Download JEE Advanced 2026 Mathematics Question Paper - 1

» Hold on briefly for download button.

Wait-starts soon.

 Download Now!

2025:Paper-1

2027:Paper-1

All the Papers

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. Consider the function $f:(0, \infty) \to (-\infty, \infty)$ given by
   $f(x)=\sqrt{x} \log_e(x)-x+1$.
   Then which one of the following statements is TRUE ?
    
   1. The derivative of the function $𝑓$ is decreasing in the interval (0, 1)
   2.   The function 𝑓 has a local maximum at some point $𝑎 ∈ (0, ∞)$
   3.   The function 𝑓 has a local minimum at some point $𝑏 ∈ (0, ∞)$
   4.   The function $𝑓$ has NEITHER a point of local maximum NOR a point of local minimum in the interval (0, ∞)
2. Let $𝑃$ be the point on the parabola $𝑦 = 𝑥^2$ such that the slope of the tangent to the parabola at the point $𝑃$ is 4 . Let $𝑄$ be the point in the first quadrant lying on the circle $𝑥^2 + 𝑦^2 = 2$ such that the slope of the tangent to the circle at the point $𝑄$ is −1 . Let $𝑅$ be the point in the first quadrant lying on the ellipse $𝑥^2 + 4𝑦^2 = 8$ such that the slope of the tangent to the ellipse at the point $𝑅$ is $−\frac{1}{2}$. Then the radius of the circle passing through the points $𝑃$, $𝑄$ and $𝑅$ is
    
   1. $\sqrt{10}$
   2.   $\sqrt{5}$
   3.  $\sqrt{\frac{5}{2}}$
   4.   $2\sqrt{5}$
3. Which one of the following matrices can be obtained by performing elementary row transformations on the 3 × 3 identity matrix ?
    
   1. $\begin{equation*} \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix} \end{equation*}$
   2.   $\begin{equation*} \begin{bmatrix} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 1 & 2 & 1 \end{bmatrix} \end{equation*}$
   3.   $\begin{equation*} \begin{bmatrix} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 2 & 5 & 8 \end{bmatrix} \end{equation*}$
   4.   $\begin{equation*} \begin{bmatrix} 1 & 1 & 1 \\ -1 & 1 & 2 \\ 0 & 2 & 3 \end{bmatrix} \end{equation*}$
4. Considering only the principal values of the inverse trigonometric functions, the value of
   $\cot^{-1}(\cot(-11))$+$10 \sin\left(2\cos^{-1}\left(\frac{1}{\sqrt{2}}\right)\right)$+$10\sin(2\tan^{-1}(2))$
   is
    
   1. $3\pi + 7$
   2.   7
   3.   $4\pi + 7$
   4.   $3\pi - 5$

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: -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. Suppose that Box I contains 6 red balls and 9 green balls, and Box II contains 8 red balls and 12 green balls. All the balls of Box I and Box II are mixed together and a ball is chosen at random from them. Let $𝐸_1$ be the event that the ball chosen belonged to Box I and let $𝐸_2$ be the event that the ball chosen belonged to Box II. Let $𝐹_1$ be the event that the ball chosen is red and let $𝐹_2$ be the event that the ball chosen is green.
   Then which of the following statements is (are) TRUE ?
    
   1. The events $𝐸_1$ and $𝐹_1$ are independent
   2.   The events $𝐸_2$ and $𝐹_2$ are dependent
   3.   The conditional probability $𝑃(𝐹_1 | 𝐸_1)$ is equal to the conditional probability $𝑃(𝐹_1 | 𝐸_2)$
   4.   The conditional probability $𝑃(𝐹_1 | 𝐸_1)$ is greater than the conditional probability $𝑃(𝐹_2 | 𝐸_2)$
6. Let $𝑃$ be the plane such that it contains the straight line $\frac{𝑥−1}{2}$=$\frac{𝑦−3}{3}$=$\frac{𝑧+2}{1}$and is perpendicular to the plane $𝑥 + 2𝑦 + 3𝑧 = 4$. Let $𝑃_1$ be the plane which passes through the point (4, 2, 2) and is parallel to $𝑃$. Then which of the following statements is (are) TRUE ?
    
   1. The equation of the plane $𝑃$ is $7𝑥 − 5𝑦 + 𝑧 = −10$
   2.   The distance between the planes $𝑃$ and $𝑃_1$ is 30
   3.   The distance of the plane $𝑃$ from the origin is $2\sqrt{3}$
   4.   The acute angle between the plane $𝑃$ and the plane $2𝑥 + 2𝑦 + 𝑧 = 3$ is $\cos^{-1}\left(\frac{1}{3\sqrt{3}}\right)$
7. Let $ℝ$ denote the set of all real numbers. Let $𝑓 ∶ ℝ → ℝ$ be an arbitrary function and let $𝑔 ∶ ℝ → ℝ$ be the function defined by
   $g(x)=xf(x)$, for all $x \in R$,
   Then which of the following statements is (are) TRUE ?
    
   1. The function $𝑔$ is always continuous at $𝑥 = 0$
   2.   If $𝑓$ is continuous at $𝑥 = 0$ , then $𝑔$ is differentiable at $𝑥 = 0$
   3.   If $𝑔$ is differentiable at $𝑥 = 0$ , then $𝑓$ is continuous at $𝑥 = 0$
   4.   If $𝑔$ is differentiable at $𝑥 = 0$, then $\lim \limits_{𝑥→0}𝑓(𝑥)$ exists
8. Consider the matrix
   $M=\begin{equation*} \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \end{equation*}$
   Let $𝑝$ , $𝑞$ , $𝑟$ , $𝑠$ , $𝑎$ , $𝑏$ , $𝑐$ and $𝑑$ be integers such that $M^{26}=\begin{equation*} \begin{bmatrix} p & q \\ r & s \end{bmatrix} \end{equation*}$ and $\sum \limits_{k=1}^{26}M^k=\begin{equation*} \begin{bmatrix} a & b \\ c & d \end{bmatrix} \end{equation*}$.
   Then which of the following statements is (are) TRUE ?
    
   1. There exists a 2 × 2 invertible matrix $𝑁$ with real entries such that $MN=N \begin{equation*} \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \end{equation*}
   2.   The value of $a$ is 378
   3.   For any two given integers $𝑚$ and $𝑛$, there exist unique integers 𝑥 and 𝑦 such that
      $𝑝𝑥 + 𝑞𝑦$ = $𝑚$ and $𝑟𝑥 + 𝑠𝑦 = 𝑛$
   4.   For each positive real number $𝑡$, the system of linear equations
      (𝑎 + 𝑡)𝑥 + 𝑏𝑦 = 1
      𝑐𝑥 + (𝑑 + 𝑡)𝑦 = −1
      has a unique solution

SECTION 3 (Maximum Marks:16)

• This section contains FOUR (04) 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.

9. Let $𝑆$ = {1, 2, 3, … , 10}. Consider the set $𝑋$ = {$𝑅 ∶ 𝑅$ is an equivalence relation on the set $𝑆$ such that $𝑅$ has exactly 42 elements} . Then the number of elements in $𝑋$ is _____________.
10. Consider the function $f: \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \to (-\infty, \infty)$ defined by $𝑓(𝑥) = (|𝑥| + |𝑥 − 1|) sin 𝑥 + [𝑥 sin 𝑥]$,where $[𝑥 sin 𝑥]$ is the greatest integer less than or equal to $𝑥 sin 𝑥$.Let $α$ be the total number of points in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ at which $𝑓$ is NOT continuous, andlet $𝛽$ be the total number of points in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ at which $𝑓$ is NOT differentiable. Then the value of $𝛼 + 𝛽$ is ___________.
11. The number of ways to distribute 10 identical red pens and 14 identical blue pens among four persons such that each person gets 6 pens, is _____________.
12. Let $\alpha=\left(1-2\cos\left(\frac{\pi}{11}\right)\right)$$\left(1-2\cos\left(\frac{3\pi}{11}\right)\right)$$\left(1-2\cos\left(\frac{9\pi}{11}\right)\right)$$\left(1-2\cos\left(\frac{27\pi}{11}\right)\right)$$\left(1-2\cos\left(\frac{81\pi}{11}\right)\right)$. Then the value of $5-\alpha^2$ is..............

SECTION 4 (Maximum Marks:16)

• This section contains FOUR (04) Matching List Sets.
• Each set has ONE Multiple Choice Question.
• Each set has TWO lists: List-I and List-II.
• List-I has Four entries (P), (Q), (R) and (S) and List-II has Five entries (1), (2), (3), (4) and (5).
• FOUR options are given in each Multiple Choice Question based on List-I and List-II and ONLY ONE of these four options satisfies the condition asked in the Multiple Choice Question.
• Answer to each question will be evaluated according to the following marking scheme:
  
  • Full Marks: +4 ONLY if the option corresponding to the correct combination 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.

13. Match each entry in List-I to the correct entry in List-II and choose the correct option.
                                                                                                        

    	List - I		List - II
    (P)	If $\alpha$ and $\beta$ are the distinct roots of the equation $𝑥^2 + 𝑥 + 1 = 0$, then the quadratic equation with roots $\frac{1}{(\alpha+1)^{2026}}$ and $\frac{1}{(\beta+1)^{2026}}$ is	(1)	$x^2+x+1$
    (Q)	If $\alpha$ and $\beta$ are the distinct roots of the equation $𝑥^2 + 𝑥 + 1 = 0$, then the quadratic equation with roots $\frac{1}{(\alpha+1)^{2027}}$ and $\frac{1}{(\beta+1)^{2027}}$ is	(2)	$x^2-x+1$
    (R)	If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2+x+1=0$, then the value of $\frac{1}{(\gamma-1)^{2026}}+\frac{1}{(\delta-1)^{2026}}$ is	(3)	$x^2+x-1$
    (S)	If $𝑝$ and $𝑟$ are the distinct roots of the equation $𝑥^2 + 𝑥 − 1 = 0$, then the value of $\frac{1}{(p+1)^3}+\frac{1}{(r+1)^3}$ is	(4)	-1
    		(5)	-4

     
    1. (P) → (1), (Q) → (2), (R) → (5), (S) → (4)
    2.   (P) → (3), (Q) → (1), (R) → (4), (S) → (5)
    3.  (P) → (1), (Q) → (2), (R) → (4), (S) → (5)
    4.   (P) → (2), (Q) → (3), (R) → (5), (S) → (4)
14. Match each entry in List-I to the correct entry in List-II and choose the correct option.
                                                                                                        

    	List - I		List - II
    (P)	The number of elements in the set {$𝑥 ∈ [−𝜋, 𝜋]$ ∶ $sin^6 𝑥 + cos^4 𝑥 = 1$}	(1)	is 1
    (Q)	The number of elements in the set {$𝑥 ∈ \left[−\frac{𝜋}{2},\frac{𝜋}{2}\right]$ ∶ $sin^2 𝑥 + cos^6 𝑥 = 1$}	(2)	is 2
    (R)	The number of elements in the set {$𝑥 ∈ [−𝜋, 𝜋]$ ∶ $cos^2(\frac{𝑥}{2}) − sin^2 𝑥 = \frac{1}{2}$}	(3)	is 3
    (S)	The number of elements in the set {$𝑥 ∈ [−2𝜋, 2𝜋]$ ∶ $6 sin^2(\frac{𝑥}{2}) − cos 3𝑥 = 3$}	(4)	is 4
    		(5)	is 5

     
    1. (P) → (2), (Q) → (5), (R) → (3), (S) → (4)
    2.   (P) → (5), (Q) → (3), (R) → (2), (S) → (4)
    3.  (P) → (5), (Q) → (4), (R) → (1), (S) → (3)
    4.  (P) → (4), (Q) → (3), (R) → (2), (S) → (5)
15. For real numbers $\alpha$, $\beta$, $\gamma$, $\delta$ and $\mu$, consider the matrix
    $M=\begin{equation*} \begin{bmatrix} \alpha & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \beta & \frac{1}{\sqrt{3}} \\ \gamma & \delta & \mu \end{bmatrix} \end{equation*}$.
    Suppose that $𝑀𝑀^𝑇 = 𝐼$, where $𝑀^𝑇$ is the transpose of the matrix $𝑀$, and $𝐼$ is the 3 × 3 identity matrix. Let
    $\vec{u}=\alpha \hat{i}+\frac{1}{\sqrt{3}}\hat{j}+\gamma \hat{k}$, $\vec{v}=\frac{1}{\sqrt{2}}\hat{i}+\beta \hat{j}+\delta \hat{k}$ and $\vec{w}=-\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{3}}\hat{j}+\mu \hat{k}$.
    Match each entry in List-I to the correct entry in List-II and choose the correct option.                                                                                                     

    	List - I		List - II
    (P)	The value of $\gamma^2$ + $\delta^2$ is	(1)	0
    (Q)	If $𝑥\vec{u}$+ $𝑦\vec{𝑣}$ + $𝑧\vec{w}$=$\hat{j}$ for some real numbers $𝑥$, $𝑦$ and $𝑧$, then the value of $𝑥$ is	(2)	1
    (R)	The value of $|\vec{u}•(\vec{v}×\vec{w})|$ is	(3)	$\frac{1}{\sqrt{2}}$
    (S)	The value of $|\vec{u}×(\vec{v}×\vec{w})|$	(4)	$\frac{1}{\sqrt{3}}$
    		(5)	$\frac{5}{6}$

     
    1. (P) → (5), (Q) → (4), (R) → (2), (S) → (1)
    2.   (P) → (4), (Q) → (5), (R) → (1), (S) → (2)
    3.  (P) → (5), (Q) → (3), (R) → (2), (S) → (1)
    4.  (P) → (5), (Q) → (4), (R) → (1), (S) → (2)
16. Match each entry in List-I to the correct entry in List-II and choose the correct option.
                                                                                                        

    	List - I		List - II
    (P)	The circle with centre (1, 2) and touching the straight line $3𝑥 + 4𝑦 = 1$, passes through	(1)	the point (1, 1)
    (Q)	The common tangent to the circle $𝑥^2 + 𝑦^2 = 2$ and the parabola $𝑦^2 = 8𝑥$ with positive slope, passes through	(2)	the point (7, 9)
    (R)	Let $𝑀$ be the end point of the latus rectum of the ellipse $3𝑥^2 + 4𝑦^2 = 48$ such that $𝑀$ lies in the first quadrant. Then the normal to the ellipse drawn at $𝑀$ passes through	(3)	the point (3, 2)
    (S)	Let $𝐻$ be the hyperbola whose centre is at the origin, one of the foci is at (5, 0), and one directrix is $5𝑥 + 16 = 0$. Then $𝐻$ passes through	(4)	the point (2, 5)
    		(5)	the point (8, 3√3 )

     
    1. (P) → (3), (Q) → (4), (R) → (1), (S) → (2)
    2.  (P) → (3), (Q) → (2), (R) → (1), (S) → (5)
    3.  (P) → (3), (Q) → (2), (R) → (4), (S) → (5)
    4.  (P) → (4), (Q) → (1), (R) → (2), (S) → (3)

2026:Paper-2

2026:Ans-Keys

All the Papers Ans-Keys