jee-advanced

Q1 3 marks Reciprocal Trig & Identities View

Considering only the principal values of the inverse trigonometric functions, the value of
$$\frac { 3 } { 2 } \cos ^ { - 1 } \sqrt { \frac { 2 } { 2 + \pi ^ { 2 } } } + \frac { 1 } { 4 } \sin ^ { - 1 } \frac { 2 \sqrt { 2 } \pi } { 2 + \pi ^ { 2 } } + \tan ^ { - 1 } \frac { \sqrt { 2 } } { \pi }$$
is $\_\_\_\_$.

Q2 3 marks Differentiating Transcendental Functions Limit involving transcendental functions View

Let $\alpha$ be a positive real number. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : ( \alpha , \infty ) \rightarrow \mathbb { R }$ be the functions defined by
$$f ( x ) = \sin \left( \frac { \pi x } { 12 } \right) \quad \text { and } \quad g ( x ) = \frac { 2 \log _ { \mathrm { e } } ( \sqrt { x } - \sqrt { \alpha } ) } { \log _ { \mathrm { e } } \left( e ^ { \sqrt { x } } - e ^ { \sqrt { \alpha } } \right) }$$
Then the value of $\lim _ { x \rightarrow \alpha ^ { + } } f ( g ( x ) )$ is $\_\_\_\_$.

Q3 3 marks Principle of Inclusion/Exclusion View

In a study about a pandemic, data of 900 persons was collected. It was found that
190 persons had symptom of fever, 220 persons had symptom of cough, 220 persons had symptom of breathing problem, 330 persons had symptom of fever or cough or both, 350 persons had symptom of cough or breathing problem or both, 340 persons had symptom of fever or breathing problem or both, 30 persons had all three symptoms (fever, cough and breathing problem). If a person is chosen randomly from these 900 persons, then the probability that the person has at most one symptom is $\_\_\_\_$.

Q4 3 marks Complex Numbers Argand & Loci Algebraic Conditions for Geometric Properties (Real, Imaginary, Collinear) View

Let $z$ be a complex number with non-zero imaginary part. If
$$\frac { 2 + 3 z + 4 z ^ { 2 } } { 2 - 3 z + 4 z ^ { 2 } }$$
is a real number, then the value of $| z | ^ { 2 }$ is $\_\_\_\_$.

Q5 3 marks Complex Numbers Argand & Loci Solving Complex Equations with Geometric Interpretation View

Let $\bar { z }$ denote the complex conjugate of a complex number $z$ and let $i = \sqrt { - 1 }$. In the set of complex numbers, the number of distinct roots of the equation
$$\bar { z } - z ^ { 2 } = i \left( \bar { z } + z ^ { 2 } \right)$$
is $\_\_\_\_$.

Q6 3 marks Arithmetic Sequences and Series Summation of Derived Sequence from AP View

Let $l _ { 1 } , l _ { 2 } , \ldots , l _ { 100 }$ be consecutive terms of an arithmetic progression with common difference $d _ { 1 }$, and let $w _ { 1 } , w _ { 2 } , \ldots , w _ { 100 }$ be consecutive terms of another arithmetic progression with common difference $d _ { 2 }$, where $d _ { 1 } d _ { 2 } = 10$. For each $i = 1,2 , \ldots , 100$, let $R _ { i }$ be a rectangle with length $l _ { i }$, width $w _ { i }$ and area $A _ { i }$. If $A _ { 51 } - A _ { 50 } = 1000$, then the value of $A _ { 100 } - A _ { 90 }$ is $\_\_\_\_$.

Q7 3 marks Permutations & Arrangements Forming Numbers with Digit Constraints View

The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits $0,2,3,4,6,7$ is $\_\_\_\_$.

Q8 3 marks Circles Circles Tangent to Each Other or to Axes View

Let $A B C$ be the triangle with $A B = 1 , A C = 3$ and $\angle B A C = \frac { \pi } { 2 }$. If a circle of radius $r > 0$ touches the sides $A B , A C$ and also touches internally the circumcircle of the triangle $A B C$, then the value of $r$ is $\_\_\_\_$.

Q9 4 marks Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View

Consider the equation
$$\int _ { 1 } ^ { e } \frac { \left( \log _ { \mathrm { e } } x \right) ^ { 1 / 2 } } { x \left( a - \left( \log _ { \mathrm { e } } x \right) ^ { 3 / 2 } \right) ^ { 2 } } d x = 1 , \quad a \in ( - \infty , 0 ) \cup ( 1 , \infty )$$
Which of the following statements is/are TRUE ?
(A) No $a$ satisfies the above equation
(B) An integer $a$ satisfies the above equation
(C) An irrational number $a$ satisfies the above equation
(D) More than one $a$ satisfy the above equation

Q10 4 marks Arithmetic Sequences and Series Sequence Defined by Recurrence with AP Connection View

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be an arithmetic progression with $a _ { 1 } = 7$ and common difference 8. Let $T _ { 1 } , T _ { 2 } , T _ { 3 } , \ldots$ be such that $T _ { 1 } = 3$ and $T _ { n + 1 } - T _ { n } = a _ { n }$ for $n \geq 1$. Then, which of the following is/are TRUE ?
(A) $T _ { 20 } = 1604$
(B) $\sum _ { k = 1 } ^ { 20 } T _ { k } = 10510$
(C) $T _ { 30 } = 3454$
(D) $\sum _ { k = 1 } ^ { 30 } T _ { k } = 35610$

Q11 4 marks Vectors: Lines & Planes Parallelism Between Line and Plane or Constraint on Parameters View

Let $P _ { 1 }$ and $P _ { 2 }$ be two planes given by
$$\begin{aligned} & P _ { 1 } : 10 x + 15 y + 12 z - 60 = 0 \\ & P _ { 2 } : \quad - 2 x + 5 y + 4 z - 20 = 0 \end{aligned}$$
Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $P _ { 1 }$ and $P _ { 2 }$ ?
(A) $\frac { x - 1 } { 0 } = \frac { y - 1 } { 0 } = \frac { z - 1 } { 5 }$
(B) $\frac { x - 6 } { - 5 } = \frac { y } { 2 } = \frac { z } { 3 }$
(C) $\frac { x } { - 2 } = \frac { y - 4 } { 5 } = \frac { z } { 4 }$
(D) $\frac { x } { 1 } = \frac { y - 4 } { - 2 } = \frac { z } { 3 }$

Q12 4 marks Vectors: Lines & Planes Perpendicular/Orthogonal Projection onto a Plane View

Let $S$ be the reflection of a point $Q$ with respect to the plane given by
$$\vec { r } = - ( t + p ) \hat { \imath } + t \hat { \jmath } + ( 1 + p ) \hat { k }$$
where $t , p$ are real parameters and $\hat { \imath } , \hat { \jmath } , \hat { k }$ are the unit vectors along the three positive coordinate axes. If the position vectors of $Q$ and $S$ are $10 \hat { \imath } + 15 \hat { \jmath } + 20 \hat { k }$ and $\alpha \hat { \imath } + \beta \hat { \jmath } + \gamma \hat { k }$ respectively, then which of the following is/are TRUE ?
(A) $3 ( \alpha + \beta ) = - 101$
(B) $3 ( \beta + \gamma ) = - 71$
(C) $3 ( \gamma + \alpha ) = - 86$
(D) $3 ( \alpha + \beta + \gamma ) = - 121$

Q13 4 marks Circles Tangent Lines and Tangent Lengths View

Consider the parabola $y ^ { 2 } = 4 x$. Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P = ( - 2,1 )$ meet the parabola at $P _ { 1 }$ and $P _ { 2 }$. Let $Q _ { 1 }$ and $Q _ { 2 }$ be points on the lines $S P _ { 1 }$ and $S P _ { 2 }$ respectively such that $P Q _ { 1 }$ is perpendicular to $S P _ { 1 }$ and $P Q _ { 2 }$ is perpendicular to $S P _ { 2 }$. Then, which of the following is/are TRUE?
(A) $\quad S Q _ { 1 } = 2$
(B) $\quad Q _ { 1 } Q _ { 2 } = \frac { 3 \sqrt { 10 } } { 5 }$
(C) $\quad P Q _ { 1 } = 3$
(D) $\quad S Q _ { 2 } = 1$

Q14 4 marks Curve Sketching Range and Image Set Determination View

Let $| M |$ denote the determinant of a square matrix $M$. Let $g : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow \mathbb { R }$ be the function defined by
$$g ( \theta ) = \sqrt { f ( \theta ) - 1 } + \sqrt { f \left( \frac { \pi } { 2 } - \theta \right) - 1 }$$
where $$f ( \theta ) = \frac { 1 } { 2 } \left| \begin{array} { c c c } 1 & \sin \theta & 1 \\ - \sin \theta & 1 & \sin \theta \\ - 1 & - \sin \theta & 1 \end{array} \right| + \left| \begin{array} { c c c } \sin \pi & \cos \left( \theta + \frac { \pi } { 4 } \right) & \tan \left( \theta - \frac { \pi } { 4 } \right) \\ \sin \left( \theta - \frac { \pi } { 4 } \right) & - \cos \frac { \pi } { 2 } & \log _ { e } \left( \frac { 4 } { \pi } \right) \\ \cot \left( \theta + \frac { \pi } { 4 } \right) & \log _ { e } \left( \frac { \pi } { 4 } \right) & \tan \pi \end{array} \right|$$
Let $p ( x )$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g ( \theta )$, and $p ( 2 ) = 2 - \sqrt { 2 }$. Then, which of the following is/are TRUE ?
(A) $p \left( \frac { 3 + \sqrt { 2 } } { 4 } \right) < 0$
(B) $p \left( \frac { 1 + 3 \sqrt { 2 } } { 4 } \right) > 0$
(C) $p \left( \frac { 5 \sqrt { 2 } - 1 } { 4 } \right) > 0$
(D) $\quad p \left( \frac { 5 - \sqrt { 2 } } { 4 } \right) < 0$

Q15 3 marks Standard trigonometric equations Matching trigonometric equations to solution set sizes View

Consider the following lists:
List-I (I) $\left\{ x \in \left[ - \frac { 2 \pi } { 3 } , \frac { 2 \pi } { 3 } \right] : \cos x + \sin x = 1 \right\}$ (II) $\left\{ x \in \left[ - \frac { 5 \pi } { 18 } , \frac { 5 \pi } { 18 } \right] : \sqrt { 3 } \tan 3 x = 1 \right\}$ (III) $\left\{ x \in \left[ - \frac { 6 \pi } { 5 } , \frac { 6 \pi } { 5 } \right] : 2 \cos ( 2 x ) = \sqrt { 3 } \right\}$ (IV) $\left\{ x \in \left[ - \frac { 7 \pi } { 4 } , \frac { 7 \pi } { 4 } \right] : \sin x - \cos x = 1 \right\}$
List-II (P) has two elements (Q) has three elements (R) has four elements (S) has five elements (T) has six elements
The correct option is:
(A) $( \mathrm { I } ) \rightarrow ( \mathrm { P } ) ; ( \mathrm { II } ) \rightarrow ( \mathrm { S } ) ; ( \mathrm { III } ) \rightarrow ( \mathrm { P } ) ; ( \mathrm { IV } ) \rightarrow ( \mathrm { S } )$
(B) (I) → (P); (II) → (P); (III) → (T); (IV) → (R)
(C) (I) → (Q); (II) → (P); (III) → (T); (IV) → (S)
(D) (I) → (Q); (II) → (S); (III) → (P); (IV) → (R)

Q16 3 marks Probability Definitions Finite Equally-Likely Probability Computation View

Two players, $P _ { 1 }$ and $P _ { 2 }$, play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let $x$ and $y$ denote the readings on the die rolled by $P _ { 1 }$ and $P _ { 2 }$, respectively. If $x > y$, then $P _ { 1 }$ scores 5 points and $P _ { 2 }$ scores 0 point. If $x = y$, then each player scores 2 points. If $x < y$, then $P _ { 1 }$ scores 0 point and $P _ { 2 }$ scores 5 points. Let $X _ { i }$ and $Y _ { i }$ be the total scores of $P _ { 1 }$ and $P _ { 2 }$, respectively, after playing the $i ^ { \text {th } }$ round.
List-I (I) Probability of $\left( X _ { 2 } \geq Y _ { 2 } \right)$ is (II) Probability of $\left( X _ { 2 } > Y _ { 2 } \right)$ is (III) Probability of $\left( X _ { 3 } = Y _ { 3 } \right)$ is (IV) Probability of $\left( X _ { 3 } > Y _ { 3 } \right)$ is
List-II (P) $\frac { 3 } { 8 }$ (Q) $\frac { 11 } { 16 }$ (R) $\frac { 5 } { 16 }$ (S) $\frac { 355 } { 864 }$ (T) $\frac { 77 } { 432 }$
The correct option is:
(A) (I) → (Q); (II) → (R); (III) → (T); (IV) → (S)
(B) (I) → (Q); (II) → (R); (III) → (T); (IV) → (T)
(C) (I) → (P); (II) → (R); (III) → (Q); (IV) → (S)
(D) (I) → (P); (II) → (R); (III) → (Q); (IV) → (T)

Q17 3 marks Simultaneous equations View

Let $p , q , r$ be nonzero real numbers that are, respectively, the $10 ^ { \text {th } } , 100 ^ { \text {th } }$ and $1000 ^ { \text {th } }$ terms of a harmonic progression. Consider the system of linear equations
$$\begin{gathered} x + y + z = 1 \\ 10 x + 100 y + 1000 z = 0 \\ q r x + p r y + p q z = 0 \end{gathered}$$
List-I (I) If $\frac { q } { r } = 10$, then the system of linear equations has (II) If $\frac { p } { r } \neq 100$, then the system of linear equations has (III) If $\frac { p } { q } \neq 10$, then the system of linear equations has (IV) If $\frac { p } { q } = 10$, then the system of linear equations has
List-II (P) $x = 0 , \quad y = \frac { 10 } { 9 } , z = - \frac { 1 } { 9 }$ as a solution (Q) $x = \frac { 10 } { 9 } , \quad y = - \frac { 1 } { 9 } , z = 0$ as a solution (R) infinitely many solutions (S) no solution (T) at least one solution
The correct option is:
(A) (I) → (T); (II) → (R); (III) → (S); (IV) → (T)
(B) (I) → (Q); (II) → (S); (III) → (S); (IV) → (R)
(C) (I) → (Q); (II) → (R); (III) → (P); (IV) → (R)
(D) (I) → (T); (II) → (S); (III) → (P); (IV) → (T)

Q18 3 marks Circles Area and Geometric Measurement Involving Circles View

Consider the ellipse
$$\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$$
Let $H ( \alpha , 0 ) , 0 < \alpha < 2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent to the ellipse at the point $E$ intersects the positive $x$-axis at a point $G$. Suppose the straight line joining $F$ and the origin makes an angle $\phi$ with the positive $x$-axis.
List-I (I) If $\phi = \frac { \pi } { 4 }$, then the area of the triangle $F G H$ is (II) If $\phi = \frac { \pi } { 3 }$, then the area of the triangle $F G H$ is (III) If $\phi = \frac { \pi } { 6 }$, then the area of the triangle $F G H$ is (IV) If $\phi = \frac { \pi } { 12 }$, then the area of the triangle $F G H$ is
List-II (P) $\frac { ( \sqrt { 3 } - 1 ) ^ { 4 } } { 8 }$ (Q) 1 (R) $\frac { 3 } { 4 }$ (S) $\frac { 1 } { 2 \sqrt { 3 } }$ (T) $\frac { 3 \sqrt { 3 } } { 2 }$
The correct option is:
(A) (I) → (R); (II) → (S); (III) → (Q); (IV) → (P)
(B) (I) → (R); (II) → (T); (III) → (S); (IV) → (P)
(C) (I) → (Q); (II) → (T); (III) → (S); (IV) → (P)
(D) (I) → (Q); (II) → (S); (III) → (Q); (IV) → (P)

jee-advanced

Papers (38)

2022 paper1