These 10 Sample quiz questions are taken from JEE 2021 Mathematics Paper (Section-B)
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Let \(\mathrm{z}=\frac{1-i \sqrt{3}}{2}, i=\sqrt{-1}\). Then the value of
\(
21+\left(\mathrm{z}+\frac{1}{\mathrm{z}}\right)^{3}+\left(\mathrm{z}^{2}+\frac{1}{\mathrm{z}^{2}}\right)^{3}+\left(\mathrm{z}^{3}+\frac{1}{\mathrm{z}^{3}}\right)^{3}+\ldots+\left(\mathrm{z}^{21}+\frac{1}{\mathrm{z}^{21}}\right)^{3}
\) is _______.
The sum of all integral values of \(k(k \neq 0)\) for which the equation \(\frac{2}{x-1}-\frac{1}{x-2}=\frac{2}{k}\) in \(x\) has no real roots, is _______.
Let the line \(L\) be the projection of the line
\(
\frac{x-1}{2}=\frac{y-3}{1}=\frac{z-4}{2}
\)
in the plane \(\mathrm{x}-2 \mathrm{y}-\mathrm{z}=3\). If \(\mathrm{d}\) is the distance of the point \((0,0,6)\) from \(\mathrm{L}\), then \(\mathrm{d}^{2}\) is equal to _______.
If \({ }^{1} \mathrm{P}_{1}+2 \cdot{ }^{2} \mathrm{P}_{2}+3 \cdot{ }^{3} \mathrm{P}_{3}+\ldots+15 \cdot{ }^{15} \mathrm{P}_{15}={ }^{\mathrm{q}} \mathrm{P}_{\mathrm{r}}-\mathrm{s}, 0 \leq \mathrm{s} \leq 1\), then \({ }^{q+s} C_{r-s}\) is equal to ________.
A wire of length \(36 \mathrm{~m}\) is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is \(\mathrm{k}\) (meter), then \(\left(\frac{4}{\pi}+1\right) \mathrm{k}\) is equal to ______.
The area of the region \(S=\left\{(x, y): 3 x^{2} \leq 4 y \leq 6 x+24\right\}\) is ______.
The locus of a point, which moves such that the sum of squares of its distances from the points \((0,0),(1,0),(0,1)(1,1)\) is 18 units, is a circle of diameter \(\mathrm{d}\). Then \(\mathrm{d}^{2}\) is equal to ______.
Let \(\mathrm{P}(\mathrm{x}, \mathrm{y})\)
\(
\begin{aligned}
&x^{2}+y^{2}+x^{2}+(y-1)^{2}+(x-1)^{2}+y^{2}+(x-1)^{2}+(y-1)^{2} \\
&\quad \Rightarrow 4\left(x^{2}+y^{2}\right)-4 y-4 x=14 \\
&\Rightarrow x^{2}+y^{2}-x-y-\frac{7}{2}=0 \\
&d=2 \sqrt{\frac{1}{4}+\frac{1}{4}+\frac{7}{2}} \\
&\Rightarrow d^{2}=16
\end{aligned}
\)
If \(\mathrm{y}=\mathrm{y}(\mathrm{x})\) is an implicit function of \(\mathrm{x}\) such that \(\log _{\mathrm{e}}(\mathrm{x}+\mathrm{y})=4 \mathrm{xy}\), then \(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\) at \(\mathrm{x}=0\) is equal to ______.
\(\begin{aligned}
&\ln (x+y)=4 x y \\
&x+y=e^{4 x y} \\
&\Rightarrow 1+\frac{d y}{d x}=e^{4 x y}\left(4 x \frac{d y}{d x}+4 y\right) \\
&\text { At } x=0 \quad \frac{d y}{d x}=3 \\
&\frac{d^{2} y}{d x^{2}}=e^{4 x y}\left(4 x \frac{d y}{d x}+4 y\right)^{2}+e^{4 x y}\left(4 x \frac{d^{2} y}{d x^{2}}+4 y\right) \\
&\text { At } x=0, \frac{d^{2} y}{d x^{2}}=e^{0}(4)^{2}+e^{0}(24) \\
&\Rightarrow \frac{d^{2} y}{d x^{2}}=40
\end{aligned}\)
The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is ________.
Let \(a, b \in \mathbf{R}, b \neq 0\), Define a function
\(
f(x)= \begin{cases}a \sin \frac{\pi}{2}(x-1), & \text { for } x \leq 0 \\ \frac{\tan 2 x-\sin 2 x}{b x^{3}}, & \text { for } x>0\end{cases}
\)
If \(\mathrm{f}\) is continuous at \(x=0\), then \(10-a b\) is equal to ______.
\(
f(x)=\left\{\begin{array}{cl}
a \sin \frac{\pi}{2}(x-1), & x \leq 0 \\
\frac{\tan 2 x-\sin 2 x}{b x^{3}}, & x>0
\end{array}\right.
\)
For continuity at ‘ 0 ‘
\(
\begin{aligned}
&\lim _{x \rightarrow 0^{+}} f(x)=f(0) \\
&\Rightarrow \lim _{x \rightarrow 0^{+}} \frac{\tan 2 x-\sin 2 x}{b x^{3}}=-a
\end{aligned}
\)
\(
\Rightarrow \lim _{x \rightarrow 0^{+}} \frac{\frac{8 x^{3}}{3}+\frac{8 x^{3}}{3 !}}{b x^{3}}=-a
\)
\(
\Rightarrow 8\left(\frac{1}{3}+\frac{1}{3 !}\right)=-a b
\)
\(
\Rightarrow 4=-a b
\)
\(
\Rightarrow 10-\mathrm{ab}=14
\)