Mathematics Test

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Mathematics Test - 50

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Question 1/10
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?

Correct

Wrong

Skipped
A
1/3

Correct

Wrong

Skipped
B
3/4

Correct

Wrong

Skipped
C
7/19

Correct

Wrong

Skipped
D
8/21

Solutions

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 $∴ n (E) = 7$
$∴ P (E) = \frac{n (E)}{n (S)} = \frac{7}{21} = \frac{1}{3}$
Question 2/10
1 / -0

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

Correct

Wrong

Skipped
A
600

Correct

Wrong

Skipped
B
120

Correct

Wrong

Skipped
C
240

Correct

Wrong

Skipped
D
60

Solutions

APPLE = 5 letters.
But two letters PP is of same kind.
Thus, required permutations,
=5!/2!
=120/2
=60
Question 3/10
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:

Correct

Wrong

Skipped
A
5 min.

Correct

Wrong

Skipped
B
9 min.

Correct

Wrong

Skipped
C
10 min.

Correct

Wrong

Skipped
D
15 min.

Solutions

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. $= 1$ $∴ x (\frac{2}{75} + \frac{1}{45}) + (30 - x) \cdot \frac{2}{75} = 1$
$\Rightarrow \frac{11 x}{225} + \frac{(60 - 2 x)}{75} = 1$
$\Rightarrow 11 x + 180 - 6 x = 225$
$\Rightarrow x = 9$
Question 4/10
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?

Correct

Wrong

Skipped
A
1/2

Correct

Wrong

Skipped
B
2/5

Correct

Wrong

Skipped
C
8/15

Correct

Wrong

Skipped
D
9/20

Solutions

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

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

Correct

Wrong

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

Correct

Wrong

Skipped
B
xe^x(2sinx + xsinx − cosx)

Correct

Wrong

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

Correct

Wrong

Skipped
D
None of these

Solutions

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

$= x e^{x} (2 \sin x + x \sin x + x \cos x)$
Question 6/10
1 / -0

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

Correct

Wrong

Skipped
A
y

Correct

Wrong

Skipped
B
y+(xⁿ/n!)

Correct

Wrong

Skipped
C
−xⁿ/n!

Correct

Wrong

Skipped
D
y−1−xⁿ/n!

Solutions

$y = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots . + \frac{x^{n}}{n!}$
$[(\log \tan x + 1) + \frac{1}{\tan x \log \tan x}]$
$\frac{d y}{d x} + \frac{x^{n}}{n!} = 1 + x + \frac{x^{2}}{2!} + \dots + \frac{x^{n}}{n!}$ $\frac{d y}{d x} = y - \frac{x^{n}}{n!}$
Question 7/10
1 / -0

Find the average of first 97 natural numbers.

Correct

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A
47

Correct

Wrong

Skipped
B
37

Correct

Wrong

Skipped
C
48

Correct

Wrong

Skipped
D
49

Solutions

$1^{st}$ Method:

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

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

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

$= \frac{(1 + 97)}{2} = 49$
or $, \frac{(2 + 96)}{2} = 49$
or $, \frac{(3 + 95)}{2} = 49$
And so on.
Question 8/10
1 / -0

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

Correct

Wrong

Skipped
A
1/√3

Correct

Wrong

Skipped
B
2/3

Correct

Wrong

Skipped
C
√3/2

Correct

Wrong

Skipped
D
1/√2

Solutions

$\sin (90^{\circ} - θ) + \cos θ = \sqrt{2} \cos (90^{\circ} - θ)$
$\Rightarrow \cos θ + \cos θ = \sqrt{2} \sin θ$
$\Rightarrow \frac{2 \cos θ}{\sin θ} = \sqrt{2}$
$\Rightarrow \cot θ = \frac{1 \to B}{\sqrt{2} \to P}$
So, $H \to$
$∴ cosec θ = \frac{H}{P} = \frac{\sqrt{3}}{\sqrt{2}}$
$= \sqrt{\frac{3}{2}}$
Question 9/10
1 / -0

log√8/log8 is equal to:

Correct

Wrong

Skipped
A
1/6

Correct

Wrong

Skipped
B
1/4

Correct

Wrong

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C
1/2

Correct

Wrong

Skipped
D
1/8

Solutions

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

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

Correct

Wrong

Skipped
A
x /log x

Correct

Wrong

Skipped
B
log x/x

Correct

Wrong

Skipped
C
(x log x)⁻¹

Correct

Wrong

Skipped
D
None of these

Solutions

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

Question 1/10

1/3

3/4

7/19

8/21

Question 2/10

600

120

240

60

Question 3/10

5 min.

9 min.

10 min.

15 min.

Question 4/10

1/2

2/5

8/15

9/20

Question 5/10

xex(2sinx + xsinx + xcosx)

xex(2sinx + xsinx − cosx)

xex(2sinx + xsinx + cosx)

None of these

Question 6/10

y

y+(xn/n!)

−xn/n!

y−1−xn/n!

Question 7/10

47

37

48

49

Question 8/10

1/√3

2/3

√3/2

1/√2

Question 9/10

1/6

1/4

1/2

1/8

Question 10/10

x /log x

log x/x

(x log x)−1

None of these

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