JEE Main 2024 (Online) 30th January Evening Shift

Let \(L_1: \vec{r}=(\hat{i}-\hat{j}+2 \hat{k})+\lambda(\hat{i}-\hat{j}+2 \hat{k}), \lambda \in \mathbb{R}\),
$$L_2: \vec{r}=(\hat{j}-\hat{k})+\mu(3 \hat{i}+\hat{j}+p \hat{k}), \mu \in \mathbb{R} \text {, and } L_3: \vec{r}=\delta(\ell \hat{...

Let a line passing through the point \((-1,2,3)\) intersect the lines \(L_1: \frac{x-1}{3}=\frac{y-2}{2}=\frac{z+1}{-2}\) at \(M(\alpha, \beta, \gamma)\) and \(L_2: \frac{x+2}{-3}=\frac{y-2}{-2}=\frac{z-1}{4}\) at \(N(a, b, c)\). Then, the ...

Let \(f(x)=(x+3)^2(x-2)^3, x \in[-4,4]\). If \(M\) and \(m\) are the maximum and minimum values of \(f\), respectively in \([-4,4]\), then the value of \(M-m\) is

The area of the region enclosed by the parabola \((y-2)^2=x-1\), the line \(x-2 y+4=0\) and the positive coordinate axes is _________.

Suppose \(2-p, p, 2-\alpha, \alpha\) are the coefficients of four consecutive terms in the expansion of \((1+x)^n\). Then the value of \(p^2-\alpha^2+6 \alpha+2 p\) equals

Let \(\alpha=\sum_\limits{k=0}^n\left(\frac{\left({ }^n C_k\right)^2}{k+1}\right)\) and \(\beta=\sum_\limits{k=0}^{n-1}\left(\frac{{ }^n C_k{ }^n C_{k+1}}{k+2}\right)\) If \(5 \alpha=6 \beta\), then \(n\) equals _______.

Consider two circles \(C_1: x^2+y^2=25\) and \(C_2:(x-\alpha)^2+y^2=16\), where \(\alpha \in(5,9)\). Let the angle between the two radii (one to each circle) drawn from one of the intersection points of \(C_1\) and \(C_2\) be $$\sin ^{-1}\l...

If \(z\) is a complex number, then the number of common roots of the equations \(z^{1985}+z^{100}+1=0\) and \(z^3+2 z^2+2 z+1=0\), is equal to

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function defined by \(f(x)=\frac{x}{\left(1+x^4\right)^{1 / 4}}\), and \(g(x)=f(f(f(f(x))))\). Then, \(18 \int_0^{\sqrt{2 \sqrt{5}}} x^2 g(x) d x\) is equal to

Let \(y=f(x)\) be a thrice differentiable function in \((-5,5)\). Let the tangents to the curve \(y=f(x)\) at \((1, f(1))\) and \((3, f(3))\) make angles \(\pi / 6\) and \(\pi / 4\), respectively with positive \(x\)-axis. If $$27 \int_\limi...

Let \(a\) and \(b\) be real constants such that the function \(f\) defined by $$f(x)=\left\{\begin{array}{ll}x^2+3 x+a & , x \leq 1 \\ b x+2 & , x>1\end{array}\right.$$ be differentiable on \(\mathbb{R}\). Then, the value of $$\int_\limits{...

Let \(\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}\) be defined as \(f(x)=a e^{2 x}+b e^x+c x\). If \(f(0)=-1, f^{\prime}\left(\log _e 2\right)=21\) and \(\int_0^{\log _e 4}(f(x)-c x) d x=\frac{39}{2}\), then the value of \(|a+b+c|\) equal...

Let \(Y=Y(X)\) be a curve lying in the first quadrant such that the area enclosed by the line \(Y-y=Y^{\prime}(x)(X-x)\) and the co-ordinate axes, where \((x, y)\) is any point on the curve, is always $$\frac{-y^2}{2 Y^{\prime}(x)}+1, Y^{\p...

Let \(f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}\) be a function satisfying \(f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}\) for all \(x, y, f(y) \neq 0\). If \(f^{\prime}(1)=2024\), then

Let \(A(\alpha, 0)\) and \(B(0, \beta)\) be the points on the line \(5 x+7 y=50\). Let the point \(P\) divide the line segment \(A B\) internally in the ratio \(7:3\). Let \(3 x-25=0\) be a directrix of the ellipse $$E: \frac{x^2}{a^2}+\fra...

If the domain of the function \(f(x)=\log _e\left(\frac{2 x+3}{4 x^2+x-3}\right)+\cos ^{-1}\left(\frac{2 x-1}{x+2}\right)\) is \((\alpha, \beta]\), then the value of \(5 \beta-4 \alpha\) is equal to

Let \(P\) be a point on the hyperbola \(H: \frac{x^2}{9}-\frac{y^2}{4}=1\), in the first quadrant such that the area of triangle formed by \(P\) and the two foci of \(H\) is \(2 \sqrt{13}\). Then, the square of the distance of \(P\) from th...

Let $$R=\left(\begin{array}{ccc}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)$$ be a non-zero \(3 \times 3\) matrix, where $$x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0,...

Consider the system of linear equations \(x+y+z=5, x+2 y+\lambda^2 z=9, x+3 y+\lambda z=\mu\), where \(\lambda, \mu \in \mathbb{R}\). Then, which of the following statement is NOT correct?

In an examination of Mathematics paper, there are 20 questions of equal marks and the question paper is divided into three sections : \(A, B\) and \(C\). A student is required to attempt total 15 questions taking at least 4 questions from e...

Bag A contains 3 white, 7 red balls and Bag B contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag A, if the ball drawn is white, is

The number of real solutions of the equation \(x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0\) is _________.

Let \(a\) and \(b\) be be two distinct positive real numbers. Let \(11^{\text {th }}\) term of a GP, whose first term is \(a\) and third term is \(b\), is equal to \(p^{\text {th }}\) term of another GP, whose first term is \(a\) and fifth ...

Let \(S_n\) be the sum to \(n\)-terms of an arithmetic progression \(3,7,11\),
If \(40<\left(\frac{6}{n(n+1)} \sum_\limits{k=1}^n S_k\right)<42\), then \(n\) equals ________.

The number of symmetric relations defined on the set \(\{1,2,3,4\}\) which are not reflexive is _________.

The variance \(\sigma^2\) of the data

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If \(x^2-y^2+2 h x y+2 g x+2 f y+c=0\) is the locus of a point, which moves such that it is always equidistant from the lines \(x+2 y+7=0\) and \(2 x-y+8=0\), then the value of \(g+c+h-f\) equals

For \(\alpha, \beta \in(0, \pi / 2)\), let \(3 \sin (\alpha+\beta)=2 \sin (\alpha-\beta)\) and a real number \(k\) be such that \(\tan \alpha=k \tan \beta\). Then, the value of \(k\) is equal to

Let \(\vec{a}=\hat{i}+\alpha \hat{j}+\beta \hat{k}, \alpha, \beta \in \mathbb{R}\). Let a vector \(\vec{b}\) be such that the angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{4}\) and \(|\vec{b}|^2=6\). If $$\vec{a} \cdot \vec{b}=3...

Let \(\vec{a}\) and \(\vec{b}\) be two vectors such that \(|\vec{b}|=1\) and \(|\vec{b} \times \vec{a}|=2\). Then \(|(\vec{b} \times \vec{a})-\vec{b}|^2\) is equal to