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Q29 Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View

If the angles $\mathrm { A } , \mathrm { B }$ and C of a triangle are in an arithmetic progression and if $\mathrm { a } , \mathrm { b }$ and c denote the lengths of the sides opposite to $\mathrm { A } , \mathrm { B }$ and C respectively, then the value of the expression $\frac { a } { c } \sin 2 C + \frac { c } { a } \sin 2 A$ is
A) $\frac { 1 } { 2 }$
B) $\frac { \sqrt { 3 } } { 2 }$
C) 1
D) $\sqrt { 3 }$

Q30 Vectors: Lines & Planes Find Cartesian Equation of a Plane View

Equation of the plane containing the straight line $\frac { x } { 2 } = \frac { y } { 3 } = \frac { z } { 4 }$ and perpendicular to the plane containing the straight lines $\frac { x } { 3 } = \frac { y } { 4 } = \frac { z } { 2 }$ and $\frac { x } { 4 } = \frac { y } { 2 } = \frac { z } { 3 }$ is
A) $x + 2 y - 2 z = 0$
B) $3 x + 2 y - 2 z = 0$
C) $x - 2 y + z = 0$
D) $5 x + 2 y - 4 z = 0$

Q31 Conditional Probability Combinatorial Conditional Probability (Counting-Based) View

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$. A fair die is thrown three times. If $r _ { 1 } , r _ { 2 }$ and $r _ { 3 }$ are the numbers obtained on the die, then the probability that $\omega ^ { r _ { 1 } } + \omega ^ { r _ { 2 } } + \omega ^ { r _ { 3 } } = 0$ is
A) $\frac { 1 } { 18 }$
B) $\frac { 1 } { 9 }$
C) $\frac { 2 } { 9 }$
D) $\frac { 1 } { 36 }$

Q32 Vectors Introduction & 2D Geometric Property Identification via Vectors View

Let $P , Q , R$ and $S$ be the points on the plane with position vectors $- 2 \hat { i } - \hat { j } , 4 \hat { i } , 3 \hat { i } + 3 \hat { j }$ and $- 3 \hat { i } + 2 \hat { j }$ respectively. The quadrilateral $P Q R S$ must be a
A) parallelogram, which is neither a rhombus nor a rectangle
B) square
C) rectangle, but not a square
D) rhombus, but not a square

Q33 Matrices Linear System and Inverse Existence View

The number of $3 \times 3$ matrices A whose entries are either 0 or 1 and for which the system $A \left[ \begin{array} { l } x \\ y \\ z \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right]$ has exactly two distinct solutions, is
A) 0
B) $2 ^ { 9 } - 1$
C) 168
D) 2

Q34 Indefinite & Definite Integrals Accumulation Function Analysis View

The value of $\lim _ { x \rightarrow 0 } \frac { 1 } { x ^ { 3 } } \int _ { 0 } ^ { x } \frac { t \ln ( 1 + t ) } { t ^ { 4 } + 4 } d t$ is
A) 0
B) $\frac { 1 } { 12 }$
C) $\frac { 1 } { 24 }$
D) $\frac { 1 } { 64 }$

Q35 Roots of polynomials Vieta's formulas: compute symmetric functions of roots View

Let $p$ and $q$ be real numbers such that $p \neq 0 , p ^ { 3 } \neq q$ and $p ^ { 3 } \neq - q$. If $\alpha$ and $\beta$ are nonzero complex numbers satisfying $\alpha + \beta = - p$ and $\alpha ^ { 3 } + \beta ^ { 3 } = q$, then a quadratic equation having $\frac { \alpha } { \beta }$ and $\frac { \beta } { \alpha }$ as its roots is
A) $\left( p ^ { 3 } + q \right) x ^ { 2 } - \left( p ^ { 3 } + 2 q \right) x + \left( p ^ { 3 } + q \right) = 0$
B) $\left( p ^ { 3 } + q \right) x ^ { 2 } - \left( p ^ { 3 } - 2 q \right) x + \left( p ^ { 3 } + q \right) = 0$
C) $\left( \mathrm { p } ^ { 3 } - \mathrm { q } \right) \mathrm { x } ^ { 2 } - \left( 5 \mathrm { p } ^ { 3 } - 2 \mathrm { q } \right) \mathrm { x } + \left( \mathrm { p } ^ { 3 } - \mathrm { q } \right) = 0$
D) $\left( p ^ { 3 } - q \right) x ^ { 2 } - \left( 5 p ^ { 3 } + 2 q \right) x + \left( p ^ { 3 } - q \right) = 0$

Q36 Stationary points and optimisation Find absolute extrema on a closed interval or domain View

Let $f , g$ and $h$ be real-valued functions defined on the interval $[ 0,1 ]$ by $f ( x ) = e ^ { x ^ { 2 } } + e ^ { - x ^ { 2 } } , g ( x ) = x e ^ { x ^ { 2 } } + e ^ { - x ^ { 2 } }$ and $h ( x ) = x ^ { 2 } e ^ { x ^ { 2 } } + e ^ { - x ^ { 2 } }$. If $a , b$ and $c$ denote, respectively, the absolute maximum of $f , g$ and $h$ on $[ 0,1 ]$, then
A) $\mathrm { a } = \mathrm { b }$ and $\mathrm { c } \neq \mathrm { b }$
B) a $=$ c and a $\neq$ b
C) $a \neq b$ and $c \neq b$
D) $a = b = c$

Q37 Conic sections Circle-Conic Interaction with Tangency or Intersection View

Let A and B be two distinct points on the parabola $\mathrm { y } ^ { 2 } = 4 \mathrm { x }$. If the axis of the parabola touches a circle of radius $r$ having $A B$ as its diameter, then the slope of the line joining A and B can be
A) $- \frac { 1 } { r }$
B) $\frac { 1 } { r }$
C) $\frac { 2 } { r }$
D) $- \frac { 2 } { \mathrm { r } }$

Q38 Sine and Cosine Rules Find an angle using the cosine rule View

Let ABC be a triangle such that $\angle \mathrm { ACB } = \frac { \pi } { 6 }$ and let $\mathrm { a } , \mathrm { b }$ and c denote the lengths of the sides opposite to $\mathrm { A } , \mathrm { B }$ and C respectively. The value(s) of x for which $\mathrm { a } = \mathrm { x } ^ { 2 } + \mathrm { x } + 1 , \mathrm {~b} = \mathrm { x } ^ { 2 } - 1$ and $\mathrm { c } = 2 \mathrm { x } + 1$ is (are)
A) $- ( 2 + \sqrt { 3 } )$
B) $1 + \sqrt { 3 }$
C) $2 + \sqrt { 3 }$
D) $4 \sqrt { 3 }$

Q39 Complex Numbers Argand & Loci Algebraic Conditions for Geometric Properties (Real, Imaginary, Collinear) View

Let $z _ { 1 }$ and $z _ { 2 }$ be two distinct complex numbers and let $z = ( 1 - t ) z _ { 1 } + t z _ { 2 }$ for some real number $t$ with $0 < t < 1$. If $\operatorname { Arg } ( w )$ denotes the principal argument of a nonzero complex number $w$, then
A) $\left| z - z _ { 1 } \right| + \left| z - z _ { 2 } \right| = \left| z _ { 1 } - z _ { 2 } \right|$
B) $\operatorname { Arg } \left( z - z _ { 1 } \right) = \operatorname { Arg } \left( z - z _ { 2 } \right)$
C) $\left| \begin{array} { c c } \mathrm { z } - \mathrm { z } _ { 1 } & \overline { \mathrm { z } } - \overline { \mathrm { z } } _ { 1 } \\ \mathrm { z } _ { 2 } - \mathrm { z } _ { 1 } & \overline { \mathrm { z } } _ { 2 } - \overline { \mathrm { z } } _ { 1 } \end{array} \right| = 0$
D) $\operatorname { Arg } \left( z - z _ { 1 } \right) = \operatorname { Arg } \left( z _ { 2 } - z _ { 1 } \right)$

Q40 Differentiating Transcendental Functions Higher-order or nth derivative computation View

Let $f$ be a real-valued function defined on the interval $( 0 , \infty )$ by $\mathrm { f } ( \mathrm { x } ) = \ell n \mathrm { x } + \int _ { 0 } ^ { \mathrm { x } } \sqrt { 1 + \sin \mathrm { t } } \mathrm { dt }$. Then which of the following statement(s) is (are) true?
A) $\mathrm { f } ^ { \prime \prime } ( \mathrm { x } )$ exists for all $\mathrm { x } \in ( 0 , \infty )$
B) $f ^ { \prime } ( x )$ exists for all $x \in ( 0 , \infty )$ and $f ^ { \prime }$ is continuous on $( 0 , \infty )$, but not differentiable on $( 0 , \infty )$
C) there exists $\alpha > 1$ such that $\left| \mathrm { f } ^ { \prime } ( \mathrm { x } ) \right| < | \mathrm { f } ( \mathrm { x } ) |$ for all $\mathrm { x } \in ( \alpha , \infty )$
D) there exists $\beta > 0$ such that $| f ( x ) | + \left| f ^ { \prime } ( x ) \right| \leq \beta$ for all $x \in ( 0 , \infty )$

Q41 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

The value(s) of $\int _ { 0 } ^ { 1 } \frac { x ^ { 4 } ( 1 - x ) ^ { 4 } } { 1 + x ^ { 2 } } d x$ is (are)
A) $\frac { 22 } { 7 } - \pi$
B) $\frac { 2 } { 105 }$
C) 0
D) $\frac { 71 } { 15 } - \frac { 3 \pi } { 2 }$

Q42 Matrices Determinant and Rank Computation View

Let p be an odd prime number and $\mathrm { T } _ { \mathrm { p } }$ be the following set of $2 \times 2$ matrices: $$\mathrm { T } _ { \mathrm { p } } = \left\{ \mathrm { A } = \left[ \begin{array} { l l } \mathrm { a } & \mathrm {~b} \\ \mathrm { c } & \mathrm { a } \end{array} \right] : \mathrm { a } , \mathrm {~b} , \mathrm { c } \in \{ 0,1,2 , \ldots , \mathrm { p } - 1 \} \right\}$$
The number of $A$ in $T _ { p }$ such that $A$ is either symmetric or skew-symmetric or both, and $\operatorname { det } ( \mathrm { A } )$ divisible by p is
A) $( p - 1 ) ^ { 2 }$
B) $2 ( p - 1 )$
C) $( p - 1 ) ^ { 2 } + 1$
D) $2 p - 1$

Q43 Matrices Determinant and Rank Computation View

Let p be an odd prime number and $\mathrm { T } _ { \mathrm { p } }$ be the following set of $2 \times 2$ matrices: $$\mathrm { T } _ { \mathrm { p } } = \left\{ \mathrm { A } = \left[ \begin{array} { l l } \mathrm { a } & \mathrm {~b} \\ \mathrm { c } & \mathrm { a } \end{array} \right] : \mathrm { a } , \mathrm {~b} , \mathrm { c } \in \{ 0,1,2 , \ldots , \mathrm { p } - 1 \} \right\}$$
The number of $A$ in $T _ { p }$ such that the trace of $A$ is not divisible by $p$ but $\operatorname { det } ( A )$ is divisible by $p$ is [Note : The trace of a matrix is the sum of its diagonal entries.]
A) $( \mathrm { p } - 1 ) \left( \mathrm { p } ^ { 2 } - \mathrm { p } + 1 \right)$
B) $\mathrm { p } ^ { 3 } - ( \mathrm { p } - 1 ) ^ { 2 }$
C) $( p - 1 ) ^ { 2 }$
D) $( \mathrm { p } - 1 ) \left( \mathrm { p } ^ { 2 } - 2 \right)$

Q44 Matrices Determinant and Rank Computation View

Let p be an odd prime number and $\mathrm { T } _ { \mathrm { p } }$ be the following set of $2 \times 2$ matrices: $$\mathrm { T } _ { \mathrm { p } } = \left\{ \mathrm { A } = \left[ \begin{array} { l l } \mathrm { a } & \mathrm {~b} \\ \mathrm { c } & \mathrm { a } \end{array} \right] : \mathrm { a } , \mathrm {~b} , \mathrm { c } \in \{ 0,1,2 , \ldots , \mathrm { p } - 1 \} \right\}$$
The number of $A$ in $T _ { p }$ such that $\operatorname { det } ( A )$ is not divisible by $p$ is
A) $2 p ^ { 2 }$
B) $p ^ { 3 } - 5 p$
C) $p ^ { 3 } - 3 p$
D) $p ^ { 3 } - p ^ { 2 }$

Q45 Circles Tangent Lines and Tangent Lengths View

The circle $x ^ { 2 } + y ^ { 2 } - 8 x = 0$ and hyperbola $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$ intersect at the points $A$ and $B$.
Equation of a common tangent with positive slope to the circle as well as to the hyperbola is
A) $2 x - \sqrt { 5 } y - 20 = 0$
B) $2 x - \sqrt { 5 } y + 4 = 0$
C) $3 x - 4 y + 8 = 0$
D) $4 x - 3 y + 4 = 0$

Q46 Circles Circle Equation Derivation View

The circle $x ^ { 2 } + y ^ { 2 } - 8 x = 0$ and hyperbola $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$ intersect at the points $A$ and $B$.
Equation of the circle with AB as its diameter is
A) $x ^ { 2 } + y ^ { 2 } - 12 x + 24 = 0$
B) $x ^ { 2 } + y ^ { 2 } + 12 x + 24 = 0$
C) $x ^ { 2 } + y ^ { 2 } + 24 x - 12 = 0$
D) $x ^ { 2 } + y ^ { 2 } - 24 x - 12 = 0$

Q47 Quadratic trigonometric equations View

The number of values of $\theta$ in the interval $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ such that $\theta \neq \frac { \mathrm { n } \pi } { 5 }$ for $\mathrm { n } = 0 , \pm 1 , \pm 2$ and $\tan \theta = \cot 5 \theta$ as well as $\sin 2 \theta = \cos 4 \theta$ is

Q48 Quadratic trigonometric equations View

The maximum value of the expression $\frac { 1 } { \sin ^ { 2 } \theta + 3 \sin \theta \cos \theta + 5 \cos ^ { 2 } \theta }$ is

Q49 Vector Product and Surfaces View

If $\overrightarrow { \mathrm { a } }$ and $\overrightarrow { \mathrm { b } }$ are vectors in space given by $\overrightarrow { \mathrm { a } } = \frac { \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } } { \sqrt { 5 } }$ and $\overrightarrow { \mathrm { b } } = \frac { 2 \hat { \mathrm { i } } + \hat { \mathrm { j } } + 3 \hat { \mathrm { k } } } { \sqrt { 14 } }$, then the value of $( 2 \vec { a } + \vec { b } ) \cdot [ ( \vec { a } \times \vec { b } ) \times ( \vec { a } - 2 \vec { b } ) ]$ is

Q50 Conic sections Eccentricity or Asymptote Computation View

The line $2 \mathrm { x } + \mathrm { y } = 1$ is tangent to the hyperbola $\frac { \mathrm { x } ^ { 2 } } { \mathrm { a } ^ { 2 } } - \frac { \mathrm { y } ^ { 2 } } { \mathrm {~b} ^ { 2 } } = 1$. If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is

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

If the distance between the plane $\mathrm { Ax } - 2 \mathrm { y } + \mathrm { z } = \mathrm { d }$ and the plane containing the lines $\frac { x - 1 } { 2 } = \frac { y - 2 } { 3 } = \frac { z - 3 } { 4 }$ and $\frac { x - 2 } { 3 } = \frac { y - 3 } { 4 } = \frac { z - 4 } { 5 }$ is $\sqrt { 6 }$, then $| d |$ is

Q52 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View

For any real number x, let $[ \mathrm { x } ]$ denote the largest integer less than or equal to x. Let $f$ be a real valued function defined on the interval $[ - 10,10 ]$ by $$f ( x ) = \left\{ \begin{array} { c c } x - [ x ] & \text { if } [ x ] \text { is odd } \\ 1 + [ x ] - x & \text { if } [ x ] \text { is even } \end{array} \right.$$ Then the value of $\frac { \pi ^ { 2 } } { 10 } \int _ { - 10 } ^ { 10 } f ( x ) \cos \pi x \, d x$ is

Q53 Matrices Determinant and Rank Computation View

Let $\omega$ be the complex number $\cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 }$. Then the number of distinct complex numbers $z$ satisfying $\left| \begin{array} { c c c } z + 1 & \omega & \omega ^ { 2 } \\ \omega & z + \omega ^ { 2 } & 1 \\ \omega ^ { 2 } & 1 & z + \omega \end{array} \right| = 0$ is equal to

Q54 Geometric Sequences and Series Sum of an Infinite Geometric Series (Direct Computation) View

Let $\mathrm { S } _ { \mathrm { k } } , \mathrm { k } = 1,2 , \ldots , 100$, denote the sum of the infinite geometric series whose first term is $\frac { \mathrm { k } - 1 } { \mathrm { k } ! }$ and the common ratio is $\frac { 1 } { \mathrm { k } }$. Then the value of $\frac { 100 ^ { 2 } } { 100 ! } + \sum _ { \mathrm { k } = 1 } ^ { 100 } \left| \left( \mathrm { k } ^ { 2 } - 3 \mathrm { k } + 1 \right) \mathrm { S } _ { \mathrm { k } } \right|$ is

Q55 Trigonometric equations in context View

The number of all possible values of $\theta$, where $0 < \theta < \pi$, for which the system of equations $$\begin{gathered} ( y + z ) \cos 3 \theta = ( x y z ) \sin 3 \theta \\ x \sin 3 \theta = \frac { 2 \cos 3 \theta } { y } + \frac { 2 \sin 3 \theta } { z } \\ ( x y z ) \sin 3 \theta = ( y + 2 z ) \cos 3 \theta + y \sin 3 \theta \end{gathered}$$ have a solution $\left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right)$ with $y _ { 0 } z _ { 0 } \neq 0$, is

Q56 Differential equations Solving Separable DEs with Initial Conditions View

Let f be a real-valued differentiable function on $\mathbf { R }$ (the set of all real numbers) such that $f ( 1 ) = 1$. If the $y$-intercept of the tangent at any point $P ( x , y )$ on the curve $y = f ( x )$ is equal to the cube of the abscissa of $P$, then the value of $f ( - 3 )$ is equal to

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