Sequences and Series Test 29

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Sequences and Series Test 29

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Weekly Quiz Competition

Question 1
1 / -0

Choose the correct answer alternatives given.
Select the missing number from the given alternatives.

A
18

B
21

C
5

D
17

Solution

80 = 8 $$\times$$ (8 + 2)
143 = 11 $$\times$$ (11 + 2)
323 = ? $$\times$$ (? + 2)
$$\therefore$$ ? = 17
Question 2
1 / -0

If $$1^{3} + 2^{3} + .... + 10^{3} = 3025$$, then the value of $$2^{3} + 4^{3} + ..... + 20^{3}$$ is

A
$$7590$$

B
$$5060$$

C
$$24200$$

D
$$12100$$

Solution

Given that, $$1^{3} + 2^{3} + .... + 10^{3} = 3025$$
Now, $$2^{3} + 4^{3} + .... + 20^{3}$$
$$= 2^{3} (1 + 2^{3} + .... + 10^{3}) = 8\times 3025 = 24200$$.
Question 3
1 / -0

If $$12\times 16 = 188$$ and $$14\times 18 = 248$$, then find the value of $$16\times 20 = ?$$

A
$$320$$

B
$$360$$

C
$$316$$

D
$$318$$

Solution

$$12\times 16 = 192 - 4 = 188$$
$$14\times 18 = 252 - 4 = 248$$
$$16\times 20 = 320 - 4 = 316$$.
Question 4
1 / -0

How many different $$4$$-person committees can be chosen form the $$100$$ members of the Senate ?

A
$$25$$

B
$$400$$

C
$$3,921,225$$

D
$$94,109,400$$

Solution

Total number of members $$=100$$.
Members has to be selected $$=4$$.
$$\therefore$$ different committees $$=^{100}C_{4}=\large{\frac{100!}{4!\times96!}}$$ $$=3,921,225$$.
Question 5
1 / -0

Arrange these numbers in ascending order.
$$756, 567, 657, 676$$

A
$$657, 567, 756, 676$$

B
$$567, 657, 676, 756$$

C
$$756, 676, 657, 567$$

D
$$676, 756, 567, 657$$

Solution

$$\Rightarrow$$ Numbers are said to be in ascending order when they are arranged from the smallest to the largest number.
$$\Rightarrow$$ The numbers which we have to arrange in ascending order are $$756,\,567,\,657$$ and $$676$$
$$\Rightarrow$$ $$567<657<676<756$$
$$\therefore$$ Ascending order $$=567,\,657,\,676,\,756$$
Question 6
1 / -0

The number of ways in which we can select 5 letters of the word INTERNATIONAL is equal to

A
$$200$$

B
$$220$$

C
$$242$$

D
$$256$$

Solution

We have thirteen letters.
$$2l, 3N, 2T, 2A$$ and $$(E, O, L,R)8$$ (types)
We can select 5(five) letters in the following manner:

1. All different $${^8C}_5 = \dfrac{8.7.5}{1.2.3} = 56$$ ways

2. 2 alike, 3 different $${^4C}_1 . {^7C}_3 = 435 = 140$$ ways (we have 4 sets of alike letters)

3. 3 alike, 2 different $${^1C}_1 . {^7C}_2 = 1.21 = 21$$ ways (we have only one set of 3 alike)

4. 3 alike and 2 alike $${^1C}_1 . {^3C}_1 = 1.3 = 3$$ ways

5.Two sets of alike and one different
$${^4C}_2 . {^6C}_1 = 6.6 = 36$$ ways

$$\therefore$$ Total number of selection is
$$56 + 140 + 21 + 3 + 36 = 256$$ ways $$\Longrightarrow$$ (d)
Question 7
1 / -0

Statement I: The function f(x) in the figure is differentiable at x = a
Statement II: The function f(x) continuous at x = a

A
Both Statement I and Statement II are true and the Statement II is the correct explanation of the Statement I.

B
Both Statement I and Statement II are true and the Statement II is not the correct explanation of the Statement I.

C
Statement I is true but Statement II is false

D
Statement I is false but Statement II is true.

Solution

$$f(x)$$ is continuous at as it can be observed from the figure there is no discontinuous at $$x=a$$ but coming to differentiability $$f(x)$$ is not differentiable at $$x=a$$ since the curve is sharp[i.e the left slope and right slope are not equal]
therefore the answer is statement I is false but statement II is true
Question 8
1 / -0

The total number of different combinations of one
or more letters which can be made from the letter of the word MISSISSIPPI is,

A
$$150$$

B
$$148$$

C
$$149$$

D
$$146$$

Solution

We have 1M, 4I ,4S , 2P
Therefore total number of selection of one or more letters=(1+1)(4+1)(4+1)(2+1)-1=149
Question 9
1 / -0

The total number of non-negative integer n satisfying the
equation $$ {n^2} = p + q\,and\,{n^3} = {p^2} + {q^2}$$, where p and q are
integers, is

A
$$1$$

B
$$2$$

C
$$3$$

D
infinite

Solution
Question 10
1 / -0

Number of arrangements of the letter $$HOLLYWOOD$$ in which all $$Os$$ are separated.

A
$$360\ ^{ 7 }{ C }_{ 3 }$$

B
$$15\ ^{ 7 }{ P }_{ 4 }$$

C
$$15\ ^{ 7 }{ C }_{ 4 }$$

D
$$30\ ^{ 7 }{ P }_{ 4 }$$

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Re-Attempt Weekly Quiz Competition

Sequences and Series Test 29

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Sequences and Series Test 29

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Rank

-

SHARING IS CARING

Question 1

18

21

5

17

Solution

Question 2

$$7590$$

$$5060$$

$$24200$$

$$12100$$

Solution

Question 3

$$320$$

$$360$$

$$316$$

$$318$$

Solution

Question 4

$$25$$

$$400$$

$$3,921,225$$

$$94,109,400$$

Solution

Question 5

$$657, 567, 756, 676$$

$$567, 657, 676, 756$$

$$756, 676, 657, 567$$

$$676, 756, 567, 657$$

Solution

Question 6

$$200$$

$$220$$

$$242$$

$$256$$

Solution

Question 7

Both Statement I and Statement II are true and the Statement II is the correct explanation of the Statement I.

Both Statement I and Statement II are true and the Statement II is not the correct explanation of the Statement I.

Statement I is true but Statement II is false

Statement I is false but Statement II is true.

Solution

Question 8

$$150$$

$$148$$

$$149$$

$$146$$

Solution

Question 9

$$1$$

$$2$$

$$3$$

infinite

Solution

Question 10

$$360\ ^{ 7 }{ C }_{ 3 }$$

$$15\ ^{ 7 }{ P }_{ 4 }$$

$$15\ ^{ 7 }{ C }_{ 4 }$$

$$30\ ^{ 7 }{ P }_{ 4 }$$

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