JEE Advanced 2021 Paper 2 Online | Properties of Triangle Question 2 | Mathematics

Consider a triangle PQR having sides of lengths p, q and r opposite to the angles P, Q and R, respectively. Then which of the following statements is (are) TRUE?

$$\cos P \ge 1 - {{{p^2}} \over {2qr}}$$

$$\cos R \ge \left( {{{q - r} \over {p + q}}} \right)\cos P + \left( {{{p - r} \over {p + q}}} \right)\cos Q$$

$${{q + r} \over p} < 2{{\sqrt {\sin q\sin R} } \over {\sin P}}$$

If p < q and p < r, then $$\cos Q > {p \over r}$$ and $$\cos R > {p \over q}$$

JEE Advanced 2020 Paper 1 Offline

MCQ (More than One Correct Answer)

-2

Let x, y and z be positive real numbers. Suppose x, y and z are the lengths of the sides of a triangle opposite to its angles X, Y, and Z, respectively. If

$$\tan {X \over 2} + \tan {Z \over 2} = {{2y} \over {x + y + z}}$$, then which of the following statements is/are TRUE?

2Y = X + Z

Y = X + Z

$$\tan {X \over 2}$$ = $${x \over {y + z}}$$

x² + z² $$-$$ y² = xz

JEE Advanced 2019 Paper 1 Offline

MCQ (More than One Correct Answer)

-1

In a non-right-angled triangle $$\Delta $$PQR, let p, q, r denote the lengths of the sides opposite to the angles At P, Q, R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at E, and RS and PE intersect at O. If p = $${\sqrt 3 }$$, q = 1, and the radius of the circumcircle of the $$\Delta $$PQR equals 1, then which of the following options is/are correct?

Length of OE = $${1 \over 6}$$

Length of RS = $${{\sqrt 7 } \over 2}$$

Area of $$\Delta $$SOE = $${{\sqrt 3 } \over {12}}$$

Radius of incircle of $$\Delta $$PQR = $${{\sqrt 3 } \over {2}}$$($${2 - \sqrt 3 }$$)

JEE Advanced 2016 Paper 1 Offline

MCQ (More than One Correct Answer)

-2

In a triangle $$\Delta $$$$XYZ$$, let $$x, y, z$$ be the lengths of sides opposite to the angles $$X, Y, Z$$ respectively, and $$2s = x + y + z$$.
If $${{s - x} \over 4} = {{s - y} \over 3} = {{s - z} \over 2}$$ and area of incircle of the triangle $$XYZ$$ is $${{8\pi } \over 3}$$, then

area of the triangle $$XYZ$$ is $$6\sqrt 6 $$

the radius of circumcircle of the triangle $$XYZ$$ is $${{35} \over 6}\sqrt 6 $$

$$\sin {X \over 2}\sin {Y \over 2}\sin {Z \over 2} = {4 \over {35}}$$

$${\sin ^2}\left( {{{X + Y} \over 2}} \right) = {3 \over 5}$$

Questions Asked from MCQ (Multiple Correct Answer)

JEE Advanced 2021 Paper 2 Online (1) JEE Advanced 2020 Paper 1 Offline (1) JEE Advanced 2019 Paper 1 Offline (1) JEE Advanced 2016 Paper 1 Offline (1) JEE Advanced 2013 Paper 2 Offline (1) IIT-JEE 2006 (1) IIT-JEE 1987 (1) IIT-JEE 1986 (1)

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