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If a point \( A(x, y) \) lies in the region bounded by the y-axis, straight lines \( 2y + x = 6 \) and \( 5x - 6y = 30 \), then the probability that \( y < 1 \) is

The slope of the tangent to a curve \( C: y=y(x) \) at any point \( (x,y) \) on it is \( \dfrac{2e^{2x}-6e^{-x}+9}{2+9e^{-2x}} \). If \( C \) passes through the points \( \left(0,\dfrac{1}{2}+\dfrac{\pi}{2\sqrt{2}}\right) \) & \( \left(\alpha,\dfrac{1}{2}e^{2\alpha}\right) \), then \( e^{\alpha} \) is equal to :

If the maximum value of \( a \), for which the function \( f_a(x)=\tan^{-1}2x-3ax+7 \) is non-decreasing in \( \left(-\dfrac{\pi}{6},\dfrac{\pi}{6}\right) \), is \( \bar a \), then \( f_{\bar a}\!\left(\dfrac{\pi}{8}\right) \) is equal to:

The number of distinct real roots of the equation
\( x^{5}\left(x^{3}-x^{2}-x+1\right)+x\left(3x^{3}-4x^{2}-2x+4\right)-1=0 \) is ________.

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi-vertical angle is \( \tan^{-1}\dfrac{3}{4} \). Water is poured in it at a constant rate of \( 6 \) cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is \( 4 \) meters, is _______.

The minimum value of the twice differentiable function \( f(x) = \displaystyle\int\limits_{0}^{x} e^{\,x-t} f'(t) \, dt - (x^{2} - x + 1) e^{x}, \ x \in \mathbb{R} \) is:

Let \( f:\mathbb{R}\to \mathbb{R} \) be a function defined by \( f(x)=(x-3)^{n_1}(x-5)^{n_2}, \, n_1,n_2 \in \mathbb{N} \). Then, which of the following is NOT true?

The number of solutions of the equation \( 2\theta - \cos^2 \theta + \sqrt{2} = 0 \) in \( R \) is equal to __________.

Let \( f(x) = 2\cos^{-1}x + 4\cot^{-1}x - 3x^{2} - 2x + 10, \, x \in [-1,1] \). If \([a,b]\) is the range of the function \( f \), then \( 4a - b \) is equal to:

Let \( S \) be the set of all the natural numbers, for which the line \( \dfrac{x}{a}+\dfrac{y}{b}=2 \) is a tangent to the curve \( \left(\dfrac{x}{a}\right)^{n}+\left(\dfrac{y}{b}\right)^{n}=2 \) at the point \( (a,b), ab\neq 0 \). Then:

The number of points where the function

\([t]\) denotes the greatest integer \( \le t \), is discontinuous is ____________.

If \( m \) and \( n \) respectively are the number of local maximum and local minimum points of the function \( f(x)=\int_{0}^{x^{2}} \frac{t^{2}-5t+4}{2+e^{t}} \, dt \), then the ordered pair \( (m,n) \) is equal to

A wire of length \( 22 \, \text{m} \) is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is:

$If\ the\ angle\ made\ by\ the\ tangent\ at\ the\ point\ (x_0, y_0)\ on\ the\ curve$ $x = 12\Big(t + \sin t \cos t\Big), y = 12\big(1 + \sin t\big)^2, 0 < t < \frac{\pi}{2},\ with\ the\ positive\ x\text{-axis}\ is\ \frac{\pi}{3},\ then\ y_0\ is\ equal\ to\ :$

The number of distinct real roots of \( x^4 - 4x + 1 = 0 \)

The sum of the maximum and minimum values of the function \( f(x) = |5x-7| + \lfloor x^2 + 2x \rfloor \) in the interval \( \left[ \dfrac{5}{4}, 2 \right] \), where \( [ t ] \) is the greatest integer \( \le t \), is ____________ .

The sum of absolute maximum and absolute minimum values of the function \( f(x) = |2x^{2} + 3x - 2| + \sin x \cos x \) in the interval \( [0, 1] \) is:

Consider a cuboid of sides \( 2x \), \( 4x \) and \( 5x \) and a closed hemisphere of radius \( r \). If the sum of their surface areas is a constant \( k \), then the ratio \( x : r \), for which the sum of their volumes is maximum, is:

Let the function \( f(x) = 2x^2 - \log_e x, \, x > 0 \), be decreasing in \( (0, a) \) and increasing in \( (a, 4) \). A tangent to the parabola \( y^2 = 4ax \) at a point \( P \) on it passes through the point \( (8a, 8a - 1) \) but does not pass through the point \( \left(-\dfrac{1}{a}, 0\right) \). If the equation of the normal at \( P \) is \( \dfrac{x}{\alpha} + \dfrac{y}{\beta} = 1 \), then \( \alpha + \beta \) is equal to ______.

If the tangent at the point \( (x_{1}, y_{1}) \) on the curve \( y = x^{3} + 3x^{2} + 5 \) passes through the origin, then \( (x_{1}, y_{1}) \) does NOT lie on the curve:

The value of \( \log_e2 \, \dfrac{d}{dx}\!\left(\log_{\cos x} \text{cosec}\, x\right) \) at \( x=\dfrac{\pi}{4} \) is

Let \( f \) and \( g \) be twice differential even functions on \( (-2,2) \) such that \( f\!\left(\frac{1}{4}\right)=0 \), \( f\!\left(\frac{1}{2}\right)=0 \), \( f(1)=1 \) and \( g\!\left(\frac{3}{4}\right)=0 \), \( g(1)=2 \). Then, the minimum number of solutions of \( f(x)g''(x)+f'(x)g'(x)=0 \) in \( (-2,2) \) is equal to ______.

Let \( f : [0,1] \to \mathbb{R} \) be a twice differentiable function in \( (0,1) \) such that \( f(0) = 3 \) and \( f(1) = 5 \). If the line \( y = 2x + 3 \) intersects the graph of \( f \) at only two distinct points in \( (0,1) \), then the least number of points \( x \in (0,1) \), at which \( f''(x) = 0 \), is __________.

The sum of the absolute maximum and absolute minimum values of the function \( f(x) = \tan^{-1}(\sin x - \cos x) \) in the interval \( [0, \pi] \) is:

If the tangent to the curve \( y=x^{3}-x^{2}+x \) at the point \( (a,b) \) is also tangent to the curve \( y=5x^{2}+2x-25 \) at the point \( (2,-1) \), then \( |2a+9b| \) is equal to ________.

The curve \( y(x) = ax^3 + bx^2 + cx + 5 \) touches the x-axis at the point \( P(-2,0) \) and cuts then y-axis at the point \( Q \), where \( y' \) is equal to \( 3 \). Then the local maximum value of \( y(x) \) is :

Let \( f(x)=3^{(x^2-2)^3+4}, \; x \in R \). Then which of the following statements are true?
\( P:\; x=0 \) is a point of local minima of \( f \)
\( Q:\; x=\sqrt{2} \) is a point of inflection of \( f \)
\( R:\; f' \) is increasing for \( x>\sqrt{2} \)

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

Let the area enclosed by the \( x \)-axis, and the tangent and normal drawn to the curve \( 4x^{3}-3xy^{2}+6x^{2}-5xy-8y^{2}+9x+14=0 \) at the point \( (-2,3) \) be \( A \). The \( 8A \) is equal to ____________ .

For the function \( f(x) = 4\log_e(x-1) - 2x^{2} + 4x + 5, \ x > 1 \), which one of the following is NOT correct?

Let \( f(x)=|(x-1)|(x^2-2x-3)|+x-3, \ x\in \mathbb{R} \). If \( m \) and \( M \) are respectively the number of points of local minimum and local maximum of \( f \) in the interval \( (0,4) \), then \( m+M \) is equal to __________.

. Then the set of all values of \( b \), for which \( f(x) \) has maximum value at \( x=1 \), is:

The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is \( 3 \) units and after \( 5 \) seconds, it becomes \( 7 \) units, then its radius after \( 9 \) seconds is :

The number of real solutions of \( x^7 + 5x^3 + 3x + 1 = 0 \) is equal to _______.

Let \( M \) and \( N \) be the number of points on the curve \( y^{5}-9xy+2x=0 \), where the tangents to the curve are parallel to \( x \)-axis , respectively. Then the value of \( M+N \) equals ______.

Let \( \lambda^{*} \) be the largest value of \( \lambda \) for which the function \( f_{\lambda}(x)=4\lambda x^{3}-36\lambda x^{2}+36x+48 \) is increasing for all \( x\in \mathbb{R} \). Then \( f_{\lambda^{*}}(1)+f_{\lambda^{*}}(-1) \) is equal to:

The function \( f(x)=x e^{x(1-x)},\; x\in \mathbb{R}, \) is:

Water is being filled at the rate of \( 1 \, \text{cm}^{3} / \text{sec} \) in a right circular conical vessel (vertex downwards) of height \( 35 \, \text{cm} \) and diameter \( 14 \, \text{cm} \). When the height of the water level is \( 10 \, \text{cm} \), the rate (in \( \text{cm}^{2} / \text{sec} \)) at which the wet conical surface area of the vessel increases is

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JEE Main 2022 27th June Morning Shift

The number of distinct real roots of \( x^4 - 4x + 1 = 0 \)