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Q41 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View

The number of distinct solutions of the equation $$\frac { 5 } { 4 } \cos ^ { 2 } 2 x + \cos ^ { 4 } x + \sin ^ { 4 } x + \cos ^ { 6 } x + \sin ^ { 6 } x = 2$$ in the interval $[ 0,2 \pi ]$ is

Q42 Circles Chord Length and Chord Properties View

Let the curve $C$ be the mirror image of the parabola $y ^ { 2 } = 4 x$ with respect to the line $x + y + 4 = 0$. If $A$ and $B$ are the points of intersection of $C$ with the line $y = - 5$, then the distance between $A$ and $B$ is

Q43 Binomial Distribution Find Minimum n for a Probability Threshold View

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96, is

Q44 Permutations & Arrangements Linear Arrangement with Constraints View

Let $n$ be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let $m$ be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of $\frac { m } { n }$ is

Q45 Circles Tangent Lines and Tangent Lengths View

If the normals of the parabola $y ^ { 2 } = 4 x$ drawn at the end points of its latus rectum are tangents to the circle $( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = r ^ { 2 }$, then the value of $r ^ { 2 }$ is

Q46 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function defined by $f ( x ) = \left\{ \begin{array} { l l } { [ x ] , } & x \leq 2 \\ 0 , & x > 2 \end{array} \right.$, where $[ x ]$ is the greatest integer less than or equal to $x$. If $I = \int _ { - 1 } ^ { 2 } \frac { x f \left( x ^ { 2 } \right) } { 2 + f ( x + 1 ) } d x$, then the value of $( 4 I - 1 )$ is

Q47 Applied differentiation Applied modeling with differentiation View

A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $V \mathrm {~mm} ^ { 3 }$, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2 mm and is of radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is 10 mm, then the value of $\frac { V } { 250 \pi }$ is

Q48 Indefinite & Definite Integrals Finding a Function from an Integral Equation View

Let $F ( x ) = \int _ { x } ^ { x ^ { 2 } + \frac { \pi } { 6 } } 2 \cos ^ { 2 } t \, d t$ for all $x \in \mathbb { R }$ and $f : \left[ 0 , \frac { 1 } { 2 } \right] \rightarrow [ 0 , \infty )$ be a continuous function. For $a \in \left[ 0 , \frac { 1 } { 2 } \right]$, if $F ^ { \prime } ( a ) + 2$ is the area of the region bounded by $x = 0 , y = 0 , y = f ( x )$ and $x = a$, then $f ( 0 )$ is

Q49 Matrices Matrix Algebra and Product Properties View

Let $X$ and $Y$ be two arbitrary, $3 \times 3$, non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$, non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?
(A) $\quad Y ^ { 3 } Z ^ { 4 } - Z ^ { 4 } Y ^ { 3 }$
(B) $X ^ { 44 } + Y ^ { 44 }$
(C) $X ^ { 4 } Z ^ { 3 } - Z ^ { 3 } X ^ { 4 }$
(D) $X ^ { 23 } + Y ^ { 23 }$

Q50 Matrices Determinant and Rank Computation View

Which of the following values of $\alpha$ satisfy the equation $$\left| \begin{array} { c c c } ( 1 + \alpha ) ^ { 2 } & ( 1 + 2 \alpha ) ^ { 2 } & ( 1 + 3 \alpha ) ^ { 2 } \\ ( 2 + \alpha ) ^ { 2 } & ( 2 + 2 \alpha ) ^ { 2 } & ( 2 + 3 \alpha ) ^ { 2 } \\ ( 3 + \alpha ) ^ { 2 } & ( 3 + 2 \alpha ) ^ { 2 } & ( 3 + 3 \alpha ) ^ { 2 } \end{array} \right| = - 648 \alpha$$?
(A) $-4$
(B) $9$
(C) $-9$
(D) $4$

Q51 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View

In $\mathbb { R } ^ { 3 }$, consider the planes $P _ { 1 } : y = 0$ and $P _ { 2 } : x + z = 1$. Let $P _ { 3 }$ be a plane, different from $P _ { 1 }$ and $P _ { 2 }$, which passes through the intersection of $P _ { 1 }$ and $P _ { 2 }$. If the distance of the point $( 0,1,0 )$ from $P _ { 3 }$ is 1 and the distance of a point $( \alpha , \beta , \gamma )$ from $P _ { 3 }$ is 2, then which of the following relations is (are) true?
(A) $2 \alpha + \beta + 2 \gamma + 2 = 0$
(B) $2 \alpha - \beta + 2 \gamma + 4 = 0$
(C) $2 \alpha + \beta - 2 \gamma - 10 = 0$
(D) $2 \alpha - \beta + 2 \gamma - 8 = 0$

Q52 Vectors: Lines & Planes Perpendicular/Orthogonal Projection onto a Plane View

In $\mathbb { R } ^ { 3 }$, let $L$ be a straight line passing through the origin. Suppose that all the points on $L$ are at a constant distance from the two planes $P _ { 1 } : x + 2 y - z + 1 = 0$ and $P _ { 2 } : 2 x - y + z - 1 = 0$. Let $M$ be the locus of the feet of the perpendiculars drawn from the points on $L$ to the plane $P _ { 1 }$. Which of the following points lie(s) on $M$?
(A) $\left( 0 , - \frac { 5 } { 6 } , - \frac { 2 } { 3 } \right)$
(B) $\left( - \frac { 1 } { 6 } , - \frac { 1 } { 3 } , \frac { 1 } { 6 } \right)$
(C) $\left( - \frac { 5 } { 6 } , 0 , \frac { 1 } { 6 } \right)$
(D) $\left( - \frac { 1 } { 3 } , 0 , \frac { 2 } { 3 } \right)$

Q53 Circles Circle Equation Derivation View

Let $P$ and $Q$ be distinct points on the parabola $y ^ { 2 } = 2 x$ such that a circle with $P Q$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $\triangle O P Q$ is $3 \sqrt { 2 }$, then which of the following is (are) the coordinates of $P$?
(A) $( 4,2 \sqrt { 2 } )$
(B) $( 9,3 \sqrt { 2 } )$
(C) $\left( \frac { 1 } { 4 } , \frac { 1 } { \sqrt { 2 } } \right)$
(D) $( 1 , \sqrt { 2 } )$

Q54 First order differential equations (integrating factor) View

Let $y ( x )$ be a solution of the differential equation $\left( 1 + e ^ { x } \right) y ^ { \prime } + y e ^ { x } = 1$. If $y ( 0 ) = 2$, then which of the following statements is (are) true?
(A) $\quad y ( - 4 ) = 0$
(B) $\quad y ( - 2 ) = 0$
(C) $\quad y ( x )$ has a critical point in the interval $( - 1,0 )$
(D) $y ( x )$ has no critical point in the interval $( - 1,0 )$

Q55 Differential equations Higher-Order and Special DEs (Proof/Theory) View

Consider the family of all circles whose centers lie on the straight line $y = x$. If this family of circles is represented by the differential equation $P y ^ { \prime \prime } + Q y ^ { \prime } + 1 = 0$, where $P , Q$ are functions of $x , y$ and $y ^ { \prime }$ (here $y ^ { \prime } = \frac { d y } { d x } , y ^ { \prime \prime } = \frac { d ^ { 2 } y } { d x ^ { 2 } }$), then which of the following statements is (are) true?
(A) $P = y + x$
(B) $P = y - x$
(C) $P + Q = 1 - x + y + y ^ { \prime } + \left( y ^ { \prime } \right) ^ { 2 }$
(D) $P - Q = x + y - y ^ { \prime } - \left( y ^ { \prime } \right) ^ { 2 }$

Q56 Differentiation from First Principles View

Let $g : \mathbb { R } \rightarrow \mathbb { R }$ be a differentiable function with $g ( 0 ) = 0 , g ^ { \prime } ( 0 ) = 0$ and $g ^ { \prime } ( 1 ) \neq 0$. Let $$f ( x ) = \left\{ \begin{array} { c c } \frac { x } { | x | } g ( x ) , & x \neq 0 \\ 0 , & x = 0 \end{array} \right.$$ and $h ( x ) = e ^ { | x | }$ for all $x \in \mathbb { R }$. Let $( f \circ h ) ( x )$ denote $f ( h ( x ) )$ and $( h \circ f ) ( x )$ denote $h ( f ( x ) )$. Then which of the following is (are) true?
(A) $f$ is differentiable at $x = 0$
(B) $\quad h$ is differentiable at $x = 0$
(C) $f \circ h$ is differentiable at $x = 0$
(D) $h \circ f$ is differentiable at $x = 0$

Q57 Function Transformations View

Let $f ( x ) = \sin \left( \frac { \pi } { 6 } \sin \left( \frac { \pi } { 2 } \sin x \right) \right)$ for all $x \in \mathbb { R }$ and $g ( x ) = \frac { \pi } { 2 } \sin x$ for all $x \in \mathbb { R }$. Let $( f \circ g ) ( x )$ denote $f ( g ( x ) )$ and $( g \circ f ) ( x )$ denote $g ( f ( x ) )$. Then which of the following is (are) true?
(A) Range of $f$ is $\left[ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right]$
(B) Range of $f \circ g$ is $\left[ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right]$
(C) $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { g ( x ) } = \frac { \pi } { 6 }$
(D) There is an $x \in \mathbb { R }$ such that $( g \circ f ) ( x ) = 1$

Q58 Vectors Introduction & 2D True/False or Multiple-Statement Verification View

Let $\triangle P Q R$ be a triangle. Let $\vec { a } = \overrightarrow { Q R } , \vec { b } = \overrightarrow { R P }$ and $\vec { c } = \overrightarrow { P Q }$. If $| \vec { a } | = 12 , | \vec { b } | = 4 \sqrt { 3 }$ and $\vec { b } \cdot \vec { c } = 24$, then which of the following is (are) true?
(A) $\frac { | \vec { c } | ^ { 2 } } { 2 } - | \vec { a } | = 12$
(B) $\frac { | \vec { c } | ^ { 2 } } { 2 } + | \vec { a } | = 30$
(C) $| \vec { a } \times \vec { b } + \vec { c } \times \vec { a } | = 48 \sqrt { 3 }$
(D) $\vec { a } \cdot \vec { b } = - 72$

Q59 Vectors Introduction & 2D Dot Product Computation View

Column I
(A) In $\mathbb { R } ^ { 2 }$, if the magnitude of the projection vector of the vector $\alpha \hat { i } + \beta \hat { j }$ on $\sqrt { 3 } \hat { i } + \hat { j }$ is $\sqrt { 3 }$ and if $\alpha = 2 + \sqrt { 3 } \beta$, then possible value(s) of $| \alpha |$ is (are)
(B) Let $a$ and $b$ be real numbers such that the function $$f ( x ) = \left\{ \begin{array} { c c } - 3 a x ^ { 2 } - 2 , & x < 1 \\ b x + a ^ { 2 } , & x \geq 1 \end{array} \right.$$ is differentiable for all $x \in \mathbb { R }$. Then possible value(s) of $a$ is (are)
(C) Let $\omega \neq 1$ be a complex cube root of unity. If $\left( 3 - 3 \omega + 2 \omega ^ { 2 } \right) ^ { 4 n + 3 } + \left( 2 + 3 \omega - 3 \omega ^ { 2 } \right) ^ { 4 n + 3 } + \left( - 3 + 2 \omega + 3 \omega ^ { 2 } \right) ^ { 4 n + 3 } = 0$, then possible value(s) of $n$ is (are)
(D) Let the harmonic mean of two positive real numbers $a$ and $b$ be 4. If $q$ is a positive real number such that $a , 5 , q , b$ is an arithmetic progression, then the value(s) of $| q - a |$ is (are) Column II (P) 1 (Q) 2 (R) 3 (S) 4 (T) 5

Q60 Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View

Column I
(A) In a triangle $\triangle X Y Z$, let $a , b$ and $c$ be the lengths of the sides opposite to the angles $X , Y$ and $Z$, respectively. If $2 \left( a ^ { 2 } - b ^ { 2 } \right) = c ^ { 2 }$ and $\lambda = \frac { \sin ( X - Y ) } { \sin Z }$, then possible values of $n$ for which $\cos ( n \pi \lambda ) = 0$ is (are)
(B) In a triangle $\triangle X Y Z$, let $a , b$ and $c$ be the lengths of the sides opposite to the angles $X , Y$ and $Z$, respectively. If $1 + \cos 2 X - 2 \cos 2 Y = 2 \sin X \sin Y$, then possible value(s) of $\frac { a } { b }$ is (are)
(C) In $\mathbb { R } ^ { 2 }$, let $\sqrt { 3 } \hat { i } + \hat { j } , \hat { i } + \sqrt { 3 } \hat { j }$ and $\beta \hat { i } + ( 1 - \beta ) \hat { j }$ be the position vectors of $X , Y$ and $Z$ with respect to the origin $O$, respectively. If the distance of $Z$ from the bisector of the acute angle of $\overrightarrow { O X }$ with $\overrightarrow { O Y }$ is $\frac { 3 } { \sqrt { 2 } }$, then possible value(s) of $| \beta |$ is (are)
(D) Suppose that $F ( \alpha )$ denotes the area of the region bounded by $x = 0 , x = 2 , y ^ { 2 } = 4 x$ and $y = | \alpha x - 1 | + | \alpha x - 2 | + \alpha x$, where $\alpha \in \{ 0,1 \}$. Then the value(s) of $F ( \alpha ) + \frac { 8 } { 3 } \sqrt { 2 }$, when $\alpha = 0$ and $\alpha = 1$, is (are) Column II (P) 1 (Q) 2 (R) 3 (S) 4 (T) 5

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