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If the equation of the normal to the curve \( y = \frac{x - a}{(x + b)(x - 2)} \) at the point \( (1, -3) \) is \( x - 4y = 13 \), then the value of \( a + b \) is equal to ______.

Let \( a \in \mathbb{Z} \) and \( [t] \) be the greatest integer \( \le t \). Then the number of points, where the function \( f(x) = [a + 13 \sin x], \; x \in (0, \pi) \) is not differentiable, is ____________.

The number of points on the curve \( y = 54x^5 - 135x^4 - 70x^3 + 180x^2 + 210x \) at which the normal lines are parallel to \( x + 90y + 2 = 0 \) is:

The number of points, where the curve \( y = x^5 - 20x^3 + 50x + 2 \) crosses the \( x \)-axis, is _________.

Let \( f : [2, 4] \to \mathbb{R} \) be a differentiable function such that \( (x \log_{e} x) f^{\prime}(x) + (\log_{e} x) f(x) + f(x) \ge 1, \, x \in [2,4] \) with \( f(2) = \dfrac{1}{2} \) and \( f(4) = \dfrac{1}{4} \). Consider the following two statements:
\( (A): \quad f(x) \le 1, \text{ for all } x \in [2, 4] \)
\( (B): \quad f(x) \ge \dfrac{1}{8}, \text{ for all } x \in [2, 4] \)
Then,

If the local maximum value of the function \( f(x)=\left(\dfrac{\sqrt{3}e}{2\sin x}\right)^{\sin^2 x},\ x\in\left(0,\dfrac{\pi}{2}\right), \) is \( \dfrac{k}{e} \), then \( \left(\dfrac{k}{e}\right)^8+\dfrac{k^8}{e^5}+k^8 \) is equal to:

A wire of length \( 20\,\text{m} \) is to the cut into two pieces. A piece of length \( l_{1} \) is bent to make a square of area \( A_{1} \) and the other piece of length \( l_{2} \) is made into a circle of area \( A_{2} \). If \( 2A_{1}+3A_{2} \) is minimum then \( (\pi l_{1}):l_{2} \) is equal to:

Let the tangent to the curve \( x^{2} + 2x - 4y + 9 = 0 \) at the point \( P(1, 3) \) on it meet the \( y \)-axis at \( A \). Let the line passing through \( P \) and parallel to the line \( x - 3y = 6 \) meet the parabola \( y^{2} = 4x \) at \( B \). If \( B \) lies on the line \( 2x - 3y = 8 \), then \( (AB)^{2} \) is equal to ____________.

Consider the triangles with vertices \( A(2,1) \), \( B(0,0) \) and \( C(t,4),\ t\in[0,4] \). If the maximum and the minimum perimeters of such triangles are obtained at \( t=\alpha \) and \( t=\beta \) respectively, then \( 6\alpha+21|\beta| \) is equal to __________.

The sum of the absolute maximum and minimum values of the function \( f(x) = \left| x^2 - 5x + 6 \right| - 3x + 2 \) in the interval \( [-1, 3] \) is equal to:

If the area of the region \( \{(x,y): |x^{2}-2| \le y \le x\} \) is \( A \), then \( 6A + 16\sqrt{2} \) is equal to ______

Let \( S \) be the set of all \( a \in \mathbb{N} \) such that the area of the triangle formed by the tangent at the point \( P(b,c) \), \( b,c \in \mathbb{N} \), on the parabola \( y^{2}=2ax \) and the lines \( x=b \), \( y=0 \) is \( 16 \, \text{unit}^2 \), then \( \sum_{a \in S} a \) is equal to ________.

A square piece of tin of side \( 30 \, \text{cm} \) is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in \( \text{cm}^2 \)) is equal to:

If the functions \( f(x)=\frac{x^{3}}{3}+2bx+\frac{ax^{2}}{2} \) and \( g(x)=\frac{x^{3}}{3}+ax+bx^{2} \), \( a \neq 2b \) have a common extreme point, then \( a+2b+7 \) is equal to:

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JEE Main 2023 30th Jan Morning Shift

The number of points on the curve \( y = 54x^5 - 135x^4 - 70x^3 + 180x^2 + 210x \) at which the normal lines are parallel to \( x + 90y + 2 = 0 \) is: