jee-advanced

Q1 Complex Numbers Argand & Loci Similarity, Rotation, and Geometric Transformations in the Complex Plane View

A particle $P$ starts from the point $z _ { 0 } = 1 + 2 i$, where $i = \sqrt { - 1 }$. It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point $z _ { 1 }$. From $z _ { 1 }$ the particle moves $\sqrt { 2 }$ units in the direction of the vector $\hat { i } + \hat { j }$ and then it moves through an angle $\frac { \pi } { 2 }$ in anticlockwise direction on a circle with centre at origin, to reach a point $z _ { 2 }$. The point $z _ { 2 }$ is given by
(A) $6 + 7 i$
(B) $- 7 + 6 i$
(C) $7 + 6 i$
(D) $- 6 + 7 i$

Q2 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View

Let the function $g : ( - \infty , \infty ) \rightarrow \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ be given by $g ( u ) = 2 \tan ^ { - 1 } \left( e ^ { u } \right) - \frac { \pi } { 2 }$. Then, $g$ is
(A) even and is strictly increasing in $(0 , \infty)$
(B) odd and is strictly decreasing in $( - \infty , \infty )$
(C) odd and is strictly increasing in $( - \infty , \infty )$
(D) neither even nor odd, but is strictly increasing in $( - \infty , \infty )$

Q3 Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View

Consider a branch of the hyperbola
$$x ^ { 2 } - 2 y ^ { 2 } - 2 \sqrt { 2 } x - 4 \sqrt { 2 } y - 6 = 0$$
with vertex at the point $A$. Let $B$ be one of the end points of its latus rectum. If $C$ is the focus of the hyperbola nearest to the point $A$, then the area of the triangle $A B C$ is
(A) $1 - \sqrt { \frac { 2 } { 3 } }$
(B) $\sqrt { \frac { 3 } { 2 } } - 1$
(C) $1 + \sqrt { \frac { 2 } { 3 } }$
(D) $\sqrt { \frac { 3 } { 2 } } + 1$

Q4 Areas Between Curves Select Correct Integral Expression View

The area of the region between the curves $y = \sqrt { \frac { 1 + \sin x } { \cos x } }$ and $y = \sqrt { \frac { 1 - \sin x } { \cos x } }$ bounded by the lines $x = 0$ and $x = \frac { \pi } { 4 }$ is
(A) $\int _ { 0 } ^ { \sqrt { 2 } - 1 } \frac { t } { \left( 1 + t ^ { 2 } \right) \sqrt { 1 - t ^ { 2 } } } d t$
(B) $\int _ { 0 } ^ { \sqrt { 2 } - 1 } \frac { 4 t } { \left( 1 + t ^ { 2 } \right) \sqrt { 1 - t ^ { 2 } } } d t$
(C) $\int _ { 0 } ^ { \sqrt { 2 } + 1 } \frac { 4 t } { \left( 1 + t ^ { 2 } \right) \sqrt { 1 - t ^ { 2 } } } d t$
(D) $\int _ { 0 } ^ { \sqrt { 2 } + 1 } \frac { t } { \left( 1 + t ^ { 2 } \right) \sqrt { 1 - t ^ { 2 } } } d t$

Q5 Straight Lines & Coordinate Geometry Collinearity and Concurrency View

Consider three points $P = ( - \sin ( \beta - \alpha ) , - \cos \beta ) , Q = ( \cos ( \beta - \alpha ) , \sin \beta )$ and $R = ( \cos ( \beta - \alpha + \theta ) , \sin ( \beta - \theta ) )$, where $0 < \alpha , \beta , \theta < \frac { \pi } { 4 }$. Then,
(A) $P$ lies on the line segment $R Q$
(B) $Q$ lies on the line segment $P R$
(C) $R$ lies on the line segment $Q P$
(D) $P , Q , R$ are non-collinear

Q6 Independent Events View

An experiment has 10 equally likely outcomes. Let $A$ and $B$ be two non-empty events of the experiment. If $A$ consists of 4 outcomes, the number of outcomes that $B$ must have so that $A$ and $B$ are independent, is
(A) 2, 4 or 8
(B) 3, 6 or 9
(C) 4 or 8
(D) 5 or 10

Q7 Vectors Introduction & 2D Optimization of a Vector Expression View

Let two non-collinear unit vectors $\hat { a }$ and $\hat { b }$ form an acute angle. A point $P$ moves so that at any time $t$ the position vector $\overrightarrow { O P }$ (where $O$ is the origin) is given by $\hat { a } \cos t + \hat { b } \sin t$. When $P$ is farthest from origin $O$, let $M$ be the length of $\overrightarrow { O P }$ and $\hat { u }$ be the unit vector along $\overrightarrow { O P }$. Then,
(A) $\hat { u } = \frac { \hat { a } + \hat { b } } { | \hat { a } + \hat { b } | }$ and $M = ( 1 + \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$
(B) $\hat { u } = \frac { \hat { a } - \hat { b } } { | \hat { a } - \hat { b } | }$ and $M = ( 1 + \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$
(C) $\hat { u } = \frac { \hat { a } + \hat { b } } { | \hat { a } + \hat { b } | }$ and $M = ( 1 + 2 \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$
(D) $\hat { u } = \frac { \hat { a } - \hat { b } } { | \hat { a } - \hat { b } | }$ and $M = ( 1 + 2 \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$

Q8 Standard Integrals and Reverse Chain Rule Reverse Chain Rule Antiderivative (MCQ) View

Let
$$I = \int \frac { e ^ { x } } { e ^ { 4 x } + e ^ { 2 x } + 1 } d x , \quad J = \int \frac { e ^ { - x } } { e ^ { - 4 x } + e ^ { - 2 x } + 1 } d x$$
Then, for an arbitrary constant $C$, the value of $J - I$ equals
(A) $\frac { 1 } { 2 } \log \left( \frac { e ^ { 4 x } - e ^ { 2 x } + 1 } { e ^ { 4 x } + e ^ { 2 x } + 1 } \right) + C$
(B) $\frac { 1 } { 2 } \log \left( \frac { e ^ { 2 x } + e ^ { x } + 1 } { e ^ { 2 x } - e ^ { x } + 1 } \right) + C$
(C) $\frac { 1 } { 2 } \log \left( \frac { e ^ { 2 x } - e ^ { x } + 1 } { e ^ { 2 x } + e ^ { x } + 1 } \right) + C$
(D) $\frac { 1 } { 2 } \log \left( \frac { e ^ { 4 x } + e ^ { 2 x } + 1 } { e ^ { 4 x } - e ^ { 2 x } + 1 } \right) + C$

Q9 Differentiating Transcendental Functions Higher-order or nth derivative computation View

Let $g ( x ) = \log f ( x )$ where $f ( x )$ is a twice differentiable positive function on $( 0 , \infty )$ such that $f ( x + 1 ) = x f ( x )$. Then, for $N = 1,2,3 , \ldots$,
$$g ^ { \prime \prime } \left( N + \frac { 1 } { 2 } \right) - g ^ { \prime \prime } \left( \frac { 1 } { 2 } \right) =$$
(A) $- 4 \left\{ 1 + \frac { 1 } { 9 } + \frac { 1 } { 25 } + \cdots + \frac { 1 } { ( 2 N - 1 ) ^ { 2 } } \right\}$
(B) $4 \left\{ 1 + \frac { 1 } { 9 } + \frac { 1 } { 25 } + \cdots + \frac { 1 } { ( 2 N - 1 ) ^ { 2 } } \right\}$
(C) $- 4 \left\{ 1 + \frac { 1 } { 9 } + \frac { 1 } { 25 } + \cdots + \frac { 1 } { ( 2 N + 1 ) ^ { 2 } } \right\}$
(D) $4 \left\{ 1 + \frac { 1 } { 9 } + \frac { 1 } { 25 } + \cdots + \frac { 1 } { ( 2 N + 1 ) ^ { 2 } } \right\}$

Q10 Geometric Sequences and Series Proof of a Structural Property of Geometric Sequences View

Suppose four distinct positive numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ are in G.P. Let $b _ { 1 } = a _ { 1 }$, $b _ { 2 } = b _ { 1 } + a _ { 2 } , b _ { 3 } = b _ { 2 } + a _ { 3 }$ and $b _ { 4 } = b _ { 3 } + a _ { 4 }$. STATEMENT-1 : The numbers $b _ { 1 } , b _ { 2 } , b _ { 3 } , b _ { 4 }$ are neither in A.P. nor in G.P. and
STATEMENT-2 : The numbers $b _ { 1 } , b _ { 2 } , b _ { 3 } , b _ { 4 }$ are in H.P.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

Q11 Discriminant and conditions for roots Root relationships and Vieta's formulas View

Let $a , b , c , p , q$ be real numbers. Suppose $\alpha , \beta$ are the roots of the equation $x ^ { 2 } + 2 p x + q = 0$ and $\alpha , \frac { 1 } { \beta }$ are the roots of the equation $a x ^ { 2 } + 2 b x + c = 0$, where $\beta ^ { 2 } \notin \{ - 1,0,1 \}$. STATEMENT-1 : $\left( p ^ { 2 } - q \right) \left( b ^ { 2 } - a c \right) \geq 0$ and STATEMENT-2 : $b \neq p a$ or $c \neq q a$
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

Q12 Circles Chord Length and Chord Properties View

Consider
$$\begin{aligned} & L _ { 1 } : 2 x + 3 y + p - 3 = 0 \\ & L _ { 2 } : 2 x + 3 y + p + 3 = 0 \end{aligned}$$
where $p$ is a real number, and $C : x ^ { 2 } + y ^ { 2 } + 6 x - 10 y + 30 = 0$. STATEMENT-1 : If line $L _ { 1 }$ is a chord of circle $C$, then line $L _ { 2 }$ is not always a diameter of circle $C$.
and
STATEMENT-2 : If line $L _ { 1 }$ is a diameter of circle $C$, then line $L _ { 2 }$ is not a chord of circle $C$.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

Q13 Differential equations Solving Separable DEs with Initial Conditions View

Let a solution $y = y ( x )$ of the differential equation
$$x \sqrt { x ^ { 2 } - 1 } d y - y \sqrt { y ^ { 2 } - 1 } d x = 0$$
satisfy $y ( 2 ) = \frac { 2 } { \sqrt { 3 } }$. STATEMENT-1 : $y ( x ) = \sec \left( \sec ^ { - 1 } x - \frac { \pi } { 6 } \right)$ and STATEMENT-2 : $y ( x )$ is given by
$$\frac { 1 } { y } = \frac { 2 \sqrt { 3 } } { x } - \sqrt { 1 - \frac { 1 } { x ^ { 2 } } }$$
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

Q14 Applied differentiation MCQ on derivative and graph interpretation View

Consider the function $f : ( - \infty , \infty ) \rightarrow ( - \infty , \infty )$ defined by
$$f ( x ) = \frac { x ^ { 2 } - a x + 1 } { x ^ { 2 } + a x + 1 } , 0 < a < 2 .$$
Which of the following is true?
(A) $( 2 + a ) ^ { 2 } f ^ { \prime \prime } ( 1 ) + ( 2 - a ) ^ { 2 } f ^ { \prime \prime } ( - 1 ) = 0$
(B) $( 2 - a ) ^ { 2 } f ^ { \prime \prime } ( 1 ) - ( 2 + a ) ^ { 2 } f ^ { \prime \prime } ( - 1 ) = 0$
(C) $f ^ { \prime } ( 1 ) f ^ { \prime } ( - 1 ) = ( 2 - a ) ^ { 2 }$
(D) $f ^ { \prime } ( 1 ) f ^ { \prime } ( - 1 ) = - ( 2 + a ) ^ { 2 }$

Q15 Stationary points and optimisation Find critical points and classify extrema of a given function View

Consider the function $f : ( - \infty , \infty ) \rightarrow ( - \infty , \infty )$ defined by
$$f ( x ) = \frac { x ^ { 2 } - a x + 1 } { x ^ { 2 } + a x + 1 } , 0 < a < 2 .$$
Which of the following is true?
(A) $f ( x )$ is decreasing on $( - 1,1 )$ and has a local minimum at $x = 1$
(B) $f ( x )$ is increasing on $( - 1,1 )$ and has a local maximum at $x = 1$
(C) $f ( x )$ is increasing on $( - 1,1 )$ but has neither a local maximum nor a local minimum at $x = 1$
(D) $f ( x )$ is decreasing on $( - 1,1 )$ but has neither a local maximum nor a local minimum at $x = 1$

Q16 Indefinite & Definite Integrals Accumulation Function Analysis View

Consider the function $f : ( - \infty , \infty ) \rightarrow ( - \infty , \infty )$ defined by
$$f ( x ) = \frac { x ^ { 2 } - a x + 1 } { x ^ { 2 } + a x + 1 } , 0 < a < 2 .$$
Let
$$g ( x ) = \int _ { 0 } ^ { e ^ { x } } \frac { f ^ { \prime } ( t ) } { 1 + t ^ { 2 } } d t$$
Which of the following is true?
(A) $g ^ { \prime } ( x )$ is positive on $( - \infty , 0 )$ and negative on $( 0 , \infty )$
(B) $g ^ { \prime } ( x )$ is negative on $( - \infty , 0 )$ and positive on $( 0 , \infty )$
(C) $g ^ { \prime } ( x )$ changes sign on both $( - \infty , 0 )$ and $( 0 , \infty )$
(D) $g ^ { \prime } ( x )$ does not change sign on $( - \infty , \infty )$

Q17 Vectors 3D & Lines Vector Algebra and Triple Product Computation View

Consider the lines
$$\begin{aligned} & L _ { 1 } : \frac { x + 1 } { 3 } = \frac { y + 2 } { 1 } = \frac { z + 1 } { 2 } \\ & L _ { 2 } : \frac { x - 2 } { 1 } = \frac { y + 2 } { 2 } = \frac { z - 3 } { 3 } \end{aligned}$$
The unit vector perpendicular to both $L _ { 1 }$ and $L _ { 2 }$ is
(A) $\frac { - \hat { i } + 7 \hat { j } + 7 \hat { k } } { \sqrt { 99 } }$
(B) $\frac { - \hat { i } - 7 \hat { j } + 5 \hat { k } } { 5 \sqrt { 3 } }$
(C) $\frac { - \hat { i } + 7 \hat { j } + 5 \hat { k } } { 5 \sqrt { 3 } }$
(D) $\frac { 7 \hat { i } - 7 \hat { j } - \hat { k } } { \sqrt { 99 } }$

Q18 Vectors 3D & Lines Shortest Distance Between Two Lines View

Consider the lines
$$\begin{aligned} & L _ { 1 } : \frac { x + 1 } { 3 } = \frac { y + 2 } { 1 } = \frac { z + 1 } { 2 } \\ & L _ { 2 } : \frac { x - 2 } { 1 } = \frac { y + 2 } { 2 } = \frac { z - 3 } { 3 } \end{aligned}$$
The shortest distance between $L _ { 1 }$ and $L _ { 2 }$ is
(A) 0
(B) $\frac { 17 } { \sqrt { 3 } }$
(C) $\frac { 41 } { 5 \sqrt { 3 } }$
(D) $\frac { 17 } { 5 \sqrt { 3 } }$

Q19 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View

Consider the lines
$$\begin{aligned} & L _ { 1 } : \frac { x + 1 } { 3 } = \frac { y + 2 } { 1 } = \frac { z + 1 } { 2 } \\ & L _ { 2 } : \frac { x - 2 } { 1 } = \frac { y + 2 } { 2 } = \frac { z - 3 } { 3 } \end{aligned}$$
The distance of the point $( 1,1,1 )$ from the plane passing through the point $( - 1 , - 2 , - 1 )$ and whose normal is perpendicular to both the lines $L _ { 1 }$ and $L _ { 2 }$ is
(A) $\frac { 2 } { \sqrt { 75 } }$
(B) $\frac { 7 } { \sqrt { 75 } }$
(C) $\frac { 13 } { \sqrt { 75 } }$
(D) $\frac { 23 } { \sqrt { 75 } }$

Q20 Straight Lines & Coordinate Geometry Collinearity and Concurrency View

Consider the lines given by
$$\begin{aligned} & L _ { 1 } : x + 3 y - 5 = 0 \\ & L _ { 2 } : 3 x - k y - 1 = 0 \\ & L _ { 3 } : 5 x + 2 y - 12 = 0 \end{aligned}$$
Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.
Column I
(A) $L _ { 1 } , L _ { 2 } , L _ { 3 }$ are concurrent, if
(B) One of $L _ { 1 } , L _ { 2 } , L _ { 3 }$ is parallel to at least one of the other two, if
(C) $L _ { 1 } , L _ { 2 } , L _ { 3 }$ form a triangle, if
(D) $L _ { 1 } , L _ { 2 } , L _ { 3 }$ do not form a triangle, if
Column II
(p) $k = - 9$
(q) $k = - \frac { 6 } { 5 }$
(r) $k = \frac { 5 } { 6 }$
(s) $k = 5$

Q21 Matrices Matrix Algebra and Product Properties View

Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.
Column I
(A) The minimum value of $\frac { x ^ { 2 } + 2 x + 4 } { x + 2 }$ is
(B) Let $A$ and $B$ be $3 \times 3$ matrices of real numbers, where $A$ is symmetric, $B$ is skew-symmetric, and $( A + B ) ( A - B ) = ( A - B ) ( A + B )$. If $( A B ) ^ { t } = ( - 1 ) ^ { k } A B$, where $( A B ) ^ { t }$ is the transpose of the matrix $A B$, then the possible values of $k$ are
(C) Let $a = \log _ { 3 } \log _ { 3 } 2$. An integer $k$ satisfying $1 < 2 ^ { \left( - k + 3 ^ { - a } \right) } < 2$, must be less than
(D) If $\sin \theta = \cos \varphi$, then the possible values of $\frac { 1 } { \pi } \left( \theta \pm \varphi - \frac { \pi } { 2 } \right)$ are
Column II
(p) 0
(q) 1
(r) 2
(s) 3

Q22 Permutations & Arrangements Word Permutations with Repeated Letters View

Consider all possible permutations of the letters of the word ENDEANOEL.
Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.
Column I
(A) The number of permutations containing the word ENDEA is
(B) The number of permutations in which the letter E occurs in the first and the last positions is
(C) The number of permutations in which none of the letters D, L, N occurs in the last five positions is
(D) The number of permutations in which the letters A, E, O occur only in odd positions is
Column II
(p) $5 !$
(q) $2 \times 5 !$
(r) $7 \times 5 !$
(s) $21 \times 5 !$

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