Download JEE Advanced 2024 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.   Considering only the principal values of the inverse trigonometric functions, the value of
   $\tan \left(\sin^{-1}\left(\frac{3}{5}\right)\right.-$$\left. 2\cos^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)$
   is
    
   1. $\frac{7}{24}$
   2. $\frac{-7}{24}$
   3. $\frac{-5}{24}$
   4. $\frac{5}{24}$
2. Let $S$=$\left\{(x, y)\in R×R:x \geq 0, y \geq 0,\right.$$\left.y^2 \leq 4x, y^2 \leq 12-2x\right.$ and $\left. 3y+\sqrt{8}x\leq5\sqrt{8}\right\}$. If the area of the region $S$ is $\alpha \sqrt{2}$, then $\alpha$ is equal to
    
   1. $\frac{17}{2}$
   2. $\frac{17}{3}$
   3. $\frac{17}{4}$
   4. $\frac{17}{5}$
3. Let $k \in R$. If $\lim \limits_{x \to 0+} (\sin (\sin kx)+\cos x + x)^{\frac{2}{x}}$=$e^6$, then the value of $k$ is
    
   1. 1
   2. 2
   3. 3
   4. 4
4.   Let $f:R\to R$ be a function defined by
   $f(x)=\left\{\begin{array}{cc}x^2 \sin \left(\frac{\pi}{x^2}\right), & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right.$
   Then which of the following statements is TRUE?
   1. $f(x)=0$ has infinitely many solutions in the interval $\left[\frac{1}{10^{10}}, \infty\right)$
   2. $f(x)=0$ has no solutions in the interval $\left[\frac{1}{\pi}, \infty\right)$
   3. The set of solutions of $f(x)=0$ in the interval $\left(0, \frac{1}{10^{10}}\right)$ is finite
   4. $f(x)=0$ has more than 25 solutions in the interval $\left(\frac{1}{\pi^2}, \frac{1}{\pi}\right)$

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 $S$ be the set of all $(\alpha, \beta) \in R×R$ such that
   $\lim \limits_{x \rightarrow \infty} \frac{\sin \left(x^2\right)\left(\log _e x\right)^\alpha \sin \left(\frac{1}{x^2}\right)}{x^{\alpha \beta}\left(\log _e(1+x)\right)^\beta}=0$.
   Then which of the following is (are) correct?
    
   1. $(-1, 3) \in S$
    
   2. $(-1, 1) \in S$
   3. $(1, -1) \in S$
   4. $(1, -2) \in S$
6.   A straight line drawn from the point $P(1, 3, 2)$, parallel to the line $\frac{x-2}{1}$=$\frac{y-4}{2}$=$\frac{z-6}{1}$, intersects the plane $L_1$:$x-y$+$3z$=6 at the point $Q$. Another straight line which passes through $Q$and is perpendicular to the plane $L_1$intersects the plane $L_2$:$2x-y$+$z$=$-4$ at the point $R$. Then which of the following statements is (are) TRUE?
    
   1. The length of the line segment $PQ$is $\sqrt{6}$
   2. The coordinates of $R$ are (1, 6, 3)
   3. The centroid of the triangle $PQR$is $\left(\frac{4}{3}, \frac{14}{3}, \frac{5}{3}\right)$
   4. The perimeter of the triangle $PQR$is $\sqrt{2}$+$\sqrt{6}$+$\sqrt{11}$
7. Let $A_1$, $B_1$, $C_1$ be three points in the $xy$ -plane. Suppose that the lines $A_1C_1$ and $B_1C_1$ are tangents to the curve $y^2$ = $8x$ at $A_1$ and $B_1$, respectively. If $O$ = (0,0) and $C_1$ = −( 4,0), then which of the following statements is (are) TRUE?
    
   1. The length of the line segment $OA_1$ is $4\sqrt{3}$
   2. The length of the line segment $A_1B_1$ is 16
   3. The orthocenter of the triangle $A_1 B_1 C_1$ is (0, 0)
   4. The orthocenter of the triangle $A_1 B_1 C_1$ is (1, 0)

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.   Let $f:R \to R$ be a function such that $f(x+y)$=$f(x)$+$f(y)$ for all $x, y \in R$, and $g: R \to (0, \infty)$ be a function such that $g(x+y)$=$g(x)$$g(y)$ for all $x, y\in R$. If $f\left(\frac{-3}{5}\right)$=12 and $g\left(\frac{-1}{3}\right)$=2, then the value of $\left(f\left(\frac{1}{4}\right)+g(-2)-8\right)g(0)$ is .........
9. A bag contains $N$ balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For $i$ =1,2,3, let $W_i$, $G_i$ and $B_i$denote the events that the ball drawn in the$i^{th}$ draw is a white ball, green ball, and blue ball, respectively. If the probability $P(W_1 \cap G_2 \cap B_3)$=$\frac{2}{5N}$ and the conditional probability $P(B_3|W_1 \cap G_2)$=$\frac{2}{9}$, then $N$ equals...........
10.   Let the function $f:R \to R$ be defined by
    $f(x)$=$\frac{\sin x}{e^{\pi x}}$$\frac{(x^{2023}+2024x+2025)}{(x^2-x+3)}$+$\frac{2}{e^{\pi x}}$$\frac{(x^{2023}+2024x+2025)}{(x^2-x+3)}$.
    Then the number of solutions of $f(x)$=0 in $R$ is ______.
11.  Let $\vec{p}$=$2\hat{i}$+$\hat{j}$+$3\hat{k}$ and $\vec{q}$=$\hat{i}-$$\hat{j}$+$\hat{k}$. If for some real numbers $\alpha$, $\beta$, and $\gamma$, we have
    $15 \hat{i}$+$10\hat{j}$+$6\hat{k}$=$\alpha(2\vec{p}+\vec{q})$+$\beta (\vec{p}-2\vec{q})$+$\gamma (\vec{p}×\vec{q})$, then the value of $\gamma$ is _______.
12.   A normal with slope $\frac{1}{\sqrt{6}}$ is drawn from the point $(0, -\alpha)$ to the parabola $x^2$=$-4ay$, where $a >0$. Let $L$ be the line passing through $(0, -\alpha)$ and parallel to the directrix of the parabola. Suppose that $L$ intersects the parabola at two points $A$ and $B$. Let $r$ denote the length of the latus rectum and $s$ denote the square of the length of the line segment $AB$. If $r:s$=1:16, then the value of $24\alpha$ is ...........
13. Let the function $f:[1, \infty) \to R$ be defined by
    $f(t)$=$\left\{\begin{array}{cc}(-1)^{n+1} 2, & \text { if } t=2 n-1, n \in \mathbb{N} \\ \frac{(2 n+1-t)}{2} f(2 n-1)+\frac{(t-(2 n-1))}{2} f(2 n+1), & \text { if } 2 n-1 < t < 2 n+1, n \in \mathbb{N}\end{array}\right.$
    Define $g(x)$=$\int \limits_{1}^{x}f(t)dt$, $x \in (1, \infty)$. Let $\alpha$ denote the number of solutions of the equation $g(x)$=0 in the interval (1, 8] and $\beta$=$\lim \limits_{x \to 1+} \frac{g(x)}{x-1}$. Then the value of $\alpha + \beta$ is equal to ..........

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"

Let $S$ ={1,2,3,4,5,6} and $X$ be the set of all relations $R$from $S$ to $S$ that satisfy both the following properties:
i.$R$ has exactly 6 elements.
ii. For each $(a, b) \in R$, we have $|a - b| \geq 2$.
Let $Y$={$R \in X$ : The range of $R$ has exactly one element}and
$Z$={$R \in X : R$ is a function from $S$ to $S$}. Let $n(A)$ denote the number of elements in a set $A$.  

14. If $n(X)$=$^mC_6$, then the value of $m$is __________.  
15. If the value of $n(Y)$+$n(Z)$ is $k^2$,then $| k |$ is _____________.  

PARAGRAPH "II"

Let $f:\left[0, \frac{\pi}{2}\right] \to [0, 1]$ be the function defined by $f(x)=\sin^2x$ and let $g:\left[0, \frac{\pi}{2}\right] \to [0, \infty)$ be the function defined by $g(x)$=$\sqrt{\frac{\pi x}{2}-x^2}$  

 
16. The value of $2 \int \limits_{0}^{\frac{\pi}{2}}f(x)g(x)dx$$-\int \limits_{0}^{\frac{\pi}{2}}g(x)dx$ is__________.  
17. The value of $\frac{16}{\pi^3}$$\int \limits_{0}^{\frac{\pi}{2}}f(x)g(x)dx$ is_____________.  

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