jee-advanced

Q1 4 marks Reciprocal Trig & Identities View

For any positive integer $n$, define $f _ { n } : ( 0 , \infty ) \rightarrow \mathbb { R }$ as
$$f _ { n } ( x ) = \sum _ { j = 1 } ^ { n } \tan ^ { - 1 } \left( \frac { 1 } { 1 + ( x + j ) ( x + j - 1 ) } \right) \text { for all } x \in ( 0 , \infty )$$
(Here, the inverse trigonometric function $\tan ^ { - 1 } x$ assumes values in $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$.) Then, which of the following statement(s) is (are) TRUE?
(A) $\sum _ { j = 1 } ^ { 5 } \tan ^ { 2 } \left( f _ { j } ( 0 ) \right) = 55$
(B) $\sum _ { j = 1 } ^ { 10 } \left( 1 + f _ { j } ^ { \prime } ( 0 ) \right) \sec ^ { 2 } \left( f _ { j } ( 0 ) \right) = 10$
(C) For any fixed positive integer $n , \lim _ { x \rightarrow \infty } \tan \left( f _ { n } ( x ) \right) = \frac { 1 } { n }$
(D) For any fixed positive integer $n , \lim _ { x \rightarrow \infty } \sec ^ { 2 } \left( f _ { n } ( x ) \right) = 1$

Q2 4 marks Circles Circle-Related Locus Problems View

Let $T$ be the line passing through the points $P ( - 2,7 )$ and $Q ( 2 , - 5 )$. Let $F _ { 1 }$ be the set of all pairs of circles ( $S _ { 1 } , S _ { 2 }$ ) such that $T$ is tangent to $S _ { 1 }$ at $P$ and tangent to $S _ { 2 }$ at $Q$, and also such that $S _ { 1 }$ and $S _ { 2 }$ touch each other at a point, say, $M$. Let $E _ { 1 }$ be the set representing the locus of $M$ as the pair ( $S _ { 1 } , S _ { 2 }$ ) varies in $F _ { 1 }$. Let the set of all straight line segments joining a pair of distinct points of $E _ { 1 }$ and passing through the point $R ( 1,1 )$ be $F _ { 2 }$. Let $E _ { 2 }$ be the set of the mid-points of the line segments in the set $F _ { 2 }$. Then, which of the following statement(s) is (are) TRUE?
(A) The point $( - 2,7 )$ lies in $E _ { 1 }$
(B) The point $\left( \frac { 4 } { 5 } , \frac { 7 } { 5 } \right)$ does NOT lie in $E _ { 2 }$
(C) The point $\left( \frac { 1 } { 2 } , 1 \right)$ lies in $E _ { 2 }$
(D) The point $\left( 0 , \frac { 3 } { 2 } \right)$ does NOT lie in $E _ { 1 }$

Q3 4 marks Simultaneous equations View

Let $S$ be the set of all column matrices $\left[ \begin{array} { l } b _ { 1 } \\ b _ { 2 } \\ b _ { 3 } \end{array} \right]$ such that $b _ { 1 } , b _ { 2 } , b _ { 3 } \in \mathbb { R }$ and the system of equations (in real variables)
$$\begin{aligned} - x + 2 y + 5 z & = b _ { 1 } \\ 2 x - 4 y + 3 z & = b _ { 2 } \\ x - 2 y + 2 z & = b _ { 3 } \end{aligned}$$
has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $\left[ \begin{array} { l } b _ { 1 } \\ b _ { 2 } \\ b _ { 3 } \end{array} \right] \in S$ ?
(A) $x + 2 y + 3 z = b _ { 1 } , 4 y + 5 z = b _ { 2 }$ and $x + 2 y + 6 z = b _ { 3 }$
(B) $x + y + 3 z = b _ { 1 } , 5 x + 2 y + 6 z = b _ { 2 }$ and $- 2 x - y - 3 z = b _ { 3 }$
(C) $- x + 2 y - 5 z = b _ { 1 } , 2 x - 4 y + 10 z = b _ { 2 }$ and $x - 2 y + 5 z = b _ { 3 }$
(D) $x + 2 y + 5 z = b _ { 1 } , 2 x + 3 z = b _ { 2 }$ and $x + 4 y - 5 z = b _ { 3 }$

Q4 4 marks Circles Tangent Lines and Tangent Lengths View

Consider two straight lines, each of which is tangent to both the circle $x ^ { 2 } + y ^ { 2 } = \frac { 1 } { 2 }$ and the parabola $y ^ { 2 } = 4 x$. Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O ( 0,0 )$ and whose semi-major axis is $O Q$. If the length of the minor axis of this ellipse is $\sqrt { 2 }$, then which of the following statement(s) is (are) TRUE?
(A) For the ellipse, the eccentricity is $\frac { 1 } { \sqrt { 2 } }$ and the length of the latus rectum is 1
(B) For the ellipse, the eccentricity is $\frac { 1 } { 2 }$ and the length of the latus rectum is $\frac { 1 } { 2 }$
(C) The area of the region bounded by the ellipse between the lines $x = \frac { 1 } { \sqrt { 2 } }$ and $x = 1$ is $\frac { 1 } { 4 \sqrt { 2 } } ( \pi - 2 )$
(D) The area of the region bounded by the ellipse between the lines $x = \frac { 1 } { \sqrt { 2 } }$ and $x = 1$ is
$$\frac { 1 } { 16 } ( \pi - 2 )$$

Q5 4 marks Complex Numbers Arithmetic True/False or Property Verification Statements View

Let $s , t , r$ be non-zero complex numbers and $L$ be the set of solutions $z = x + i y ( x , y \in \mathbb { R } , i = \sqrt { - 1 } )$ of the equation $s z + t \bar { z } + r = 0$, where $\bar { z } = x - i y$. Then, which of the following statement(s) is (are) TRUE?
(A) If $L$ has exactly one element, then $| s | \neq | t |$
(B) If $| s | = | t |$, then $L$ has infinitely many elements
(C) The number of elements in $L \cap \{ z : | z - 1 + i | = 5 \}$ is at most 2
(D) If $L$ has more than one element, then $L$ has infinitely many elements

Q6 4 marks Differential equations Solving Separable DEs with Initial Conditions View

Let $f : ( 0 , \pi ) \rightarrow \mathbb { R }$ be a twice differentiable function such that
$$\lim _ { t \rightarrow x } \frac { f ( x ) \sin t - f ( t ) \sin x } { t - x } = \sin ^ { 2 } x \text { for all } x \in ( 0 , \pi )$$
If $f \left( \frac { \pi } { 6 } \right) = - \frac { \pi } { 12 }$, then which of the following statement(s) is (are) TRUE?
(A) $f \left( \frac { \pi } { 4 } \right) = \frac { \pi } { 4 \sqrt { 2 } }$
(B) $f ( x ) < \frac { x ^ { 4 } } { 6 } - x ^ { 2 }$ for all $x \in ( 0 , \pi )$
(C) There exists $\alpha \in ( 0 , \pi )$ such that $f ^ { \prime } ( \alpha ) = 0$
(D) $f ^ { \prime \prime } \left( \frac { \pi } { 2 } \right) + f \left( \frac { \pi } { 2 } \right) = 0$

Q7 3 marks Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View

The value of the integral
$$\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 + \sqrt { 3 } } { \left( ( x + 1 ) ^ { 2 } ( 1 - x ) ^ { 6 } \right) ^ { \frac { 1 } { 4 } } } d x$$
is $\_\_\_\_$ .

Q8 3 marks 3x3 Matrices Range of Determinant Values for Constrained Matrices View

Let $P$ be a matrix of order $3 \times 3$ such that all the entries in $P$ are from the set $\{ - 1,0,1 \}$. Then, the maximum possible value of the determinant of $P$ is $\_\_\_\_$ .

Q9 3 marks Combinations & Selection Counting Functions or Mappings with Constraints View

Let $X$ be a set with exactly 5 elements and $Y$ be a set with exactly 7 elements. If $\alpha$ is the number of one-one functions from $X$ to $Y$ and $\beta$ is the number of onto functions from $Y$ to $X$, then the value of $\frac { 1 } { 5 ! } ( \beta - \alpha )$ is $\_\_\_\_$ .

Q10 3 marks Differential equations Solving Separable DEs with Initial Conditions View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a differentiable function with $f ( 0 ) = 0$. If $y = f ( x )$ satisfies the differential equation
$$\frac { d y } { d x } = ( 2 + 5 y ) ( 5 y - 2 )$$
then the value of $\lim _ { x \rightarrow - \infty } f ( x )$ is $\_\_\_\_$ .

Q11 3 marks Exponential Functions Functional Equation with Exponentials View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a differentiable function with $f ( 0 ) = 1$ and satisfying the equation
$$f ( x + y ) = f ( x ) f ^ { \prime } ( y ) + f ^ { \prime } ( x ) f ( y ) \text { for all } x , y \in \mathbb { R }$$
Then, the value of $\log _ { e } ( f ( 4 ) )$ is $\_\_\_\_$ .

Q12 3 marks Vectors: Lines & Planes Perpendicular/Orthogonal Projection onto a Plane View

Let $P$ be a point in the first octant, whose image $Q$ in the plane $x + y = 3$ (that is, the line segment $P Q$ is perpendicular to the plane $x + y = 3$ and the mid-point of $P Q$ lies in the plane $x + y = 3$ ) lies on the $z$-axis. Let the distance of $P$ from the $x$-axis be 5 . If $R$ is the image of $P$ in the $x y$-plane, then the length of $P R$ is $\_\_\_\_$ .

Q13 3 marks Vectors: Cross Product & Distances View

Consider the cube in the first octant with sides $O P , O Q$ and $O R$ of length 1 , along the $x$-axis, $y$-axis and $z$-axis, respectively, where $O ( 0,0,0 )$ is the origin. Let $S \left( \frac { 1 } { 2 } , \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$ be the centre of the cube and $T$ be the vertex of the cube opposite to the origin $O$ such that $S$ lies on the diagonal $O T$. If $\vec { p } = \overrightarrow { S P } , \vec { q } = \overrightarrow { S Q } , \vec { r } = \overrightarrow { S R }$ and $\vec { t } = \overrightarrow { S T }$, then the value of $| ( \vec { p } \times \vec { q } ) \times ( \vec { r } \times \vec { t } ) |$ is $\_\_\_\_$ .

Q14 3 marks Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View

Let
$$X = \left( { } ^ { 10 } C _ { 1 } \right) ^ { 2 } + 2 \left( { } ^ { 10 } C _ { 2 } \right) ^ { 2 } + 3 \left( { } ^ { 10 } C _ { 3 } \right) ^ { 2 } + \cdots + 10 \left( { } ^ { 10 } C _ { 10 } \right) ^ { 2 }$$
where ${ } ^ { 10 } C _ { r } , r \in \{ 1,2 , \cdots , 10 \}$ denote binomial coefficients. Then, the value of $\frac { 1 } { 1430 } X$ is $\_\_\_\_$.

Q15 3 marks Composite & Inverse Functions Determine Domain or Range of a Composite Function View

Let $E _ { 1 } = \left\{ x \in \mathbb { R } : x \neq 1 \right.$ and $\left. \frac { x } { x - 1 } > 0 \right\}$ and $E _ { 2 } = \left\{ x \in E _ { 1 } : \sin ^ { - 1 } \left( \log _ { e } \left( \frac { x } { x - 1 } \right) \right) \right.$ is a real number $\}$. (Here, the inverse trigonometric function $\sin ^ { - 1 } x$ assumes values in $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$.) Let $f : E _ { 1 } \rightarrow \mathbb { R }$ be the function defined by $f ( x ) = \log _ { e } \left( \frac { x } { x - 1 } \right)$ and $g : E _ { 2 } \rightarrow \mathbb { R }$ be the function defined by $g ( x ) = \sin ^ { - 1 } \left( \log _ { e } \left( \frac { x } { x - 1 } \right) \right)$.
LIST-I P. The range of $f$ is Q. The range of $g$ contains R. The domain of $f$ contains S. The domain of $g$ is
LIST-II

$\left( - \infty , \frac { 1 } { 1 - e } \right] \cup \left[ \frac { e } { e - 1 } , \infty \right)$
$( 0,1 )$
$\left[ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right]$
$( - \infty , 0 ) \cup ( 0 , \infty )$
$\left( - \infty , \frac { e } { e - 1 } \right]$
$( - \infty , 0 ) \cup \left( \frac { 1 } { 2 } , \frac { e } { e - 1 } \right]$

The correct option is:
(A) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 2 } ; \mathbf { R } \rightarrow \mathbf { 1 } ; \mathbf { S } \rightarrow \mathbf { 1 }$
(B) $\mathbf { P } \rightarrow \mathbf { 3 } ; \mathbf { Q } \rightarrow \mathbf { 3 } ; \mathbf { R } \rightarrow \mathbf { 6 } ; \mathbf { S } \rightarrow \mathbf { 5 }$
(C) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 2 } ; \mathbf { R } \rightarrow \mathbf { 1 } ; \mathbf { S } \rightarrow \mathbf { 6 }$
(D) $\mathrm { P } \rightarrow 4 ; \mathrm { Q } \rightarrow 3 ; \mathrm { R } \rightarrow 6 ; \mathrm { S } \rightarrow 5$

Q16 3 marks Combinations & Selection Selection with Group/Category Constraints View

In a high school, a committee has to be formed from a group of 6 boys $M _ { 1 } , M _ { 2 } , M _ { 3 } , M _ { 4 } , M _ { 5 } , M _ { 6 }$ and 5 girls $G _ { 1 } , G _ { 2 } , G _ { 3 } , G _ { 4 } , G _ { 5 }$.
(i) Let $\alpha _ { 1 }$ be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls.
(ii) Let $\alpha _ { 2 }$ be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls.
(iii) Let $\alpha _ { 3 }$ be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls.
(iv) Let $\alpha _ { 4 }$ be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls and such that both $M _ { 1 }$ and $G _ { 1 }$ are NOT in the committee together.
LIST-I P. The value of $\alpha _ { 1 }$ is Q. The value of $\alpha _ { 2 }$ is R. The value of $\alpha _ { 3 }$ is S. The value of $\alpha _ { 4 }$ is
LIST-II

136
189
192
200
381
461

The correct option is:
(A) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 6 ; ~ } \mathbf { R } \rightarrow \mathbf { 2 ; } \mathbf { S } \rightarrow \mathbf { 1 }$
(B) $\mathbf { P } \rightarrow \mathbf { 1 } ; \mathbf { Q } \rightarrow \mathbf { 4 } ; \mathbf { R } \rightarrow \mathbf { 2 } ; \mathbf { S } \rightarrow \mathbf { 3 }$
(C) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 6 } ; \mathbf { R } \rightarrow \mathbf { 5 } ; \mathbf { S } \rightarrow \mathbf { 2 }$
(D) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 2 } ; \mathbf { R } \rightarrow \mathbf { 3 } ; \mathbf { S } \rightarrow \mathbf { 1 }$

Q17 3 marks Conic sections Eccentricity or Asymptote Computation View

Let $H$ : $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$, where $a > b > 0$, be a hyperbola in the $x y$-plane whose conjugate axis $L M$ subtends an angle of $60 ^ { \circ }$ at one of its vertices $N$. Let the area of the triangle $L M N$ be $4 \sqrt { 3 }$.
LIST-I P. The length of the conjugate axis of $H$ is Q. The eccentricity of $H$ is R. The distance between the foci of $H$ is S. The length of the latus rectum of $H$ is
LIST-II

8
$\frac { 4 } { \sqrt { 3 } }$
$\frac { 2 } { \sqrt { 3 } }$
4

The correct option is:
(A) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 2 } ; \mathbf { R } \rightarrow \mathbf { 1 } ; \mathbf { S } \rightarrow \mathbf { 3 }$
(B) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 3 } ; \mathbf { R } \rightarrow \mathbf { 1 } ; \mathbf { S } \rightarrow \mathbf { 2 }$
(C) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 1 } ; \mathbf { R } \rightarrow \mathbf { 3 } ; \mathbf { S } \rightarrow \mathbf { 2 }$
(D) $\mathbf { P } \rightarrow \mathbf { 3 } ; \mathbf { Q } \rightarrow \mathbf { 4 } ; \mathbf { R } \rightarrow \mathbf { 2 } ; \mathbf { S } \rightarrow \mathbf { 1 }$

Q18 3 marks Differentiating Transcendental Functions Piecewise function analysis with transcendental components View

Let $f _ { 1 } : \mathbb { R } \rightarrow \mathbb { R } , f _ { 2 } : \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right) \rightarrow \mathbb { R } , f _ { 3 } : \left( - 1 , e ^ { \frac { \pi } { 2 } } - 2 \right) \rightarrow \mathbb { R }$ and $f _ { 4 } : \mathbb { R } \rightarrow \mathbb { R }$ be functions defined by
(i) $\quad f _ { 1 } ( x ) = \sin \left( \sqrt { 1 - e ^ { - x ^ { 2 } } } \right)$,
(ii) $\quad f _ { 2 } ( x ) = \left\{ \begin{array} { c c } \frac { | \sin x | } { \tan ^ { - 1 } x } & \text { if } x \neq 0 \\ 1 & \text { if } x = 0 \end{array} \right.$, where the inverse trigonometric function $\tan ^ { - 1 } x$ assumes values in $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$,
(iii) $\quad f _ { 3 } ( x ) = \left[ \sin \left( \log _ { e } ( x + 2 ) \right) \right]$, where, for $t \in \mathbb { R } , [ t ]$ denotes the greatest integer less than or equal to $t$,
(iv) $\quad f _ { 4 } ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } \sin \left( \frac { 1 } { x } \right) & \text { if } x \neq 0 \\ 0 & \text { if } x = 0 \end{array} \right.$.
LIST-I P. The function $f _ { 1 }$ is Q. The function $f _ { 2 }$ is R. The function $f _ { 3 }$ is S. The function $f _ { 4 }$ is
LIST-II

NOT continuous at $x = 0$
continuous at $x = 0$ and NOT differentiable at $x = 0$
differentiable at $x = 0$ and its derivative is NOT continuous at $x = 0$
differentiable at $x = 0$ and its derivative is continuous at $x = 0$

The correct option is:
(A) $\mathbf { P } \rightarrow \mathbf { 2 ; } \mathbf { Q } \rightarrow \mathbf { 3 ; } \mathbf { R } \rightarrow \mathbf { 1 ; } \mathbf { S } \rightarrow \mathbf { 4 }$
(B) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 1 } ; \mathbf { R } \rightarrow \mathbf { 2 } ; \mathbf { S } \rightarrow \mathbf { 3 }$
(C) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 2 } ; \mathbf { R } \rightarrow \mathbf { 1 } ; \mathbf { S } \rightarrow \mathbf { 3 }$
(D) $\mathbf { P } \rightarrow \mathbf { 2 } ; \mathbf { Q } \rightarrow \mathbf { 1 } ; \mathbf { R } \rightarrow \mathbf { 4 } ; \mathbf { S } \rightarrow \mathbf { 3 }$

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