LFM Pure and Mechanics

jee-advanced 2014 Q56 Find tangent line equation at a given point View

The slope of the tangent to the curve $\left(y - x^5\right)^2 = x\left(1 + x^2\right)^2$ at the point $(1, 3)$ is

jee-advanced 2016 Q44 Find tangent line equation at a given point View

Let $a , b \in \mathbb { R }$ and $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by $f ( x ) = a \cos \left( \left| x ^ { 3 } - x \right| \right) + b | x | \sin \left( \left| x ^ { 3 } + x \right| \right)$. Then $f$ is
(A) differentiable at $x = 0$ if $a = 0$ and $b = 1$
(B) differentiable at $x = 1$ if $a = 1$ and $b = 0$
(C) NOT differentiable at $x = 0$ if $a = 1$ and $b = 0$
(D) NOT differentiable at $x = 1$ if $a = 1$ and $b = 1$

jee-main 2013 Q82 Find tangent line with a specified slope or from an external point View

The intercepts on the $x$-axis made by tangents to the curve, $y = \int_0^x |t|\, dt, x \in R$, which are parallel to the line $y = 2x$, are equal to
(1) $\pm 3$
(2) $\pm 4$
(3) $\pm 1$
(4) $\pm 2$

jee-main 2014 Q82 Common tangent line to two curves View

The slope of the line touching both the parabolas $y ^ { 2 } = 4 x$ and $x ^ { 2 } = - 32 y$ is
(1) $\frac { 1 } { 8 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 3 } { 2 }$

jee-main 2016 Q81 Normal or perpendicular line problems View

Let C be a curve given by $y ( x ) = 1 + \sqrt { 4 x - 3 } , x > \frac { 3 } { 4 }$. If $P$ is a point on C , such that the tangent at $P$ has slope $\frac { 2 } { 3 }$, then a point through which the normal at $P$ passes, is :
(1) $( 1,7 )$
(2) $( 3 , - 4 )$
(3) $( 4 , - 3 )$
(4) $( 2,3 )$

jee-main 2016 Q83 Normal or perpendicular line problems View

Consider $f(x) = \tan^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right)$, $x \in \left(0, \frac{\pi}{2}\right)$. A normal to $y = f(x)$ at $x = \frac{\pi}{6}$ also passes through the point:
(1) $(0, 0)$
(2) $\left(0, \frac{2\pi}{3}\right)$
(3) $\left(\frac{\pi}{6}, 0\right)$
(4) $\left(\frac{\pi}{4}, 0\right)$

jee-main 2016 Q86 Normal or perpendicular line problems View

Consider $f(x) = \tan^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right)$, $x \in \left(0, \frac{\pi}{2}\right)$. A normal to $y = f(x)$ at $x = \frac{\pi}{6}$ also passes through the point: (1) $(0, 0)$ (2) $\left(0, \frac{2\pi}{3}\right)$ (3) $\left(\frac{\pi}{6}, 0\right)$ (4) $\left(\frac{\pi}{4}, 0\right)$

jee-main 2017 Q71 Normal or perpendicular line problems View

The normal to the curve $y ( x - 2 ) ( x - 3 ) = x + 6$ at the point where the curve intersects the $y$-axis passes through the point:
(1) $\left( \frac { 1 } { 2 } , - \frac { 1 } { 3 } \right)$
(2) $\left( \frac { 1 } { 2 } , \frac { 1 } { 3 } \right)$
(3) $\left( - \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$
(4) $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$

jee-main 2017 Q81 Normal or perpendicular line problems View

The normal to the curve $y(x - 2)(x - 3) = x + 6$ at the point where the curve intersects the $y$-axis passes through the point:
(1) $\left(-\dfrac{1}{2}, -\dfrac{1}{2}\right)$
(2) $\left(\dfrac{1}{2}, \dfrac{1}{2}\right)$
(3) $\left(\dfrac{1}{2}, -\dfrac{1}{3}\right)$
(4) $\left(\dfrac{1}{2}, \dfrac{1}{3}\right)$

jee-main 2017 Q81 Geometric properties of tangent lines (intersections, lengths, areas) View

The tangent at the point $( 2 , - 2 )$ to the curve, $x ^ { 2 } y ^ { 2 } - 2 x = 4 ( 1 - y )$ does not pass through the point:
(1) $( - 2 , - 7 )$
(2) $( 8,5 )$
(3) $( - 4 , - 9 )$
(4) $\left( 4 , \frac { 1 } { 3 } \right)$

jee-main 2018 Q70 Normal or perpendicular line problems View

If $\beta$ is one of the angles between the normals to the ellipse $x ^ { 2 } + 3 y ^ { 2 } = 9$ at the points $( 3 \cos \theta , \sqrt { 3 } \sin \theta )$ and $( - 3 \sin \theta , \sqrt { 3 } \cos \theta ) ; \theta \in \left( 0 , \frac { \pi } { 2 } \right) ;$ then $\frac { 2 \cot \beta } { \sin 2 \theta }$ is equal to :
(1) $\frac { 1 } { \sqrt { 3 } }$
(2) $\frac { \sqrt { 3 } } { 4 }$
(3) $\frac { 2 } { \sqrt { 3 } }$
(4) $\sqrt { 2 }$

jee-main 2018 Q71 Normal or perpendicular line problems View

If $\beta$ is one of the angles between the normals to the ellipse, $x ^ { 2 } + 3 y ^ { 2 } = 9$ at the points ( $3 \cos \theta , \sqrt { 3 } \sin \theta$ ) and $( - 3 \sin \theta , \sqrt { 3 } \cos \theta ) ; \in \left( 0 , \frac { \pi } { 2 } \right) ;$ then $\frac { 2 \cot \beta } { \sin 2 \theta }$ is equal to
(1) $\sqrt { 2 }$
(2) $\frac { 2 } { \sqrt { 3 } }$
(3) $\frac { 1 } { \sqrt { 3 } }$
(4) $\frac { \sqrt { 3 } } { 4 }$

jee-main 2019 Q80 Find tangent line with a specified slope or from an external point View

The tangent to the curve $y = x ^ { 2 } - 5 x + 5$, parallel to the line $2 y = 4 x + 1$, also passes through the point :
(1) $\left( \frac { 1 } { 4 } , \frac { 7 } { 2 } \right)$
(2) $\left( \frac { 7 } { 2 } , \frac { 1 } { 4 } \right)$
(3) $\left( - \frac { 1 } { 8 } , 7 \right)$
(4) $\left( \frac { 1 } { 8 } , - 7 \right)$

jee-main 2019 Q81 Determine unknown parameters from tangent conditions View

If the tangent to the curve, $y = x ^ { 3 } + a x - b$ at the point $( 1 , - 5 )$ is perpendicular to the line, $- x + y + 4 = 0$, then which one of the following points lies on the curve?
(1) $( 2 , - 2 )$
(2) $( 2 , - 1 )$
(3) $( - 2,1 )$
(4) $( - 2,2 )$

jee-main 2020 Q64 Find tangent line with a specified slope or from an external point View

If the tangent to the curve $y = x + \sin y$ at a point $(a, b)$ is parallel to the line joining $\left(0, \frac{3}{2}\right)$ and $\left(\frac{1}{2}, 2\right)$, then
(1) $b = a$
(2) $|b - a| = 1$
(3) $|a + b| = 1$
(4) $b = \frac{\pi}{2} + a$

jee-main 2020 Q65 Normal or perpendicular line problems View

The equation of the normal to the curve $y = ( 1 + x ) ^ { 2 y } + \cos ^ { 2 } \left( \sin ^ { - 1 } x \right)$, at $x = 0$ is
(1) $y + 4 x = 2$
(2) $y = 4 x + 2$
(3) $x + 4 y = 8$
(4) $2 y + x = 4$

jee-main 2020 Q66 Normal or perpendicular line problems View

Let $P(h, k)$ be a point on the curve $y = x^{2} + 7x + 2$, nearest to the line, $y = 3x - 3$. Then the equation of the normal to the curve at $P$ is
(1) $x + 3y + 26 = 0$
(2) $x + 3y - 62 = 0$
(3) $x - 3y - 11 = 0$
(4) $x - 3y + 22 = 0$

jee-main 2020 Q73 Determine unknown parameters from tangent conditions View

If the lines $x + y = a$ and $x - y = b$ touch the curve $y = x^2 - 3x + 2$ at the points where the curve intersects the $x$-axis, then $\frac{a}{b}$ is equal to ...

jee-main 2021 Q86 Normal or perpendicular line problems View

If the curves $x = y ^ { 4 }$ and $x y = k$ cut at right angles, then $( 4 k ) ^ { 6 }$ is equal to $\underline{\hspace{1cm}}$.

jee-main 2022 Q74 Find tangent line with a specified slope or from an external point View

If the tangent at the point $\left( x _ { 1 } , y _ { 1 } \right)$ on the curve $y = x ^ { 3 } + 3 x ^ { 2 } + 5$ passes through the origin, then $\left( x _ { 1 } , y _ { 1 } \right)$ does NOT lie on the curve

jee-main 2022 Q74 Find tangent line with a specified slope or from an external point View

If the line $y = 4 + kx$, $k > 0$, is the tangent to the parabola $y = x - x^2$ at the point $P$ and $V$ is the vertex of the parabola, then the slope of the line through $P$ and $V$ is
(1) $\frac{3}{2}$
(2) $\frac{26}{9}$
(3) $\frac{5}{2}$
(4) $\frac{23}{6}$

jee-main 2022 Q74 Prove a given line is tangent to a curve View

Let $S$ be the set of all the natural numbers, for which the line $\frac { x } { a } + \frac { y } { b } = 2$ is a tangent to the curve $\left( \frac { x } { a } \right) ^ { n } + \left( \frac { y } { b } \right) ^ { n } = 2$ at the point $( a , b ) , ab \neq 0$. Then
(1) $S = \phi$
(2) $n ( S ) = 1$
(3) $S = \{ 2k : k \in N \}$
(4) $S = N$

jee-main 2022 Q76 Normal or perpendicular line problems View

Consider a curve $y = y ( x )$ in the first quadrant as shown in the figure. Let the area $A _ { 1 }$ is twice the area $A _ { 2 }$. Then the normal to the curve perpendicular to the line $2 x - 12 y = 15$ does NOT pass through the point
(1) $( 6,21 )$
(2) $( 8,9 )$
(3) $( 10 , - 4 )$
(4) $( 12 , - 15 )$

jee-main 2022 Q87 Normal or perpendicular line problems View

Let the function $f ( x ) = 2 x ^ { 2 } - \log _ { e } x , x > 0$, be decreasing in $( 0 , a )$ and increasing in $( a , 4 )$. A tangent to the parabola $y ^ { 2 } = 4 a x$ at a point $P$ on it passes through the point $( 8 a , 8 a - 1 )$ but does not pass through the point $\left( - \frac { 1 } { a } , 0 \right)$. If the equation of the normal at $P$ is $\frac { x } { \alpha } + \frac { y } { \beta } = 1$, then $\alpha + \beta$ is equal to $\_\_\_\_$.

jee-main 2022 Q88 Find tangent line with a specified slope or from an external point View

Let $M$ and $N$ be the number of points on the curve $y ^ { 5 } - 9 x y + 2 x = 0$, where the tangents to the curve are parallel to $x$-axis and $y$-axis, respectively. Then the value of $M + N$ equals $\_\_\_\_$ .

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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