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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 _______.

The angle of elevation of a jet plane from a point A on the ground is \( 60^\circ \). After a flight of \( 20 \, \text{seconds} \) at the speed of \( 432 \, \text{km/hour} \), the angle of elevation changes to \( 30^\circ \). If the jet plane is flying at a constant height, then its height is:

Let \( P(x) \) be a real polynomial of degree \( 3 \) which vanishes at \( x = -3 \). Let \( P(x) \) have local minima at \( x = 1 \), local maxima at \( x = -1 \) and \( \displaystyle\int\limits_{-1}^{1} P(x)\,dx = 18 \), then the sum of all the coefficients of the polynomials \( P(x) \) is equal to:

Let \( f(x) \) be a cubic polynomial with \( f(1)=-10 \), \( f(-1)=6 \), and has a local minima at \( x=1 \), and \( f'(x) \) has a local minima at \( x=-1 \). Then \( f(3) \) is equal to ______.

Let \( f:\mathbb{R}\to\mathbb{R} \) be defined as
Then \( f \) is increasing function in the interval.

A box open from top is made from a rectangular sheet of dimension \( a \times b \) by cutting squares each of side \( x \) from each of the four corners and folding up the flaps. If the volume of the box is maximum, then \( x \) is equal to :

A wire of length \( 36 \, \text{m} \) is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum and the circumference of the circle is \( k \) (\( \text{meter} \)), then \( \left(\frac{4}{\pi} + 1\right)k \) is equal to ________.

The number of distinct real roots of the equation \( 3x^4 + 4x^3 - 12x^2 + 4 = 0 \) is \( \_\_\_\_\_\_\_\_\_\_ \).

A wire of length \( 20 \, \text{m} \) is to be cut into \( 2 \) pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is:

If the curve \( y = ax^{2} + bx + c, \, x \in \mathbb{R} \), passes through the point \( (1,2) \) and the tangent line to this curve at origin is \( y = x \), then the possible values of \( a, b, c \), are:

Let \(f\) be a twice differentiable function defined on \(\mathbb{R}\) such that \(f(0)=1\), \(f'(0)=2\) and \(f'(x)\neq 0\) for all \(x\in \mathbb{R}\). If \(\begin{vmatrix} f(x) & f'(x) \\ f'(x) & f''(x) \end{vmatrix}=0\), for all \(x\in \mathbb{R}\), then the value of \(f(1)\) lies in the interval:

Let \( f(x)=3\sin^{4}x+10\sin^{3}x+6\sin^{2}x-3, \ x \in \left[-\dfrac{\pi}{6},\dfrac{\pi}{2}\right] \). Then, \( f \) is:

Let ‘\( a \)’ be a real number such that the function \( f(x)=ax^{2}+6x-15,\ x\in \mathbb{R} \) is increasing in \( \left(-\infty,\dfrac{3}{4}\right) \) and decreasing in \( \left(\dfrac{3}{4},\infty\right) \). Then the function \( g(x)=ax^{2}-6x+15,\ x\in \mathbb{R} \) has a:

Consider the function \( f : R \rightarrow R \) defined by . Then \( f \) is :

The local maximum value of the function \( f(x) = \left(\frac{2}{x}\right)^{x^{2}}, \; x>0 \), is:

Let \( a \in \mathbb{R} \) be such that the function is continuous at \( x=0 \), where \( \{x\} = x - [ x ], \, [ x ] \) is the greatest integer less than or equal to \( x \). Then:

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JEE Main 2021 25th July Evening Shift