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Let \( A, B \) be points on the two half-lines \( x - \sqrt{3}\,|y| = \alpha, \; \alpha > 0 \) at a distance of \( \alpha \) from their point of intersection \( P \). The line segment \( AB \) meets the angle bisector of the given half-lines at the point \( Q \). If \( PQ = \dfrac{9}{2} \) and \( R \) is the radius of the circumcircle of \( \triangle PAB \), then \( \dfrac{\alpha^{2}}{R} \) is equal to ______.

A circle is inscribed in an equilateral triangle of side of length \( 12 \). If the area and perimeter of any square inscribed in this circle are \( m \) and \( n \), respectively, then \( m + n^2 \) equal to:

The lengths of the sides of a triangle are \( 10 + x^2 \), \( 10 + x^2 \) and \( 20 - 2x^2 \). If for \( x = k \), the area of the triangle is maximum, then \( 3k^2 \) is equal to:

If in a triangle \( ABC \), \( AB=5 \) units, \( \angle B=\cos^{-1}\!\left(\dfrac{3}{5}\right) \) and radius of circumcircle of \( \triangle ABC \) is \( 5 \) units, then the area (in sq. units) of \( \triangle ABC \) is:

The angle of elevation of the top \( P \) of a vertical tower \( PQ \) of height \( 10 \) from a point \( A \) on the horizontal ground is \( 45^\circ \). Let \( R \) be a point on \( AQ \) and from a point \( B \), vertically above \( R \), the angle of elevation of \( P \) is \( 60^\circ \). If \( \angle BAQ = 30^\circ, AB = d \) and the area of the trapezium \( PQRB \) is \( \alpha \), then the ordered pair \( (d,\alpha) \) is:

Let \( a,b \) and \( c \) be the length of sides of a triangle \( ABC \) such that \( \dfrac{a+b}{7}=\dfrac{b+c}{8}=\dfrac{c+a}{9} \). If \( r \) and \( R \) the radius of incircle and radius of circumcircle of the triangle \( ABC \), respectively, then the value of \( \dfrac{R}{r} \) is equal to:

A tower \( PQ \) stands on a horizontal ground with base \( Q \) on the ground. The point \( R \) divides the tower in two parts such that \( QR = 15 \, \text{m} \). If from a point \( A \) on the ground the angle of elevation of \( R \) is \( 60^\circ \) and the part \( PR \) of the tower subtends an angle of \( 15^\circ \) at \( A \), then the height of the tower is :

Let \( \dfrac{\sin A}{\sin B} = \dfrac{\sin(A - C)}{\sin(C - B)} \) where \( A, B, C \) are angles of a triangle \( ABC \). If the lengths of the sides opposite these angles are \( a, b, c \) respectively, then:

For a triangle \( \triangle ABC \), 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?

In \( \triangle ABC \), the lengths of sides \( AC \) and \( AB \) are \( 12 \, \text{cm} \) and \( 5 \, \text{cm} \), respectively. If the area of \( \triangle ABC \) is \( 30 \, \text{cm}^{2} \) and \( R \) and \( r \) are respectively the radii of circumcircle and incircle of \( \triangle ABC \), then the value of \( 2R+r \) (in \( \text{cm} \)) is equal to ____________.

From the top \( A \) of a vertical wall \( AB \) of height \( 30 \, \text{m} \), the angles of depression of the top \( P \) and bottom \( Q \) of a vertical tower \( PQ \) are \( 15^\circ \) and \( 60^\circ \) respectively. \( B \) and \( Q \) are on the same horizontal level. If \( C \) is a point on \( AB \) such that \( CB = PQ \), then the area (in \( \text{m}^2 \)) of the quadrilateral \( BCPQ \) is equal to:

A spherical gas balloon of radius \( 16 \) meter subtends an angle \( 60^\circ \) at the eye of the observer \( A \) while the angle of elevation of its center from the eye of \( A \) is \( 75^\circ \). Then the height (in meter) of the top most point of the balloon from the level of the observer's eye is:

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JEE Main 2021 20th July Morning Shift

If in a triangle \( ABC \), \( AB=5 \) units, \( \angle B=\cos^{-1}\!\left(\dfrac{3}{5}\right) \) and radius of circumcircle of \( \triangle ABC \) is \( 5 \) units, then the area (in sq. units) of \( \triangle ABC \) is: