LFM Pure

taiwan-gsat 2024 Q9 6 marks View

Let $a, b, c, d$ be real numbers. It is known that the augmented matrices of two systems of linear equations $\left\{\begin{array}{l} ax + by = 2 \\ cx + dy = 1 \end{array}\right.$ and $\left\{\begin{array}{l} ax + by = -1 \\ cx + dy = -1 \end{array}\right.$, after the same row operations, become $\left[\begin{array}{cc|c} 1 & -1 & 3 \\ 0 & 1 & 2 \end{array}\right]$ and $\left[\begin{array}{cc|c} 1 & -1 & 2 \\ 0 & 1 & -1 \end{array}\right]$ respectively. Then the solution to the system of equations $\left\{\begin{array}{l} ax + by = 0 \\ cx + dy = 1 \end{array}\right.$ is $x = $ 9-1, 9-2, $y = $ 9-3.

turkey-yks 2010 Q8 View

$$\frac { 1 } { 2 } - 3 a = \frac { 1 } { 8 } + 3 b$$
Given this, what is the sum $\mathbf { a } + \mathbf { b }$?
A) $\frac { 3 } { 4 }$
B) $\frac { 5 } { 6 }$
C) $\frac { 1 } { 8 }$
D) $\frac { 5 } { 8 }$
E) $\frac { 4 } { 9 }$

turkey-yks 2010 Q10 View

$$\begin{aligned} & x ^ { 3 } - 2 y = 7 \\ & x ^ { 4 } - 2 x y = 21 \end{aligned}$$
Given this, what is $\mathbf { x }$?
A) 3
B) 5
C) 7
D) 9
E) 11

turkey-yks 2010 Q12 View

Let $\mathrm { a } , \mathrm { b } , \mathrm { x }$ and y be positive numbers such that
$$\begin{aligned} & \frac { x } { a } \cdot \frac { b } { y } = 2 \\ & \frac { a ^ { 2 } } { x ^ { 2 } } + \frac { b ^ { 2 } } { y ^ { 2 } } = 20 \end{aligned}$$
Given this, which of the following is the value of x in terms of a?
A) $\frac { a } { 2 }$
B) $\frac { 3 a } { 4 }$
C) $\frac { 3 a } { 5 }$
D) $\frac { 4 a } { 5 }$
E) $\frac { 5 a } { 6 }$

turkey-yks 2010 Q19 View

The sum of a three-digit number $ABC$ and a two-digit number $AB$ is 392.
Accordingly, what is the sum $\mathrm { A } + \mathrm { B } + \mathrm { C }$?
A) 7
B) 9
C) 11
D) 15
E) 19

turkey-yks 2011 Q8 View

$\frac{a - 1}{b} = \frac{c}{a}$
$$\frac{a}{c - 2} = \frac{b + 3}{a - 1}$$
Given that, what is the value of the expression $3c - 2b$?
A) 8 B) 9 C) 6 D) 3 E) 4

turkey-yks 2011 Q11 View

For distinct numbers a and b
$$\frac{a^{2}}{b} - \frac{b^{2}}{a} = b - a$$
Given that, what is the value of the expression $\frac{a}{b} + \frac{b}{a}$?
A) $-2$ B) $-1$ C) 0 D) 1 E) 4

turkey-yks 2012 Q6 View

$$\frac { a - 1 } { a - 3 } = \frac { a - 5 } { a - 4 }$$
Given that, what is a?
A) $\frac { 8 } { 5 }$
B) $\frac { 13 } { 4 }$
C) $\frac { 9 } { 4 }$

turkey-yks 2012 Q7 View

Let x and y be real numbers such that
$$\begin{aligned} & x ^ { 2 } - 4 y = - 7 \\ & y ^ { 2 } - 2 x = 2 \end{aligned}$$
Given this, what is the sum $x + y$?
A) 3
B) 4
C) 5
D) $\frac { 4 } { 3 }$
E) $\frac { 5 } { 3 }$

turkey-yks 2012 Q10 View

For real numbers $x , y$ and $z$
$$\begin{aligned} & x \cdot y = 14 \\ & x \cdot z = 20 \\ & 3x + 2y + z = 24 \end{aligned}$$
Given that, what is x?
A) $\frac { 8 } { 3 }$
B) $\frac { 14 } { 5 }$
C) 3

turkey-yks 2012 Q38 View

$$\lim _ { x \rightarrow 0 } \frac { \sin 3 x } { 2 - \sqrt { 4 - x } }$$
What is the value of this limit?
A) 3
B) 9
C) 12
D) 15
E) 16

turkey-yks 2013 Q7 View

$\mathbf { a } , \mathbf { b } , \mathbf { c }$ are non-zero real numbers and $\mathbf { a } + \mathbf { b } + \mathbf { c } = \mathbf { a b }$. Given this,
$$\frac { a b + a c + b c + c ^ { 2 } } { a b c }$$
Which of the following is this expression equal to?
A) $\frac { a + 1 } { a }$
B) $\frac { b + 1 } { b }$
C) $\frac { c + 1 } { c }$
D) $\frac { b } { a }$
E) $\frac { b } { c }$

turkey-yks 2013 Q22 View

If Ahmet's salary is increased by half of Deniz's salary, then the sum of their salaries becomes 2 times Ahmet's initial salary.
If Ahmet's salary is A TL and Deniz's salary is D TL, what is the relationship between $A$ and $D$?
A) $5A = 8D$
B) $5A = 6D$
C) $4A = 5D$
D) $3A = 4D$
E) $2A = 3D$

turkey-yks 2013 Q24 View

In a laboratory, the following are known about a drug experiment conducted on male and female guinea pig mice.

Male mice were given 1 tablet of medicine every 12 hours, and female mice were given 1 tablet every 8 hours.
Male mice were given 0.5 gram tablets, and female mice were given 1 gram tablets.
A total of 85 grams of medicine was given to these mice in one day, in the form of 95 tablets.

Accordingly, how many mice were used in the experiment?
A) 20
B) 25
C) 30
D) 35
E) 40

turkey-yks 2013 Q25 View

Stationery materials are to be distributed to students in a class. Enough materials are brought to the class so that each of the 36 students receives one pencil, one pencil sharpener, and one eraser. However, on the distribution day, since some students are absent, each student present in the class is given 3 pencils, 2 pencil sharpeners, and 1 eraser.
If a total of 42 items of these materials remain after distribution, how many pencil sharpeners are left?
A) 10
B) 11
C) 12
D) 13
E) 14

turkey-yks 2014 Q10 View

Let $\mathbf { k }$ be a nonzero real number such that
$$\begin{aligned} & x ^ { 2 } + y ^ { 2 } = ( 6 k ) ^ { 2 } \\ & ( x - 2 k ) ^ { 2 } + y ^ { 2 } = ( 2 k \sqrt { 5 } ) ^ { 2 } \end{aligned}$$
Accordingly, which of the following is the equivalent of $x ^ { 2 } - y ^ { 2 }$ in terms of $k$?
A) $13 \mathrm { k } ^ { 2 }$
B) $14 \mathrm { k } ^ { 2 }$
C) $15 k ^ { 2 }$

turkey-yks 2017 Q15 View

Sets $A$, $B$, and $C$ are defined as $$\begin{aligned}& A = \{ ( x , x ) : x \in \mathbb { R } \} \\& B = \{ ( x , 3 - x ) : x \in \mathbb { R } \} \\& C = \{ ( x , x + 4 ) : x \in \mathbb { R } \}\end{aligned}$$ Given that $( p , q ) \in A \cap B$ and $( r , s ) \in B \cap C$, $$\frac { p - r } { q + s }$$ what is the value of this expression?\ A) $\frac { 1 } { 3 }$\ B) $\frac { 1 } { 4 }$\ C) $\frac { 3 } { 4 }$\ D) $\frac { 4 } { 5 }$\ E) $\frac { 2 } { 5 }$

turkey-yks 2020 Q24 View

Boxes A, B, C, and D contain a certain number of balls. The number of balls in box A is:

equal to 2 times the number of balls in box B,
equal to 3 times the number of balls in box C,
equal to 4 times the number of balls in box D.

If one of the boxes contains 8 balls, how many balls are there in total in these boxes?
A) 30
B) 36
C) 40
D) 44
E) 50

turkey-yks 2020 Q25 View

Districts A, B, and C and the roads between these districts are shown in the figure below.
The road distances of points D and E, which are on these roads, to some districts are given in the tables on the signs.
Accordingly, what is the difference between the road distance from district C to district B and the road distance from district C to district A in km?
A) 6
B) 8
C) 10
D) 12
E) 14

turkey-yks 2020 Q26 View

\c{C}\i{}nar has a total of 78 pens, some of which are blue. He distributed these pens among three pen holders as follows.

The number of pens in the pen holders is directly proportional to 3, 4, and 6.
The number of blue pens in each pen holder is equal to each other.
In one of the pen holders, the ratio of the number of blue pens to the total number of pens in that holder is $\frac{1}{2}$; in another pen holder, this ratio is $\frac{1}{3}$.

Accordingly, how many blue pens does \c{C}\i{}nar have in total?
A) 18
B) 24
C) 27
D) 30
E) 36

turkey-yks 2020 Q27 View

A group of students, each 7 years old, visited a botanical garden in 2015; another group of students, each 10 years old, visited in 2020. The official who guided the groups through the garden said about the same historical tree in the garden to both groups: ``The age of this tree is equal to the sum of all of your ages.''
From these two groups, if the number of students in the first group is 10 more than the number of students in the second group, how old is this tree in 2020?
A) 220
B) 230
C) 240
D) 250
E) 260

turkey-yks 2020 Q33 View

A rectangular towel has one side blue and the other side white. This towel is hung on a straight rack such that the short sides of the towel are parallel to the rack. The length of the non-overlapping part of the towel's sides is 6 cm when the towel is hung as in Figure 1; and 12 cm when hung as in Figure 2.
The ratio of the area of the blue side of the towel visible in Figure 1 to the area visible in Figure 2 is $\frac{5}{4}$.
Accordingly, what is the length of the long side of the towel in cm?
A) 24
B) 28
C) 30
D) 36
E) 40

turkey-yks 2020 Q34 View

The front view of a cabinet consisting of five compartments shown in the figure is square-shaped. The door of each compartment is a rectangle with equal areas.
One compartment has been filled with only shirts as shown in the figure.
Accordingly, how many times is the long side of the lid of the compartment containing the shirts compared to its short side?
A) $\frac{4}{3}$
B) $\frac{5}{3}$
C) $\frac{7}{4}$

turkey-yks 2020 Q38 View

A wooden piece in the shape of a square right prism with a square base has a base edge length equal to 2 times its height. When a cube with an edge length equal to the height of the wooden piece is removed from inside this wooden piece, the surface area of the resulting shape in the final state is 8 square units more than the surface area of the wooden piece in the initial state.
Accordingly, what is the volume of the wooden piece in the initial state in cubic units?
A) 32
B) 80
C) 108
D) 144
E) 256

turkey-yks 2020 Q39 View

The volume of a rectangular prism is equal to the product of its base area and height.
A closed glass container in the shape of a rectangular prism contains 360 cubic units of water. When the container is placed on a flat surface with different faces completely touching the surface, the height of the water is 2 units, 4 units, and 5 units respectively.
Accordingly, what is the volume of the container in cubic units?
A) 540
B) 720
C) 840
D) 960
E) 4080

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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