LFM Pure

turkey-yks 2017 Q59 Area Computation in Coordinate Geometry View

Below are given squares $\mathrm { ABCD }$, $\mathrm { BLPR }$, and KLMN with side lengths of 3, 2, and 1 units respectively.
In the figure, points $\mathrm { A }$, $\mathrm { B }$, $\mathrm { K }$, and L are collinear.\ Accordingly, what is the area of triangle DNP in square units?\ A) 3\ B) 4\ C) 5\ D) 6\ E) 8

turkey-yks 2017 Q60 Section Ratio and Division of Segments View

The square ABCD given above is divided into four rectangles of equal area.
Accordingly, what is the ratio $\frac { | AE | } { | AD | }$?\ A) $\frac { 2 } { 3 }$\ B) $\frac { 3 } { 4 }$\ C) $\frac { 3 } { 5 }$\ D) $\frac { 5 } { 8 }$\ E) $\frac { 9 } { 16 }$

turkey-yks 2018 Q31 Area Computation in Coordinate Geometry View

In the Cartesian coordinate plane; a triangle with one vertex at the origin and the other vertices on the lines $y = x$ and $y = - x$ has its medians intersecting at point $(2,4)$.
Accordingly, what is the area of this triangle in square units?
A) 18 B) 24 C) 27 D) $9 \sqrt { 2 }$ E) $18 \sqrt { 2 }$

turkey-yks 2018 Q33 Line Equation and Parametric Representation View

The square shown in the figure in the Cartesian coordinate plane is divided into two regions of equal area by a line with slope $\frac { - 1 } { 4 }$.
If this line intersects the x-axis at point $(a, 0)$, what is a?
A) 12 B) 14 C) 16 D) 18 E) 20

turkey-yks 2019 Q6 Point-to-Line Distance Computation View

In the figure below, three points indicating the positions of apple, pear, and walnut trees in a garden located between a main street and a side street that intersect perpendicularly with each other and have straight edges are shown.
Of the trees in this garden, the one closest to the main street is the apple tree, and the one farthest is the pear tree.
Accordingly, which of the following is the correct ordering from the tree closest to the side street to the one farthest?
A) Pear - Walnut - Apple
B) Pear - Apple - Walnut
C) Walnut - Pear - Apple
D) Apple - Pear - Walnut
E) Apple - Walnut - Pear

turkey-yks 2019 Q34 Collinearity and Concurrency View

Let m be a real number. In the rectangular coordinate plane,

the slope of a line passing through the point $( 0,1 )$ is $m$,
the slope of a line passing through the point $( 0,0 )$ is $2 m$,
the slope of a line passing through the point $( 1,0 )$ is $3 m$, and these three lines intersect at one point.

Accordingly, what is the value of $m$?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 3 } { 5 }$
E) $\frac { 4 } { 5 }$

turkey-yks 2019 Q38 Line Equation and Parametric Representation View

Emre marks a point on the x-axis of the Cartesian coordinate plane in a mathematics class activity. Then, by decreasing the x-coordinate of this marked point by 1 unit and increasing the y-coordinate by 3 units, he obtains a second point, and when he applies the same operation to the second point, he obtains a third point on the y-axis.
What is the sum of the coordinates of the fourth point that Emre will obtain by applying the same operation to the third point?
A) 4
B) 5
C) 6
D) 7
E) 8

turkey-yks 2020 Q31 Geometric Figure on Coordinate Plane View

5 identical isosceles right triangles with right-angled side lengths of 1 unit are arranged as shown in Figure 1 such that their hypotenuses are on the same line and the vertices of adjacent triangles coincide.
Then triangle $ABC$ is rotated around point $A$ by some amount, and as shown in Figure 2, points B, C, and D become collinear.
Accordingly, what is the distance between points C and D in the final position in units?
A) 4
B) 5
C) 6
D) $3\sqrt{2}$
E) $4\sqrt{2}$

turkey-yks 2020 Q37 Geometric Figure on Coordinate Plane View

In the rectangular coordinate plane, a square $ABCD$ with two vertices at $A(0, a)$ and $B(0, b)$ is given. The vertex $C$ of square $ABCD$ lies on the line $y = \frac{x}{3}$. If $a + b = 15$, what is the sum of the coordinates of point $D$?
A) 14
B) 18
C) 21
D) 24
E) 27

turkey-yks 2020 Q38 Line Equation and Parametric Representation View

In the rectangular coordinate plane, it is known that a line $d$ passes through point $A(-4, 1)$ and is perpendicular to the line $2x - y = 5$. If the point where line $d$ intersects the x-axis is $(a, 0)$ and the point where it intersects the y-axis is $(0, b)$, what is the sum $a + b$?
A) -3
B) -1
C) 0
D) 1
E) 3

turkey-yks 2020 Q39 Area Computation in Coordinate Geometry View

In the rectangular coordinate plane, points $A(2, 7)$ and $B(-1, 4)$ are translated 3 units in the positive direction along the x-axis to obtain points $D$ and $C$ respectively.
Accordingly, what is the area of the quadrilateral with vertices at points A, B, C, and D in square units?
A) 9
B) 10
C) 11
D) 12
E) 13

turkey-yks 2021 Q32 Triangle Properties and Special Points View

In the rectangular coordinate plane, one vertex of a triangle is at the origin, its centroid is at the point $( 0,6 )$, and its orthocenter is at the point $( 0,8 )$.
Accordingly, what is the area of this triangle in square units?
A) 18
B) 21
C) 24
D) 27
E) 30

turkey-yks 2021 Q33 Line Equation and Parametric Representation View

In the rectangular coordinate plane, points A and B lie on the line $y = x + 2$, and the distance between them is 3 units.
Given that the coordinates of the midpoint of segment [AB] are $( -1, 1 )$, in which regions of the analytic plane are points A and B located?
A) Both in region II
B) Both in region III
C) One in region I, the other in region II
D) One in region I, the other in region III
E) One in region II, the other in region III

turkey-yks 2021 Q34 Area Computation in Coordinate Geometry View

In the rectangular coordinate plane, two lines that intersect perpendicularly at point $A ( 3,4 )$ have slopes whose sum is $\frac { 3 } { 2 }$.
If the points where these two lines intersect the x-axis are points B and C, what is the area of triangle ABC in square units?
A) 24
B) 20
C) 16
D) 12
E) 8

turkey-yks 2021 Q38 Reflection and Image in a Line View

In the rectangular coordinate plane, the symmetric point of $( 4,4 )$ with respect to a line passing through $( 1,0 )$ is $( a , 0 )$. Accordingly, what is the product of the values that $a$ can take?
A) $-24$
B) $-16$
C) $-8$
D) $16$
E) $32$

turkey-yks 2023 Q38 Slope and Angle Between Lines View

In the rectangular coordinate plane, the line $2x + y = 12$ and a line d intersect at point $\mathrm{A}(4,4)$. These two lines divide every circle centered at point $\mathrm{A}(4,4)$ into four equal areas.
Accordingly, which of the following is the equation of line d?
A) $-2x + y = -4$ B) $x - 3y = -8$ C) $3x + y = 16$ D) $x + 2y = 12$ E) $x - 2y = -4$

turkey-yks 2024 Q35 Triangle Properties and Special Points View

In a rectangular coordinate plane, points $A(9,2)$, $B(10,1)$, $C$, $D(4,13)$, $E(3,6)$ and $F$ are given.
Given that the centroid of triangle $ABC$ and the centroid of triangle $DEF$ are the same point, what is the distance between points $C$ and $F$ in units?
A) 10 B) 13 C) 15 D) 17 E) 20

turkey-yks 2024 Q36 Geometric Figure on Coordinate Plane View

In the rectangular coordinate plane below, a red square with one side on the $y$-axis and a blue square with one side on the $x$-axis share a common vertex.
One vertex of each of the red and blue squares lies on the line $\dfrac{x}{2} + \dfrac{y}{3} = 1$.
According to this, what is the side length of the red square in units?
A) $\dfrac{14}{15}$ B) $\dfrac{15}{16}$ C) $\dfrac{16}{17}$ D) $\dfrac{17}{18}$ E) $\dfrac{18}{19}$

turkey-yks 2024 Q38 Reflection and Image in a Line View

In a rectangular coordinate plane, point $A(a, b)$; its reflection with respect to point $B(3, 0)$ is point $C$, and its reflection with respect to the $y$-axis is point $D$.
Given that the equation of the line passing through points $C$ and $D$ is $y = -x - 1$, what is the sum $a + b$?
A) 7 B) 13 C) 15 D) 19 E) 24

turkey-yks 2025 Q35 Area Computation in Coordinate Geometry View

In the rectangular coordinate plane, a triangle $OAB$ with one vertex at the origin and the other two vertices on the axes, and the line segment $[PR]$ connecting the points $P(6, -3)$ and $R(-2, 9)$ are drawn. The line segment $[PR]$ passes through the midpoints of both $[OA]$ and $[OB]$.
According to this, what is the area of triangle $OAB$ in square units?
A) 36 B) 42 C) 48 D) 54 E) 60

turkey-yks 2025 Q36 Geometric Figure on Coordinate Plane View

Let $a$ and $b$ be positive real numbers. In the rectangular coordinate plane, the region between the lines $y = -\sqrt{3}x$ and $y = ax + b$ and the $x$-axis forms an equilateral triangle with area $9\sqrt{3}$ square units.
Accordingly, what is the product $a \cdot b$?
A) 18 B) 24 C) 27 D) 30 E) 36

turkey-yks 2025 Q38 Line Equation and Parametric Representation View

In the rectangular coordinate plane, when point $A$ is translated 15 units in the negative direction along the $x$-axis, the resulting point lies on the line $d: 4x - 3y + 24 = 0$.
Accordingly, if point $A$ is translated how many units in the positive direction along the $y$-axis, the resulting point will lie on line $d$?
A) 9 B) 12 C) 16 D) 20 E) 25

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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