1
JEE Main 2024 (Online) 31st January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

$$\text { If } f(x)=\left|\begin{array}{ccc} x^3 & 2 x^2+1 & 1+3 x \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2 \end{array}\right| \text { for all } x \in \mathbb{R} \text {, then } 2 f(0)+f^{\prime}(0) \text { is equal to }$$

A
24
B
18
C
42
D
48
2
JEE Main 2024 (Online) 31st January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

If one of the diameters of the circle $$x^2+y^2-10 x+4 y+13=0$$ is a chord of another circle $$\mathrm{C}$$, whose center is the point of intersection of the lines $$2 x+3 y=12$$ and $$3 x-2 y=5$$, then the radius of the circle $$\mathrm{C}$$ is :

A
4
B
3$$\sqrt2$$
C
6
D
$$\sqrt{20}$$
3
JEE Main 2024 (Online) 31st January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

If the foci of a hyperbola are same as that of the ellipse $$\frac{x^2}{9}+\frac{y^2}{25}=1$$ and the eccentricity of the hyperbola is $$\frac{15}{8}$$ times the eccentricity of the ellipse, then the smaller focal distance of the point $$\left(\sqrt{2}, \frac{14}{3} \sqrt{\frac{2}{5}}\right)$$ on the hyperbola, is equal to

A
$$14 \sqrt{\frac{2}{5}}-\frac{4}{3}$$
B
$$7 \sqrt{\frac{2}{5}}+\frac{8}{3}$$
C
$$7 \sqrt{\frac{2}{5}}-\frac{8}{3}$$
D
$$14 \sqrt{\frac{2}{5}}-\frac{16}{3}$$
4
JEE Main 2024 (Online) 31st January Morning Shift
Numerical
+4
-1
Change Language

If $$\alpha$$ denotes the number of solutions of $$|1-i|^x=2^x$$ and $$\beta=\left(\frac{|z|}{\arg (z)}\right)$$, where $$z=\frac{\pi}{4}(1+i)^4\left[\frac{1-\sqrt{\pi} i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi} i}\right], i=\sqrt{-1}$$, then the distance of the point $$(\alpha, \beta)$$ from the line $$4 x-3 y=7$$ is __________.

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