iCON Education HYD, 79930 92826, 73309 72826JEE Main 2025 (Online) 23rd January Morning Shift
Let the area of a $\triangle P Q R$ with vertices $P(5,4), Q(-2,4)$ and $R(a, b)$ be 35 square units. If its orthocenter and centroid are $O\left(2, \frac{14}{5}\right)$ and $C(c, d)$ respectively, then $c+2 d$ is equal to
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2024 (Online) 9th April Evening Shift
Two vertices of a triangle $\mathrm{ABC}$ are $\mathrm{A}(3,-1)$ and $\mathrm{B}(-2,3)$, and its orthocentre is $\mathrm{P}(1,1)$. If the coordinates of the point $\mathrm{C}$ are $(\alpha, \beta)$ and the centre of the of the circle circumscribing the triangle $\mathrm{PAB}$ is $(\mathrm{h}, \mathrm{k})$, then the value of $(\alpha+\beta)+2(\mathrm{~h}+\mathrm{k})$ equals
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2024 (Online) 29th January Morning Shift
Let $\left(5, \frac{a}{4}\right)$ be the circumcenter of a triangle with vertices $\mathrm{A}(a,-2), \mathrm{B}(a, 6)$ and $C\left(\frac{a}{4},-2\right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha+\beta+\gamma$ is
A.
60
B.
62
C.
53
D.
30
Correct Answer: C
Explanation:
$\begin{aligned}
& A(a,-2), B(a, 6), C\left(\frac{a}{4},-2\right), O\left(5, \frac{a}{4}\right) \\
& A O=B O \\
& (a-5)^2+\left(\frac{a}{4}+2\right)^2=(a-5)^2+\left(\frac{a}{4}-6\right)^2 \\
& a=8 \\
& A B=8, A C=6, B C=10 \\
& \alpha=5, \beta=24, \gamma=24
\end{aligned}$
2023
JEE Mains
MCQ
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2023 (Online) 12th April Morning Shift
In a triangle ABC, if $\cos \mathrm{A}+2 \cos \mathrm{B}+\cos C=2$ and the lengths of the sides opposite to the angles A and C are 3 and 7 respectively, then $\mathrm{\cos A-\cos C}$ is equal to
$
\begin{aligned}
\cos A & -\cos C=\frac{b^2+c^2-a^2}{2 b c}-\frac{a^2+b^2-c^2}{2 \mathrm{ab}} \\\\
& =\frac{25+49-9}{70}-\frac{9+25-49}{30} \\\\
& =\frac{65}{70}+\frac{1}{2}=\frac{20}{14}=\frac{10}{7}
\end{aligned}
$
2023
JEE Mains
MCQ
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2023 (Online) 1st February Morning Shift
For a triangle $ABC$, the value of $\cos 2A + \cos 2B + \cos 2C$ is least. If its inradius is 3 and incentre is M, then which of the following is NOT correct?
Area of $\triangle \mathrm{ABC}=\frac{\sqrt{3}}{4} a^2=27 \sqrt{3}$
2023
JEE Mains
MCQ
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2023 (Online) 30th January Morning Shift
A straight line cuts off the intercepts $\mathrm{OA}=\mathrm{a}$ and $\mathrm{OB}=\mathrm{b}$ on the positive directions of $x$-axis and $y$ axis respectively. If the perpendicular from origin $O$ to this line makes an angle of $\frac{\pi}{6}$ with positive direction of $y$-axis and the area of $\triangle \mathrm{OAB}$ is $\frac{98}{3} \sqrt{3}$, then $\mathrm{a}^{2}-\mathrm{b}^{2}$ is equal to :
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2022 (Online) 25th June Morning Shift
Let a, b and c be the length of sides of a triangle ABC such that ${{a + b} \over 7} = {{b + c} \over 8} = {{c + a} \over 9}$. If r and R are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the value of ${R \over r}$ is equal to :
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2021 (Online) 27th August Morning Shift
Let ${{\sin A} \over {\sin B}} = {{\sin (A - C)} \over {\sin (C - B)}}$, where A, B, C are angles of triangle ABC. If the lengths of the sides opposite these angles are a, b, c respectively, then :
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2021 (Online) 20th July Morning Shift
If in a triangle ABC, AB = 5 units, $\angle B = {\cos ^{ - 1}}\left( {{3 \over 5}} \right)$ and radius of circumcircle of $\Delta$ABC is 5 units, then the area (in sq. units) of $\Delta$ABC is :
A.
$10 + 6\sqrt 2 $
B.
$8 + 2\sqrt 2 $
C.
$6 + 8\sqrt 3 $
D.
$4 + 2\sqrt 3 $
Correct Answer: C
Explanation:
As, $\cos B = {3 \over 5} \Rightarrow B = 53^\circ $
As, $R = 5 \Rightarrow {c \over {\sin c}} = 2R$
$ \Rightarrow {5 \over {10}} = \sin c \Rightarrow C = 30^\circ $
Sign change of ${{dA} \over {dx}}$ at $x = {r \over 2}$
$ \Rightarrow $ A has maximum at $x = {r \over 2}$
$BC = 2\sqrt {{r^2} - {x^2}} = \sqrt 3 r$,
$AM = r + {1 \over 2}r$ = ${3 \over 2}r$
$ \Rightarrow AB = AC = \sqrt 3 r$
2021
JEE Mains
MCQ
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2021 (Online) 24th February Evening Shift
Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices (a, c), (2, b) and (a, b) be $\left( {{{10} \over 3},{7 \over 3}} \right)$. If $\alpha$, $\beta$ are the roots of the equation $a{x^2} + bx + 1 = 0$, then the value of ${\alpha ^2} + {\beta ^2} - \alpha \beta $ is :
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2020 (Online) 4th September Morning Slot
A triangle ABC lying in the first quadrant has two vertices as A(1, 2) and B(3, 1). If $\angle BAC = {90^o}$ and area$\left( {\Delta ABC} \right) = 5\sqrt 5 $ s units, then the abscissa of the vertex C is :
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2019 (Online) 8th April Evening Slot
If the lengths of the sides of a triangle are in A.P.
and the greatest angle is double the smallest, then
a ratio of lengths of the sides of this triangle is :
A.
5 : 9 : 13
B.
5 : 6 : 7
C.
4 : 5 : 6
D.
3 : 4 : 5
Correct Answer: C
Explanation:
Let smallest angle $\angle A$ = $\theta $
and largest angle $\angle C$ is double of $\angle A$.
$ \therefore $ $\angle C$ = 2$\theta $
$ \therefore $ $\angle B$ = $\pi $ - 3$\theta $
Let length of sides are
a, b, c
where a < b < c
As a, b, c are in A.P then
2b = a + c
From sin rule we can say,
2 sin B = sin A + sin C
$ \Rightarrow $ 2 sin ($\pi $ – 3$\theta $) = sin $\theta $ + sin 2$\theta $
$ \Rightarrow $ 2 sin (3$\theta $) = sin $\theta $ + sin 2$\theta $
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2019 (Online) 11th January Morning Slot
In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. If x2 – c2 = y, where c is the length of the third side of the triangle, then the circumradius of the triangle is :
A.
${y \over {\sqrt 3 }}$
B.
${c \over 3}$
C.
${c \over {\sqrt 3 }}$
D.
${3 \over 2}$y
Correct Answer: C
Explanation:
Given a + b = x and ab = y
If x2 $-$ c2 = y $ \Rightarrow $ (a + b)2 $-$ c2 = ab
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2019 (Online) 10th January Evening Slot
With the usual notation, in $\Delta $ABC, if $\angle A + \angle B$ = 120o, a = $\sqrt 3 $ $+$ 1, b = $\sqrt 3 $ $-$ 1 then the ratio $\angle A:\angle B,$ is :
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2018 (Offline)
Let the orthocentre and centroid of a triangle be A(-3, 5) and B(3, 3) respectively. If C is the circumcentre
of this triangle, then the radius of the circle having line segment AC as diameter, is :
A.
${{3\sqrt 5 } \over 2}$
B.
$\sqrt {10} $
C.
$2\sqrt {10} $
D.
$3\sqrt {{5 \over 2}} $
Correct Answer: D
Explanation:
In a triangle, orthocentre, centroid and circumcenter are collinear and centroid divides orthocenter and circumcenter in 2 : 1 ratio.
In a $\Delta PQR,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} $ If $3{\mkern 1mu} \sin {\mkern 1mu} P + 4{\mkern 1mu} \cos {\mkern 1mu} Q = 6$ and $4\sin Q + 3\cos P = 1,$ then the angle R is equal to :
A.
${{5\pi } \over 6}$
B.
${{\pi } \over 6}$
C.
${{\pi } \over 4}$
D.
${{3\pi } \over 4}$
Correct Answer: B
Explanation:
Given $3$ $\sin \,P + 4\cos Q = 6$ $\,\,\,\,\,\,\,\,...\left( i \right)$
$4\sin Q + 3\cos P = 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)$
The sides of a triangle are $\sin \alpha ,\,\cos \alpha $ and $\sqrt {1 + \sin \alpha \cos \alpha } $ for some $0 < \alpha < {\pi \over 2}$. Then the greatest angle of the triangle is :
A.
${150^ \circ }$
B.
${90^ \circ }$
C.
${120^ \circ }$
D.
${60^ \circ }$
Correct Answer: C
Explanation:
Let $a = \sin \alpha ,b = \cos \alpha $
and $c = \sqrt {1 + \sin \alpha \cos \alpha } $
Clearly $a$ and $b < 1$ but $c > 1$
as $\,\,\,\sin \alpha > 0$ and $\cos \alpha > 0$
$\therefore$ $c$ is the greatest side and greatest angle is $C$
If in a $\Delta ABC$ $a\,{\cos ^2}\left( {{C \over 2}} \right) + c\,{\cos ^2}\left( {{A \over 2}} \right) = {{3b} \over 2},$ then the sides $a, b$ and $c$ :
In a triangle $ABC$, medians $AD$ and $BE$ are drawn. If $AD=4$,
$\angle DAB = {\pi \over 6}$ and $\angle ABE = {\pi \over 3}$, then the area of the $\angle \Delta ABC$ is :
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2025 (Online) 22nd January Evening Shift
Let $\mathrm{A}(6,8), \mathrm{B}(10 \cos \alpha,-10 \sin \alpha)$ and $\mathrm{C}(-10 \sin \alpha, 10 \cos \alpha)$, be the vertices of a triangle. If $L(a, 9)$ and $G(h, k)$ be its orthocenter and centroid respectively, then $(5 a-3 h+6 k+100 \sin 2 \alpha)$ is equal to ___________.
Correct Answer: 145
Explanation:
All the three points $A, B, C$ lie on the circle $x^2+y^2=100$ so circumcentre is $(0,0)$
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2024 (Online) 6th April Evening Shift
In a triangle $\mathrm{ABC}, \mathrm{BC}=7, \mathrm{AC}=8, \mathrm{AB}=\alpha \in \mathrm{N}$ and $\cos \mathrm{A}=\frac{2}{3}$. If $49 \cos (3 \mathrm{C})+42=\frac{\mathrm{m}}{\mathrm{n}}$, where $\operatorname{gcd}(m, n)=1$, then $m+n$ is equal to _________.
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2024 (Online) 4th April Evening Shift
Consider a triangle $\mathrm{ABC}$ having the vertices $\mathrm{A}(1,2), \mathrm{B}(\alpha, \beta)$ and $\mathrm{C}(\gamma, \delta)$ and angles $\angle A B C=\frac{\pi}{6}$ and $\angle B A C=\frac{2 \pi}{3}$. If the points $\mathrm{B}$ and $\mathrm{C}$ lie on the line $y=x+4$, then $\alpha^2+\gamma^2$ is equal to _______.
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2023 (Online) 10th April Evening Shift
In the figure, $\theta_{1}+\theta_{2}=\frac{\pi}{2}$ and $\sqrt{3}(\mathrm{BE})=4(\mathrm{AB})$. If the area of $\triangle \mathrm{CAB}$ is $2 \sqrt{3}-3$ unit ${ }^{2}$, when $\frac{\theta_{2}}{\theta_{1}}$ is the largest, then the perimeter (in unit) of $\triangle \mathrm{CED}$ is equal to _________.
Correct Answer: 6
Explanation:
We have, $\theta_1+\theta_2=\frac{\pi}{2}$ and $\sqrt{3}(B E)=4 A B$
Let $A B=x$ unit
$
\begin{aligned}
& A C=x \tan \theta_1 \\\\
& E D=x \tan \theta_2 \\\\
& B E=B D+D E
\end{aligned}
$
$
\begin{array}{rlrl}
& \Rightarrow \frac{4}{\sqrt{3}} x =x\left(\tan \theta_1+\tan \theta_2\right) {[\because \sqrt{3} B E=4 A B]} \\\\
& \Rightarrow \frac{4}{\sqrt{3}}=\tan \theta_1+\tan \left(\frac{\pi}{2}-\theta_1\right) {\left[\because \theta_1+\theta_2=\frac{\pi}{2}\right]}
\end{array}
$
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2021 (Online) 25th July Evening Shift
If a rectangle is inscribed in an equilateral triangle of side length $2\sqrt 2 $ as shown in the figure, then the square of the largest area of such a rectangle is _____________.
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2021 (Online) 16th March Evening Shift
In $\Delta$ABC, the lengths of sides AC and AB are 12 cm and 5 cm, respectively. If the area of $\Delta$ABC is 30 cm2 and R and r are respectively the radii of circumcircle and incircle of $\Delta$ABC, then the value of 2R + r (in cm) is equal to ___________.
iCON Education HYD, 79930 92826, 73309 72826JEE Main 2021 (Online) 16th March Morning Shift
Let ABCD be a square of side of unit length. Let a circle C1 centered at A with unit radius is drawn. Another circle C2 which touches C1 and the lines AD and AB are tangent to it, is also drawn. Let a tangent line from the point C to the circle C2 meet the side AB at E. If the length of EB is $\alpha$ + ${\sqrt 3 }$ $\beta$, where $\alpha$, $\beta$ are integers, then $\alpha$ + $\beta$ is equal to ____________.
