LFM Pure and Mechanics

csat-suneung 2026 Q9 4 marks Find tangent line with a specified slope or from an external point View

For a positive number $a$, let the function $f ( x )$ be $$f ( x ) = x ^ { 3 } + 3 a x ^ { 2 } - 9 a ^ { 2 } x + 4$$ When the line $y = 5$ is tangent to the curve $y = f ( x )$, what is the value of $f ( 2 )$? [4 points]
(1) 11
(2) 12
(3) 13
(4) 14
(5) 15

csat-suneung 2026 Q13 4 marks Geometric properties of tangent lines (intersections, lengths, areas) View

For the function $f ( x ) = x ^ { 2 } - 4 x - 3$, let $l$ be the tangent line to the curve $y = f ( x )$ at the point $( 1 , - 6 )$, and for the function $g ( x ) = \left( x ^ { 3 } - 2 x \right) f ( x )$, let $m$ be the tangent line to the curve $y = g ( x )$ at the point $( 1,6 )$. What is the area of the figure enclosed by the two lines $l , m$ and the $y$-axis? [4 points]
(1) 21
(2) 28
(3) 35
(4) 42
(5) 49

gaokao 2015 Q15 Normal or perpendicular line problems View

15. The tangent line to the curve $y = e ^ { x }$ at the point $( 0,1 )$ is perpendicular to the tangent line to the curve $y = \frac { 1 } { x } ( x > 0 )$ at point P. The coordinates of P are $\_\_\_\_$

gaokao 2015 Q18 Find tangent line equation at a given point View

18. (This question is worth 13 points) Given the function $f ( x ) = \ln \frac { 1 + x } { 1 - x }$. (I) Find the equation of the tangent line to the curve $y = f ( x )$ at the point $( 0 , f ( 0 ) )$; (II) Prove: When $x \in ( 0,1 )$, $f ( x ) > 2 \left( x + \frac { x ^ { 3 } } { 3 } \right)$; (III) Let the real number $k$ be such that $f ( x ) > k \left( x + \frac { x ^ { 3 } } { 3 } \right)$ holds for all $x \in ( 0,1 )$. Find the maximum value of $k$.

gaokao 2017 Q20 12 marks Find tangent line equation at a given point View

(12 points)
Let $A$ and $B$ be two points on the curve $C: y = \frac{x^2}{4}$, and the sum of the $x$-coordinates of $A$ and $B$ is 4.
(1) Find the slope of line $AB$;
(2) Find the equation of line $AB$.

gaokao 2018 Q5 5 marks Determine unknown parameters from tangent conditions View

Let $f ( x ) = x ^ { 3 } + ( a - 1 ) x ^ { 2 } + a x$. If $f ( x )$ is an odd function, then the equation of the tangent line to the curve $y = f ( x )$ at the point $( 0,0 )$ is
A. $y = - 2 x$
B. $y = - x$
C. $y = 2 x$
D. $y = x$

gaokao 2018 Q6 5 marks Determine unknown parameters from tangent conditions View

Let $f ( x ) = x ^ { 3 } + ( a - 1 ) x ^ { 2 } + a x$. If $f ( x )$ is an odd function, then the equation of the tangent line to $y = f ( x )$ at the point $( 0,0 )$ is
A. $y = - 2 x$
B. $y = - x$
C. $y = 2 x$
D. $y = x$

gaokao 2018 Q13 5 marks Find tangent line equation at a given point View

The equation of the tangent line to the curve $y = 2 \ln x$ at the point $( 1,0 )$ is \_\_\_\_.

gaokao 2018 Q13 5 marks Find tangent line equation at a given point View

The equation of the tangent line to the curve $y = 2 \ln ( x + 1 )$ at the point $( 0,0 )$ is $\_\_\_\_$.

gaokao 2019 Q6 5 marks Determine unknown parameters from tangent conditions View

The tangent line to the curve $y = a \mathrm { e } ^ { x } + x \ln x$ at the point $( 1 , a \mathrm { e } )$ has equation $y = 2 x + b$ . Then
A. $a = \mathrm { e } , b = - 1$
B. $a = \mathrm { e } , b = 1$
C. $a = \mathrm { e } ^ { - 1 } , b = 1$
D. $a = \mathrm { e } ^ { - 1 } , b = - 1$

gaokao 2019 Q6 Determine unknown parameters from tangent conditions View

6. The tangent line to the curve $y = a \mathrm { e } ^ { x } + x \ln x$ at the point $( 1 , a \mathrm { e } )$ has equation $y = 2 x + b$ . Then
A. $a = \mathrm { e } , \quad b = - 1$
B. $a = \mathrm { e } , b = 1$
C. $a = \mathrm { e } ^ { - 1 } , b = 1$
D. $a = \mathrm { e } ^ { - 1 } , b = - 1$

gaokao 2019 Q7 Determine unknown parameters from tangent conditions View

7. The tangent line to the curve $y = a \mathrm { e } ^ { x } + x \ln x$ at the point $( 1 , a e )$ has equation $y = 2 x + b$ . Then
A. $a = \mathrm { e } , b = - 1$
B. $a = \mathrm { e } , b = 1$
C. $a = \mathrm { e } ^ { - 1 } , b = 1$
D. $a = \mathrm { e } ^ { - 1 } , b = - 1$

gaokao 2019 Q10 Geometric properties of tangent lines (intersections, lengths, areas) View

10. The system of inequalities $\left\{ \begin{array} { l } x - 1 \geq 0 , \\ k x - y \leq 0 , \\ x + \sqrt { 3 } y - 3 \sqrt { 3 } \leq 0 \end{array} \right.$ represents a planar region that is an equilateral triangle. The minimum value of $z = x + 3 y$ is
A. $2 + 3 \sqrt { 3 }$ B. $1 + 3 \sqrt { 3 }$ C. $2 + \sqrt { 3 }$ D. $1 + \sqrt { 3 }$

gaokao 2019 Q10 Find tangent line equation at a given point View

10. The equation of the tangent line to the curve $y = 2 \sin x + \cos x$ at the point $( \pi , - 1 )$ is
A. $x - y - \pi - 1 = 0$
B. $2 x - y - 2 \pi - 1 = 0$
C. $2 x + y - 2 \pi + 1 = 0$
D. $x + y - \pi + 1 = 0$

gaokao 2020 Q6 5 marks Find tangent line equation at a given point View

The equation of the tangent line to the graph of $f ( x ) = x ^ { 4 } - 2 x ^ { 3 }$ at the point $( 1 , f ( 1 ) )$ is
A. $y = - 2 x - 1$
B. $y = - 2 x + 1$
C. $y = 2 x - 3$
D. $y = 2 x + 1$

gaokao 2020 Q10 5 marks Common tangent line to two curves View

If line $l$ is tangent to both the curve $y = \sqrt { x }$ and the circle $x ^ { 2 } + y ^ { 2 } = \frac { 1 } { 5 }$ , then the equation of $l$ is
A. $y = 2 x + 1$
B. $y = 2 x + \frac { 1 } { 2 }$
C. $y = \frac { 1 } { 2 } x + 1$
D. $y = \frac { 1 } { 2 } x + \frac { 1 } { 2 }$

gaokao 2020 Q15 5 marks Find tangent line with a specified slope or from an external point View

A tangent line to the curve $y = \ln x + x + 1$ has slope 2. The equation of this tangent line is $\_\_\_\_$.

gaokao 2021 Q13 Find tangent line equation at a given point View

13. The equation of the tangent line to the curve $y = \frac{2x - 1}{x + 2}$ at the point $(-1, -3)$ is $\_\_\_\_$.

gaokao 2022 Q15 Existence or count of tangent lines with given properties View

15. If the curve $y = ( x + a ) \mathrm { e } ^ { x }$ has two tangent lines passing through the origin, then the range of $a$ is $\_\_\_\_$ .

gaokao 2024 Q13 5 marks Common tangent line to two curves View

If the tangent line to the curve $y = \mathrm { e } ^ { x } + x$ at the point $( 0,1 )$ is also a tangent line to the curve $y = \ln ( x + 1 ) + a$ , then $a = $ $\_\_\_\_$ .

gaokao 2025 Q12 5 marks Determine unknown parameters from tangent conditions View

If the line $y = 2x + 5$ is tangent to the curve $y = \mathrm{e}^x + x + a$, then $a = $ $\_\_\_\_$ .

gaokao 2025 Q12 5 marks Determine unknown parameters from tangent conditions View

If the line $y = 2x + 5$ is tangent to the curve $y = e^x + x + a$, then $a = $ $\_\_\_\_$ .

isi-entrance 2006 Q1 Geometric properties of tangent lines (intersections, lengths, areas) View

If $x^{2/3} + y^{2/3} = a^{1/3}$, find the equation of the tangent to the curve at a point, and show that the length of the tangent intercepted between the axes is constant.

jee-advanced 2007 Q54 Geometric properties of tangent lines (intersections, lengths, areas) View

The tangent to the curve $y = e^x$ drawn at the point $(c, e^c)$ intersects the line joining the points $(c-1, e^{c-1})$ and $(c+1, e^{c+1})$
(A) on the left of $x = c$
(B) on the right of $x = c$
(C) at no point
(D) at all points

jee-advanced 2014 Q52 Normal or perpendicular line problems View

If $st = 1$, then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is
(A) $\frac{(t^2+1)^2}{2t^3}$
(B) $\frac{a(t^2+1)^2}{2t^3}$
(C) $\frac{a(t^2+1)^2}{t^3}$
(D) $\frac{a(t^2+2)^2}{t^3}$

Load questions...

LFM Pure and Mechanics

Indices and Surds 107

Exponential Functions 100

Laws of Logarithms 200

Exponential Equations & Modelling 58

Arithmetic Sequences and Series 271

Geometric Sequences and Series 203

Differentiation from First Principles 37

Tangents, normals and gradients 93

Stationary points and optimisation 384

Applied differentiation 164

Indefinite & Definite Integrals 533

Areas Between Curves 83

Areas by integration 109

Volumes of Revolution 61

Numerical integration 32

Vectors Introduction & 2D 204

Vectors 3D & Lines 233

Travel graphs 11

SUVAT in 2D & Gravity 6

Constant acceleration (SUVAT) 32

Variable acceleration (1D) 40

Variable acceleration (vectors) 26

Forces, equilibrium and resultants 9

Newton's laws and connected particles 18

Pulley systems 5

Motion on a slope 4

Friction 10

Momentum and Collisions 15

Moments 13

Parametric curves and Cartesian conversion 11

Parametric differentiation 15

Parametric integration 4

Projectiles 40