jee-advanced

Q1 Complex Numbers Argand & Loci True/False or Multiple-Statement Verification View

For a non-zero complex number $z$, let $\arg ( z )$ denote the principal argument with $- \pi < \arg ( z ) \leq \pi$. Then, which of the following statement(s) is (are) FALSE?
(A) $\arg ( - 1 - i ) = \frac { \pi } { 4 }$, where $i = \sqrt { - 1 }$
(B) The function $f : \mathbb { R } \rightarrow ( - \pi , \pi ]$, defined by $f ( t ) = \arg ( - 1 + i t )$ for all $t \in \mathbb { R }$, is continuous at all points of $\mathbb { R }$, where $i = \sqrt { - 1 }$
(C) For any two non-zero complex numbers $z _ { 1 }$ and $z _ { 2 }$, $$\arg \left( \frac { z _ { 1 } } { z _ { 2 } } \right) - \arg \left( z _ { 1 } \right) + \arg \left( z _ { 2 } \right)$$ is an integer multiple of $2 \pi$
(D) For any three given distinct complex numbers $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$, the locus of the point $z$ satisfying the condition $$\arg \left( \frac { \left( z - z _ { 1 } \right) \left( z _ { 2 } - z _ { 3 } \right) } { \left( z - z _ { 3 } \right) \left( z _ { 2 } - z _ { 1 } \right) } \right) = \pi$$ lies on a straight line

Q2 Sine and Cosine Rules Multi-step composite figure problem View

In a triangle $P Q R$, let $\angle P Q R = 30 ^ { \circ }$ and the sides $P Q$ and $Q R$ have lengths $10 \sqrt { 3 }$ and 10, respectively. Then, which of the following statement(s) is (are) TRUE?
(A) $\angle Q P R = 45 ^ { \circ }$
(B) The area of the triangle $P Q R$ is $25 \sqrt { 3 }$ and $\angle Q R P = 120 ^ { \circ }$
(C) The radius of the incircle of the triangle $P Q R$ is $10 \sqrt { 3 } - 15$
(D) The area of the circumcircle of the triangle $P Q R$ is $100 \pi$

Q3 Vectors 3D & Lines Multi-Part 3D Geometry Problem View

Let $P _ { 1 } : 2 x + y - z = 3$ and $P _ { 2 } : x + 2 y + z = 2$ be two planes. Then, which of the following statement(s) is (are) TRUE?
(A) The line of intersection of $P _ { 1 }$ and $P _ { 2 }$ has direction ratios $1,2 , - 1$
(B) The line $$\frac { 3 x - 4 } { 9 } = \frac { 1 - 3 y } { 9 } = \frac { z } { 3 }$$ is perpendicular to the line of intersection of $P _ { 1 }$ and $P _ { 2 }$
(C) The acute angle between $P _ { 1 }$ and $P _ { 2 }$ is $60 ^ { \circ }$
(D) If $P _ { 3 }$ is the plane passing through the point $( 4,2 , - 2 )$ and perpendicular to the line of intersection of $P _ { 1 }$ and $P _ { 2 }$, then the distance of the point $( 2,1,1 )$ from the plane $P _ { 3 }$ is $\frac { 2 } { \sqrt { 3 } }$

Q4 Stationary points and optimisation Existence or properties of extrema via abstract/theoretical argument View

For every twice differentiable function $f : \mathbb { R } \rightarrow [ - 2,2 ]$ with $( f ( 0 ) ) ^ { 2 } + \left( f ^ { \prime } ( 0 ) \right) ^ { 2 } = 85$, which of the following statement(s) is (are) TRUE?
(A) There exist $r , s \in \mathbb { R }$, where $r < s$, such that $f$ is one-one on the open interval ( $r , s$ )
(B) There exists $x _ { 0 } \in ( - 4,0 )$ such that $\left| f ^ { \prime } \left( x _ { 0 } \right) \right| \leq 1$
(C) $\lim _ { x \rightarrow \infty } f ( x ) = 1$
(D) There exists $\alpha \in ( - 4,4 )$ such that $f ( \alpha ) + f ^ { \prime \prime } ( \alpha ) = 0$ and $f ^ { \prime } ( \alpha ) \neq 0$

Q5 Exponential Functions True/False or Multiple-Statement Verification View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be two non-constant differentiable functions. If $$f ^ { \prime } ( x ) = \left( e ^ { ( f ( x ) - g ( x ) ) } \right) g ^ { \prime } ( x ) \text { for all } x \in \mathbb { R }$$ and $f ( 1 ) = g ( 2 ) = 1$, then which of the following statement(s) is (are) TRUE?
(A) $f ( 2 ) < 1 - \log _ { \mathrm { e } } 2$
(B) $f ( 2 ) > 1 - \log _ { \mathrm { e } } 2$
(C) $g ( 1 ) > 1 - \log _ { \mathrm { e } } 2$
(D) $g ( 1 ) < 1 - \log _ { e } 2$

Q6 Differential equations Integral Equations Reducible to DEs View

Let $f : [ 0 , \infty ) \rightarrow \mathbb { R }$ be a continuous function such that $$f ( x ) = 1 - 2 x + \int _ { 0 } ^ { x } e ^ { x - t } f ( t ) d t$$ for all $x \in [ 0 , \infty )$. Then, which of the following statement(s) is (are) TRUE?
(A) The curve $y = f ( x )$ passes through the point $( 1,2 )$
(B) The curve $y = f ( x )$ passes through the point $( 2 , - 1 )$
(C) The area of the region $\left\{ ( x , y ) \in [ 0,1 ] \times \mathbb { R } : f ( x ) \leq y \leq \sqrt { 1 - x ^ { 2 } } \right\}$ is $\frac { \pi - 2 } { 4 }$
(D) The area of the region $\left\{ ( x , y ) \in [ 0,1 ] \times \mathbb { R } : f ( x ) \leq y \leq \sqrt { 1 - x ^ { 2 } } \right\}$ is $\frac { \pi - 1 } { 4 }$

Q7 Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View

The value of $$\left( \left( \log _ { 2 } 9 \right) ^ { 2 } \right) ^ { \frac { 1 } { \log _ { 2 } \left( \log _ { 2 } 9 \right) } } \times ( \sqrt { 7 } ) ^ { \frac { 1 } { \log _ { 4 } 7 } }$$ is $\_\_\_\_$.

Q8 Solving quadratics and applications Counting solutions or configurations satisfying a quadratic system View

The number of 5 digit numbers which are divisible by 4, with digits from the set $\{ 1,2,3,4,5 \}$ and the repetition of digits is allowed, is $\_\_\_\_$.

Q9 Arithmetic Sequences and Series Counting or Combinatorial Problems on APs View

Let $X$ be the set consisting of the first 2018 terms of the arithmetic progression $1,6,11 , \ldots$, and $Y$ be the set consisting of the first 2018 terms of the arithmetic progression $9,16,23 , \ldots$. Then, the number of elements in the set $X \cup Y$ is $\_\_\_\_$.

Q10 Geometric Sequences and Series Geometric Series with Trigonometric or Functional Terms View

The number of real solutions of the equation $$\sin ^ { - 1 } \left( \sum _ { i = 1 } ^ { \infty } x ^ { i + 1 } - x \sum _ { i = 1 } ^ { \infty } \left( \frac { x } { 2 } \right) ^ { i } \right) = \frac { \pi } { 2 } - \cos ^ { - 1 } \left( \sum _ { i = 1 } ^ { \infty } \left( - \frac { x } { 2 } \right) ^ { i } - \sum _ { i = 1 } ^ { \infty } ( - x ) ^ { i } \right)$$ lying in the interval $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$ is $\_\_\_\_$. (Here, the inverse trigonometric functions $\sin ^ { - 1 } x$ and $\cos ^ { - 1 } x$ assume values in $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ and $[ 0 , \pi ]$, respectively.)

Q11 Geometric Sequences and Series Determine the Limit of a Sequence via Geometric Series View

For each positive integer $n$, let $$y _ { n } = \frac { 1 } { n } ( ( n + 1 ) ( n + 2 ) \cdots ( n + n ) ) ^ { \frac { 1 } { n } }$$ For $x \in \mathbb { R }$, let $[ x ]$ be the greatest integer less than or equal to $x$. If $\lim _ { n \rightarrow \infty } y _ { n } = L$, then the value of $[ L ]$ is $\_\_\_\_$.

Q12 Vectors Introduction & 2D Angle or Cosine Between Vectors View

Let $\vec { a }$ and $\vec { b }$ be two unit vectors such that $\vec { a } \cdot \vec { b } = 0$. For some $x , y \in \mathbb { R }$, let $\vec { c } = x \vec { a } + y \vec { b } + ( \vec { a } \times \vec { b } )$. If $| \vec { c } | = 2$ and the vector $\vec { c }$ is inclined at the same angle $\alpha$ to both $\vec { a }$ and $\vec { b }$, then the value of $8 \cos ^ { 2 } \alpha$ is $\_\_\_\_$.

Q13 Addition & Double Angle Formulae Trigonometric Equation Solving via Identities View

Let $a , b , c$ be three non-zero real numbers such that the equation $$\sqrt { 3 } a \cos x + 2 b \sin x = c , x \in \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$$ has two distinct real roots $\alpha$ and $\beta$ with $\alpha + \beta = \frac { \pi } { 3 }$. Then, the value of $\frac { b } { a }$ is $\_\_\_\_$.

Q14 Areas by integration View

A farmer $F _ { 1 }$ has a land in the shape of a triangle with vertices at $P ( 0,0 ) , Q ( 1,1 )$ and $R ( 2,0 )$. From this land, a neighbouring farmer $F _ { 2 }$ takes away the region which lies between the side $P Q$ and a curve of the form $y = x ^ { n } ( n > 1 )$. If the area of the region taken away by the farmer $F _ { 2 }$ is exactly $30 \%$ of the area of $\triangle P Q R$, then the value of $n$ is $\_\_\_\_$.

Q15 Circles Circle-Related Locus Problems View

Let $S$ be the circle in the $x y$-plane defined by the equation $x ^ { 2 } + y ^ { 2 } = 4$. Let $E _ { 1 } E _ { 2 }$ and $F _ { 1 } F _ { 2 }$ be the chords of $S$ passing through the point $P _ { 0 } ( 1,1 )$ and parallel to the $x$-axis and the $y$-axis, respectively. Let $G _ { 1 } G _ { 2 }$ be the chord of $S$ passing through $P _ { 0 }$ and having slope $- 1$. Let the tangents to $S$ at $E _ { 1 }$ and $E _ { 2 }$ meet at $E _ { 3 }$, the tangents to $S$ at $F _ { 1 }$ and $F _ { 2 }$ meet at $F _ { 3 }$, and the tangents to $S$ at $G _ { 1 }$ and $G _ { 2 }$ meet at $G _ { 3 }$. Then, the points $E _ { 3 } , F _ { 3 }$, and $G _ { 3 }$ lie on the curve
(A) $x + y = 4$
(B) $( x - 4 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 16$
(C) $( x - 4 ) ( y - 4 ) = 4$
(D) $x y = 4$

Q16 Circles Circle-Related Locus Problems View

Let $S$ be the circle in the $x y$-plane defined by the equation $x ^ { 2 } + y ^ { 2 } = 4$. Let $P$ be a point on the circle $S$ with both coordinates being positive. Let the tangent to $S$ at $P$ intersect the coordinate axes at the points $M$ and $N$. Then, the mid-point of the line segment $M N$ must lie on the curve
(A) $( x + y ) ^ { 2 } = 3 x y$
(B) $x ^ { 2 / 3 } + y ^ { 2 / 3 } = 2 ^ { 4 / 3 }$
(C) $x ^ { 2 } + y ^ { 2 } = 2 x y$
(D) $x ^ { 2 } + y ^ { 2 } = x ^ { 2 } y ^ { 2 }$

Q17 Permutations & Arrangements Probability via Permutation Counting View

There are five students $S _ { 1 } , S _ { 2 } , S _ { 3 } , S _ { 4 }$ and $S _ { 5 }$ in a music class and for them there are five seats $R _ { 1 } , R _ { 2 } , R _ { 3 } , R _ { 4 }$ and $R _ { 5 }$ arranged in a row, where initially the seat $R _ { i }$ is allotted to the student $S _ { i } , i = 1,2,3,4,5$. But, on the examination day, the five students are randomly allotted the five seats. The probability that, on the examination day, the student $S _ { 1 }$ gets the previously allotted seat $R _ { 1 }$, and NONE of the remaining students gets the seat previously allotted to him/her is
(A) $\frac { 3 } { 40 }$
(B) $\frac { 1 } { 8 }$
(C) $\frac { 7 } { 40 }$
(D) $\frac { 1 } { 5 }$

Q18 Permutations & Arrangements Probability via Permutation Counting View

There are five students $S _ { 1 } , S _ { 2 } , S _ { 3 } , S _ { 4 }$ and $S _ { 5 }$ in a music class and for them there are five seats $R _ { 1 } , R _ { 2 } , R _ { 3 } , R _ { 4 }$ and $R _ { 5 }$ arranged in a row, where initially the seat $R _ { i }$ is allotted to the student $S _ { i } , i = 1,2,3,4,5$. But, on the examination day, the five students are randomly allotted the five seats. For $i = 1,2,3,4$, let $T _ { i }$ denote the event that the students $S _ { i }$ and $S _ { i + 1 }$ do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $T _ { 1 } \cap T _ { 2 } \cap T _ { 3 } \cap T _ { 4 }$ is
(A) $\frac { 1 } { 15 }$
(B) $\frac { 1 } { 10 }$
(C) $\frac { 7 } { 60 }$
(D) $\frac { 1 } { 5 }$

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