LFM Pure

jee-main 2019 Q80 Geometric or applied optimisation problem View

The shortest distance between the line $y = x$ and the curve $y^2 = x - 2$ is
(1) $\frac{7}{4\sqrt{2}}$
(2) $\frac{7}{8}$
(3) $\frac{11}{4\sqrt{2}}$
(4) 2

jee-main 2020 Q52 Circle Equation and Properties via Complex Number Manipulation View

If $\operatorname { Re } \left( \frac { z - 1 } { 2 z + i } \right) = 1$, where $z = x + i y$, then the point $(x, y)$ lies on a
(1) circle whose centre is at $\left( - \frac { 1 } { 2 } , - \frac { 3 } { 2 } \right)$
(2) straight line whose slope is $- \frac { 2 } { 3 }$
(3) straight line whose slope is $\frac { 3 } { 2 }$
(4) circle whose diameter is $\frac { \sqrt { 5 } } { 2 }$

jee-main 2020 Q55 Triangle Properties and Special Points View

If a $\triangle ABC$ has vertices $A ( - 1,7 ) , B ( - 7,1 )$ and $C ( 5 , - 5 )$, then its orthocentre has coordinates:
(1) $( - 3,3 )$
(2) $( 3 , - 3 )$
(3) $\left( - \frac { 3 } { 5 } , \frac { 3 } { 5 } \right)$
(4) $\left( \frac { 3 } { 5 } , - \frac { 3 } { 5 } \right)$

jee-main 2020 Q55 Line Equation and Parametric Representation View

If the perpendicular bisector of the line segment joining the points $P ( 1,4 )$ and $Q ( k , 3 )$ has $y$-intercept equal to $-4$, then a value of $k$ is:
(1) $-2$
(2) $-4$
(3) $\sqrt { 14 }$
(4) $\sqrt { 15 }$

jee-main 2020 Q55 Reflection and Image in a Line View

Let $L$ denote the line in the $xy$-plane with $x$ and $y$ intercepts as 3 and 1 respectively. Then the image of the point $(-1,-4)$ in the line is:
(1) $\left(\frac{11}{5},\frac{28}{5}\right)$
(2) $\left(\frac{29}{5},\frac{8}{5}\right)$
(3) $\left(\frac{8}{5},\frac{29}{5}\right)$
(4) $\left(\frac{29}{5},\frac{11}{5}\right)$

jee-main 2020 Q56 Area Computation in Coordinate Geometry View

A triangle $ABC$ lying in the first quadrant has two vertices as $A ( 1,2 )$ and $B ( 3,1 )$. If $\angle BAC = 90 ^ { \circ }$, and $\operatorname { ar } ( \Delta \mathrm { ABC } ) = 5 \sqrt { 5 }$ sq. units, then the abscissa of the vertex C is :
(1) $1 + \sqrt { 5 }$
(2) $1 + 2 \sqrt { 5 }$
(3) $2 + \sqrt { 5 }$
(4) $2 \sqrt { 5 } - 1$

jee-main 2020 Q56 Reflection and Image in a Line View

A ray of light coming from the point $( 2,2 \sqrt { 3 } )$ is incident at an angle $30 ^ { \circ }$ on the line $x = 1$ at the point $A$. The ray gets reflected on the line $x = 1$ and meets $x$-axis at the point $B$. Then, the line $A B$ passes through the point
(1) $\left( 3 , - \frac { 1 } { \sqrt { 3 } } \right)$
(2) $\left( 4 , - \frac { \sqrt { 3 } } { 2 } \right)$
(3) $( 3 , - \sqrt { 3 } )$
(4) $( 4 , - \sqrt { 3 } )$

jee-main 2020 Q57 Locus Determination View

The locus of the mid-points of the perpendiculars drawn from points on the line $x = 2 y$, to the line $x = y$, is.
(1) $2 x - 3 y = 0$
(2) $5 x - 7 y = 0$
(3) $3 x - 2 y = 0$
(4) $7 x - 5 y = 0$

jee-main 2020 Q57 Solve trigonometric inequality View

The set of all possible values of $\theta$ in the interval $( 0 , \pi )$ for which the points $( 1,2 )$ and $( \sin \theta , \cos \theta )$ lie on the same side of the line $x + y = 1$ is?
(1) $\left( 0 , \frac { \pi } { 2 } \right)$
(2) $\left( \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } \right)$
(3) $\left( 0 , \frac { 3 \pi } { 4 } \right)$
(4) $\left( 0 , \frac { \pi } { 4 } \right)$

jee-main 2020 Q69 Triangle Properties and Special Points View

Let $D$ be the centroid of the triangle with vertices $( 3 , - 1 ) , ( 1,3 )$ and $( 2,4 )$. Let P be the point of intersection of the lines $x + 3 y - 1 = 10$ and $3 x - y + 1 = 0$. Then, the line passing through the points $D$ and P also passes through the point:
(1) $( - 9 , - 6 )$
(2) $( 9,7 )$
(3) $( 7,6 )$
(4) $( - 9 , - 7 )$

jee-main 2021 Q63 Area Computation in Coordinate Geometry View

Let $A ( - 1,1 ) , B ( 3,4 )$ and $C ( 2,0 )$ be given three points. A line $y = m x , m > 0$, intersects lines $AC$ and $BC$ at point $P$ and $Q$ respectively. Let $A _ { 1 }$ and $A _ { 2 }$ be the areas of $\triangle ABC$ and $\triangle PQC$ respectively, such that $A _ { 1 } = 3 A _ { 2 }$, then the value of $m$ is equal to :
(1) $\frac { 4 } { 15 }$
(2) 1
(3) 2
(4) 3

jee-main 2021 Q63 Area Computation in Coordinate Geometry View

Let $A ( a , 0 ) , B ( b , 2 b + 1 )$ and $C ( 0 , b ) , b \neq 0 , | b | \neq 1$, be points such that the area of triangle $A B C$ is 1 sq. unit, then the sum of all possible values of $a$ is: (1) $\frac { - 2 b } { b + 1 }$ (2) $\frac { 2 b ^ { 2 } } { b + 1 }$ (3) $\frac { - 2 b ^ { 2 } } { b + 1 }$ (4) $\frac { 2 b } { b + 1 }$

jee-main 2021 Q64 Line Equation and Parametric Representation View

A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\frac { 1 } { 4 }$. Three stones $A , B$ and $C$ are placed at the points $1,1,2,2$ and $4,4$ respectively. Then which of these stones is / are on the path of the man?
(1) $C$ only
(2) All the three
(3) $B$ only
(4) $A$ only

jee-main 2021 Q64 Point-to-Line Distance Computation View

If $p$ and $q$ are the lengths of the perpendiculars from the origin on the lines, $x \operatorname { cosec } \alpha - y \sec \alpha = k \cot 2 \alpha$ and $x \sin \alpha + y \cos \alpha = k \sin 2 \alpha$ respectively, then $k ^ { 2 }$ is equal to $:$
(1) $2 p ^ { 2 } + q ^ { 2 }$
(2) $p ^ { 2 } + 2 q ^ { 2 }$
(3) $4 q ^ { 2 } + p ^ { 2 }$
(4) $4 p ^ { 2 } + q ^ { 2 }$

jee-main 2021 Q65 Sketching a Curve from Analytical Properties View

Let $P$ be a variable point on the parabola $y = 4 x ^ { 2 } + 1$. Then, the locus of the mid-point of the point $P$ and the foot of the perpendicular drawn from the point $P$ to the line $y = x$ is:
(1) $( 3 x - y ) ^ { 2 } + ( x - 3 y ) + 2 = 0$
(2) $2 ( 3 x - y ) ^ { 2 } + ( x - 3 y ) + 2 = 0$
(3) $( 3 x - y ) ^ { 2 } + 2 ( x - 3 y ) + 2 = 0$
(4) $2 ( x - 3 y ) ^ { 2 } + ( 3 x - y ) + 2 = 0$

jee-main 2021 Q67 Evaluate trigonometric expression given a constraint View

Two poles $A B$ of length $a$ metres and $C D$ of length $a + b ( b \neq a )$ metres are erected at the same horizontal level with bases at $B$ and $D$. If $B D = x$ and $\tan \angle A C B = \frac { 1 } { 2 }$, then: (1) $x ^ { 2 } + 2 ( a + 2 b ) x - b ( a + b ) = 0$ (2) $x ^ { 2 } + 2 ( a + 2 b ) x + a ( a + b ) = 0$ (3) $x ^ { 2 } - 2 a x + b ( a + b ) = 0$ (4) $x ^ { 2 } - 2 a x + a ( a + b ) = 0$

jee-main 2021 Q68 Slope and Angle Between Lines View

Let the equation of the pair of lines, $y = p x$ and $y = q x$, can be written as $( y - p x ) ( y - q x ) = 0$. Then the equation of the pair of the angle bisectors of the lines $x ^ { 2 } - 4 x y - 5 y ^ { 2 } = 0$ is:
(1) $x ^ { 2 } - 3 x y + y ^ { 2 } = 0$
(2) $x ^ { 2 } + 4 x y - y ^ { 2 } = 0$
(3) $x ^ { 2 } + 3 x y - y ^ { 2 } = 0$
(4) $x ^ { 2 } - 3 x y - y ^ { 2 } = 0$

jee-main 2021 Q69 Slope and Angle Between Lines View

The equation of one of the straight lines which passes through the point $(1, 3)$ and makes an angle $\tan ^ { - 1 } ( \sqrt { 2 } )$ with the straight line, $y + 1 = 3 \sqrt { 2 } x$ is
(1) $4 \sqrt { 2 } x + 5 y - ( 15 + 4 \sqrt { 2 } ) = 0$
(2) $5 \sqrt { 2 } x + 4 y - ( 15 + 4 \sqrt { 2 } ) = 0$
(3) $4 \sqrt { 2 } x + 5 y - 4 \sqrt { 2 } = 0$
(4) $4 \sqrt { 2 } x - 5 y - ( 5 + 4 \sqrt { 2 } ) = 0$

jee-main 2021 Q84 Inscribed/Circumscribed Circle Computations View

Consider a triangle having vertices $A ( - 2,3 ) , B ( 1,9 )$ and $C ( 3,8 )$. If a line $L$ passing through the circumcentre of triangle $ABC$, bisects line $BC$, and intersects $y$-axis at point $\left( 0 , \frac { \alpha } { 2 } \right)$, then the value of real number $\alpha$ is $\underline{\hspace{1cm}}$.

jee-main 2022 Q63 Triangle Properties and Special Points View

Let the circumcentre of a triangle with vertices $A ( a , 3 ) , B ( b , 5 )$ and $C ( a , b ) , a b > 0$ be $P ( 1,1 )$. If the line $A P$ intersects the line $B C$ at the point $Q \left( k _ { 1 } , k _ { 2 } \right)$, then $k _ { 1 } + k _ { 2 }$ is equal to
(1) 2
(2) $\frac { 4 } { 7 }$
(3) $\frac { 2 } { 7 }$
(4) 4

jee-main 2022 Q63 Area Computation in Coordinate Geometry View

Let $R$ be the point $( 3,7 )$ and let $P$ and $Q$ be two points on the line $x + y = 5$ such that $PQR$ is an equilateral triangle. Then the area of $\triangle PQR$ is
(1) $\frac { 25 } { 4 \sqrt { 3 } }$
(2) $\frac { 25 \sqrt { 3 } } { 2 }$
(3) $\frac { 25 } { \sqrt { 3 } }$
(4) $\frac { 25 } { 2 \sqrt { 3 } }$

jee-main 2022 Q64 Collinearity and Concurrency View

Let the area of the triangle with vertices $A ( 1 , \alpha ) , B ( \alpha , 0 )$ and $C ( 0 , \alpha )$ be 4 sq. units. If the points $( \alpha , - \alpha ) , ( - \alpha , \alpha )$ and $\left( \alpha ^ { 2 } , \beta \right)$ are collinear, then $\beta$ is equal to
(1) 64
(2) - 8
(3) - 64
(4) 512

jee-main 2022 Q64 Triangle Properties and Special Points View

In an isosceles triangle $ABC$, the vertex $A$ is $( 6,1 )$ and the equation of the base $BC$ is $2x + y = 4$. Let the point $B$ lie on the line $x + 3y = 7$. If $( \alpha , \beta )$ is the centroid of the triangle $ABC$, then $15( \alpha + \beta )$ is equal to

jee-main 2022 Q64 Slope and Angle Between Lines View

The distance between the two points $A$ and $A ^ { \prime }$ which lie on $y = 2$ such that both the line segments $A B$ and $A ^ { \prime } B$ (where $B$ is the point $( 2,3 )$ ) subtend angle $\frac { \pi } { 4 }$ at the origin, is equal to
(1) 10
(2) $\frac { 48 } { 5 }$
(3) $\frac { 52 } { 5 }$
(4) 3

jee-main 2022 Q64 Line Equation and Parametric Representation View

A line, with the slope greater than one, passes through the point $A(4,3)$ and intersects the line $x - y - 2 = 0$ at the point $B$. If the length of the line segment $AB$ is $\frac { \sqrt { 29 } } { 3 }$, then $B$ also lies on the line
(1) $2x + y = 9$
(2) $3x - 2y = 7$
(3) $x + 2y = 6$
(4) $2x - 3y = 3$

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LFM Pure

Straight Lines & Coordinate Geometry 242

Circles 702

Simultaneous equations 134

Proof 868

Proof by induction 36

Sine and Cosine Rules 185

Radians, Arc Length and Sector Area 22

Trig Graphs & Exact Values 95

Standard trigonometric equations 142

Trig Proofs 92

Quadratic trigonometric equations 8

Trigonometric equations in context 11

Modulus function 37

Matrices 1085

Linear transformations 72

Invariant lines and eigenvalues and vectors 339

Reciprocal Trig & Identities 63

Addition & Double Angle Formulae 88

Harmonic Form 14

Small angle approximation 14

Chain Rule 281

Product & Quotient Rules 13

Differentiating Transcendental Functions 160

Connected Rates of Change 48

Standard Integrals and Reverse Chain Rule 82

Integration by Substitution 213

Integration by Parts 44

Implicit equations and differentiation 27

Differential equations 308

3x3 Matrices 164