Permutations and Combinations Test 34

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Permutations and Combinations Test 34

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Weekly Quiz Competition

Question 1
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 2
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 3
1 / -0

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

A
$$116$$

B
$$129$$

C
$$108$$

D
$$111$$

Solution
Question 4
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$$=3\times 6\times 6=108$$
Question 5
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 6
1 / -0

Find the area of the triangle whose vertices are $$(-5, -1), (3, -5), (5, 2)$$

A
$$31 sq$$ $$units$$

B
$$32sq$$ $$units$$

C
$$33 sq$$ $$units$$

D
$$\text{no triangle can be formed}$$

Solution

Area of a triangle whose vertices are $$A(x_1,y_1), B(x_2,y_2), C(x_3,y_3)$$ is given as,
$$Area=(\dfrac{1}{2})[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)] $$
Let $$A$$ be the required area
$$A=(\dfrac{1}{2})[(-5)(-5-2)+3(2+1)+5(-1+5)]\\ (\dfrac{1}{2})[35+9+20]\\=32sq\>unit$$
So, the correct option is (B)
Question 7
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 8
1 / -0

The sides AB, BC, CA of a triangle ABC have $$3,4$$ and $$5$$ interior points respectively on them. The number of triangles that can be constructed using these points as vertices is-

A
$$205$$

B
$$210$$

C
$$315$$

D
$$216$$

Solution

$$\rightarrow $$Suppose we had $$12$$ points, none of them collinear, number of possible triangles $$=^{ 12 }{ C }_{ 3 }$$
$$\rightarrow $$If out of these $$3,4$$ and $$5$$ points were chosen separately to lie on each side of a triangle ABC, then number of possible triangles
$$=^{ 12 }{ C }_{ 3 }-^{ 3 }{ C }_{ 3 }-^{ 5 }{ C }_{ 3 }-^{ 4 }{ C }_{ 3 }$$
$$=220-1-10-4$$
$$=205$$
Question 9
1 / -0

Ten students of the physics department decided to go on a educational trip.They hired a mini bus for the trip, but the bus can only carry eight students at a time and each student goes at least once. Find the minimum number of trips the bus has to make so that each students can go for equal number of trips.

A
$$5$$

B
$$4$$

C
$$6$$

D
$$8$$

Solution

Ten students $$\longrightarrow$$ Eight students in bus $$2$$ students are excluded/not taken on a trip so we have exclude every students once, so that no. of trips made by each students are equal so no. of trips $$= \dfrac{10}{2} =5$$
$$A$$ is correct.
Question 10
1 / -0

Number of rectangles in the grid shown which are not squares is?

A
$$160$$

B
$$162$$

C
$$170$$

D
$$185$$

Solution

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Permutations and Combinations Test 34

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Permutations and Combinations Test 34

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Question 1

$$17$$

$$33$$

$$50$$

$$147$$

Solution

Question 2

$$50000$$

$$50100$$

$$50300$$

$$50400$$

Solution

Question 3

$$116$$

$$129$$

$$108$$

$$111$$

Solution

Question 4

108

98

72

112

Solution

Question 5

$$125$$

$$25$$

$$625$$

$$156$$

Solution

Question 6

$$31 sq$$ $$units$$

$$32sq$$ $$units$$

$$33 sq$$ $$units$$

$$\text{no triangle can be formed}$$

Solution

Question 7

$$4$$

$$8$$

$$1$$

$$2$$

Solution

Question 8

$$205$$

$$210$$

$$315$$

$$216$$

Solution

Question 9

$$5$$

$$4$$

$$6$$

$$8$$

Solution

Question 10

$$160$$

$$162$$

$$170$$

$$185$$

Solution

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