LFM Pure

isi-entrance 2013 Q28 4 marks Intersection of Circles or Circle with Conic View

Let $a$ be a real number. The number of distinct solutions $(x, y)$ of the system of equations $(x - a)^2 + y^2 = 1$ and $x^2 = y^2$, can only be
(A) $0, 1, 2, 3, 4$ or 5
(B) 0, 1 or 3
(C) $0, 1, 2$ or 4
(D) $0, 2, 3$, or 4

isi-entrance 2013 Q39 4 marks Chord Length and Chord Properties View

Consider a circle with centre $O$. Two chords $AB$ and $CD$ extended intersect at a point $P$ outside the circle. If $\angle AOC = 43^\circ$ and $\angle BPD = 18^\circ$, then the value of $\angle BOD$ is
(A) $36^\circ$
(B) $29^\circ$
(C) $7^\circ$
(D) $25^\circ$

isi-entrance 2013 Q50 4 marks Circle-Related Locus Problems View

A triangle $ABC$ has a fixed base $BC$. If $AB : AC = 1 : 2$, then the locus of the vertex $A$ is
(A) a circle whose centre is the midpoint of $BC$
(B) a circle whose centre is on the line $BC$ but not the midpoint of $BC$
(C) a straight line
(D) none of the above.

isi-entrance 2013 Q51 4 marks Circle-Related Locus Problems View

Let $P$ be a variable point on a circle $C$ and $Q$ be a fixed point outside $C$. If $R$ is the mid-point of the line segment $PQ$, then the locus of $R$ is
(A) a circle
(B) an ellipse
(C) a line segment
(D) segment of a parabola

isi-entrance 2014 Q11 Inscribed/Circumscribed Circle Computations View

A circle of radius $r$ is inscribed in a circular sector. The chord of the sector has length $a$. If the circle touches the chord and the two radii of the sector, find the relation between $a$ and $r$.
(A) $a = 8r/5$ (B) $a = 5r/8$ (C) $a = 4r/3$ (D) $a = 3r/4$

isi-entrance 2015 QB3 Circles Tangent to Each Other or to Axes View

An isosceles triangle with base 6 cms. and base angles $30 ^ { \circ }$ each is inscribed in a circle. A second circle, which is situated outside the triangle, touches the first circle and also touches the base of the triangle at its midpoint. Find its radius.

isi-entrance 2015 QB5 Intersection of Circles or Circle with Conic View

If a circle intersects the hyperbola $y = 1 / x$ at four distinct points $\left( x _ { i } , y _ { i } \right) , i = 1,2,3,4$, then prove that $x _ { 1 } x _ { 2 } = y _ { 3 } y _ { 4 }$.

isi-entrance 2015 QB3 Circles Tangent to Each Other or to Axes View

An isosceles triangle with base 6 cms. and base angles $30 ^ { \circ }$ each is inscribed in a circle. A second circle, which is situated outside the triangle, touches the first circle and also touches the base of the triangle at its midpoint. Find its radius.

isi-entrance 2015 QB5 Intersection of Circles or Circle with Conic View

If a circle intersects the hyperbola $y = 1 / x$ at four distinct points $\left( x _ { i } , y _ { i } \right) , i = 1,2,3,4$, then prove that $x _ { 1 } x _ { 2 } = y _ { 3 } y _ { 4 }$.

isi-entrance 2015 Q17 4 marks Intersection of Circles or Circle with Conic View

Let $a$ be a real number. The number of distinct solutions $( x , y )$ of the system of equations $( x - a ) ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } = y ^ { 2 }$, can only be
(a) $0,1,2,3,4$ or 5
(b) 0, 1 or 3
(c) 0, 1, 2 or 4
(d) $0,2,3$, or 4

isi-entrance 2015 Q17 4 marks Intersection of Circles or Circle with Conic View

Let $a$ be a real number. The number of distinct solutions $( x , y )$ of the system of equations $( x - a ) ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } = y ^ { 2 }$, can only be
(a) $0,1,2,3,4$ or 5
(b) 0, 1 or 3
(c) 0, 1, 2 or 4
(d) $0,2,3$, or 4

isi-entrance 2015 Q26 4 marks Circle-Related Locus Problems View

Let $P$ be a variable point on a circle $C$ and $Q$ be a fixed point outside $C$. If $R$ is the mid-point of the line segment $P Q$, then the locus of $R$ is
(a) a circle
(b) an ellipse
(c) a line segment
(d) segment of a parabola.

isi-entrance 2015 Q26 4 marks Circle-Related Locus Problems View

Let $P$ be a variable point on a circle $C$ and $Q$ be a fixed point outside $C$. If $R$ is the mid-point of the line segment $P Q$, then the locus of $R$ is
(a) a circle
(b) an ellipse
(c) a line segment
(d) segment of a parabola.

isi-entrance 2016 Q21 4 marks Area and Geometric Measurement Involving Circles View

Let $n$ be a positive integer. Consider a square $S$ of side $2n$ units with sides parallel to the coordinate axes. Divide $S$ into $4n^2$ unit squares by drawing $2n-1$ horizontal and $2n-1$ vertical lines one unit apart. A circle of diameter $2n-1$ is drawn with its centre at the intersection of the two diagonals of the square $S$. How many of these unit squares contain a portion of the circumference of the circle?
(A) $4n - 2$
(B) $4n$
(C) $8n - 4$
(D) $8n - 2$

isi-entrance 2016 Q21 4 marks Area and Geometric Measurement Involving Circles View

Let $n$ be a positive integer. Consider a square $S$ of side $2n$ units with sides parallel to the coordinate axes. Divide $S$ into $4 n ^ { 2 }$ unit squares by drawing $2n - 1$ horizontal and $2n - 1$ vertical lines one unit apart. A circle of diameter $2n - 1$ is drawn with its centre at the intersection of the two diagonals of the square $S$. How many of these unit squares contain a portion of the circumference of the circle?
(A) $4 n - 2$
(B) $4 n$
(C) $8 n - 4$
(D) $8 n - 2$

isi-entrance 2016 Q23 4 marks Circles Tangent to Each Other or to Axes View

An isosceles triangle with base 6 cms. and base angles $30^\circ$ each is inscribed in a circle. A second circle touches the first circle and also touches the base of the triangle at its midpoint. If the second circle is situated outside the triangle, then its radius (in cms.) is
(A) $3\sqrt{3}/2$
(B) $\sqrt{3}/2$
(C) $\sqrt{3}$
(D) $4/\sqrt{3}$

isi-entrance 2016 Q23 4 marks Circles Tangent to Each Other or to Axes View

An isosceles triangle with base 6 cms. and base angles $30 ^ { \circ }$ each is inscribed in a circle. A second circle touches the first circle and also touches the base of the triangle at its midpoint. If the second circle is situated outside the triangle, then its radius (in cms.) is
(A) $3 \sqrt { 3 } / 2$
(B) $\sqrt { 3 } / 2$
(C) $\sqrt { 3 }$
(D) $4 / \sqrt { 3 }$

isi-entrance 2016 Q39 4 marks Chord Length and Chord Properties View

Consider a circle with centre $O$. Two chords $AB$ and $CD$ extended intersect at a point $P$ outside the circle. If $\angle AOC = 43^\circ$ and $\angle BPD = 18^\circ$, then the value of $\angle BOD$ is
(A) $36^\circ$
(B) $29^\circ$
(C) $7^\circ$
(D) $25^\circ$

isi-entrance 2016 Q39 4 marks Chord Length and Chord Properties View

Consider a circle with centre $O$. Two chords $A B$ and $C D$ extended intersect at a point $P$ outside the circle. If $\angle A O C = 43 ^ { \circ }$ and $\angle B P D = 18 ^ { \circ }$, then the value of $\angle B O D$ is
(A) $36 ^ { \circ }$
(B) $29 ^ { \circ }$
(C) $7 ^ { \circ }$
(D) $25 ^ { \circ }$

isi-entrance 2016 Q50 4 marks Circle-Related Locus Problems View

A triangle $ABC$ has a fixed base $BC$. If $AB : AC = 1 : 2$, then the locus of the vertex $A$ is
(A) a circle whose centre is the midpoint of $BC$
(B) a circle whose centre is on the line $BC$ but not the midpoint of $BC$
(C) a straight line
(D) none of the above

isi-entrance 2016 Q50 4 marks Circle-Related Locus Problems View

A triangle $A B C$ has a fixed base $B C$. If $A B : A C = 1 : 2$, then the locus of the vertex $A$ is
(A) a circle whose centre is the midpoint of $B C$
(B) a circle whose centre is on the line $B C$ but not the midpoint of $B C$
(C) a straight line
(D) none of the above

isi-entrance 2016 Q51 4 marks Circle-Related Locus Problems View

Let $P$ be a variable point on a circle $C$ and $Q$ be a fixed point outside $C$. If $R$ is the mid-point of the line segment $PQ$, then the locus of $R$ is
(A) a circle
(B) an ellipse
(C) a line segment
(D) segment of a parabola

isi-entrance 2016 Q51 4 marks Circle-Related Locus Problems View

Let $P$ be a variable point on a circle $C$ and $Q$ be a fixed point outside $C$. If $R$ is the mid-point of the line segment $P Q$, then the locus of $R$ is
(A) a circle
(B) an ellipse
(C) a line segment
(D) segment of a parabola

isi-entrance 2016 Q60 4 marks Circles Tangent to Each Other or to Axes View

Let $n \geq 3$ be an integer. Assume that inside a big circle, exactly $n$ small circles of radius $r$ can be drawn so that each small circle touches the big circle and also touches both its adjacent small circles. Then, the radius of the big circle is
(A) $r \operatorname{cosec} \frac{\pi}{n}$
(B) $r \left( 1 + \operatorname{cosec} \frac{2\pi}{n} \right)$
(C) $r \left( 1 + \operatorname{cosec} \frac{\pi}{2n} \right)$
(D) $r \left( 1 + \operatorname{cosec} \frac{\pi}{n} \right)$

isi-entrance 2016 Q60 4 marks Circles Tangent to Each Other or to Axes View

Let $n \geq 3$ be an integer. Assume that inside a big circle, exactly $n$ small circles of radius $r$ can be drawn so that each small circle touches the big circle and also touches both its adjacent small circles. Then, the radius of the big circle is
(A) $r \operatorname { cosec } \frac { \pi } { n }$
(B) $r \left( 1 + \operatorname { cosec } \frac { 2 \pi } { n } \right)$
(C) $r \left( 1 + \operatorname { cosec } \frac { \pi } { 2 n } \right)$
(D) $r \left( 1 + \operatorname { cosec } \frac { \pi } { n } \right)$

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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