Mathematics Test

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Mathematics Test - 50

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Weekly Quiz Competition

Question 1
1 / -0

In a box, there are 8 red, 7 blue and 6 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green?

A
1/3

B
3/4

C
7/19

D
8/21

Solution

Total number of balls
$=(8+7+6)$
$=21$
Let $E=$ event that the ball drawn is neither red nor green
= event that the ball drawn is blue $\therefore n(E)=7$
$\therefore P(E)=\frac{n(E)}{n(S)}=\frac{7}{21}=\frac{1}{3}$
Question 2
1 / -0

How many Permutations of the letters of the word APPLE are there?

A
600

B
120

C
240

D
60

Solution

APPLE = 5 letters.
But two letters PP is of same kind.
Thus, required permutations,
=5!/2!
=120/2
=60
Question 3
1 / -0

Two pipes A and B can fill a cistern in $37 \frac{1}{2}$ minutes and 45 minutes respectively. Both pipes are opened. The cistern will be filled in just half an hour, if the B is turned off after:

A
5 min.

B
9 min.

C
10 min.

D
15 min.

Solution

Let $B$ be turned off after $x$ minutes.
Then, Part filled by $(A+B)$ in $x$ min. $+$ Part filled by $A$ in $(30-x)$ min. $=1$ $\therefore x\left(\frac{2}{75}+\frac{1}{45}\right)+(30-x) \cdot \frac{2}{75}=1$
$\Rightarrow \frac{11 x}{225}+\frac{(60-2 x)}{75}=1$
$\Rightarrow 11 x+180-6 x=225$
$\Rightarrow x=9$
Question 4
1 / -0

Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 5?

A
1/2

B
2/5

C
8/15

D
9/20

Solution

Here, $\mathrm{S}=\{1,2,3,4, \ldots, 19,20\}$
Let $E=$ event of getting a multiple of 3 or 5 $=\{3,6,9,12,15,18,5,10,20\}$
$\therefore P(E)=\frac{n(E)}{n(S)}=\frac{9}{20}$
Question 5
1 / -0

$\frac{d}{d x}\left(x^{2} e^{x} \sin x\right)=$

A
xe^x(2sinx + xsinx + xcosx)

B
xe^x(2sinx + xsinx − cosx)

C
xe^x(2sinx + xsinx + cosx)

D
None of these

Solution

$\frac{d}{d x}\left(x^{2} e^{x} \sin x\right)=x^{2} \frac{d}{d x}\left(e^{x} \sin x\right)+e^{x} \sin x \frac{d}{d x}\left(x^{2}\right)$

$=x e^{x}(2 \sin x+x \sin x+x \cos x)$
Question 6
1 / -0

If $y=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\ldots . .+\frac{x^{n}}{n !},$ then $\frac{d y}{d x}=$

A
y

B
y+(xⁿ/n!)

C
−xⁿ/n!

D
y−1−xⁿ/n!

Solution

$y=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\ldots .+\frac{x^{n}}{n !}$
$\left[(\log \tan x+1)+\frac{1}{\tan x \log \tan x}\right] \quad$
$\frac{d y}{d x}+\frac{x^{n}}{n !}=1+x+\frac{x^{2}}{2 !}+\ldots+\frac{x^{n}}{n !}$ $\frac{d y}{d x}=y-\frac{x^{n}}{n !}$
Question 7
1 / -0

Find the average of first 97 natural numbers.

A
47

B
37

C
48

D
49

Solution

$1^{\text {st }}$ Method:

Average of $1^{\text {st }}$ n natural number is given by $=\frac{\left(\frac{|\mathrm{n} \times(\mathrm{n}+1)|}{2}\right)}{\mathrm{n}}$

Average of $1^{\text {st }}$ 97 natural number is given by
\[
\left.\begin{array}{l}
\left\{\frac{\left(\frac{|97 \times(97+1)|}{2}\right)}{97}\right\} \\
=49
\end{array}\right\}
\]

$2^{\text {st }}$ Method:
These numbers are in AP series so average $=\frac{(\text { sum of corresponding term })}{2}$

$=\frac{(1+97)}{2}=49$
or $, \frac{(2+96)}{2}=49$
or $, \frac{(3+95)}{2}=49$
And so on.
Question 8
1 / -0

If sin (90^∘−θ) + cosθ = √2cos(90^∘−θ), then the value of cosecθ is?

A
1/√3

B
2/3

C
√3/2

D
1/√2

Solution

$\sin \left(90^{\circ}-\theta\right)+\cos \theta=\sqrt{2} \cos \left(90^{\circ}-\theta\right)$
$\Rightarrow \cos \theta+\cos \theta=\sqrt{2} \sin \theta$
$\Rightarrow \frac{2 \cos \theta}{\sin \theta}=\sqrt{2}$
$\Rightarrow \cot \theta=\frac{1 \rightarrow B}{\sqrt{2} \rightarrow P}$
So, $\mathrm{H} \rightarrow$
$\therefore \operatorname{cosec} \theta=\frac{\mathrm{H}}{\mathrm{P}}=\frac{\sqrt{3}}{\sqrt{2}}$
$=\sqrt{\frac{3}{2}}$
Question 9
1 / -0

log√8/log8 is equal to:

A
1/6

B
1/4

C
1/2

D
1/8

Solution

$\frac{\log \sqrt{8}}{\log 8}=\frac{\log (8)^{\frac{1}{2}}}{\log 8}$
$=\frac{\frac{1}{2} \log 8}{\log 8}$
$=\frac{1}{2}$
Question 10
1 / -0

$\frac{d}{d x} \log (\log x)=$

A
x /log x

B
log x/x

C
(x log x)⁻¹

D
None of these

Solution

$\frac{d}{d x} \log (\log x)=\frac{1}{x} \cdot \frac{1}{\log x}=(x \log x)^{-1}$

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Re-Attempt Weekly Quiz Competition

Mathematics Test - 50

Result

Mathematics Test - 50

Score

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Rank

-

SHARING IS CARING

Question 1

1/3

3/4

7/19

8/21

Solution

Question 2

600

120

240

60

Solution

Question 3

5 min.

9 min.

10 min.

15 min.

Solution

Question 4

1/2

2/5

8/15

9/20

Solution

Question 5

xex(2sinx + xsinx + xcosx)

xex(2sinx + xsinx − cosx)

xex(2sinx + xsinx + cosx)

None of these

Solution

Question 6

y

y+(xn/n!)

−xn/n!

y−1−xn/n!

Solution

Question 7

47

37

48

49

Solution

Question 8

1/√3

2/3

√3/2

1/√2

Solution

Question 9

1/6

1/4

1/2

1/8

Solution

Question 10

x /log x

log x/x

(x log x)−1

None of these

Solution

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xe^x(2sinx + xsinx + xcosx)

xe^x(2sinx + xsinx − cosx)

xe^x(2sinx + xsinx + cosx)

y+(xⁿ/n!)

−xⁿ/n!

y−1−xⁿ/n!

(x log x)⁻¹