jee-advanced

Q1 Matrices Matrix Algebra and Product Properties View

Let $$P_1 = I = \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right], \quad P_2 = \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right], \quad P_3 = \left[\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right],$$ $$P_4 = \left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right], \quad P_5 = \left[\begin{array}{lll}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{array}\right], \quad P_6 = \left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]$$ and $X = \sum_{k=1}^{6} P_k \left[\begin{array}{lll}2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1\end{array}\right] P_k^T$ where $P_k^T$ denotes the transpose of the matrix $P_k$. Then which of the following options is/are correct?
(A) If $X\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right] = \alpha\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$, then $\alpha = 30$
(B) $X$ is a symmetric matrix
(C) The sum of diagonal entries of $X$ is 18
(D) $X - 30I$ is an invertible matrix

Q2 Matrices Eigenvalue and Characteristic Polynomial Analysis View

Let $x \in \mathbb{R}$ and let $$P = \left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 3\end{array}\right], \quad Q = \left[\begin{array}{ccc}2 & x & x \\ 0 & 4 & 0 \\ x & x & 6\end{array}\right] \text{ and } R = PQP^{-1}$$
Then which of the following options is/are correct?
(A) There exists a real number $x$ such that $PQ = QP$
(B) $\det R = \det\left[\begin{array}{lll}2 & x & x \\ 0 & 4 & 0 \\ x & x & 5\end{array}\right] + 8$, for all $x \in \mathbb{R}$
(C) For $x = 0$, if $R\left[\begin{array}{l}1 \\ a \\ b\end{array}\right] = 6\left[\begin{array}{l}1 \\ a \\ b\end{array}\right]$, then $a + b = 5$
(D) For $x = 1$, there exists a unit vector $\alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}$ for which $R\left[\begin{array}{l}\alpha \\ \beta \\ \gamma\end{array}\right] = \left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$

Q3 Reciprocal Trig & Identities View

For non-negative integers $n$, let $$f(n) = \frac{\sum_{k=0}^{n} \sin\left(\frac{k+1}{n+2}\pi\right)\sin\left(\frac{k+2}{n+2}\pi\right)}{\sum_{k=0}^{n} \sin^2\left(\frac{k+1}{n+2}\pi\right)}$$
Assuming $\cos^{-1}x$ takes values in $[0, \pi]$, which of the following options is/are correct?
(A) $f(4) = \frac{\sqrt{3}}{2}$
(B) $\lim_{n\rightarrow\infty} f(n) = \frac{1}{2}$
(C) If $\alpha = \tan(\cos^{-1}f(6))$, then $\alpha^2 + 2\alpha - 1 = 0$
(D) $\sin(7\cos^{-1}f(5)) = 0$

Q4 Differentiation from First Principles View

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function. We say that $f$ has
$$\text{PROPERTY 1 if } \lim_{h\rightarrow 0} \frac{f(h) - f(0)}{\sqrt{|h|}} \text{ exists and is finite, and}$$
PROPERTY 2 if $\lim_{h\rightarrow 0} \frac{f(h) - f(0)}{h^2}$ exists and is finite.
Then which of the following options is/are correct?
(A) $f(x) = |x|$ has PROPERTY 1
(B) $f(x) = x^{2/3}$ has PROPERTY 1
(C) $f(x) = x|x|$ has PROPERTY 2
(D) $f(x) = \sin x$ has PROPERTY 2

Q5 Stationary points and optimisation Find critical points and classify extrema of a given function View

Let $$f(x) = \frac{\sin\pi x}{x^2}, \quad x > 0$$
Let $x_1 < x_2 < x_3 < \cdots < x_n < \cdots$ be all the points of local maximum of $f$ and $y_1 < y_2 < y_3 < \cdots < y_n < \cdots$ be all the points of local minimum of $f$. Then which of the following options is/are correct?
(A) $x_1 < y_1$
(B) $x_{n+1} - x_n > 2$ for every $n$
(C) $x_n \in \left(2n, 2n + \frac{1}{2}\right)$ for every $n$
(D) $|x_n - y_n| > 1$ for every $n$

Q6 Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View

For $a \in \mathbb{R}$, $|a| > 1$, let $$\lim_{n\rightarrow\infty}\left(\frac{1 + \sqrt[3]{2} + \cdots + \sqrt[3]{n}}{n^{7/3}\left(\frac{1}{(an+1)^2} + \frac{1}{(an+2)^2} + \cdots + \frac{1}{(an+n)^2}\right)}\right) = 54$$
Then the possible value(s) of $a$ is/are
(A) $-9$
(B) $-6$
(C) $7$
(D) $8$

Q7 Stationary points and optimisation Find critical points and classify extrema of a given function View

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = (x-1)(x-2)(x-5)$. Define $$F(x) = \int_0^x f(t)\,dt, \quad x > 0$$
Then which of the following options is/are correct?
(A) $F$ has a local minimum at $x = 1$
(B) $F$ has a local maximum at $x = 2$
(C) $F$ has two local maxima and one local minimum in $(0, \infty)$
(D) $F(x) \neq 0$ for all $x \in (0, 5)$

Q8 Vectors 3D & Lines MCQ: Point Membership on a Line View

Three lines $$\begin{array}{ll} L_1: & \vec{r} = \lambda\hat{i}, \lambda \in \mathbb{R}, \\ L_2: & \vec{r} = \hat{k} + \mu\hat{j}, \mu \in \mathbb{R} \text{ and} \\ L_3: & \vec{r} = \hat{i} + \hat{j} + v\hat{k}, v \in \mathbb{R} \end{array}$$ are given. For which point(s) $Q$ on $L_2$ can we find a point $P$ on $L_1$ and a point $R$ on $L_3$ so that $P$, $Q$ and $R$ are collinear?
(A) $\hat{k} - \frac{1}{2}\hat{j}$
(B) $\hat{k}$
(C) $\hat{k} + \frac{1}{2}\hat{j}$
(D) $\hat{k} + \hat{j}$

Q9 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View

Suppose $$\det\left[\begin{array}{cc}\sum_{k=0}^{n} k & \sum_{k=0}^{n} {}^nC_k k^2 \\ \sum_{k=0}^{n} {}^nC_k & \sum_{k=0}^{n} {}^nC_k 3^k\end{array}\right] = 0$$ holds for some positive integer $n$. Then $\sum_{k=0}^{n} \frac{{}^nC_k}{k+1}$ equals\_\_\_\_

Q10 Permutations & Arrangements Circular Arrangement View

Five persons $A$, $B$, $C$, $D$ and $E$ are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is\_\_\_\_

Q11 Probability Definitions Combinatorial Counting (Non-Probability) View

Let $|X|$ denote the number of elements in a set $X$. Let $S = \{1,2,3,4,5,6\}$ be a sample space, where each element is equally likely to occur. If $A$ and $B$ are independent events associated with $S$, then the number of ordered pairs $(A, B)$ such that $1 \leq |B| < |A|$, equals\_\_\_\_

Q12 Reciprocal Trig & Identities View

The value of $$\sec^{-1}\left(\frac{1}{4}\sum_{k=0}^{10}\sec\left(\frac{7\pi}{12} + \frac{k\pi}{2}\right)\sec\left(\frac{7\pi}{12} + \frac{(k+1)\pi}{2}\right)\right)$$ in the interval $\left[-\frac{\pi}{4}, \frac{3\pi}{4}\right]$ equals

Q13 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View

The value of the integral $$\int_0^{\pi/2} \frac{3\sqrt{\cos\theta}}{(\sqrt{\cos\theta} + \sqrt{\sin\theta})^5}\,d\theta$$ equals

Q14 Vectors Introduction & 2D Optimization of a Vector Expression View

Let $\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + \hat{k}$ be two vectors. Consider a vector $\vec{c} = \alpha\vec{a} + \beta\vec{b}$, $\alpha, \beta \in \mathbb{R}$. If the projection of $\vec{c}$ on the vector $(\vec{a} + \vec{b})$ is $3\sqrt{2}$, then the minimum value of $(\vec{c} - (\vec{a} \times \vec{b})) \cdot \vec{c}$ equals

Q15 Trig Graphs & Exact Values View

Let $f(x) = \sin(\pi\cos x)$ and $g(x) = \cos(2\pi\sin x)$ be two functions defined for $x > 0$. Define the following sets whose elements are written in the increasing order: $$\begin{array}{ll} X = \{x : f(x) = 0\}, & Y = \{x : f'(x) = 0\} \\ Z = \{x : g(x) = 0\}, & W = \{x : g'(x) = 0\} \end{array}$$
List-I contains the sets $X$, $Y$, $Z$ and $W$. List-II contains some information regarding these sets.
List-I: (I) $X$ (II) $Y$ (III) $Z$ (IV) $W$
List-II: (P) $\supseteq \left\{\frac{\pi}{2}, \frac{3\pi}{2}, 4\pi, 7\pi\right\}$ (Q) an arithmetic progression (R) NOT an arithmetic progression (S) $\supseteq \left\{\frac{\pi}{6}, \frac{7\pi}{6}, \frac{13\pi}{6}\right\}$ (T) $\supseteq \left\{\frac{\pi}{3}, \frac{2\pi}{3}, \pi\right\}$ (U) $\supseteq \left\{\frac{\pi}{6}, \frac{3\pi}{4}\right\}$
Which of the following is the only CORRECT combination?
(A) (I), (P), (R)
(B) (II), (Q), (T)
(C) (I), (Q), (U)
(D) (II), (R), (S)

Q16 Trig Graphs & Exact Values View

Let $f(x) = \sin(\pi\cos x)$ and $g(x) = \cos(2\pi\sin x)$ be two functions defined for $x > 0$. Define the following sets whose elements are written in the increasing order: $$\begin{array}{ll} X = \{x : f(x) = 0\}, & Y = \{x : f'(x) = 0\} \\ Z = \{x : g(x) = 0\}, & W = \{x : g'(x) = 0\} \end{array}$$
List-I contains the sets $X$, $Y$, $Z$ and $W$. List-II contains some information regarding these sets.
List-I: (I) $X$ (II) $Y$ (III) $Z$ (IV) $W$
List-II: (P) $\supseteq \left\{\frac{\pi}{2}, \frac{3\pi}{2}, 4\pi, 7\pi\right\}$ (Q) an arithmetic progression (R) NOT an arithmetic progression (S) $\supseteq \left\{\frac{\pi}{6}, \frac{7\pi}{6}, \frac{13\pi}{6}\right\}$ (T) $\supseteq \left\{\frac{\pi}{3}, \frac{2\pi}{3}, \pi\right\}$ (U) $\supseteq \left\{\frac{\pi}{6}, \frac{3\pi}{4}\right\}$
Which of the following is the only CORRECT combination?
(A) (III), (R), (U)
(B) (IV), (P), (R), (S)
(C) (III), (P), (Q), (U)
(D) (IV), (Q), (T)

Q17 Circles Intersection of Circles or Circle with Conic View

Let the circles $C_1 : x^2 + y^2 = 9$ and $C_2 : (x-3)^2 + (y-4)^2 = 16$, intersect at the points $X$ and $Y$. Suppose that another circle $C_3 : (x-h)^2 + (y-k)^2 = r^2$ satisfies the following conditions: (i) centre of $C_3$ is collinear with the centres of $C_1$ and $C_2$, (ii) $C_1$ and $C_2$ both lie inside $C_3$, and (iii) $C_3$ touches $C_1$ at $M$ and $C_2$ at $N$.
Let the line through $X$ and $Y$ intersect $C_3$ at $Z$ and $W$, and let a common tangent of $C_1$ and $C_3$ be a tangent to the parabola $x^2 = 8\alpha y$.
List-I: (I) $2h + k$ (II) $\frac{\text{Length of } ZW}{\text{Length of } XY}$ (III) $\frac{\text{Area of triangle } MZN}{\text{Area of triangle } ZMW}$ (IV) $\alpha$
List-II: (P) $6$ (Q) $\sqrt{6}$ (R) $\frac{5}{4}$ (S) $\frac{21}{5}$ (T) $2\sqrt{6}$ (U) $\frac{10}{3}$
Which of the following is the only CORRECT combination?
(A) (I), (S)
(B) (I), (U)
(C) (II), (Q)
(D) (II), (T)

Q18 Circles Intersection of Circles or Circle with Conic View

Let the circles $C_1 : x^2 + y^2 = 9$ and $C_2 : (x-3)^2 + (y-4)^2 = 16$, intersect at the points $X$ and $Y$. Suppose that another circle $C_3 : (x-h)^2 + (y-k)^2 = r^2$ satisfies the following conditions: (i) centre of $C_3$ is collinear with the centres of $C_1$ and $C_2$, (ii) $C_1$ and $C_2$ both lie inside $C_3$, and (iii) $C_3$ touches $C_1$ at $M$ and $C_2$ at $N$.
Let the line through $X$ and $Y$ intersect $C_3$ at $Z$ and $W$, and let a common tangent of $C_1$ and $C_3$ be a tangent to the parabola $x^2 = 8\alpha y$.
List-I: (I) $2h + k$ (II) $\frac{\text{Length of } ZW}{\text{Length of } XY}$ (III) $\frac{\text{Area of triangle } MZN}{\text{Area of triangle } ZMW}$ (IV) $\alpha$
List-II: (P) $6$ (Q) $\sqrt{6}$ (R) $\frac{5}{4}$ (S) $\frac{21}{5}$ (T) $2\sqrt{6}$ (U) $\frac{10}{3}$
Which of the following is the only INCORRECT combination?
(A) (I), (P)
(B) (IV), (U)
(C) (III), (R)
(D) (IV), (S)

jee-advanced

Papers (38)

2019 paper2