LFM Pure

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) $2x - 3y = 3$

jee-main 2022 Q64 Area Computation in Coordinate Geometry View

Let $A ( 1,1 ) , B ( - 4,3 ) , C ( - 2 , - 5 )$ be vertices of a triangle $A B C , P$ be a point on side $B C$, and $\Delta _ { 1 }$ and $\Delta _ { 2 }$ be the areas of triangle $A P B$ and $A B C$ respectively. If $\Delta _ { 1 } : \Delta _ { 2 } = 4 : 7$, then the area enclosed by the lines $A P , A C$ and the $x$-axis is
(1) $\frac { 1 } { 4 }$
(2) $\frac { 3 } { 4 }$
(3) $\frac { 1 } { 2 }$
(4) 1

jee-main 2022 Q65 Point-to-Line Distance Computation View

Let the point $P(\alpha, \beta)$ be at a unit distance from each of the two lines $L_1: 3x - 4y + 12 = 0$, and $L_2: 8x + 6y + 11 = 0$. If $P$ lies below $L_1$ and above $L_2$, then $100(\alpha + \beta)$ is equal to
(1) $-14$
(2) 42
(3) $-22$
(4) 14

jee-main 2022 Q65 Triangle Properties and Special Points View

For $t \in (0, 2\pi)$, if $ABC$ is an equilateral triangle with vertices $A(\sin t, -\cos t)$, $B(\cos t, \sin t)$ and $C(a, b)$ such that its orthocentre lies on a circle with centre $\left(1, \frac{1}{3}\right)$, then $a^2 - b^2$ is equal to
(1) $\frac{8}{3}$
(2) $8$
(3) $\frac{77}{9}$
(4) $\frac{80}{9}$

jee-main 2022 Q65 Triangle Properties and Special Points View

The equations of the sides $A B , B C$ and $C A$ of a triangle $A B C$ are $2 x + y = 0 , x + p y = 39$ and $x - y = 3$ respectively and $P ( 2,3 )$ is its circumcentre. Then which of the following is NOT true
(1) $( A C ) ^ { 2 } = 9 p$
(2) $( A C ) ^ { 2 } + p ^ { 2 } = 136$
(3) $32 <$ area $( \triangle A B C ) < 36$
(4) $34 <$ area $( \triangle A B C ) < 38$

jee-main 2022 Q66 Geometric Figure on Coordinate Plane View

Let $m_1, m_2$ be the slopes of two adjacent sides of a square of side $a$ such that $a^2 + 11a + 3(m_1^2 + m_2^2) = 220$. If one vertex of the square is $(10\cos\alpha - \sin\alpha, 10\sin\alpha + \cos\alpha)$, where $\alpha \in \left(0, \frac{\pi}{2}\right)$ and the equation of one diagonal is $(\cos\alpha - \sin\alpha)x + (\sin\alpha + \cos\alpha)y = 10$, then $72(\sin^4\alpha + \cos^4\alpha) + a^2 - 3a + 13$ is equal to
(1) 119
(2) 128
(3) 145
(4) 155

jee-main 2022 Q85 Triangle Properties and Special Points View

The equations of the sides $AB , BC$ and $CA$ of a triangle $ABC$ are $2 x + y = 0 , x + p y = 15 a$ and $x - y = 3$ respectively. If its orthocentre is $( 2 , a ) , - \frac { 1 } { 2 } < a < 2$, then $p$ is equal to $\_\_\_\_$.

jee-main 2023 Q66 Triangle Properties and Special Points View

Let $( \alpha , \beta )$ be the centroid of the triangle formed by the lines $15 x - y = 82$, $6 x - 5 y = - 4$ and $9 x + 4 y = 17$. Then $\alpha + 2 \beta$ and $2 \alpha - \beta$ are the roots of the equation
(1) $x ^ { 2 } - 7 x + 12 = 0$
(2) $x ^ { 2 } - 14 x + 48 = 0$
(3) $x ^ { 2 } - 13 x + 42 = 0$
(4) $x ^ { 2 } - 10 x + 25 = 0$

jee-main 2023 Q66 Triangle Properties and Special Points View

If $( \alpha , \beta )$ is the orthocenter of the triangle $ABC$ with vertices $A ( 3 , - 7 ) , B ( - 1,2 )$ and $C ( 4,5 )$, then $9 \alpha - 6 \beta + 60$ is equal to
(1) 25
(2) 35
(3) 30
(4) 40

jee-main 2023 Q66 Triangle Properties and Special Points View

If the orthocentre of the triangle, whose vertices are $(1,2)$, $(2,3)$ and $(3,1)$ is $(\alpha, \beta)$, then the quadratic equation whose roots are $\alpha + 4\beta$ and $4\alpha + \beta$, is
(1) $x^2 - 19x + 90 = 0$
(2) $x^2 - 18x + 80 = 0$
(3) $x^2 - 22x + 120 = 0$
(4) $x^2 - 20x + 99 = 0$

jee-main 2023 Q68 Triangle Properties and Special Points View

The equations of the sides $AB , BC \& CA$ of a triangle $ABC$ are $2x + y = 0 , x + py = 21a ( a \neq 0 )$ and $x - y = 3$ respectively. Let $P ( 2 , a )$ be the centroid of the triangle $ABC$, then $( BC ) ^ { 2 }$ is equal to

jee-main 2023 Q68 Locus Determination View

If the point $\left( \alpha , \frac { 7 \sqrt { 3 } } { 3 } \right)$ lies on the curve traced by the mid-points of the line segments of the lines $x \cos \theta + y \sin \theta = 7 , \theta \in \left( 0 , \frac { \pi } { 2 } \right)$ between the co-ordinates axes, then $\alpha$ is equal to
(1) - 7
(2) $- 7 \sqrt { 3 }$
(3) $7 \sqrt { 3 }$
(4) 7

jee-main 2023 Q69 Triangle Properties and Special Points View

Let $A ( 0,1 ) , B ( 1,1 )$ and $C ( 1,0 )$ be the mid-points of the sides of a triangle with incentre at the point $D$. If the focus of the parabola $y ^ { 2 } = 4 a x$ passing through $D$ is $( \alpha + \beta \sqrt { 2 } , 0 )$, where $\alpha$ and $\beta$ are rational numbers, then $\frac { \alpha } { \beta ^ { 2 } }$ is equal to
(1) 8
(2) 12
(3) 6
(4) $\frac { 9 } { 2 }$

jee-main 2023 Q69 Point-to-Line Distance Computation View

The distance of the point $( 6 , - 2 \sqrt { 2 } )$ from the common tangent $y = m x + c , m > 0$, of the curves $x = 2 y ^ { 2 }$ and $x = 1 + y ^ { 2 }$ is
(1) $\frac { 1 } { 3 }$
(2) 5
(3) $\frac { 14 } { 3 }$
(4) $5 \sqrt { 3 }$

jee-main 2023 Q69 Reflection and Image in a Line View

A light ray emits from the origin making an angle $30 ^ { \circ }$ with the positive $x$-axis. After getting reflected by the line $\mathrm { x } + \mathrm { y } = 1$, if this ray intersects x-axis at Q , then the abscissa of Q is
(1) $\frac { 2 } { ( \sqrt { 3 } - 1 ) }$
(2) $\frac { 2 } { 3 + \sqrt { 3 } }$
(3) $\frac { 2 } { 3 - \sqrt { 3 } }$
(4) $\frac { \sqrt { 3 } } { 2 ( \sqrt { 3 } + 1 ) }$

jee-main 2023 Q69 Triangle Properties and Special Points View

Let $C ( \alpha , \beta )$ be the circumcentre of the triangle formed by the lines $4 x + 3 y = 69$, $4 y - 3 x = 17$, and $x + 7 y = 61$. Then $( \alpha - \beta ) ^ { 2 } + \alpha + \beta$ is equal to
(1) 18
(2) 17
(3) 15
(4) 16

jee-main 2023 Q70 Triangle Properties and Special Points View

The equations of sides $AB$ and $AC$ of a triangle $ABC$ are $( \lambda + 1 ) x + \lambda y = 4$ and $\lambda x + ( 1 - \lambda ) y + \lambda = 0$ respectively. Its vertex $A$ is on the $y$-axis and its orthocentre is $( 1,2 )$. The length of the tangent from the point $C$ to the part of the parabola $y ^ { 2 } = 6 x$ in the first quadrant is
(1) $\sqrt { 6 }$
(2) $2 \sqrt { 2 }$
(3) 2
(4) 4

jee-main 2023 Q70 Area Computation in Coordinate Geometry View

Let $B$ and $C$ be the two points on the line $y + x = 0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on $\mathrm { y } - 2 \mathrm { x } = 2$ such that $\triangle A B C$ is an equilateral triangle. Then, the area of the $\triangle A B C$ is
(1) $3 \sqrt { 3 }$
(2) $2 \sqrt { 3 }$
(3) $\frac { 8 } { \sqrt { 3 } }$
(4) $\frac { 10 } { \sqrt { 3 } }$

jee-main 2024 Q64 Line Equation and Parametric Representation View

A line passing through the point $A ( 9,0 )$ makes an angle of $30 ^ { \circ }$ with the positive direction of $x$-axis. If this line is rotated about $A$ through an angle of $15 ^ { \circ }$ in the clockwise direction, then its equation in the new position is
(1) $\frac { y } { \sqrt { 3 } - 2 } + x = 9$
(2) $\frac { x } { \sqrt { 3 } - 2 } + y = 9$
(3) $\frac { x } { \sqrt { 3 } + 2 } + y = 9$
(4) $\frac { y } { \sqrt { 3 } + 2 } + x = 9$

jee-main 2024 Q64 Geometric Figure on Coordinate Plane View

Let $\alpha , \quad \beta , \quad \gamma , \quad \delta \in Z$ and let $A(\alpha , \beta)$, $B(1, 0)$, $C(\gamma , \delta)$ and $D(1, 2)$ be the vertices of a parallelogram $ABCD$. If $AB = \sqrt { 10 }$ and the points $A$ and $C$ lie on the line $3 y = 2 x + 1$, then $2 \alpha + \beta + \gamma + \delta$ is equal to
(1) 10
(2) 5
(3) 12
(4) 8

jee-main 2024 Q64 Area Computation in Coordinate Geometry View

Let a variable line of slope $m > 0$ passing through the point $( 4 , - 9 )$ intersect the coordinate axes at the points $A$ and $B$. The minimum value of the sum of the distances of $A$ and $B$ from the origin is
(1) 30
(2) 25
(3) 15
(4) 10

jee-main 2024 Q65 Slope and Angle Between Lines View

The portion of the line $4 x + 5 y = 20$ in the first quadrant is trisected by the lines $\mathrm { L } _ { 1 }$ and $\mathrm { L } _ { 2 }$ passing through the origin. The tangent of an angle between the lines $L _ { 1 }$ and $L _ { 2 }$ is:
(1) $\frac { 8 } { 5 }$
(2) $\frac { 25 } { 41 }$
(3) $\frac { 2 } { 5 }$
(4) $\frac { 30 } { 41 }$

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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