LFM Pure

jee-main 2019 Q68 Point-to-Line Distance Computation View

A point on the straight line, $3x + 5y = 15$ which is equidistant from the coordinate axes will lie only in:
(1) $1^{\text{st}}$ and $2^{\text{nd}}$ quadrants
(2) $1^{\text{st}}$, $2^{\text{nd}}$ and $4^{\text{th}}$ quadrants
(3) $1^{\text{st}}$ quadrant
(4) $4^{\text{th}}$ quadrant

jee-main 2019 Q68 Slope and Angle Between Lines View

Slope of a line passing through $P ( 2,3 )$ and intersecting the line $x + y = 7$ at a distance of 4 units from $P$, is
(1) $\frac { \sqrt { 7 } - 1 } { \sqrt { 7 } + 1 }$
(2) $\frac { 1 - \sqrt { 7 } } { 1 + \sqrt { 7 } }$
(3) $\frac { \sqrt { 5 } - 1 } { \sqrt { 5 } + 1 }$
(4) $\frac { 1 - \sqrt { 5 } } { 1 + \sqrt { 5 } }$

jee-main 2019 Q68 Point-to-Line Distance Computation View

If the two lines $x + ( a - 1 ) y = 1$ and $2 x + a ^ { 2 } y = 1 , ( a \in R - \{ 0,1 \} )$ are perpendicular, then the distance of their point of intersection from the origin is
(1) $\frac { 2 } { \sqrt { 5 } }$
(2) $\frac { \sqrt { 2 } } { 5 }$
(3) $\frac { 2 } { 5 }$
(4) $\sqrt { \frac { 2 } { 5 } }$

jee-main 2019 Q69 Locus Determination View

A point $P$ moves on the line $2 x - 3 y + 4 = 0$. If $Q ( 1,4 )$ and $R ( 3 , - 2 )$ are fixed points, then the locus of the centroid of $\triangle P Q R$ is a line:
(1) with slope $\frac { 2 } { 3 }$
(2) with slope $\frac { 3 } { 2 }$
(3) parallel to $y$-axis
(4) parallel to $x$-axis

jee-main 2019 Q70 Triangle Properties and Special Points View

Let the equations of two sides of a triangle be $3x - 2y + 6 = 0$ and $4x + 5y - 20 = 0$. If the orthocenter of this triangle is at $(1,1)$ then the equation of its third side is:
(1) $122y + 26x + 1675 = 0$
(2) $26x - 122y - 1675 = 0$
(3) $26x + 61y + 1675 = 0$
(4) $122y - 26x - 1675 = 0$

jee-main 2019 Q75 Perspective, Projection, and Applied Geometry View

Two vertical poles of height, $20 m$ and $80 m$ stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, from this horizontal plane is:
(1) 16
(2) 12
(3) 18
(4) 15

jee-main 2019 Q78 Line Equation and Parametric Representation View

If the system of linear equations $$\begin{aligned} & x - 2 y + k z = 1 \\ & 2 x + y + z = 2 \\ & 3 x - y - k z = 3 \end{aligned}$$ has a solution $( x , y , z )$, $z \neq 0$, then $( x , y )$ lies on the straight line whose equation is:
(1) $4 x - 3 y - 4 = 0$
(2) $3 x - 4 y - 4 = 0$
(3) $3 x - 4 y - 1 = 0$
(4) $4 x - 3 y - 1 = 0$

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 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 2020 Q72 Triangle Properties and Special Points View

Let $A(1,0)$, $B(6,2)$ and $C \left( \frac { 3 } { 2 } , 6 \right)$ be the vertices of a triangle $ABC$. If $P$ is a point inside the triangle $ABC$ such that the triangles $APC$, $APB$ and $BPC$ have equal areas, then the length of the line segment $PQ$, where $Q$ is the point $\left( - \frac { 7 } { 6 } , - \frac { 1 } { 3 } \right)$, is

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 Triangle Properties and Special Points View

Let the centroid of an equilateral triangle $ABC$ be at the origin. Let one of the sides of the equilateral triangle be along the straight line $x + y = 3$. If $R$ and $r$ be the radius of circumcircle and incircle respectively of $\triangle ABC$, then $( R + r )$ is equal to:
(1) $\frac { 9 } { \sqrt { 2 } }$
(2) $7 \sqrt { 2 }$
(3) $2 \sqrt { 2 }$
(4) $3 \sqrt { 2 }$

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 Q68 Line Equation and Parametric Representation View

The number of integral values of $m$ so that the abscissa of point of intersection of lines $3 x + 4 y = 9$ and $y = m x + 1$ is also an integer, is:
(1) 1
(2) 2
(3) 3
(4) 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 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) $\frac { 25 } { 2 \sqrt { 3 } }$

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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