LFM Pure

gaokao 2019 Q3 5 marks Point-to-Line Distance Computation View

The parametric equation of line $l$ is $\left\{ \begin{array} { l } x = 1 + 3 t , \\ y = 2 + 4 t \end{array} \right.$ ($t$ is the parameter). The distance from point $(1,0)$ to line $l$ is (A) $\frac { 1 } { 5 }$ (B) $\frac { 2 } { 5 }$ (C) $\frac { 4 } { 5 }$ (D) $\frac { 6 } { 5 }$

gaokao 2020 Q8 5 marks Point-to-Line Distance Computation View

The maximum distance from the point $( 0 , - 1 )$ to the line $y = k ( x + 1 )$ is
A. 1
B. $\sqrt { 2 }$
C. $\sqrt { 3 }$
D. 2

gaokao 2020 Q14 5 marks Line Equation and Parametric Representation View

Given that the equation of line $l$ is $3 x - 4 y + 1 = 0$, which of the following is a parametric equation of $l$? ( )
A. $\left\{ \begin{array} { l } x = 4 + 3 t \\ y = 3 - 4 t \end{array} \right.$
B. $\left\{ \begin{array} { l } x = 4 + 3 t \\ y = 3 + 4 t \end{array} \right.$
C. $\left\{ \begin{array} { l } x = 1 - 4 t \\ y = 1 + 3 t \end{array} \right.$
D. $\left\{ \begin{array} { l } x = 1 + 4 t \\ y = 1 + 3 t \end{array} \right.$

grandes-ecoles 2015 QI.B.1 Line Equation and Parametric Representation View

Draw a graph of $\Delta\left(0, \vec{e}_1\right)$ and $\Delta\left(2, \frac{\vec{e}_1 + \vec{e}_2}{\sqrt{2}}\right)$.

grandes-ecoles 2015 QI.B.2 Line Equation and Parametric Representation View

Determine a Cartesian equation of $\Delta\left(q, \vec{u}_\theta\right)$.

grandes-ecoles 2015 QI.B.3 Line Equation and Parametric Representation View

Show that a parametrization of $\Delta\left(q, \vec{u}_\theta\right)$ is given by $\left\{ \begin{array}{l} x(t) = q\cos\theta - t\sin\theta \\ y(t) = q\sin\theta + t\cos\theta \end{array} \right.$ when $t$ ranges over $\mathbb{R}$.

grandes-ecoles 2015 QI.B.4 Collinearity and Concurrency View

Under what condition are the lines $\Delta(q, \vec{u})$ and $\Delta(r, \vec{v})$ identical?

grandes-ecoles 2015 QI.C.1 Matrix and Linear Algebra Approach to Lines View

We denote by $\mathcal{D}$ the set of affine lines of the plane and we consider the application $\Psi : \left\{ \begin{array}{cll} G & \rightarrow & \mathcal{D} \\ M(A, \vec{b}) & \mapsto \Delta\left(\left\langle A\vec{e}_1, \vec{b}\right\rangle, A\vec{e}_1\right) \end{array} \right.$.
Draw $\Psi(M(A, \vec{b}))$ in the case $A = R_{\pi/6}$ and $\vec{b} = \binom{1}{2}$.

grandes-ecoles 2015 QII.A Geometric Figure on Coordinate Plane View

We take for $\Omega$ (only in this question) the interior of the equilateral triangle with vertices $(1,0), (-1/2, \sqrt{3}/2)$ and $(-1/2, -\sqrt{3}/2)$. We define, for all $\lambda \in \mathbb{R}^*$ and all pairs $(x_0, y_0) \in \mathbb{R}^2$: $$\Omega_{x_0, y_0, \lambda} = \left\{ \lambda(x,y) + (x_0, y_0) \mid (x,y) \in \Omega \right\}$$ Draw a figure on which both $\Omega$ and $\Omega_{2,1,1/2}$ appear.

isi-entrance 2009 Q9 Point-to-Line Distance Computation View

Let $P_0 = (0,0)$, $P_1 = (0,4)$, $P_2 = (4,0)$, $P_3 = (-4,-4)$, $P_4 = (2,4)$, $P_5 = (4,6)$ (or similar points). Find the region of all points closer to $P_0$ than to any of $P_1, P_2, P_3, P_4, P_5$, and compute its perimeter.

isi-entrance 2010 Q8 Perspective, Projection, and Applied Geometry View

Consider a rectangular cardboard box of height 3, breadth 4 and length 10 units. There is a lizard in one corner $A$ of the box and an insect in the corner $B$ which is farthest from $A$. The length of the shortest path between the lizard and the insect along the surface of the box is
(a) $\sqrt{5^{2} + 10^{2}}$
(b) $\sqrt{7^{2} + 10^{2}}$
(c) $4 + \sqrt{3^{2} + 10^{2}}$
(d) $3 + \sqrt{4^{2} + 10^{2}}$

isi-entrance 2012 Q1 Locus Determination View

A rod slides with its ends on two coordinate axes. A point $P$ divides the rod in the ratio $1:2$. Find the locus of $P$.

isi-entrance 2012 Q26 Perspective, Projection, and Applied Geometry View

A room is in the shape of a rectangular box. The shortest path along the surface from one corner $A$ to the opposite corner $B$ has length $\sqrt{29}$ (given the relevant dimensions are $5$ and $2$). Find this shortest distance.

isi-entrance 2013 Q5 4 marks Geometric Figure on Coordinate Plane View

Let $A B C D$ be a unit square. Four points $E , F , G$ and $H$ are chosen on the sides $A B , B C , C D$ and $D A$ respectively. The lengths of the sides of the quadrilateral $E F G H$ are $\alpha , \beta , \gamma$ and $\delta$. Which of the following is always true?
(A) $1 \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 2 \sqrt { 2 }$
(B) $2 \sqrt { 2 } \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 4 \sqrt { 2 }$
(C) $2 \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 4$
(D) $\sqrt { 2 } \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 2 + \sqrt { 2 }$

isi-entrance 2013 Q10 4 marks Locus Determination View

Let $A$ be the fixed point $(0,4)$ and $B$ be a moving point $(2t, 0)$. Let $M$ be the mid-point of $AB$ and let the perpendicular bisector of $AB$ meet the $y$-axis at $R$. The locus of the mid-point $P$ of $MR$ is
(A) $y + x ^ { 2 } = 2$
(B) $x ^ { 2 } + ( y - 2 ) ^ { 2 } = 1 / 4$
(C) $( y - 2 ) ^ { 2 } - x ^ { 2 } = 1 / 4$
(D) none of the above

isi-entrance 2013 Q46 4 marks Slope and Angle Between Lines View

Suppose $ABCD$ is a quadrilateral such that $\angle BAC = 50^\circ, \angle CAD = 60^\circ, \angle CBD = 30^\circ$ and $\angle BDC = 25^\circ$. If $E$ is the point of intersection of $AC$ and $BD$, then the value of $\angle AEB$ is
(A) $75^\circ$
(B) $85^\circ$
(C) $95^\circ$
(D) $110^\circ$

isi-entrance 2013 Q48 4 marks Locus Determination View

Let $L$ be the point $(t, 2)$ and $M$ be a point on the $y$-axis such that $LM$ has slope $-t$. Then the locus of the midpoint of $LM$, as $t$ varies over all real values, is
(A) $y = 2 + 2x^2$
(B) $y = 1 + x^2$
(C) $y = 2 - 2x^2$
(D) $y = 1 - x^2$

isi-entrance 2013 Q70 4 marks Reflection and Image in a Line View

The equation $x^3 y + x y^3 + x y = 0$ represents
(A) a circle
(B) a circle and a pair of straight lines
(C) a rectangular hyperbola
(D) a pair of straight lines

isi-entrance 2014 Q6 Reflection and Image in a Line View

A ray of light is incident along the line $x = 2y$ and hits a mirror. The angle of incidence equals the angle of reflection. Find the equation of the reflected ray passing through the point $(2, 1)$.
(A) $4x - 3y = 5$ (B) $3x - 4y = 2$ (C) $x - y = 1$ (D) $2x - y = 3$

isi-entrance 2014 Q14 Area Computation in Coordinate Geometry View

Let $A = (h, k)$, $B = (2, 6)$, $C = (5, 2)$ be vertices of a triangle with area 12. Find the minimum distance from $A$ to the origin.
(A) $\dfrac{16}{\sqrt{5}}$ (B) $\dfrac{8}{\sqrt{5}}$ (C) $\dfrac{32}{\sqrt{5}}$ (D) $\dfrac{16}{\sqrt{5}}$

isi-entrance 2015 Q25 4 marks Line Equation and Parametric Representation View

The equation $x ^ { 3 } y + x y ^ { 3 } + x y = 0$ represents
(a) a circle
(b) a circle and a pair of straight lines
(c) a rectangular hyperbola
(d) a pair of straight lines.

isi-entrance 2015 Q25 4 marks Line Equation and Parametric Representation View

The equation $x ^ { 3 } y + x y ^ { 3 } + x y = 0$ represents
(a) a circle
(b) a circle and a pair of straight lines
(c) a rectangular hyperbola
(d) a pair of straight lines.

isi-entrance 2016 Q5 4 marks Geometric Figure on Coordinate Plane View

Let $A B C D$ be a unit square. Four points $E , F , G$ and $H$ are chosen on the sides $A B , B C , C D$ and $D A$ respectively. The lengths of the sides of the quadrilateral $E F G H$ are $\alpha , \beta , \gamma$ and $\delta$. Which of the following is always true?
(A) $1 \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 2 \sqrt { 2 }$
(B) $2 \sqrt { 2 } \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 4 \sqrt { 2 }$
(C) $2 \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 4$
(D) $\sqrt { 2 } \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 2 + \sqrt { 2 }$

isi-entrance 2016 Q5 4 marks Geometric Figure on Coordinate Plane View

Let $A B C D$ be a unit square. Four points $E , F , G$ and $H$ are chosen on the sides $A B , B C , C D$ and $D A$ respectively. The lengths of the sides of the quadrilateral $E F G H$ are $\alpha , \beta , \gamma$ and $\delta$. Which of the following is always true?
(A) $1 \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 2 \sqrt { 2 }$
(B) $2 \sqrt { 2 } \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 4 \sqrt { 2 }$
(C) $2 \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 4$
(D) $\sqrt { 2 } \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 2 + \sqrt { 2 }$

isi-entrance 2016 Q10 4 marks Locus Determination View

Let $A$ be the fixed point $(0,4)$ and $B$ be a moving point $(2t, 0)$. Let $M$ be the mid-point of $AB$ and let the perpendicular bisector of $AB$ meet the $y$-axis at $R$. The locus of the mid-point $P$ of $MR$ is
(A) $y + x ^ { 2 } = 2$
(B) $x ^ { 2 } + ( y - 2 ) ^ { 2 } = 1 / 4$
(C) $( y - 2 ) ^ { 2 } - x ^ { 2 } = 1 / 4$
(D) none of the above

1
2
3
4
5
6
7
8
9
10

Load questions...

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