jee-advanced

Q20 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View

For $\mathrm { r } = 0,1 , \ldots , 10$, let $\mathrm { A } _ { \mathrm { r } } , \mathrm { B } _ { \mathrm { r } }$ and $\mathrm { C } _ { \mathrm { r } }$ denote, respectively, the coefficient of $\mathrm { x } ^ { \mathrm { r } }$ in the expansions of $( 1 + \mathrm { x } ) ^ { 10 } , ( 1 + \mathrm { x } ) ^ { 20 }$ and $( 1 + \mathrm { x } ) ^ { 30 }$. Then
$$\sum _ { r = 1 } ^ { 10 } A _ { r } \left( B _ { 10 } B _ { r } - C _ { 10 } A _ { r } \right)$$
is equal to
A) $\mathrm { B } _ { 10 } - \mathrm { C } _ { 10 }$
B) $\mathrm { A } _ { 10 } \left( \mathrm {~B} _ { 10 } ^ { 2 } - \mathrm { C } _ { 10 } \mathrm {~A} _ { 10 } \right)$
C) O
D) $\mathrm { C } _ { 10 } - \mathrm { B } _ { 10 }$

Q21 Combinations & Selection Subset Counting with Set-Theoretic Conditions View

Let $S = \{ 1,2,3,4 \}$. The total number of unordered pairs of disjoint subsets of $S$ is equal to
A) 25
B) 34
C) 42
D) 41

Q22 Composite & Inverse Functions Derivative of an Inverse Function View

Let $f$ be a real-valued function defined on the interval $( - 1,1 )$ such that $e ^ { - x } f ( x ) = 2 + \int _ { 0 } ^ { x } \sqrt { t ^ { 4 } + 1 } d t$, for all $x \in ( - 1,1 )$, and let $f ^ { - 1 }$ be the inverse function of $f$. Then $\left( f ^ { - 1 } \right) ^ { \prime } ( 2 )$ is equal to
A) 1
B) $\frac { 1 } { 3 }$
C) $\frac { 1 } { 2 }$
D) $\frac { 1 } { e }$

Q23 Vectors: Lines & Planes Perpendicular/Orthogonal Projection onto a Plane View

If the distance of the point $\mathrm { P } ( 1 , - 2,1 )$ from the plane $\mathrm { x } + 2 \mathrm { y } - 2 z = \alpha$, where $\alpha > 0$, is 5 , then the foot of the perpendicular from $P$ to the plane is
A) $\left( \frac { 8 } { 3 } , \frac { 4 } { 3 } , - \frac { 7 } { 3 } \right)$
B) $\left( \frac { 4 } { 3 } , - \frac { 4 } { 3 } , \frac { 1 } { 3 } \right)$
C) $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } , \frac { 10 } { 3 } \right)$
D) $\left( \frac { 2 } { 3 } , - \frac { 1 } { 3 } , \frac { 5 } { 2 } \right)$

Q24 Vectors 3D & Lines Vector Algebra and Triple Product Computation View

Two adjacent sides of a parallelogram ABCD are given by $\overrightarrow { \mathrm { AB } } = 2 \hat { \mathrm { i } } + 10 \hat { \mathrm { j } } + 11 \hat { \mathrm { k } }$ and $\overrightarrow { \mathrm { AD } } = - \hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + 2 \hat { \mathrm { k } }$
The side AD is rotated by an acute angle $\alpha$ in the plane of the parallelogram so that AD becomes $\mathrm { AD } ^ { \prime }$. If $\mathrm { AD } ^ { \prime }$ makes a right angle with the side AB , then the cosine of the angle $\alpha$ is given by
A) $\frac { 8 } { 9 }$
B) $\frac { \sqrt { 17 } } { 9 }$
C) $\frac { 1 } { 9 }$
D) $\frac { 4 \sqrt { 5 } } { 9 }$

Q25 Conditional Probability Sequential/Multi-Stage Conditional Probability View

A signal which can be green or red with probability $\frac { 4 } { 5 }$ and $\frac { 1 } { 5 }$ respectively, is received by station A and then transmitted to station B. The probability of each station receiving the signal correctly is $\frac { 3 } { 4 }$. If the signal received at station $B$ is green, then the probability that the original signal was green is
A) $\frac { 3 } { 5 }$
B) $\frac { 6 } { 7 }$
C) $\frac { 20 } { 23 }$
D) $\frac { 9 } { 20 }$

Q26 Radians, Arc Length and Sector Area View

Two parallel chords of a circle of radius 2 are at a distance $\sqrt { 3 } + 1$ apart. If the chords subtend at the center, angles of $\frac { \pi } { k }$ and $\frac { 2 \pi } { k }$, where $k > 0$, then the value of $[ k ]$ is [Note : [k] denotes the largest integer less than or equal to k]

Q27 Sine and Cosine Rules Circumradius or incircle radius computation View

Consider a triangle ABC and let $\mathrm { a } , \mathrm { b }$ and c denote the lengths of the sides opposite to vertices $\mathrm { A } , \mathrm { B }$ and C respectively. Suppose $\mathrm { a } = 6 , \mathrm {~b} = 10$ and the area of the triangle is $15 \sqrt { 3 }$. If $\angle \mathrm { ACB }$ is obtuse and if r denotes the radius of the incircle of the triangle, then $r ^ { 2 }$ is equal to

Q28 Stationary points and optimisation Find critical points and classify extrema of a given function View

Let f be a function defined on $\mathbf { R }$ (the set of all real numbers) such that $\mathrm { f } ^ { \prime } ( \mathrm { x } ) = 2010 ( \mathrm { x } - 2009 ) ( \mathrm { x } - 2010 ) ^ { 2 } ( \mathrm { x } - 2011 ) ^ { 3 } ( \mathrm { x } - 2012 ) ^ { 4 }$, for all $\mathrm { x } \in \mathbf { R }$.
If $g$ is a function defined on $\mathbf { R }$ with values in the interval $( 0 , \infty )$ such that
$$\mathrm { f } ( \mathrm { x } ) = \ell n ( \mathrm {~g} ( \mathrm { x } ) ) \text {, for all } \mathrm { x } \in \mathbf { R } \text {, }$$
then the number of points in $\mathbf { R }$ at which $g$ has a local maximum is

Q29 Arithmetic Sequences and Series Summation of Derived Sequence from AP View

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots , a _ { 11 }$ be real numbers satisfying $\mathrm { a } _ { 1 } = 15 , \quad 27 - 2 \mathrm { a } _ { 2 } > 0$ and $\mathrm { a } _ { \mathrm { k } } = 2 \mathrm { a } _ { \mathrm { k } - 1 } - \mathrm { a } _ { \mathrm { k } - 2 }$ for $\mathrm { k } = 3,4 , \ldots , 11$.
If $\frac { a _ { 1 } ^ { 2 } + a _ { 2 } ^ { 2 } + \ldots + a _ { 11 } ^ { 2 } } { 11 } = 90$, then the value of $\frac { a _ { 1 } + a _ { 2 } + \ldots + a _ { 11 } } { 11 }$ is equal to

Q30 3x3 Matrices Determinant of Parametric or Structured Matrix View

Let k be a positive real number and let
$$A = \left[ \begin{array} { c c c } 2 k - 1 & 2 \sqrt { k } & 2 \sqrt { k } \\ 2 \sqrt { k } & 1 & - 2 k \\ - 2 \sqrt { k } & 2 k & - 1 \end{array} \right] \text { and } B = \left[ \begin{array} { c c c } 0 & 2 k - 1 & \sqrt { k } \\ 1 - 2 k & 0 & 2 \sqrt { k } \\ - \sqrt { k } & - 2 \sqrt { k } & 0 \end{array} \right]$$
If $\operatorname { det } ( \operatorname { adj } \mathrm { A } ) + \operatorname { det } ( \operatorname { adj } \mathrm { B } ) = 10 ^ { 6 }$, then $[ \mathrm { k } ]$ is equal to [Note : adj M denotes the adjoint of a square matrix M and $[ \mathrm { k } ]$ denotes the largest integer less than or equal to k].

Q31 Curve Sketching Number of Solutions / Roots via Curve Analysis View

Consider the polynomial
$$f ( x ) = 1 + 2 x + 3 x ^ { 2 } + 4 x ^ { 3 }$$
Let s be the sum of all distinct real roots of $\mathrm { f } ( \mathrm { x } )$ and let $\mathrm { t } = | \mathrm { s } |$.
The real number $s$ lies in the interval
A) $\left( - \frac { 1 } { 4 } , 0 \right)$
B) $\left( - 11 , - \frac { 3 } { 4 } \right)$
C) $\left( - \frac { 3 } { 4 } , - \frac { 1 } { 2 } \right)$
D) $\left( 0 , \frac { 1 } { 4 } \right)$

Q32 Areas Between Curves Compute Area Directly (Numerical Answer) View

Consider the polynomial
$$f ( x ) = 1 + 2 x + 3 x ^ { 2 } + 4 x ^ { 3 }$$
Let s be the sum of all distinct real roots of $\mathrm { f } ( \mathrm { x } )$ and let $\mathrm { t } = | \mathrm { s } |$.
The area bounded by the curve $y = f ( x )$ and the lines $x = 0 , y = 0$ and $x = t$, lies in the interval
A) $\left( \frac { 3 } { 4 } , 3 \right)$
B) $\left( \frac { 21 } { 64 } , \frac { 11 } { 16 } \right)$
C) $( 9,10 )$
D) $\left( 0 , \frac { 21 } { 64 } \right)$

Q33 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View

Consider the polynomial
$$f ( x ) = 1 + 2 x + 3 x ^ { 2 } + 4 x ^ { 3 }$$
Let s be the sum of all distinct real roots of $\mathrm { f } ( \mathrm { x } )$ and let $\mathrm { t } = | \mathrm { s } |$.
The function $f ^ { \prime } ( x )$ is
A) increasing in $\left( - t , - \frac { 1 } { 4 } \right)$ and decreasing in $\left( - \frac { 1 } { 4 } , t \right)$
B) decreasing in $\left( - t , - \frac { 1 } { 4 } \right)$ and increasing in $\left( - \frac { 1 } { 4 } , t \right)$
C) increasing in (-t, t)
D) decreasing in (-t, t)

Q34 Conic sections Tangent and Normal Line Problems View

Tangents are drawn from the point $P ( 3,4 )$ to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ touching the ellipse at points A and B.
The coordinates of $A$ and $B$ are
A) $( 3,0 )$ and $( 0,2 )$
B) $\left( - \frac { 8 } { 5 } , \frac { 2 \sqrt { 161 } } { 15 } \right)$ and $\left( - \frac { 9 } { 5 } , \frac { 8 } { 5 } \right)$
C) $\left( - \frac { 8 } { 5 } , \frac { 2 \sqrt { 161 } } { 15 } \right)$ and $( 0,2 )$
D) $(3, 0)$ and $\left( - \frac { 9 } { 5 } , \frac { 8 } { 5 } \right)$

Q35 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View

Tangents are drawn from the point $P ( 3,4 )$ to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ touching the ellipse at points A and B.
The orthocenter of the triangle $P A B$ is
A) $\left( 5 , \frac { 8 } { 7 } \right)$
B) $\left( \frac { 7 } { 5 } , \frac { 25 } { 8 } \right)$
C) $\left( \frac { 11 } { 5 } , \frac { 8 } { 5 } \right)$
D) $\left( \frac { 8 } { 25 } , \frac { 7 } { 5 } \right)$

Q36 Circles Circle-Related Locus Problems View

Tangents are drawn from the point $P ( 3,4 )$ to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ touching the ellipse at points A and B.
The equation of the locus of the point whose distances from the point $P$ and the line AB are equal, is
A) $9 x ^ { 2 } + y ^ { 2 } - 6 x y - 54 x - 62 y + 241 = 0$
B) $x ^ { 2 } + 9 y ^ { 2 } + 6 x y - 54 x + 62 y - 241 = 0$
C) $9 x ^ { 2 } + 9 y ^ { 2 } - 6 x y - 54 x - 62 y - 241 = 0$
D) $x ^ { 2 } + y ^ { 2 } - 2 x y + 27 x + 31 y - 120 = 0$

Q37 Complex Numbers Argand & Loci Locus Identification from Modulus/Argument Equation View

Match the statements in Column-I with those in Column-II. [Note: Here $z$ takes values in the complex plane and $\operatorname { Im } z$ and $\operatorname { Re } z$ denote, respectively, the imaginary part and the real part of $z$.]
Column I
A) The set of points $z$ satisfying $| z - i | z \| = | z + i | z \mid$ is contained in or equal to
B) The set of points $z$ satisfying $| z + 4 | + | z - 4 | = 10$ is contained in or equal to
C) If $| w | = 2$, then the set of points $z = w - \frac { 1 } { w }$ is contained in or equal to
D) If $| w | = 1$, then the set of points $z = w + \frac { 1 } { w }$ is contained in or equal to
Column II p) an ellipse with eccentricity $\frac { 4 } { 5 }$ q) the set of points $z$ satisfying $\operatorname { Im } z = 0$ r) the set of points $z$ satisfying $| \operatorname { Im } z | \leq 1$ s) the set of points $z$ satisfying $| \operatorname { Re } z | \leq 2$ t) the set of points $z$ satisfying $| z | \leq 3$

Q38 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View

Match the statements in Column-I with the values in Column-II.
Column I
A) A line from the origin meets the lines $\frac { x - 2 } { 1 } = \frac { y - 1 } { - 2 } = \frac { z + 1 } { 1 }$ and $\frac { x - \frac { 8 } { 3 } } { 2 } = \frac { y + 3 } { - 1 } = \frac { z - 1 } { 1 }$ at $P$ and $Q$ respectively. If length $\mathrm { PQ } = d$, then $d ^ { 2 }$ is
B) The values of $x$ satisfying $\tan ^ { - 1 } ( x + 3 ) - \tan ^ { - 1 } ( x - 3 ) = \sin ^ { - 1 } \left( \frac { 3 } { 5 } \right)$ are
C) Non-zero vectors $\vec { a } , \vec { b }$ and $\vec { c }$ satisfy $\vec { a } \cdot \vec { b } = 0$, $( \overrightarrow { \mathrm { b } } - \overrightarrow { \mathrm { a } } ) \cdot ( \overrightarrow { \mathrm { b } } + \overrightarrow { \mathrm { c } } ) = 0$ and $2 | \overrightarrow { \mathrm {~b} } + \overrightarrow { \mathrm { c } } | = | \overrightarrow { \mathrm { b } } - \overrightarrow { \mathrm { a } } |$. If $\vec { a } = \mu \vec { b } + 4 \vec { c }$, then the possible values of $\mu$ are
D) Let f be the function on $[ - \pi , \pi ]$ given by $f ( 0 ) = 9$ and $f ( x ) = \sin \left( \frac { 9 x } { 2 } \right) / \sin \left( \frac { x } { 2 } \right)$ for $x \neq 0$. The value of $\frac { 2 } { \pi } \int _ { - \pi } ^ { \pi } f ( x ) d x$ is
Column II p) $-4$ q) $0$ r) $4$ s) $-1$ (or as given in paper) t) $6$

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