LFM Pure

cmi-entrance 2023 Q13 10 marks Linear Transformation and Endomorphism Properties View

(A) (5 marks) Let $n \geq 2$ be an integer. Let $V$ be the $\mathbb { R }$-vector-space of homogeneous real polynomials in three variables $X , Y , Z$ of degree $n$. Let $p = ( 1,0,0 )$. Let
$$W = \left\{ f \in V \left\lvert \, f ( p ) = \frac { \partial f } { \partial X } ( p ) \right. \right\}$$
Determine the dimension of $V / W$.
(B) (5 marks) A linear transformation $T : \mathbb { R } ^ { 9 } \longrightarrow \mathbb { R } ^ { 9 }$ is defined on the standard basis $e _ { 1 } , \ldots , e _ { 9 }$ by
$$\begin{aligned} & T e _ { i } = e _ { i - 1 } , \quad i = 3 , \ldots , 9 \\ & T e _ { 2 } = e _ { 3 } \\ & T e _ { 1 } = e _ { 1 } + e _ { 3 } + e _ { 8 } . \end{aligned}$$
Determine the nullity of $T$.

cmi-entrance 2023 Q17 Linear Transformation and Endomorphism Properties View

Denote by $V$ the $\mathbb { Q }$-vector-space $\mathbb { Q } [ X ]$ (polynomial ring in one variable $X$ ). Show that $V ^ { * }$ is not isomorphic to $V$.

cmi-entrance 2024 Q5 True/False or Multiple-Select Conceptual Reasoning View

Let $p \geq 3$ be a prime number and $V$ be an $n$-dimensional vector space over $\mathbb { F } _ { p }$. Let $T : V \rightarrow V$ be a linear transformation. Select all the true statement(s) from below.
(A) $T$ has an eigenvalue in $\mathbb { F } _ { p }$.
(B) If $T ^ { p - 1 } = I$, then the minimal polynomial of $T$ has distinct roots in $\mathbb { F } _ { p }$.
(C) If $T \neq I$ and $T ^ { p - 1 } = I$, then the characteristic polynomial of $T$ has distinct roots in $\mathbb { F } _ { p }$.
(D) If $T ^ { p - 1 } = I$, then $T$ is diagonalizable over $\mathbb { F } _ { p }$.

cmi-entrance 2025 Q1 4 marks True/False or Multiple-Select Conceptual Reasoning View

Let $T : \mathbb { R } ^ { 3 } \longrightarrow \mathbb { R } ^ { 3 }$ be a linear transformation such that $T \neq 0$ and $T ^ { 4 } = 0$. Pick the correct statement(s) from below.
(A) $T ^ { 3 } = 0$.
(B) $\operatorname { Image } ( T ) \neq \operatorname { Image } \left( T ^ { 2 } \right)$.
(C) $\operatorname { rank } \left( T ^ { 2 } \right) \leq 1$.
(D) $\operatorname { rank } ( T ) = 2$.

cmi-entrance 2025 Q2 4 marks Projection and Orthogonality View

Let $W = \left\{ ( a , b , c , d ) \in \mathbb { R } ^ { 4 } \mid 3 a - b + 6 c = 0 \right\}$ and $T : \mathbb { R } ^ { 4 } \longrightarrow W$ be a linear map with $T ^ { 2 } = T$. Suppose $T$ is onto. Pick the correct statement(s) from below.
(A) $T ( u + v ) = T ( u ) + v$ for all $u \in \mathbb { R } ^ { 4 } , v \in W$.
(B) $\operatorname { ker } ( T - I )$ contains three linearly independent vectors.
(C) $( 1,3,0,2 ) \in \operatorname { ker } ( T )$.
(D) If $v _ { 1 } , v _ { 2 } \in \operatorname { ker } ( T )$ are nonzero, then $v _ { 1 } = c v _ { 2 }$ for some $c \in \mathbb { R }$.

csat-suneung 2005 Q2 2 marks Linear System and Inverse Existence View

For two matrices $A = \left( \begin{array} { l l } 1 & 2 \\ 2 & 5 \end{array} \right) , B = \left( \begin{array} { l l } 2 & - 3 \\ 1 & - 2 \end{array} \right)$, what is the sum of all components of matrix $X$ that satisfies $A X = B$? [2 points]
(1) $- 2$
(2) $- 1$
(3) 0
(4) 1
(5) 2

csat-suneung 2005 Q2 2 marks Linear System and Inverse Existence View

For two matrices $A = \left( \begin{array} { l l } 1 & 2 \\ 2 & 5 \end{array} \right) , B = \left( \begin{array} { l l } 2 & - 3 \\ 1 & - 2 \end{array} \right)$, what is the sum of all components of matrix $X$ that satisfies $A X = B$? [2 points]
(1) - 2
(2) - 1
(3) 0
(4) 1
(5) 2

csat-suneung 2005 Q6 3 marks Matrix Algebra and Product Properties View

For square matrices $A$ and $B$ of order 2, select all statements that are always true from . (Here, $E$ is the identity matrix and $O$ is the zero matrix.) [3 points]
ㄱ. $( A + B ) ^ { 2 } = A ^ { 2 } + 2 A B + B ^ { 2 }$ ㄴ. If $A ^ { 2 } + A - 2 E = O$, then $A$ has an inverse matrix. ㄷ. If $A \neq O$ and $A ^ { 2 } = A$, then $A = E$.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄴ, ㄷ

csat-suneung 2005 Q8 3 marks Matrix Algebra and Product Properties View

The following table shows the manufacturing cost per unit, selling price, and sales volume for two products A and B produced by a company last year.

Category	Product A	Product B
Manufacturing Cost	$a _ { 11 }$	$a _ { 12 }$
Selling Price	$a _ { 21 }$	$a _ { 22 }$

Sales Volume	First Half	Second Half
A	$b _ { 11 }$	$b _ { 12 }$
B	$b _ { 21 }$	$b _ { 22 }$

Represent the above tables as matrices $A = \left( \begin{array} { l l } a _ { 11 } & a _ { 12 } \\ a _ { 21 } & a _ { 22 } \end{array} \right)$ and $B = \left( \begin{array} { l l } b _ { 11 } & b _ { 12 } \\ b _ { 21 } & b _ { 22 } \end{array} \right)$ respectively, and let the product of these two matrices be $A B = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$. When the profit per unit is defined as the selling price minus the manufacturing cost, select all correct statements from . [3 points]
ㄱ. $a + b$ is the total manufacturing cost of products sold in the first half of last year. ㄴ. $c + d$ is the total selling amount of products sold throughout last year. ㄷ. $d - b$ is the total profit from products sold in the second half of last year.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

csat-suneung 2005 Q18 Matrix Algebra and Product Properties View

For the quadratic equation $x ^ { 2 } - 4 x - 1 = 0$ with roots $\alpha$ and $\beta$, find the sum of all components of the product of two matrices $\left( \begin{array} { l l } \alpha & \beta \\ 0 & \alpha \end{array} \right) \left( \begin{array} { l l } \beta & \alpha \\ 0 & \beta \end{array} \right)$.

csat-suneung 2005 Q22 4 marks Matrix Entry and Coefficient Identities View

Natural numbers are arranged at regular intervals on the sides and vertices of squares with side lengths $1, 3, 5, \cdots, 2 n - 1, \cdots$ as shown in the figure below. In each square, 1 is placed directly above the lower left vertex.
Let the $2 \times 2$ matrices with the natural numbers at the four vertices of each square as components be $A _ { 1 } , A _ { 2 } , A _ { 3 } , \cdots , A _ { n } , \cdots$ in order. For example, $A _ { 1 } = \left( \begin{array} { l l } 1 & 2 \\ 4 & 3 \end{array} \right) , A _ { 2 } = \left( \begin{array} { c c } 3 & 6 \\ 12 & 9 \end{array} \right)$. Find the sum of all components of matrix $A _ { 15 }$. [4 points]

csat-suneung 2006 Q2 2 marks Matrix Algebra and Product Properties View

For two matrices $A = \left( \begin{array} { l l } 1 & 1 \\ 1 & 0 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 2 \\ 3 & 4 \end{array} \right)$, what is the matrix $X$ that satisfies $2 A + X = A B$? [2 points]
(1) $\left( \begin{array} { r r } 1 & 5 \\ 3 & - 1 \end{array} \right)$
(2) $\left( \begin{array} { r r } 2 & 4 \\ - 1 & 2 \end{array} \right)$
(3) $\left( \begin{array} { l l } 2 & 5 \\ 7 & 0 \end{array} \right)$
(4) $\left( \begin{array} { l l } 2 & 7 \\ 4 & 5 \end{array} \right)$
(5) $\left( \begin{array} { l l } 4 & 6 \\ 1 & 2 \end{array} \right)$

csat-suneung 2006 Q2 2 marks Matrix Algebra and Product Properties View

For two matrices $A = \left( \begin{array} { l l } 1 & 1 \\ 1 & 0 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 2 \\ 3 & 4 \end{array} \right)$, what is the matrix $X$ that satisfies $2 A + X = A B$? [2 points]
(1) $\left( \begin{array} { r r } 1 & 5 \\ 3 & - 1 \end{array} \right)$
(2) $\left( \begin{array} { r r } 2 & 4 \\ - 1 & 2 \end{array} \right)$
(3) $\left( \begin{array} { l l } 2 & 5 \\ 7 & 0 \end{array} \right)$
(4) $\left( \begin{array} { l l } 2 & 7 \\ 4 & 5 \end{array} \right)$
(5) $\left( \begin{array} { l l } 4 & 6 \\ 1 & 2 \end{array} \right)$

csat-suneung 2006 Q6 3 marks Matrix Algebra and Product Properties View

For all non-zero $2 \times 2$ square matrices $A , B$ satisfying the following three conditions, which matrix is always equal to $B ^ { 3 } + 2 B A ^ { 3 }$? (Here, $E$ is the identity matrix.) [3 points] (가) $A B = B A$ (나) $( E - B ) ^ { 2 } = E - B$ (다) $A B = - B$
(1) $2 A$
(2) $- A$
(3) $E$
(4) $2 B$
(5) $- B$

csat-suneung 2006 Q27 4 marks Matrix Power Computation and Application View

For two matrices $A = \left( \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$, define sets $S , T$ as $$\begin{aligned} & S = \left\{ \binom { x } { y } \left\lvert \, \binom { x } { y } = A ^ { n } \binom { 1 } { 1 } \right. , n \text { is a natural number} \right\} \\ & T = \left\{ \binom { x } { y } \left\lvert \, \binom { x } { y } = B ^ { n } \binom { 1 } { 1 } \right. , n \text { is a natural number} \right\} \end{aligned}$$ Which of the following in are correct? [4 points] 〈Remarks〉 ㄱ. If $\binom { a } { b } \in S$, then $\binom { b } { a } \in T$. ㄴ. If $\binom { a } { b } \in S , \binom { c } { d } \in S$, then $\binom { a + c } { b + d } \in S$. ㄷ. If $\binom { a } { b } \in S , \binom { p } { q } \in T$, then the matrix $\left( \begin{array} { l l } a & p \\ b & q \end{array} \right)$ has an inverse matrix.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

csat-suneung 2007 Q2 2 marks Linear System and Inverse Existence View

For two matrices $A = \left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right) , B = \left( \begin{array} { l l } 2 & 1 \\ 3 & 3 \end{array} \right)$, what is the sum of all components of the matrix $( A + B ) ^ { - 1 }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

csat-suneung 2007 Q2 2 marks Linear System and Inverse Existence View

For two matrices $A = \left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right) , B = \left( \begin{array} { l l } 2 & 1 \\ 3 & 3 \end{array} \right)$, what is the sum of all components of the matrix $( A + B ) ^ { - 1 }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

csat-suneung 2007 Q12 3 marks True/False or Multiple-Select Conceptual Reasoning View

Two $2 \times 2$ square matrices $A , B$ satisfy $A ^ { 2 } = E , B ^ { 2 } = B$. Which of the following statements in the given options are always true? (Note: $E$ is the identity matrix.) [3 points]
Given Options ㄱ. If matrix $B$ has an inverse matrix, then $B = E$. ㄴ. $( E - A ) ^ { 5 } = 2 ^ { 4 } ( E - A )$ ㄷ. $( E - A B A ) ^ { 2 } = E - A B A$
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

csat-suneung 2007 Q21 3 marks Linear System and Inverse Existence View

For a $2 \times 2$ square matrix $A$ satisfying $( A + E ) ^ { 2 } = A$ and a matrix $\binom { p } { q }$, $$\left( A + A ^ { - 1 } \right) \binom { p } { q } = \binom { 3 } { - 7 }$$ holds. Find the value of $p ^ { 2 } + q ^ { 2 }$. (Note: $E$ is the identity matrix.) [3 points]

csat-suneung 2007 Q30 4 marks Determinant and Rank Computation View

For a $2 \times 2$ square matrix $X = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$, $$D ( X ) = a d - b c$$ is defined. For a $2 \times 2$ square matrix $A = \left( \begin{array} { l l } 1 & 1 \\ 0 & p \end{array} \right)$, $$D \left( A ^ { 2 } \right) = D ( 5 A )$$ Find the sum of all constants $p$ that satisfy this condition. [4 points]

csat-suneung 2008 Q2 2 marks Matrix Algebra and Product Properties View

For two matrices $A = \left( \begin{array} { r r } 1 & - 2 \\ 3 & 0 \end{array} \right) , B = \left( \begin{array} { r r } 2 & 0 \\ 1 & - 1 \end{array} \right)$, what is the matrix $X$ that satisfies $A = 2 B - X$? [2 points]
(1) $\left( \begin{array} { r r } 3 & 2 \\ - 1 & - 2 \end{array} \right)$
(2) $\left( \begin{array} { r r } 3 & - 2 \\ 1 & 2 \end{array} \right)$
(3) $\left( \begin{array} { r r } - 1 & - 2 \\ 3 & 2 \end{array} \right)$
(4) $\left( \begin{array} { r r } - 2 & - 1 \\ 2 & 3 \end{array} \right)$
(5) $\left( \begin{array} { l l } - 3 & 1 \\ - 2 & 2 \end{array} \right)$

csat-suneung 2008 Q2 2 marks Matrix Algebra and Product Properties View

For two matrices $A = \left( \begin{array} { r r } 1 & - 2 \\ 3 & 0 \end{array} \right) , B = \left( \begin{array} { r r } 2 & 0 \\ 1 & - 1 \end{array} \right)$, what is the matrix $X$ that satisfies $A = 2 B - X$? [2 points]
(1) $\left( \begin{array} { r r } 3 & 2 \\ - 1 & - 2 \end{array} \right)$
(2) $\left( \begin{array} { r r } 3 & - 2 \\ 1 & 2 \end{array} \right)$
(3) $\left( \begin{array} { r r } - 1 & - 2 \\ 3 & 2 \end{array} \right)$
(4) $\left( \begin{array} { r r } - 2 & - 1 \\ 2 & 3 \end{array} \right)$
(5) $\left( \begin{array} { l l } - 3 & 1 \\ - 2 & 2 \end{array} \right)$

csat-suneung 2008 Q5 3 marks Linear System and Inverse Existence View

For the matrix $A = \left( \begin{array} { c c } 2 n & - 7 \\ - 1 & n \end{array} \right)$, what is the natural number $n$ such that all components of the inverse matrix $A ^ { - 1 }$ are natural numbers? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

csat-suneung 2008 Q15 4 marks True/False or Multiple-Select Conceptual Reasoning View

For two non-zero real numbers $a , b$, two square matrices $A , B$ satisfy $AB = \left( \begin{array} { l l } a & 0 \\ 0 & b \end{array} \right)$. Which of the following in are correct? [4 points]
ㄱ. If $a = b$, then the inverse matrix $A ^ { - 1 }$ of $A$ exists. ㄴ. If $a = b$, then $A B = B A$. ㄷ. If $a \neq b$ and $A = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$, then $A B = B A$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

csat-suneung 2008 Q20 3 marks Matrix Power Computation and Application View

For the matrix $A = \left( \begin{array} { l l } 1 & 0 \\ 3 & 1 \end{array} \right)$ with $A ^ { 8 } = \left( \begin{array} { l l } 1 & 0 \\ a & 1 \end{array} \right)$, find the value of $a$. [3 points]

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LFM Pure

Straight Lines & Coordinate Geometry 247

Circles 443

Simultaneous equations 79

Proof 711

Proof by induction 25

Sine and Cosine Rules 195

Radians, Arc Length and Sector Area 35

Trig Graphs & Exact Values 75

Standard trigonometric equations 106

Trig Proofs 53

Quadratic trigonometric equations 36

Trigonometric equations in context 18

Modulus function 49

Matrices 1553

Linear transformations 26

Invariant lines and eigenvalues and vectors 188

Reciprocal Trig & Identities 44

Addition & Double Angle Formulae 98

Harmonic Form 12

Small angle approximation 20

Chain Rule 113

Product & Quotient Rules 18

Differentiating Transcendental Functions 183

Connected Rates of Change 56

Standard Integrals and Reverse Chain Rule 43

Integration by Substitution 116

Integration by Parts 74

Implicit equations and differentiation 43

Differential equations 352

3x3 Matrices 144