Straight Lines Test 33

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Straight Lines Test 33

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Weekly Quiz Competition

Question 1
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 2
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 3
1 / -0

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

A
$$160$$

B
$$162$$

C
$$170$$

D
$$185$$

Solution
Question 4
1 / -0

Number of ways in which $$7$$ green bottles and $$8$$ blue bottles can be arranged in a row if exactly $$1$$ pair of green bottles is side by side is (Assume all bottles to be a like except for the colour).

A
$$84$$

B
$$360$$

C
$$504$$

D
None of the above

Solution

Consider the two green bottles as one entity. Let's call it $$GG$$.

We will also refer to the green bottle as: $$G$$, and to the blue bottle as: $$B$$.

So now we have $$14$$ units to arrange ( 1 green pair bottle + the remaining 5 green bottles + the 8 blue bottles):

$$GG, G, G, G, G, G, B, B, B, B, B, B, B, B$$

Now place the 8 blue bottles in a row, such that between each bottle has a gap before and after it:

_B_B_B_B_B_B_B_B_

Now we need to take the six green units (the 1 pair and 5 bottles) and place them in six of the nine gaps.

Therefore, the solution is: $$^9C_6 = 84 ways$$.
Question 5
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 6
1 / -0

If the letters of the word "VARUN" are written in all possible ways and then are arranged as in a dictionary, then the rank of the word VARUN is?

A
$$98$$

B
$$99$$

C
$$100$$

D
$$101$$

Solution

$${\textbf{Step 1}}\;:\;{\mathbf{Finding}}\;{\mathbf{number}}\;{\mathbf{of}}\;{\mathbf{words}}\;{\mathbf{starting}}\;{\mathbf{with}}\;{\mathbf{the}}\;{\mathbf{alphabets}}\;{\mathbf{in}}\;{\mathbf{the}}\;{\mathbf{given}}\;{\mathbf{word}}$$

$${\text{No}}.{\text{ of words beginning with A }} = 4! = 4 \times 3 \times 2 = 24$$

$${\text{No}}.{\text{ of words beginning with N }} = 4! = 4 \times 3 \times 2 = 24$$

$${\text{No}}.{\text{ of words beginning with R }} = 4! = 4 \times 3 \times 2 = 24$$

$${\text{No}}.{\text{ of words beginning with U }} = 4! = 4 \times 3 \times 2 = 24$$

$${\text{So}},{\text{ in total }}96{\text{ words will be formed while beginning with letter A}},{\text{ N}},{\text{ R and U}}.$$

$${\textbf{Step 2}}\;\;:\;{\mathbf{Finding}}\;{\mathbf{the}}\;{\mathbf{order}}\;{\mathbf{of}}\;{\mathbf{the}}\;{\mathbf{word}}$$

$${\text{Order of }} 97{\text { th word }} - {\text{ VANRU}}$$

$${\text{Order of }}98{\text{th word }} - {\text{ VANUR}}$$

$${\text{Order of }}99{\text{th word }} - {\text{ VARNU}}$$

$${\text{Order of }}100{\text{th word}} - {\text{ VARUN}}.$$

$${\mathbf{Therefore}},\;{\mathbf{rank}}\;{\mathbf{of}}\;{\mathbf{word}}\;'{\mathbf{VARUN}}'\;{\textbf{is 100. Option C is correct.}}$$
Question 7
1 / -0

If the vertices of a triangle are $$(1,2),(4,-6)$$ and $$(3,5)$$, then its area is

A
$$\cfrac{23}{2}$$ sq. units

B
$$\cfrac{25}{2}$$ sq. units

C
$$12$$ sq.unit

D
none of these

Solution

Area of triangle having vertices $$(x_1,y_1), (x_2,y_2)$$ and $$(x_3,y_3)$$ is given by
Area $$ = \dfrac{1}{2} \times [ x_1 (y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) ] $$
Therefore, area of required triangle is given by
Area $$ = \dfrac{1}{2} \times [ 1 (-6 - 5) + 4(5 - 2) + 3(2 + 6) ] $$
$$=\dfrac{1}{2}[-11+12+24]$$
$$=\dfrac{1}{2}\times25$$
$$=\dfrac{25}{2}\ sq.unit$$
Question 8
1 / -0

The unit's place digit in $$(1446)^{4n + 3}$$ is

A
$$n$$

B
$$0$$

C
$$6$$

D
None of these

Solution
Question 9
1 / -0

From 0 to 9 , four digited numbers can be formed such that
the digits are in ascending order is

A
$${}^{10}{P_4}$$

B
$${}^{10}{C_4}$$

C
$${}^{10}{P_4} - {}^9{P_3}$$

D
$${}^{10}{C_4} - {}^9{C_3}$$

Solution

Selection of$$ 4$$ digits out of$$ 10$$ (including 0)=$$ ^{10}C_4$$
Question 10
1 / -0

The last three digits in $$10!$$ are ?

A
$$800$$

B
$$700$$

C
$$500$$

D
$$600$$

Solution

The last three digits in$$ 10!$$ are,
The $$10!$$ is,
$$=10\times 9\times 8\times 7 \times 6\times 5\times 4\times 3\times 2\times 1\\$$
$$=3628800.$$

So, the last three digits of$$ 10!$$ is
$$=3628(800),$$
$$=800$$.

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Re-Attempt Weekly Quiz Competition

Straight Lines Test 33

Result

Straight Lines Test 33

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SHARING IS CARING

Question 1

$$31 sq$$ $$units$$

$$32sq$$ $$units$$

$$33 sq$$ $$units$$

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

Solution

Question 2

$$5$$

$$4$$

$$6$$

$$8$$

Solution

Question 3

$$160$$

$$162$$

$$170$$

$$185$$

Solution

Question 4

$$84$$

$$360$$

$$504$$

None of the above

Solution

Question 5

$$205$$

$$210$$

$$315$$

$$216$$

Solution

Question 6

$$98$$

$$99$$

$$100$$

$$101$$

Solution

Question 7

$$\cfrac{23}{2}$$ sq. units

$$\cfrac{25}{2}$$ sq. units

$$12$$ sq.unit

none of these

Solution

Question 8

$$n$$

$$0$$

$$6$$

None of these

Solution

Question 9

$${}^{10}{P_4}$$

$${}^{10}{C_4}$$

$${}^{10}{P_4} - {}^9{P_3}$$

$${}^{10}{C_4} - {}^9{C_3}$$

Solution

Question 10

$$800$$

$$700$$

$$500$$

$$600$$

Solution

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