Sequences and Series Test 30

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Sequences and Series Test 30

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Weekly Quiz Competition

Question 1
1 / -0

Pam likes the numbers 1689 and 6891. Knowing this, which pair of numbers will she like among the ones below?

A
1981 and 1891

B
19 and 91

C
190 and 160

D
1198911 and 1168611

Solution
Question 2
1 / -0

Function f(x)=$$\left| {x - 2} \right| - 2\left| {x - 4} \right|\,is$$ discontinous at:

A
$$x=2,4$$

B
$$x=2$$

C
No where

D
Except $$x=2$$

Solution

$$f(x)=\mid x-2 \mid -2\mid x-4 \mid$$
$$\therefore f\left(x \right)=\begin{Bmatrix} -(x-2)+2(x-4),\qquad x<2\\ (x-2)+2(x-4), \qquad 2\le x \le 4\\ (x-2)-2(x-4) \qquad x\gt 4 \\ \end{Bmatrix}$$
At $$x=2$$
L.H.L(Left Hand Limit)
$$\displaystyle\lim_{x\to\ 2^{-}}=-(2-2)+2(2-4) = -4$$
R.H.LRight Hand Limit/0
$$\displaystyle\lim_{x\to\ 2^{+}}=(2-2)+2(2-4)=-4$$
$$\therefore \displaystyle\lim_{x\to\ 2^{-}}f(x) = \displaystyle\lim_{x\to\ 2^{+}}f(x)$$
At $$x=4$$
L.H.L(Left Hand Limit)
$$\displaystyle\lim_{x\to\ 4^{-}}=(4-2)+2(4-4)=2$$
R.H.L(Right Hand Limit)
$$\displaystyle\lim_{x\to\ 4^{+}}=(4-2)-2(4-4)=2$$
$$\therefore \displaystyle\lim_{x\to\ 4^{-}}f(x) = \displaystyle\lim_{x\to\ 4^{+}}f(x)$$
$$\therefore \ f(x)$$ is continuous at everywhere.
Question 3
1 / -0

$$67,84,95,.,133,158$$

A
$$116$$

B
$$129$$

C
$$108$$

D
$$111$$

Solution
Question 4
1 / -0

How many committee of five persons with a chairperson can be selected from $$12$$ persons.

A
$$924$$

B
$$825$$

C
$$736$$

D
$$643$$

Solution

No of Persons $$=5+1$$(Chairperson)

$$\therefore$$ No of committees that can be selected $$={}^{12}C_6=924$$
Question 5
1 / -0

Number of identical terms in the sequence $$2, 5, 8, 11,$$___ upto $$100$$ terms and $$3, 5, 7, 9, 11$$____ upto $$100$$ terms are

A
$$17$$

B
$$33$$

C
$$50$$

D
$$147$$

Solution

Fierst series: $$2, 5, 8, 11.........$$
$$a_1 =2;\ d=3;\ n=100$$
$$l=20(100-1)^*5=2+99^*3=2+297=299$$
so, series $$=2, 5, 8, 11,.....,197, 197,200,203,....,299$$
second series; $$3, 5, 9, 11,.....$$
$$a_1=3;\ d=2;\ n=100$$
$$l=3+(100-1)^*2=3+99^*2=3+198=201$$
so series $$=3, 5, 7, 9, 11, ...., 201$$
series having similar terms; $$5, 11, 17, ....., 197$$
$$a_1=5;\ d=6;\ l=197$$
$$l=a_1+(n-1)d$$
$$197=5+(n-1)6$$
$$192/6=n-1$$
$$32+1=n$$
$$n=33$$
Question 6
1 / -0

Find the value of ?.

A
$$125$$

B
$$25$$

C
$$625$$

D
$$156$$

Solution

$${\left( {3 + 1} \right)^2} = {\left( 4 \right)^2} = 16$$

$${\left( {15 + 6} \right)^2} = {\left( {21} \right)^2} = 441$$

$${\left( {10 + 5} \right)^2} = {\left( {15} \right)^2} = 225$$

$$so,\,{\left( {12 + 13} \right)^2} = {\left( {25} \right)^2} = 625$$

therefore the answer is $$625$$
Question 7
1 / -0

The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabet ?

A
$$50000$$

B
$$50100$$

C
$$50300$$

D
$$50400$$

Solution

$$2$$ vowels can be chosen in $$5{c}_{2}$$ ways.
$$2$$ consonants can be chosen in $$21{c}_{2}$$ ways.
$$4$$ letter can be arranged in $$4$$ ways.
$$\therefore$$ The no of words containing $$2$$ vowels and $$2$$ consonants $$=5{c}_{2}\times 21{c}_{2}\times 4!=10\times 210\times 24=50400$$
Question 8
1 / -0

If $$P,\,Q$$ be two arithmetic means between $$\dfrac{1}{3}$$ and $$\dfrac{1}{24}$$, then their value are

A
$$\dfrac{7}{72},\dfrac{5}{36}$$

B
$$\dfrac{17}{72},\dfrac{5}{36}$$

C
$$\dfrac{7}{36},\dfrac{5}{72}$$

D
$$\dfrac{5}{72},\dfrac{17}{72}$$

Solution

P and Q are the arithmetic means between $$\dfrac{1}{3} $$ and $$\dfrac{1}{24}$$

So the resulting sequence formed an AP
$$\dfrac{1}{3},P,Q, \dfrac{1}{24}$$ are in AP

$$\Rightarrow a_4=a+(4-1)d$$
$$\Rightarrow\dfrac{1}{24}=\dfrac{1}{3}+(4-1)d$$

$$\Rightarrow \dfrac{-7}{24}=3d$$

$$\Rightarrow d=\dfrac{-7}{72}$$

$$\therefore P=a+d=\dfrac{1}{3}+\dfrac{-7}{72}$$

=$$\dfrac{24-7}{72}$$

=$$\dfrac{17}{72}$$

and $$Q=a+2d=\dfrac{1}{3}+\dfrac{-14}{72}$$

=$$\dfrac{24-14}{72}$$

=$$\dfrac{5}{36}$$
Question 9
1 / -0

Find the area of the triangle formed by joining the mid points of the sides of the triangle whose vertices are $$(0.-1), (2, 1) and (0, 3)$$

A
$$4$$

B
$$8$$

C
$$1$$

D
$$2$$

Solution

Area of triangle having vertices $$(x_1,y_1), (x_2,y_2)$$ and $$(x_3,y_3)$$ is given by
Area $$ = \dfrac{1}{2} \times [ x_1 (y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) ] $$
Also, we know $$\\Area\>of\>triangle\>=4\times\>of\>triangle\>formed\>using\>mid-point\>\\=4\times(\frac{1}{2})[x_1(y_2-y_3)+x_2(y_3+y_1)+x_3(y_1-y_2)]\\=2[0+2(3-1)+0]=8sq\>unit$$
Question 10
1 / -0

How many $$3$$-digit even numbers can be formed from the digits $$1, 2, 3, 4, 5, 6$$ if the digits can be repeated?

A
108

B
98

C
72

D
112

Solution

Only $$3$$ numbers are possible at units place $$(2,4,6)$$ as we need even numbers. But at $$10's$$ place and $$100's$$ place all $$6$$ are possible.
No. of digit even no's$$=3\times 6\times 6=108$$

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Sequences and Series Test 30

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Sequences and Series Test 30

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SHARING IS CARING

Question 1

1981 and 1891

19 and 91

190 and 160

1198911 and 1168611

Solution

Question 2

$$x=2,4$$

$$x=2$$

No where

Except $$x=2$$

Solution

Question 3

$$116$$

$$129$$

$$108$$

$$111$$

Solution

Question 4

$$924$$

$$825$$

$$736$$

$$643$$

Solution

Question 5

$$17$$

$$33$$

$$50$$

$$147$$

Solution

Question 6

$$125$$

$$25$$

$$625$$

$$156$$

Solution

Question 7

$$50000$$

$$50100$$

$$50300$$

$$50400$$

Solution

Question 8

$$\dfrac{7}{72},\dfrac{5}{36}$$

$$\dfrac{17}{72},\dfrac{5}{36}$$

$$\dfrac{7}{36},\dfrac{5}{72}$$

$$\dfrac{5}{72},\dfrac{17}{72}$$

Solution

Question 9

$$4$$

$$8$$

$$1$$

$$2$$

Solution

Question 10

108

98

72

112

Solution

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