10}$, whose mean is 5 and variance is 7.If the mean and the variance of the first 8 observations $𝑥_1$,  $𝑥_2$, … , $𝑥_8$ are 4 and 3.5 , respectively, and $𝑥_9 < 𝑥_{10}$, then the value of $3𝑥_9 + 2𝑥_{10}$ is ___________.
13. Consider the ellipse $E$ given by $\frac{x^2}{18}$+$\frac{y^2}{12}$=1. Let $𝐻$ be the hyperbola whose eccentricity is the reciprocal of the eccentricity of $𝐸$ and whose foci are the same as that of $𝐸$. Let $𝑃$ and $𝑄$ be the points of intersection of $𝐻$ and the parabola $\sqrt{5} y=x^2$ in the first quadrant. Let $d$ be the distance between $P$ and $Q$.
    If $a$ and $b$ are the integers such that $d^2=a+b\sqrt{5}$, then the value of $a-b$ is ...........
14. For a real number $\alpha$, let $[\alpha]$ denote the greatest integer less than or equal to $\alpha$. For a finite set $𝑆$, let $|𝑆|$ denote the number of elements in the set $𝑆$. Consider the functions $𝑓 ∶ (−3, 3) → (−∞, ∞)$ and $𝑔 ∶ (−3, 3) → (−∞, ∞)$ defined by $𝑓(𝑥)$ = $[𝑥^3]log_𝑒(1 + sin^2(𝜋(𝑥 − [𝑥])))$ and $𝑔(𝑥) = 𝑥^3 sin^2(𝜋 log_𝑒(1 + 𝑥 − [𝑥]))$. Let 𝐴 = {$𝑥 ∈ (−3, 3)$ ∶ $𝑓$ is discontinuous at $𝑥$} and 𝐵 = {$𝑥 ∈ (−3, 3)$ ∶ $𝑔$ is discontinuous at $𝑥$}. Then the value of $|𝐴| + 2|𝐵| − |𝐴 ∩ 𝐵|$ is ___________.

SECTION 4 (Maximum Marks:8)

• 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: +2 If ONLY the correct numerical value is entered in the designated place;
  • Zero Marks: 0 In all other cases.

Question Stem for Question Nos. 15 to 16

Consider the curve $C_1$ given by
$y=e^{-x}$ for $x \in [0, 10\pi]$,
and the curve $C_2$ given by $y=e^{-x}(\sin x+\cos x)$ for $x \in [0, 10\pi]$.
Let $𝑛$ be the total number of points of intersection of the curves $𝐶_1$ and $𝐶_2$.
Suppose that $\alpha_1$, $\alpha_2$, … , $\alpha_𝑛 \in [0, 10\pi]$ are the $𝑥 -$ coordinates of the points of intersection of the curves $𝐶_1$ and $𝐶_2$ such that $\alpha_1$ < $\alpha_2$ < ⋯ < $\alpha_𝑛$.  

 
15.  The value of $𝑛$ is ___________.
16.  Let $𝛽$ be the area of the region enclosed between the curves $𝐶_1$, $𝐶_2$, and the lines$𝑥 = \alpha_1$ and $𝑥 = \alpha_4$. Then the value of
    $-\frac{1}{\pi}\log_e(\beta-2e^{-\frac{\pi}{2}})$
    is....................

Question Stem for Question Nos. 17 to 18

Consider the ellipse given by
$x^2+4y^2=1$ and $4x^2+y^2=1$  

 
17.  Let $𝑃$ be the point in the first quadrant where the given ellipses intersect. If $𝜃$ is the acute angle between the tangents to the given ellipses at the point $𝑃$, then the value of $4 tan 𝜃$ is ___________.
18.  If $\alpha$ is the area of the common region that lies inside both the given ellipses, then the value of $\cot \alpha$ is ___________.

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