Mathematics Test

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Mathematics Test - 51

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Weekly Quiz Competition

Question 1
1 / -0

The average temperature for Wednesday, Thursday and Friday was 40°C. The average for Thursday, Friday and Saturday was 41° C. If temperature on Saturday was 42° C, what was the temperature on Wednesday?

A
39° C

B
44° C

C
38° C

D
41° C

Solution

Average temperature for Wednesday, Thursday and Friday = 40° C
Total temperature = 3 × 40 = 120° C
Average temperature for Thursday, Friday and Saturday = 41° C
Total temperature = 41 × 3 = 123° C
Temperature on Saturday = 42° C
Now,
(Thursday + Friday + Saturday) - (Wednesday + Thursday + Friday) = 123 - 120;
Saturday - Wednesday = 3
Wednesday = 42 - 3 = 39° C
Question 2
1 / -0

How many different words can be formed using all the letters of the word ALLAHABAD?
(a) When vowels occupy the even positions.
(b) Both L do not occur together.

A
7560,60,1680

B
7890,120,650

C
7650,200,4444

D
None of these

Solution

ALLAHABAD = 9 letters. Out of these 9 letters there is 4 A's and 2 L's are there.
So, permutations = 9!/4!.2! = 7560
(a) There are 4 vowels and all are alike i.e. 4A's.
_2^nd _4^th _6^th _8^th _
These even places can be occupied by 4 vowels. In 4!4! = 1 Way.
In other five places 5 other letter can be occupied of which two are alike i.e. 2L's.
Number of ways = 5!/2! Ways.
Hence, total number of ways in which vowels occupy the even places = 5!/2!×1 = 60 ways.
(b) Taking both L's together and treating them as one letter we have 8 letters out of which A repeats 4 times and others are distinct.
These 8 letters can be arranged in 8!/4! = 1680 ways.
Also two L can be arranged themselves in 2! ways.
So, Total no. of ways in which L are together = 1680 × 2 = 3360 ways.
Now,
Total arrangement in which L never occur together,
= Total arrangement - Total no. of ways in which L occur together.
= 7560 - 3360
= 4200 ways
Question 3
1 / -0

A bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue?

A
10/21

B
11/21

C
2/7

D
5/7

Solution

Total number of balls
\(=(2+3+2)\)
\(=7\)

Let S be the sample space Then, \(n(S)=\) Number of ways of drawing 2 balls out of 7

\(\mathrm{n}(\mathrm{S})={ }^{7} C_{2}\)
\(\mathrm{n}(\mathrm{S})=\frac{(7 \times 6)}{(2 \times 1)}\)
\(\mathrm{n}(\mathrm{S})=21\)

Let \(E=\) Event of 2 balls, none of which is blue
\(\therefore \mathrm{n}(\mathrm{E})=\) Number of ways of drawing 2 balls out of \((2+3)\) balls \(\mathbf{n}(\mathbf{E})={ }^{5} C_{2}\)
\(\mathbf{n}(\mathbf{E})=\frac{(5 \times 4)}{(2 \times 1)}\)
\(\mathbf{n}(\mathbf{E})=\mathbf{1 0}\)
\(\therefore P(E)=\frac{n(E)}{n(S)}=\frac{10}{21}\)
Question 4
1 / -0

\(\frac{d}{d x}\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^{2}=\)

A

\(1-\frac{1}{x^{2}}\)

B

\(1+\frac{1}{x^{2}}\)

C

\(1-\frac{1}{2x^{}}\)

D
None of these

Solution

\(\frac{d}{d x}\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^{2}=\frac{d}{d x}\left[x+\frac{1}{x}+1\right]=1-\frac{1}{x^{2}}\)
Question 5
1 / -0

In how many ways can 8 Indians and, 4 American and 4 Englishmen can be seated in a row so that all person of the same nationality sit together?

A
3! 4! 8! 4!

B
3! 8!

C
4! 4!

D
8! 4! 4!

Solution

Taking all person of same nationality as one person, then we will have only three people.
These three person can be arranged themselves in 3! Ways.
8 Indians can be arranged themselves in 8! Way.
4 American can be arranged themselves in 4! Ways.
4 Englishman can be arranged themselves in 4! Ways.
Hence, required number of ways = 3! 8! 4! 4! Ways.
Question 6
1 / -0

In how many ways 4 boys and 3 girls can be seated in a row so that they are alternate.

A
144

B
288

C
12

D
256

Solution

Let the Arrangement be,
B G B G B G B
4 boys can be seated in 4! Ways
Girl can be seated in 3! Ways
Required number of ways,
= 4! × 3!
= 144
Question 7
1 / -0

In how many ways 2 students can be chosen from the class of 20 students?

A
190

B
180

C
240

D
390

Solution

Number of ways
=20C₂
=20!/2!×18!
=20×19/2
=190
Question 8
1 / -0

If sin A−cosA = √3–1/2, then the value of sin A.cosA is

A
1/3

B
√3/2

C
1/4

D
√3/4

Solution

\(\sin A-\cos A=\frac{\sqrt{3}-1}{2}\)
Shortcutmethod :
Put, \(\theta=60^{\circ}\)
\(\Rightarrow \sin A-\cos A=\frac{\sqrt{3}-1}{2}\)
\(\Rightarrow \sin 60^{\circ}-\cos 60^{\circ}=\frac{\sqrt{3}-1}{2}\)
\(\Rightarrow \frac{\sqrt{3}}{2}-\frac{1}{2}=\frac{\sqrt{3}-1}{2}\)
\(\Rightarrow \frac{\sqrt{3}-1}{2}=\frac{\sqrt{3}-1}{2}(\text { Matched })\)
Hence, \(\sin A \cdot \cos A\)
\(\Rightarrow \frac{\sqrt{3}}{2} \times \frac{1}{2}\)
\(\Rightarrow \frac{\sqrt{3}}{4}\)

Alternate :
\(\sin A-\cos A=\frac{\sqrt{3}-1}{2}\)
Squaring both side, \(\Rightarrow \sin ^{2} A+\cos ^{2} A-2 . \sin A . \cos A=\left(\frac{\sqrt{3}-1}{2}\right)^{2}\)
\(\Rightarrow 1-2 \sin \mathrm{A} \cdot \cos \mathrm{A}=\frac{3+1-2 \sqrt{3}}{4}\)
\(\Rightarrow 2 \sin \mathrm{A} \cdot \cos \mathrm{A}=1-2 \frac{(2-\sqrt{3})}{4}\)
\(\Rightarrow 2 \sin A \cdot \cos A=\frac{2-2+\sqrt{3}}{2}\)
\(\Rightarrow \sin A \cdot \cos A=\frac{\sqrt{3}}{4}\)
Question 9
1 / -0

Three pipes A, B and C can fill a tank from empty to full in 30 minutes, 20 minutes, and 10 minutes respectively. When the tank is empty, all the three pipes are opened. A, B and C discharge chemical solutions P,Q and R respectively. What is the proportion of the solution R in the liquid in the tank after 3 minutes?

A
5/11

B
6/11

C
7/11

D
8/11

Solution

Part filled by \((A+B+C)\) in 3 minutes \(=3\left(\frac{1}{30}+\frac{1}{20}+\frac{1}{10}\right)\)
\(=\left(3 \times \frac{11}{60}\right)=\frac{11}{20}\)
Part filled by \(\mathrm{C}\) in \(3 \mathrm{minutes}=\frac{3}{10}\)
\(\therefore\) Required ratio \(=\left(\frac{3}{10} \times \frac{20}{11}\right)=\frac{6}{11}\)
Question 10
1 / -0

Which of the following statements is not correct?

A
log₁₀ 10 = 1

B
log (2 + 3) = log (2 x 3)

C
log₁₀ 1 = 0

D
log (1 + 2 + 3) = log 1 + log 2 + log 3

Solution

\(\mathbf{A}=\) since \(\log _{a} a=1,\) so \(\log _{10} 10=1\)
\(B=\log (2+3)=5\)
and \(\log (2 \times 3)=\log 6\)
\(=\log 2+\log 3\)
\(\therefore \log (2+3) \neq \log (2 \times 3)\)
\(C=\) since \(\log _{a} 1=0,\) so \(\log _{10} 1=0\)
\(\mathrm{D}=\log (1+2+3)=\log 6\)
\(=\log (1 \times 2 \times 3)\)
\(=\log 1+\log 2+\log 3\)
So (2) is incorrect

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Re-Attempt Weekly Quiz Competition

Mathematics Test - 51

Result

Mathematics Test - 51

Score

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SHARING IS CARING

Question 1

39° C

44° C

38° C

41° C

Solution

Question 2

7560,60,1680

7890,120,650

7650,200,4444

None of these

Solution

Question 3

10/21

11/21

2/7

5/7

Solution

Question 4

\(1-\frac{1}{x^{2}}\)

\(1+\frac{1}{x^{2}}\)

\(1-\frac{1}{2x^{}}\)

None of these

Solution

Question 5

3! 4! 8! 4!

3! 8!

4! 4!

8! 4! 4!

Solution

Question 6

144

288

12

256

Solution

Question 7

190

180

240

390

Solution

Question 8

1/3

√3/2

1/4

√3/4

Solution

Question 9

5/11

6/11

7/11

8/11

Solution

Question 10

log10 10 = 1

log (2 + 3) = log (2 x 3)

log10 1 = 0

log (1 + 2 + 3) = log 1 + log 2 + log 3

Solution

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log₁₀ 10 = 1

log₁₀ 1 = 0