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2022 paper2

18 maths questions

Q1 3 marks Addition & Double Angle Formulae Multi-Step Composite Problem Using Identities View
Let $\alpha$ and $\beta$ be real numbers such that $- \frac { \pi } { 4 } < \beta < 0 < \alpha < \frac { \pi } { 4 }$. If $\sin ( \alpha + \beta ) = \frac { 1 } { 3 }$ and $\cos ( \alpha - \beta ) = \frac { 2 } { 3 }$, then the greatest integer less than or equal to
$$\left( \frac { \sin \alpha } { \cos \beta } + \frac { \cos \beta } { \sin \alpha } + \frac { \cos \alpha } { \sin \beta } + \frac { \sin \beta } { \cos \alpha } \right) ^ { 2 }$$
is $\_\_\_\_$ .
Q2 3 marks Differential equations Solving Separable DEs with Initial Conditions View
If $y ( x )$ is the solution of the differential equation
$$x d y - \left( y ^ { 2 } - 4 y \right) d x = 0 \text { for } x > 0 , \quad y ( 1 ) = 2$$
and the slope of the curve $y = y ( x )$ is never zero, then the value of $10 y ( \sqrt { 2 } )$ is $\_\_\_\_$ .
Q3 3 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
The greatest integer less than or equal to
$$\int _ { 1 } ^ { 2 } \log _ { 2 } \left( x ^ { 3 } + 1 \right) d x + \int _ { 1 } ^ { \log _ { 2 } 9 } \left( 2 ^ { x } - 1 \right) ^ { \frac { 1 } { 3 } } d x$$
is $\_\_\_\_$ .
Q4 3 marks Exponential Equations & Modelling Solve Exponential Equation for Unknown Variable View
The product of all positive real values of $x$ satisfying the equation
$$x ^ { \left( 16 \left( \log _ { 5 } x \right) ^ { 3 } - 68 \log _ { 5 } x \right) } = 5 ^ { - 16 }$$
is $\_\_\_\_$.
Q5 3 marks Taylor series Limit evaluation using series expansion or exponential asymptotics View
If
$$\beta = \lim _ { x \rightarrow 0 } \frac { e ^ { x ^ { 3 } } - \left( 1 - x ^ { 3 } \right) ^ { \frac { 1 } { 3 } } + \left( \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } - 1 \right) \sin x } { x \sin ^ { 2 } x }$$
then the value of $6 \beta$ is $\_\_\_\_$ .
Q6 3 marks 3x3 Matrices Determinant of Parametric or Structured Matrix View
Let $\beta$ be a real number. Consider the matrix
$$A = \left( \begin{array} { c c c } \beta & 0 & 1 \\ 2 & 1 & - 2 \\ 3 & 1 & - 2 \end{array} \right)$$
If $A ^ { 7 } - ( \beta - 1 ) A ^ { 6 } - \beta A ^ { 5 }$ is a singular matrix, then the value of $9 \beta$ is $\_\_\_\_$ .
Q7 3 marks Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
Consider the hyperbola
$$\frac { x ^ { 2 } } { 100 } - \frac { y ^ { 2 } } { 64 } = 1$$
with foci at $S$ and $S _ { 1 }$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $\angle S P S _ { 1 } = \alpha$, with $\alpha < \frac { \pi } { 2 }$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S _ { 1 } P$ at $P _ { 1 }$. Let $\delta$ be the distance of $P$ from the straight line $S P _ { 1 }$, and $\beta = S _ { 1 } P$. Then the greatest integer less than or equal to $\frac { \beta \delta } { 9 } \sin \frac { \alpha } { 2 }$ is $\_\_\_\_$ .
Q8 3 marks Areas Between Curves Area Involving Piecewise or Composite Functions View
Consider the functions $f , g : \mathbb { R } \rightarrow \mathbb { R }$ defined by
$$f ( x ) = x ^ { 2 } + \frac { 5 } { 12 } \quad \text { and } \quad g ( x ) = \begin{cases} 2 \left( 1 - \frac { 4 | x | } { 3 } \right) , & | x | \leq \frac { 3 } { 4 } \\ 0 , & | x | > \frac { 3 } { 4 } \end{cases}$$
If $\alpha$ is the area of the region
$$\left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : | x | \leq \frac { 3 } { 4 } , 0 \leq y \leq \min \{ f ( x ) , g ( x ) \} \right\}$$
then the value of $9 \alpha$ is $\_\_\_\_$ .
Q9 4 marks Sine and Cosine Rules Multi-step composite figure problem View
Let $P Q R S$ be a quadrilateral in a plane, where $Q R = 1 , \angle P Q R = \angle Q R S = 70 ^ { \circ } , \angle P Q S = 15 ^ { \circ }$ and $\angle P R S = 40 ^ { \circ }$. If $\angle R P S = \theta ^ { \circ } , P Q = \alpha$ and $P S = \beta$, then the interval(s) that contain(s) the value of $4 \alpha \beta \sin \theta ^ { \circ }$ is/are
(A) $( 0 , \sqrt { 2 } )$
(B) $( 1,2 )$
(C) $( \sqrt { 2 } , 3 )$
(D) $( 2 \sqrt { 2 } , 3 \sqrt { 2 } )$
Q10 4 marks Geometric Sequences and Series Geometric Series with Trigonometric or Functional Terms View
Let
$$\alpha = \sum _ { k = 1 } ^ { \infty } \sin ^ { 2 k } \left( \frac { \pi } { 6 } \right) .$$
Let $g : [ 0,1 ] \rightarrow \mathbb { R }$ be the function defined by
$$g ( x ) = 2 ^ { \alpha x } + 2 ^ { \alpha ( 1 - x ) }$$
Then, which of the following statements is/are TRUE ?
(A) The minimum value of $g ( x )$ is $2 ^ { \frac { 7 } { 6 } }$
(B) The maximum value of $g ( x )$ is $1 + 2 ^ { \frac { 1 } { 3 } }$
(C) The function $g ( x )$ attains its maximum at more than one point
(D) The function $g ( x )$ attains its minimum at more than one point
Q11 4 marks Complex Numbers Argand & Loci Modulus Inequalities and Triangle Inequality Applications View
Let $\bar { z }$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of
$$( \bar { z } ) ^ { 2 } + \frac { 1 } { z ^ { 2 } }$$
are integers, then which of the following is/are possible value(s) of $| z |$ ?
(A) $\left( \frac { 43 + 3 \sqrt { 205 } } { 2 } \right) ^ { \frac { 1 } { 4 } }$
(B) $\left( \frac { 7 + \sqrt { 33 } } { 4 } \right) ^ { \frac { 1 } { 4 } }$
(C) $\left( \frac { 9 + \sqrt { 65 } } { 4 } \right) ^ { \frac { 1 } { 4 } }$
(D) $\left( \frac { 7 + \sqrt { 13 } } { 6 } \right) ^ { \frac { 1 } { 4 } }$
Q12 4 marks Circles Circles Tangent to Each Other or to Axes View
Let $G$ be a circle of radius $R > 0$. Let $G _ { 1 } , G _ { 2 } , \ldots , G _ { n }$ be $n$ circles of equal radius $r > 0$. Suppose each of the $n$ circles $G _ { 1 } , G _ { 2 } , \ldots , G _ { n }$ touches the circle $G$ externally. Also, for $i = 1,2 , \ldots , n - 1$, the circle $G _ { i }$ touches $G _ { i + 1 }$ externally, and $G _ { n }$ touches $G _ { 1 }$ externally. Then, which of the following statements is/are TRUE ?
(A) If $n = 4$, then $( \sqrt { 2 } - 1 ) r < R$
(B) If $n = 5$, then $r < R$
(C) If $n = 8$, then $( \sqrt { 2 } - 1 ) r < R$
(D) If $n = 12$, then $\sqrt { 2 } ( \sqrt { 3 } + 1 ) r > R$
Q13 4 marks Vectors Introduction & 2D True/False or Multiple-Statement Verification View
Let $\hat { \imath } , \hat { \jmath }$ and $\hat { k }$ be the unit vectors along the three positive coordinate axes. Let
$$\begin{aligned} & \vec { a } = 3 \hat { \imath } + \hat { \jmath } - \hat { k } , \\ & \vec { b } = \hat { \imath } + b _ { 2 } \hat { \jmath } + b _ { 3 } \hat { k } , \quad b _ { 2 } , b _ { 3 } \in \mathbb { R } , \\ & \vec { c } = c _ { 1 } \hat { \imath } + c _ { 2 } \hat { \jmath } + c _ { 3 } \hat { k } , \quad c _ { 1 } , c _ { 2 } , c _ { 3 } \in \mathbb { R } \end{aligned}$$
be three vectors such that $b _ { 2 } b _ { 3 } > 0 , \vec { a } \cdot \vec { b } = 0$ and
$$\left( \begin{array} { r c r } 0 & - c _ { 3 } & c _ { 2 } \\ c _ { 3 } & 0 & - c _ { 1 } \\ - c _ { 2 } & c _ { 1 } & 0 \end{array} \right) \left( \begin{array} { l } 1 \\ b _ { 2 } \\ b _ { 3 } \end{array} \right) = \left( \begin{array} { r } 3 - c _ { 1 } \\ 1 - c _ { 2 } \\ - 1 - c _ { 3 } \end{array} \right)$$
Then, which of the following is/are TRUE ?
(A) $\vec { a } \cdot \vec { c } = 0$
(B) $\vec { b } \cdot \vec { c } = 0$
(C) $| \vec { b } | > \sqrt { 10 }$
(D) $| \vec { c } | \leq \sqrt { 11 }$
Q14 4 marks Differential equations Qualitative Analysis of DE Solutions View
For $x \in \mathbb { R }$, let the function $y ( x )$ be the solution of the differential equation
$$\frac { d y } { d x } + 12 y = \cos \left( \frac { \pi } { 12 } x \right) , \quad y ( 0 ) = 0$$
Then, which of the following statements is/are TRUE ?
(A) $y ( x )$ is an increasing function
(B) $y ( x )$ is a decreasing function
(C) There exists a real number $\beta$ such that the line $y = \beta$ intersects the curve $y = y ( x )$ at infinitely many points
(D) $y ( x )$ is a periodic function
Q15 3 marks Combinations & Selection Selection with Group/Category Constraints View
Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen ?
(A) 21816
(B) 85536
(C) 12096
(D) 156816
Q16 3 marks Matrices Matrix Power Computation and Application View
If $M = \left( \begin{array} { r r } \frac { 5 } { 2 } & \frac { 3 } { 2 } \\ - \frac { 3 } { 2 } & - \frac { 1 } { 2 } \end{array} \right)$, then which of the following matrices is equal to $M ^ { 2022 }$ ?
(A) $\quad \left( \begin{array} { r r } 3034 & 3033 \\ - 3033 & - 3032 \end{array} \right)$
(B) $\quad \left( \begin{array} { l l } 3034 & - 3033 \\ 3033 & - 3032 \end{array} \right)$
(C) $\quad \left( \begin{array} { r r } 3033 & 3032 \\ - 3032 & - 3031 \end{array} \right)$
(D) $\quad \left( \begin{array} { r r } 3032 & 3031 \\ - 3031 & - 3030 \end{array} \right)$
Q17 3 marks Conditional Probability Sequential/Multi-Stage Conditional Probability View
Suppose that
Box-I contains 8 red, 3 blue and 5 green balls, Box-II contains 24 red, 9 blue and 15 green balls, Box-III contains 1 blue, 12 green and 3 yellow balls, Box-IV contains 10 green, 16 orange and 6 white balls.
A ball is chosen randomly from Box-I; call this ball $b$. If $b$ is red then a ball is chosen randomly from Box-II, if $b$ is blue then a ball is chosen randomly from Box-III, and if $b$ is green then a ball is chosen randomly from Box-IV. The conditional probability of the event 'one of the chosen balls is white' given that the event 'at least one of the chosen balls is green' has happened, is equal to
(A) $\frac { 15 } { 256 }$
(B) $\frac { 3 } { 16 }$
(C) $\frac { 5 } { 52 }$
(D) $\frac { 1 } { 8 }$
Q18 3 marks Sequences and series, recurrence and convergence Series convergence and power series analysis View
For positive integer $n$, define
$$f ( n ) = n + \frac { 16 + 5 n - 3 n ^ { 2 } } { 4 n + 3 n ^ { 2 } } + \frac { 32 + n - 3 n ^ { 2 } } { 8 n + 3 n ^ { 2 } } + \frac { 48 - 3 n - 3 n ^ { 2 } } { 12 n + 3 n ^ { 2 } } + \cdots + \frac { 25 n - 7 n ^ { 2 } } { 7 n ^ { 2 } }$$
Then, the value of $\lim _ { n \rightarrow \infty } f ( n )$ is equal to
(A) $3 + \frac { 4 } { 3 } \log _ { e } 7$
(B) $4 - \frac { 3 } { 4 } \log _ { e } \left( \frac { 7 } { 3 } \right)$
(C) $4 - \frac { 4 } { 3 } \log _ { e } \left( \frac { 7 } { 3 } \right)$
(D) $3 + \frac { 3 } { 4 } \log _ { e } 7$