LFM Pure

isi-entrance 2013 Q13 4 marks Determine an angle or side from a trigonometric identity/equation View

If $A , B , C$ are the angles of a triangle and $\sin ^ { 2 } A + \sin ^ { 2 } B = \sin ^ { 2 } C$, then $C$ is equal to
(A) $30 ^ { \circ }$
(B) $90 ^ { \circ }$
(C) $45 ^ { \circ }$
(D) none of the above

isi-entrance 2014 Q8 Compute area of a triangle or related figure View

A regular pentagon is inscribed in a circle of radius $r$ and another regular pentagon is circumscribed about the same circle. Find the ratio of the area of the inscribed pentagon to the area of the circumscribed pentagon.
(A) $\sin^2 36^\circ$ (B) $\cos^2 36^\circ$ (C) $\tan^2 36^\circ$ (D) $\cos^2 54^\circ$

isi-entrance 2014 Q22 Circumradius or incircle radius computation View

In a triangle $ABC$, the circumradius is $r$ and $BC = r/2$. The circumcentre $O$ lies on $AD$ where $D$ is the midpoint of $BC$. Find the ratio $BC : AD$.
(A) $\sqrt{3} : \sqrt{2}$ (B) $\sqrt{2} : \sqrt{3}$ (C) $1 : \sqrt{3}$ (D) $\sqrt{3} : 1$

isi-entrance 2016 Q11 4 marks Find an angle using the cosine rule View

The sides of a triangle are given to be $x ^ { 2 } + x + 1, 2 x + 1$ and $x ^ { 2 } - 1$. Then the largest of the three angles of the triangle is
(A) $75 ^ { \circ }$
(B) $\left( \frac { x } { x + 1 } \pi \right)$ radians
(C) $120 ^ { \circ }$
(D) $135 ^ { \circ }$

isi-entrance 2016 Q11 4 marks Find an angle using the cosine rule View

The sides of a triangle are given to be $x ^ { 2 } + x + 1, 2 x + 1$ and $x ^ { 2 } - 1$. Then the largest of the three angles of the triangle is
(A) $75 ^ { \circ }$
(B) $\left( \frac { x } { x + 1 } \pi \right)$ radians
(C) $120 ^ { \circ }$
(D) $135 ^ { \circ }$

isi-entrance 2016 Q12 4 marks Heights and distances / angle of elevation problem View

Two poles, $AB$ of length two metres and $CD$ of length twenty metres are erected vertically with bases at $B$ and $D$. The two poles are at a distance not less than twenty metres. It is observed that $\tan \angle ACB = 2/77$. The distance between the two poles is
(A) $72 m$
(B) 68 m
(C) 24 m
(D) 24.27 m

isi-entrance 2016 Q12 4 marks Heights and distances / angle of elevation problem View

Two poles, $A B$ of length two metres and $C D$ of length twenty metres are erected vertically with bases at $B$ and $D$. The two poles are at a distance not less than twenty metres. It is observed that $\tan \angle A C B = 2 / 77$. The distance between the two poles is
(A) $72 m$
(B) 68 m
(C) 24 m
(D) 24.27 m

isi-entrance 2016 Q13 4 marks Determine an angle or side from a trigonometric identity/equation View

If $A , B , C$ are the angles of a triangle and $\sin ^ { 2 } A + \sin ^ { 2 } B = \sin ^ { 2 } C$, then $C$ is equal to
(A) $30 ^ { \circ }$
(B) $90 ^ { \circ }$
(C) $45 ^ { \circ }$
(D) none of the above

isi-entrance 2016 Q13 4 marks Determine an angle or side from a trigonometric identity/equation View

If $A , B , C$ are the angles of a triangle and $\sin ^ { 2 } A + \sin ^ { 2 } B = \sin ^ { 2 } C$, then $C$ is equal to
(A) $30 ^ { \circ }$
(B) $90 ^ { \circ }$
(C) $45 ^ { \circ }$
(D) none of the above

isi-entrance 2016 Q46 4 marks Multi-step composite figure problem 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 2016 Q46 4 marks Multi-step composite figure problem View

Suppose $A B C D$ is a quadrilateral such that $\angle B A C = 50 ^ { \circ } , \angle C A D = 60 ^ { \circ } , \angle C B D = 30 ^ { \circ }$ and $\angle B D C = 25 ^ { \circ }$. If $E$ is the point of intersection of $A C$ and $B D$, then the value of $\angle A E B$ is
(A) $75 ^ { \circ }$
(B) $85 ^ { \circ }$
(C) $95 ^ { \circ }$
(D) $110 ^ { \circ }$

isi-entrance 2017 Q21 Determine an angle or side from a trigonometric identity/equation View

In a triangle $ABC$, $3\sin A + 4\cos B = 6$ and $4\sin B + 3\cos A = 1$ hold. Then the angle $C$ equals
(A) $30^\circ$
(B) $60^\circ$
(C) $120^\circ$
(D) $150^\circ$.

isi-entrance 2019 Q3 Compute area of a triangle or related figure View

Consider a paper in the shape of an equilateral triangle $A B C$ with circumcenter $O$ and perimeter 9 units. If we fold the paper in such a way that each of the vertices $A , B , C$ gets identified with $O$, then the area of the resulting shape in square units is:
(A) $\frac { 3 \sqrt { 3 } } { 4 }$
(B) $\frac { 4 } { \sqrt { 3 } }$
(C) $\frac { 3 \sqrt { 3 } } { 2 }$
(D) $3 \sqrt { 3 }$.

isi-entrance 2022 Q25 Circumradius or incircle radius computation View

A triangle has sides of lengths $\sqrt { 5 } , 2 \sqrt { 2 } , \sqrt { 3 }$ units. Then, the radius of its inscribed circle is:
(A) $\frac { \sqrt { 5 } + \sqrt { 3 } + 2 \sqrt { 2 } } { 2 }$
(B) $\frac { \sqrt { 5 } + \sqrt { 3 } + 2 \sqrt { 2 } } { 3 }$
(C) $\sqrt { 5 } + \sqrt { 3 } + 2 \sqrt { 2 }$
(D) $\frac { \sqrt { 5 } + \sqrt { 3 } - 2 \sqrt { 2 } } { 2 }$

isi-entrance 2023 Q5 Multi-step composite figure problem View

Consider a right angled triangle $\triangle A B C$ whose hypotenuse $A C$ is of length 1. The bisector of $\angle A C B$ intersects $A B$ at $D$. If $B C$ is of length $x$, then what is the length of $CD$?
(A) $\sqrt { \frac { 2 x ^ { 2 } } { 1 + x } }$
(B) $\frac { 1 } { \sqrt { 2 + 2 x } }$
(C) $\sqrt { \frac { x } { 1 + x } }$
(D) $\frac { x } { \sqrt { 1 - x ^ { 2 } } }$

isi-entrance 2024 Q9 Multi-step composite figure problem View

In $\triangle ABC$, $CD$ is the median and $BE$ is the altitude. Given that $\overline{CD} = \overline{BE}$, what is the value of $\angle ACD$?
(A) $\pi/3$
(B) $\pi/4$
(C) $\pi/5$
(D) $\pi/6$

isi-entrance 2026 Q11 Ambiguous case and triangle existence/uniqueness View

The number of triplets $( a , b , c )$ of integers such that $a < b < c$ and $a , b , c$ are sides of a triangle with perimeter 21 is
(a) 7 .
(B) 8.
(C) 11 .
(D) 12 .

isi-entrance 2026 Q14 Multi-step composite figure problem View

Suppose $A B C D$ is a quadrilateral such that $\angle B A C = 50 ^ { \circ } , \angle C A D = 60 ^ { \circ } , \angle C B D = 30 ^ { \circ }$ and $\angle B D C = 25 ^ { \circ }$. If $E$ is the point of intersection of $A C$ and $B D$, then the value of $\angle A E B$ is
(a) $75 ^ { \circ }$.
(B) $85 ^ { \circ }$.
(C) $95 ^ { \circ }$.
(D) $110 ^ { \circ }$.

jee-advanced 2009 Q31 Determine an angle or side from a trigonometric identity/equation View

In a triangle $A B C$ with fixed base $B C$, the vertex $A$ moves such that
$$\cos B + \cos C = 4 \sin ^ { 2 } \frac { A } { 2 }$$
If $a , b$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A , B$ and $C$, respectively, then
(A) $b + c = 4 a$
(B) $b + c = 2 a$
(C) locus of point $A$ is an ellipse
(D) locus of point $A$ is a pair of straight lines

jee-advanced 2009 Q33 Ambiguous case and triangle existence/uniqueness View

Let $ABC$ and $ABC^{\prime}$ be two non-congruent triangles with sides $AB=4$, $AC=AC^{\prime}=2\sqrt{2}$ and angle $B=30^{\circ}$. The absolute value of the difference between the areas of these triangles is

jee-advanced 2010 Q27 Circumradius or incircle radius computation View

Consider a triangle ABC and let $\mathrm { a } , \mathrm { b }$ and c denote the lengths of the sides opposite to vertices $\mathrm { A } , \mathrm { B }$ and C respectively. Suppose $\mathrm { a } = 6 , \mathrm {~b} = 10$ and the area of the triangle is $15 \sqrt { 3 }$. If $\angle \mathrm { ACB }$ is obtuse and if r denotes the radius of the incircle of the triangle, then $r ^ { 2 }$ is equal to

jee-advanced 2010 Q29 Determine an angle or side from a trigonometric identity/equation View

If the angles $\mathrm { A } , \mathrm { B }$ and C of a triangle are in an arithmetic progression and if $\mathrm { a } , \mathrm { b }$ and c denote the lengths of the sides opposite to $\mathrm { A } , \mathrm { B }$ and C respectively, then the value of the expression $\frac { a } { c } \sin 2 C + \frac { c } { a } \sin 2 A$ is
A) $\frac { 1 } { 2 }$
B) $\frac { \sqrt { 3 } } { 2 }$
C) 1
D) $\sqrt { 3 }$

jee-advanced 2010 Q38 Find an angle using the cosine rule View

Let ABC be a triangle such that $\angle \mathrm { ACB } = \frac { \pi } { 6 }$ and let $\mathrm { a } , \mathrm { b }$ and c denote the lengths of the sides opposite to $\mathrm { A } , \mathrm { B }$ and C respectively. The value(s) of x for which $\mathrm { a } = \mathrm { x } ^ { 2 } + \mathrm { x } + 1 , \mathrm {~b} = \mathrm { x } ^ { 2 } - 1$ and $\mathrm { c } = 2 \mathrm { x } + 1$ is (are)
A) $- ( 2 + \sqrt { 3 } )$
B) $1 + \sqrt { 3 }$
C) $2 + \sqrt { 3 }$
D) $4 \sqrt { 3 }$

jee-advanced 2013 Q48 Circumradius or incircle radius computation View

In a triangle $P Q R$, $P$ is the largest angle and $\cos P = \frac { 1 } { 3 }$. Further the incircle of the triangle touches the sides $P Q , Q R$ and $R P$ at $N , L$ and $M$ respectively, such that the lengths of $P N , Q L$ and $R M$ are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)
(A) 16
(B) 18
(C) 24
(D) 22

jee-advanced 2014 Q47 Determine an angle or side from a trigonometric identity/equation View

In a triangle the sum of two sides is $x$ and the product of the same two sides is $y$. If $x^2 - c^2 = y$, where $c$ is the third side of the triangle, then the ratio of the in-radius to the circum-radius of the triangle is
(A) $\frac{3y}{2x(x+c)}$
(B) $\frac{3y}{2c(x+c)}$
(C) $\frac{3y}{4x(x+c)}$
(D) $\frac{3y}{4c(x+c)}$

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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