LFM Pure and Mechanics

tmua 2016 Q3 1 marks Normal or perpendicular line problems View

A line is drawn normal to the curve $y = \frac { 2 } { x ^ { 2 } }$ at the point on the curve where $x = 1$. This line cuts the $x$-axis at $P$ and the $y$-axis at $Q$. The length of $P Q$ is
A $\frac { 3 \sqrt { 5 } } { 2 }$ B $\frac { 3 \sqrt { 17 } } { 4 }$ C $\frac { 7 \sqrt { 17 } } { 4 }$ D $\frac { 35 } { 4 }$ E $\frac { 35 \sqrt { 5 } } { 2 }$ F $\frac { 3 \sqrt { 17 } } { 2 }$

tmua 2018 Q11 1 marks Normal or perpendicular line problems View

The line $y = m x + 5$, where $m > 0$, is normal to the curve $y = 10 - x ^ { 2 }$ at the point ( $p , q$ ).
What is the value of $p$ ?
A $\frac { \sqrt { 2 } } { 6 }$
B $- \frac { \sqrt { 2 } } { 6 }$
C $\frac { 3 \sqrt { 2 } } { 2 }$
D $- \frac { 3 \sqrt { 2 } } { 2 }$
E $\sqrt { 5 }$
F $- \sqrt { 5 }$

turkey-yks 2010 Q40 Determine unknown parameters from tangent conditions View

For the function $f(x) = 2x^{3} - ax^{2} + 3$, what should $a$ be so that the equation of the tangent line to the curve at some point is $y = 4$?
A) $-3$
B) $-1$
C) $0$
D) $1$
E) $3$

turkey-yks 2010 Q43 Find tangent line with a specified slope or from an external point View

The tangent line drawn from a point $A(x, y)$ on the parabola $y^{2} = 4x$ has slope 1.
Accordingly, what is $x + y$, the sum of the coordinates of point $A$?
A) 1
B) 2
C) 3
D) 4
E) 5

turkey-yks 2010 Q45 Determine unknown parameters from tangent conditions View

If the line tangent to the parabola $y = x^{2} + bx + c$ at the point $x = 2$ is $y = x$, what is the sum $b + c$?
A) $-2$
B) $-1$
C) $0$
D) $1$
E) $2$

turkey-yks 2011 Q42 Geometric properties of tangent lines (intersections, lengths, areas) View

At what point does the tangent line to the curve $\mathbf { y } = \sin ( \pi \mathrm { x } ) + \mathrm { e } ^ { \mathrm { x } }$ at $\mathrm { x } = 1$ intersect the y-axis?
A) $- \pi$
B) - 1
C) 0
D) $e - 1$
E) $\pi$

turkey-yks 2013 Q43 Determine unknown parameters from tangent conditions View

Given that the function $f$ has derivative $f ^ { \prime } ( x ) = 3 x ^ { 2 }$ and the tangent line at the point $x = a ( a > 0 )$ is the line $y - 12 x + 14 = 0$, what is the value of $f ( 1 )$?
A) $- 2$
B) 0
C) 1
D) 3
E) 5

turkey-yks 2014 Q41 Find tangent line equation at a given point View

The tangent line drawn to the graph of the function $y = f ( x )$ at the point $( 2,4 )$ passes through the point $( - 1,3 )$.
Accordingly, what is the value of $f ^ { \prime } ( 2 )$?
A) $\frac { 1 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 4 } { 3 }$
E) $\frac { 3 } { 5 }$

turkey-yks 2014 Q44 Find tangent line with a specified slope or from an external point View

$$x ^ { 2 } - y ^ { 2 } = 1$$
What is the distance between the points where the lines tangent to the hyperbola curve and having slope 3 intersect the y-axis, in units?
A) $\sqrt { 2 }$
B) $2 \sqrt { 2 }$
C) $4 \sqrt { 2 }$
D) $\sqrt { 3 }$
E) $2 \sqrt { 3 }$

turkey-yks 2015 Q44 Prove a given line is tangent to a curve View

The line $y = 4 x - 2$ is tangent to the graph of the function $f ( x ) = x ^ { 4 } + 1$ at the point $\mathrm { P } ( \mathrm { a } , \mathrm { b } )$.
Accordingly, what is the sum $a + b$?
A) 3
B) 4
C) 5
D) 6
E) 7

turkey-yks 2016 Q42 Determine unknown parameters from tangent conditions View

Let a and b be real numbers. In the rectangular coordinate plane, the parabola
$$y = a x ^ { 2 } + b x$$
passes through the point $( 1,2 )$, and the tangent line to the parabola at this point intersects the y-axis at the point $( 0,1 )$.
Accordingly, what is the product $a \cdot b$?
A) - 3
B) - 2
C) - 1
D) 2
E) 4

turkey-yks 2017 Q42 Determine unknown parameters from tangent conditions View

Let a and b be real numbers, and $$f ( x ) = a \cdot \ln x + b \cdot x ^ { 2 } + 3$$ The equation of the tangent line drawn to the graph of the function at the point $(1, f(1))$ is given as $y - 2x + 1 = 0$.
Accordingly, what is the product $\mathbf{a} \cdot \mathbf{b}$?\ A) $- 18$\ B) $- 16$\ C) $- 12$\ D) $- 8$\ E) $- 6$

turkey-yks 2018 Q21 Determine unknown parameters from tangent conditions View

Let a, b and c be real numbers. The equation of the tangent line to the curve
$$y = \frac { a } { x + a }$$
at point $P ( a , b )$ is given in the form
$$y = \frac { - x } { 8 } + c$$
Accordingly, what is the sum $a + b + c$?
A) $\frac { 7 } { 4 }$ B) $\frac { 11 } { 4 }$ C) $\frac { 13 } { 4 }$ D) 2 E) 3

turkey-yks 2019 Q25 Ordering and Comparing Exponential Values View

Below; the graphs of linear functions $f$, $g$ and $h$ are shown in Figure 1 on a rectangular coordinate plane divided into unit squares, and the derivatives of these functions are shown in Figure 2.
Accordingly; what is the correct ordering of $f ( 0 ) , g ( 0 )$ and $h ( 0 )$?
A) $\mathrm { f } ( 0 ) < \mathrm { h } ( 0 ) < \mathrm { g } ( 0 )$
B) $g ( 0 ) < f ( 0 ) < h ( 0 )$
C) $g ( 0 ) < h ( 0 ) < f ( 0 )$
D) $h ( 0 ) < f ( 0 ) < g ( 0 )$
E) $h ( 0 ) < g ( 0 ) < f ( 0 )$

turkey-yks 2019 Q26 Common tangent line to two curves View

In the rectangular coordinate plane, the tangent line drawn to the graph of the function $f ( x ) = x ^ { 2 } + a x$ at the point $( 2 , f ( 2 ) )$ is tangent to the graph of the function $g ( x ) = b x ^ { 3 }$ at the point $( 1 , g ( 1 ) )$. Accordingly, what is the product $\mathbf { a } \cdot \mathbf { b }$?
A) 2
B) 4
C) 6
D) 8
E) 10

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LFM Pure and Mechanics

Indices and Surds 96

Exponential Functions 197

Laws of Logarithms 242

Exponential Equations & Modelling 28

Arithmetic Sequences and Series 363

Geometric Sequences and Series 159

Differentiation from First Principles 45

Tangents, normals and gradients 140

Stationary points and optimisation 505

Applied differentiation 76

Indefinite & Definite Integrals 502

Areas Between Curves 71

Areas by integration 221

Volumes of Revolution 68

Numerical integration 49

Vectors Introduction & 2D 211

Vectors 3D & Lines 524

Travel graphs 14

SUVAT in 2D & Gravity 31

Constant acceleration (SUVAT) 78

Variable acceleration (1D) 55

Variable acceleration (vectors) 32

Forces, equilibrium and resultants 24

Newton's laws and connected particles 21

Pulley systems 6

Motion on a slope 6

Friction 8

Momentum and Collisions 34

Moments 28

Parametric curves and Cartesian conversion 9

Parametric differentiation 13

Parametric integration 6

Projectiles 49