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If the function \( f(x)=2x^3-9ax^2+12a^2x+1, \ a>0 \) has a local maximum at \( x=\alpha \) and a local minimum at \( x=\alpha^2 \), then \( \alpha \) and \( \alpha^2 \) are the roots of the equation:

If the term independent of \( x \) in the expansion of \( \left(\sqrt{a}\,x^2+\frac{1}{2x^3}\right)^{10} \) is \( 105 \), then \( a^2 \) is equal to:

The area of the region in the first quadrant inside the circle \( x^2+y^2=8 \) and outside the parabola \( y^2=2x \) is equal to:

Let \( \vec{a}=4\hat{i}-\hat{j}+\hat{k} \) and \( \vec{b}=11\hat{i}-\hat{j}-\hat{k} \) and \( \vec{c} \) be a vector such that \( (\vec{a}+\vec{b})\times\vec{c}=\vec{c}\times(2\vec{a}+3\vec{b}) \). If \( (2\vec{a}+3\vec{b}).\\vec{c}=1670 \), then \( |\vec{c}|^2 \) is equal to:

The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to:

If the system of equations \( x+4y-z=\lambda, \ 7x+9y+\mu z=-3, \ 5x+y+2z=-1 \) has infinitely many solutions, then \( (2\mu+3\lambda )\) is equal to:

where \( a>0 \) and \( g(x):=(f(|x|)-|f(x|))/2 \). Then the function \( g:[-a,a]\to[-a,a] \) is:

Let \( y=y(x) \) be the solution curve of the differential equation \( \sec y\,\dfrac{dy}{dx}+2x\sin y=x^{3}\cos y,\; y(1)=0 \). Then \( y(\sqrt{3}) \) is equal to:

If the line segment joining the point \( (5,2) \) and \( (2,a) \) subtends an angle \( \dfrac{\pi}{4} \) at the origin, then the absolute value of the product of all possible values of \( a \) is:

If \( \alpha \ne a,\ \beta \ne b,\ \gamma \ne c \) and then \( \dfrac{a}{\alpha-a}+\dfrac{b}{\beta-b}+\dfrac{\gamma}{\gamma-c} \) is equal to:

If the image of the point \( (-4,5) \) in the line \( x+2y=2 \) lies on the circle \( (x+4)^{2}+(y-3)^{2}=r^{2} \), then \( r \) is equal to:

If the value of \(\displaystyle \frac{3\cos 36^\circ + 5\sin 18^\circ}{5\cos 36^\circ - 3\sin 18^\circ} \) is \(\displaystyle \frac{a\sqrt{5} - b}{c} \), where \( a \), \( b \), \( c \) are natural numbers and \( \gcd(a,c)=1 \), then \( a+b+c \) is equal to :

Let \( \vec{a}= \hat{\imath}+2\hat{\jmath}+3\hat{k} \), \( \vec{b}=2\hat{\imath}+3\hat{\jmath}-5\hat{k} \) and \( \vec{c}=3\hat{\imath}-\hat{\jmath}+\lambda \hat{k} \) be three vectors. Let \( \vec{r} \) unit vector along \( \vec{b}+\vec{c} \). If \( \vec{r}\cdot \vec{a}=3 \), then \( 3\lambda \) is equal to :

There are three bags \( X \), \( Y \) and \( Z \). Bag \( X \) contains \( 5 \) one–rupee coins and \( 4 \) five-rupee coins; Bag \( Y \) contains \( 4 \) one-rupee coins and \( 5 \) five-rupee coins and Bag \( Z \) contains \( 3 \) one-rupee coins and \( 6 \) five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag \( Y \), is:

Let \( A=[2,3,6,8,9,11] \) and \( B=[1,4,5,10,15] \). Let \( R \) be a relation on \( A\times B \) defined by \( (a,b)R(c,d) \) if and only if \( 3ad-7bc \) is an even integer. Then the relation \( R \) is:

Let \( \displaystyle \int_{\alpha}^{\log_{e}4}\frac{dx}{\sqrt{e^{x}-1}}=\frac{\pi}{6} \). Then \( e^{\alpha} \) and \( e^{-\alpha} \) are the roots of the equation:

The sum of all possible values of \( \theta \in [-\pi, 2\pi] \), for which \(\displaystyle \frac{1 + i\cos\theta}{1 - 2i\cos\theta} \) is purely imaginary, is equal to:

If the shortest distance between the lines \( \frac{x-\lambda}{2}=\frac{y-4}{4}=\frac{z-3}{4} \) and \( \frac{x-2}{4}=\frac{y-4}{6}=\frac{z-7}{8} \) is \( \frac{13}{\sqrt{29}} \), then a value of \( \lambda \) is:

For \( a,b>0 \), let be a continuous function at \( x=0 \). Then \( \dfrac{b}{a} \) is equal to:

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is \( \frac{70}{3} \) and the product of the third and fifth terms is \( 49 \). Then the sum of the \( 4^{\text{th}} \), \( 6^{\text{th}} \) and \( 8^{\text{th}} \) terms is equal to:

Let \( \alpha |x|=|y|e^{xy-\beta},\ \alpha,\beta\in \mathbb{N} \) be the solution of the differential equation \( x\,dy-y\,dx+xy(x\,dy+y\,dx)=0,\ y(1)=2 \). Then \( \alpha+\beta \) is equal to _________.

Let \( S \) be the focus of the hyperbola \( \dfrac{x^{2}}{3}-\dfrac{y^{2}}{5}=1 \), on the positive \( x \)-axis. Let \( C \) be the circle with its centre at \( A(\sqrt{6},\sqrt{5}) \) and passing through the point \( S \). If \( O \) is the origin and \( SAB \) is a diameter of \( C \), then the square of the area of the triangle \( OSB \) is equal to ________.

Let \( a,b,c \in \mathbb{N} \) and \( a <b <c\). Let the mean , the mean deviation about the mean and the variance of the 5 observation 9, 25 \(a,b,c\) be 18, 4 and \(\dfrac{136}{5},\) respectively. Then \(2a+b-c\) is equal to _________.

Let \( P(\alpha,\beta,\gamma) \) be the image of the point \( Q(1,6,4) \) in the line \(\displaystyle \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3} \). Then \( 2\alpha+\beta+\gamma \) equal to ________.

An arithmetic progression is written in the following way
The sum of all the terms of the \( 10^{\text{th}} \) row is __________.

If \(\displaystyle \alpha = \lim_{x \to 0^+} \left( \frac{e^{\sqrt{\tan x}} - e^{\sqrt{x}}}{\sqrt{\tan x} - \sqrt{x}} \right)\) and \(\displaystyle \beta=\lim_{x\to 0}(1+\sin x)^{\frac{1}{2}\cot x} \) are the roots of the quadratic equation \( ax^2+bx-\sqrt{e}=0 \), then \( 12\log_{e}(a+b) \) is equal to __________.

Let a ray of light passing through the point \( (3,10) \) reflects on the line \( 2x+y=6 \) and the reflected ray passes through the point \( (7,2) \). If the equation of the incident ray is \( ax+by+1=0 \), then \( a^2+b^2+3ab \) is equal to ________.

Let \( A \) be the region enclosed by the parabola \( y^2=2x \) and the line \( x=24 \). Then the maximum area of the rectangle inscribed in the region \( A \) is __________.

If \(\displaystyle \int \frac{1}{\sqrt[5]{(x-1)^4 (x+3)^6}} \, dx = A \left( \frac{\alpha x - 1}{\beta x + 3} \right)^B + C \), where \( C \) is the constant of integration, then the value of \( \alpha + \beta + 20AB \) is __________.

The number of distinct real roots of the equation \( |x+1|\,|x+3|-4|x+2|+5=0 \), is __________.

JEE Main 2024 8th April Evening Shift

Mathematics Physics Chemistry

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An arithmetic progression is written in the following way
The sum of all the terms of the \( 10^{\text{th}} \) row is __________.