Mathematics Test

Result

Mathematics Test - 23

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Weekly Quiz Competition

Question 1
1 / -0

If $\alpha$ is a root of $x^{4}=1$ with negative principal argument, then principal argument of $\omega$ where

A
$5 \pi / 4$

B
$-3 \pi / 4$

C
$\pi / 4$

D
$-\pi / 4$

Solution

Obviously $\left.\alpha=-\mathrm{i} \text { (where } i^{2}=\sqrt{-1}\right)$

taking $\alpha^{n}$ common from $R_{2}$ and $\frac{1}{\alpha^{n}}$ common from $R_{3}$

So, $\omega=\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & -i & i \\ i & 1 & 0\end{array}\right|=-1-i$

$\Rightarrow \arg (\omega)=-\frac{3 \pi}{4}$
Question 2
1 / -0

The integrating factor of the differential equation is given by

A

B

C

D
x

Solution
Question 3
1 / -0

If $x, y, z$ are integers in $A . P,$ lying between 1 and $9,$ and $x 51, y 41$ and $z 31$ are three digit numbers then the value of $\left|\begin{array}{lll}5 & 4 & 3 \\ x 51 & y 41 & z 31 \\ x & y & z\end{array}\right|$ is

A
$x + y + z$

B
0

C
$x - y + z$

D
$x - y - z$

Solution

$\left|\begin{array}{lll}5 & 4 & 3 \\ 100 x+51 & 100 y+41 & 100 y+31 \\ x & y & z\end{array}\right|$

$\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\left(100 \mathrm{R}_{3}\right)$

$\Delta=\left|\begin{array}{lll}5 & 4 & 3 \\ 51 & 41 & 31 \\ \mathrm{x} & \mathrm{y} & \mathrm{z}\end{array}\right| \Rightarrow \Delta=0$
Question 4
1 / -0

A
True

B
False

C
May be true or false

D
Data insufficient

Solution
Question 5
1 / -0

$A$ and $B$ are two independent events such that $\mathrm{PA} \cup \mathrm{B}^{\prime}=0.8$ and $\mathrm{PA}=0.3 .$ Then, $P(B)$ is

A
2/7

B
2/3

C
3/7

D
2/8

Solution
Question 6
1 / -0

Assertion [A]: From 0 to π, area of region bounded by y = sin⁡x and x -axis is greater then the area bounded by y = sin²⁡x, x -axis.

A
Both $[A]$ and $[R]$ art true, and $[R]$ is the correct explanation of $[A]$

B
Both $[A]$ and $[R]$ are true but $[R]$ is not the correct explanation of $[A]$

C
$[A]$ is true but $[R]$ is false.

D
$[A]$ is false but $[R]$ is true.

Solution

If the curve is symmetrical about a coordinate axis (or a line or origin), then we find the area of one symmetrical portion and multiply it by the number of symmetrical portions to get the required area.
Question 7
1 / -0

Let $A$ and $B$ be two fixed points and $P$, another point in the plane, moves in such a way that $k_{1} P A+k_{2} P B=k_{3}$ where $k_{1}, k_{2}$ and $k_{3}$ are real constants. Then which one of the following is not the locus of $P$ -

A
a circle, if $k_{1}=0$ and $k 2, k 3>0$

B
a circle, if $k_{1}>0$ and $b$ and $k_{3}=0$

C
an ellipse, if and and

D
an hyperbola, if $k_{2}=-1$ and $k 1, k 3>0$

Solution

$\Rightarrow P B=\frac{k_{3}}{k_{2}}>0$

$\Rightarrow P$ describes a circle with $B$ as centre and radius $=\frac{k_{3}}{k_{2}}$

If $\mathrm{k}_{3}=0,$ then $\mathrm{k}_{1} \mathrm{PA}+\mathrm{k}_{2} \mathrm{PB}=0$

$\Rightarrow \frac{P A}{P B}=\frac{k_{2}}{-k_{1}}=k>0$

$\Rightarrow P$ described a circle with $\mathrm{P}_{1} \mathrm{P}_{2}$ as its diameter, $\mathrm{P}_{1}$ and $\mathrm{P}_{2}$ being the points which divide $\mathrm{AB}$ internally and externally in the ratio $\mathrm{k}: 1$

If $\mathrm{k}_{1}=\mathrm{k}_{2}>0$ and $\mathrm{k}_{3}>0,$ then

$\mathrm{PA}+\mathrm{PB}=\frac{\mathrm{k}_{3}}{\mathrm{k}_{1}}=\mathrm{k}>0$

$\Rightarrow \mathrm{P}$ describes an ellipse with $\mathrm{A}$ and $\mathrm{B}$ as its foci.
Question 8
1 / -0

A
Infinitely many solutions

B
No solution

C
One solution

D
None of these

Solution

Undefined control sequence \ therefore system of equation has infinitely many solutions.
Question 9
1 / -0

A
${(1 + \log x)}^{- 1}$

B
${(1 + \log x)}^{- 2}$

C
$(\log x) {(1 + \log x)}^{- 2}$

D
None of these

Solution
Question 10
1 / -0

A
0

B
-24

C
3600

D
-345

Solution