Mathematics Test

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Mathematics Test - 62

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Weekly Quiz Competition

Question 1
1 / -0

The value of \(1 \div\left[1 \frac{1}{8} \div \frac{\left(3 \frac{1}{5}+\frac{3}{5}\right) \div \frac{8}{5}}{\left\{\frac{5}{8}+\left(\frac{1}{8} \div \frac{1}{3}\right)\right\}}\right]\)

A
19/16

B
19/7

C
19/9

D
19/64

Solution

\(\Rightarrow 1 \div\left[1 \frac{1}{8} \div \frac{\left(3 \frac{1}{5}+\frac{3}{5}\right) \div \frac{8}{5}}{\left\{\frac{5}{8}+\left(\frac{1}{8} \div \frac{1}{3}\right)\right\}}\right]\)
\(\Rightarrow 1 \div\left[\frac{9}{8} \div \frac{\left(\frac{16}{5}+\frac{3}{5}\right) \div \frac{8}{5}}{\left\{\frac{5}{8}+\left(\frac{1}{8} \times \frac{3}{1}\right)\right\}}\right]\)
\(\Rightarrow 1 \div\left[\frac{9}{8} \div \frac{\frac{19}{5} \times \frac{5}{8}}{\frac{5}{8}+\frac{3}{8}}\right]\)
\(\Rightarrow 1 \div\left[\frac{9}{8} \times \frac{8}{19}\right.\)
\(\Rightarrow 1+[9 / 19]\)
\(\Rightarrow 19 / 9\)
Question 2
1 / -0

In a lottery, there are 10 prizes and 25 blanks. A lottery is drawn at random. What is the probability of getting a prize?

A
1/10

B
2/5

C
2/7

D
5/7

Solution

\(P\) (getting a prize)
\(=\frac{10}{10+25}\)
\(=\frac{10}{35}\)
\(=\frac{2}{7}\)
Question 3
1 / -0

Two dice are tossed. The probability that the total score is a prime number is:

A
1/6

B
5/12

C
1/2

D
7/9

Solution

Clearly, \(n(S)=(6 \times 6)=36\)
Let \(E=\) Event that the sum is a prime number.Then

\(E=\{(1,1),(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),(3,2),(3,4),\)

(4,1),(4,3),(5,2),(5,6),(6,1),(6,5)\}

\(\therefore n(E)=15\)
\(\therefore P(E)=\frac{n(E)}{n(S)}\)
\(=\frac{15}{36}\)
\(=\frac{5}{12}\)
Question 4
1 / -0

If \(\frac{x-a^{2}}{b+c}+\frac{x-b^{2}}{c+a}+\frac{x-c^{2}}{a+b}=4(a+b+c),\) then \(x=?\)

A
(a + b + c)²

B
a² + b²+ c²

C
ab + bc + ca

D
a²+ b² + c² - ab - bc - ca

Solution

\(\frac{x-a^{2}}{b+c}+\frac{x-b^{2}}{c+a}+\frac{x-c^{2}}{a+b}=4(a+b+c)\)
Note : In such type of question to save your valuable time assume values as per your need which make your calculation easier.
Assume \(a=1, b=0, c=1\)
Make sure there will be no \(\left(\frac{0}{0}\right)\) form
\(\therefore \frac{x-1}{1+0}+\frac{x-0}{1+1}+\frac{x-1}{1+0}=4\)
\(\Rightarrow x-1+\frac{x}{2}+x-1=4 \times 2\)
\(\Rightarrow x+\frac{x}{2}+x=8+2\)
\(\Rightarrow \frac{5 x}{2}=10\)
\(\Rightarrow x=4\)
Now put values in options take option
Question 5
1 / -0

The average monthly salary of 660 workers in a factory is Rs. 380. The average monthly salary of officers is Rs. 2100 and the average monthly salary of the other workers is Rs. 340. Find the number of other workers.

A
645

B
650

C
640

D
642

Solution

Total salary of 660 workers
\(=660 \times 380\)
\(=R s .250800\)
If other workers be \(x ;\) then,
\(=[(660-x) \times 2100]+340 x\)
\(=250800\)
Or, \(1386000-2100 x+340 x=250800\)
\(1760 x=1135200\)
Hence, \(x=\frac{1135200}{1760}=645\)
Number of other workers \(=645\)
Question 6
1 / -0

A tank is filled in 5 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

A
20 hours

B
25 hours

C
35 hours

D
Cannot be determined

Solution

Suppose pipe \(A\) alone takes \(x\) hours to fill the tank. Then, pipes \(B\) and \(C\) will take \(\frac{x}{2}\) and \(\frac{x}{4}\) hours respectively to fill the tank. \(\therefore \frac{1}{x}+\frac{2}{x}+\frac{4}{x}=\frac{1}{5}\)
\(\Rightarrow \frac{7}{x}=\frac{1}{5}\)
\(\Rightarrow x=35\) hrs
Question 7
1 / -0

If \(x \neq 0, y \neq 0\) and \(z \neq 0\) and \(\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=\frac{1}{x y}+\frac{1}{y z}+\frac{1}{z x}\) then the relation among \(x, y, z\) is?

A
x + y + y = 0

B
x + y = z

C
1/x+1/y+1/z=0

D
x = y = z

Solution

\(\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=\frac{1}{x y}+\frac{1}{y z}+\frac{1}{z x}\)
Go through option 4 take \(x=y=z\) \(\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=\frac{1}{x y}+\frac{1}{y z}+\frac{1}{z x}\)
\(\therefore\) Option ' 4 ' is right
Question 8
1 / -0

A bag contains 4 white, 5 red and 6 blue balls. Three balls are drawn at random from the bag. The probability that all of them are red, is:

A
1/22

B
3/22

C
2/91

D
2/77

Solution

Let \(S\) be the sample space Then, \(n(S)=\) number of ways of drawing 3 balls out of 15

\(={ }^{15} C_{3}\)
\(=\frac{(15 \times 14 \times 13)}{(3 \times 2 \times 1)}\)
\(=455\)

Let \(E=\) event of getting all the 3 red balls

\(\therefore n(E)={ }^{5} C_{3}={ }^{5} C_{2}\)
\(=\frac{(5 \times 4)}{(2 \times 1)}=10\)
\(\therefore P(E)=\frac{n(E)}{n(S)}\)
\(=\frac{10}{455}\)
\(=\frac{2}{91}\)
Question 9
1 / -0

If y=a sinx+b cosx,then y²+(dy/dx)²is a

A
Function of x

B
Function of y

C
Function of x and y

D
Constant

Solution

\(y=a \sin x+b \cos x \quad\) Differentiating with respect
to \(x,\) we get \(\frac{d y}{d x}=a \cos x-b \sin x \quad\) Now
\(\left(\frac{d y}{d x}\right)^{2}=(a \cos x-b \sin x)^{2}\)
\(=a^{2} \cos ^{2} x+b^{2} \sin ^{2} x-2 a b \sin x \cos x \quad\) and
\(y^{2}=(a \sin x+b \cos x)^{2}\)
\(=a^{2} \sin ^{2} x+b^{2} \cos ^{2} x+2 a b \sin x \cos x \quad\) So,
\(\begin{aligned}\left(\frac{d y}{d x}\right)^{2}+y^{2}=a^{2}\left(\sin ^{2} x+\cos ^{2} x\right)+b^{2}\left(\sin ^{2} x+\cos ^{2} x\right) & \text { ( ) } \\ \text { Hence }\left(\frac{d y}{d x}\right)^{2}+y^{2}=\left(a^{2}+b^{2}\right)=\text { constant. } \end{aligned}\)
Question 10
1 / -0

A man travels equal distances of his journey at 40, 30 and 15 km/h. respectively. Find his average speed for whole journey.

A
24

B
25

C
27

D
28

Solution

Required average speed \(=\left[\frac{(3 \times 40 \times 30 \times 15)}{\{(40 \times 30)+(40 \times 15)+(30 \times 15)\}}\right]\)

\(=24 \mathrm{km} / \mathrm{hr}\)

Alternatively Time taken to traveled \(\frac{1}{3}\) distance of journey with speed \(40 \mathrm{kmph}\) \(=\frac{\left(\frac{1}{3}\right)}{40}=\frac{1}{120}\)
Time taken to traveled \(\frac{1}{3}\) distance of journey with speed \(30 \mathrm{kmph}\) \(=\frac{\left(\frac{1}{3}\right)}{30}=\frac{1}{90}\)
Time taken to traveled \(\frac{1}{3}\) distance of journey with speed \(15 \mathrm{kmph}\) \(=\frac{\left(\frac{1}{3}\right)}{15}=\frac{1}{45}\)
Total time taken
\(=\left(\frac{1}{120}\right)+\left(\frac{1}{90}\right)+\left(\frac{1}{45}\right)\)
\(=\frac{45}{1080}\)
Average speed \(=\frac{\text { Total distance traveled }}{\text { Total time taken }}\)

\(=\frac{1}{\left(\frac{45}{1080}\right)}\)
\(=24 \mathrm{kmph}\)

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Re-Attempt Weekly Quiz Competition

Mathematics Test - 62

Result

Mathematics Test - 62

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SHARING IS CARING

Question 1

19/16

19/7

19/9

19/64

Solution

Question 2

1/10

2/5

2/7

5/7

Solution

Question 3

1/6

5/12

1/2

7/9

Solution

Question 4

(a + b + c)2

a2 + b2 + c2

ab + bc + ca

a2 + b2 + c2 - ab - bc - ca

Solution

Question 5

645

650

640

642

Solution

Question 6

20 hours

25 hours

35 hours

Cannot be determined

Solution

Question 7

x + y + y = 0

x + y = z

1/x+1/y+1/z=0

x = y = z

Solution

Question 8

1/22

3/22

2/91

2/77

Solution

Question 9

Function of x

Function of y

Function of x and y

Constant

Solution

Question 10

24

25

27

28

Solution

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(a + b + c)²

a² + b²+ c²

a²+ b² + c² - ab - bc - ca