Permutations and Combinations Test 28

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Permutations and Combinations Test 28

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Weekly Quiz Competition

Question 1
1 / -0

Consider an incomplete pyramid of balls on a square base having $$18$$ layers; and having $$13$$ balls on each side of the top layer. Then the total number $$N$$ of balls in that pyramid satisfies

A
$$9000 < N < 10000$$

B
$$8000 < N < 9000$$

C
$$7000 < N < 8000$$

D
$$10000 < N < 12000$$

Solution

The top layer has $$(13\times 13)$$ balls the layer below it will have $$(14\times 14)$$ balls

We have $$18$$ layers

So the total number of balls

$$N=(13\times 13)+(14\times 14)+.......(30\times 30)\\ N={ 13 }^{ 2 }+{ 14 }^{ 2 }+.....{ 30 }^{ 2 }$$

Sum of squares of first $$n$$ natural numbers is $$\dfrac { n(n+1)(2n+1) }{ 6 } \\ $$

$$\therefore N=$$ sum of first $$30$$ $$-$$ sum of first $$12$$

$$N=\dfrac { 30\times 31\times 61 }{ 6 } -\dfrac { 12\times 13\times 25 }{ 6 } \\ N=8805\\ $$

$$\Rightarrow 8000<N<9000$$
Hence, option B is correct.
Question 2
1 / -0

With the help of match-sticks, Zalak prepared a pattern as shown below. When $$97$$ matchsticks are used, the serial number of the figure will be ...........

A
Figure 32

B
Figure 95

C
Figure 49

D
Figure 48

Solution

Pattern associated with the number of matchsticks:
Figure 1: No. of sticks $$=3=3+0$$
Figure 2: No. of sticks $$=5=3+2$$
Figure 3: No. of sticks $$=7=3+2+2$$
Figure 4: No. of sticks $$=9=3+2+2+2$$
----------------------------------------
The rule associated with it can be given as no. of sticks $$=3+2(n-1)$$ where $$n$$ is the number of the figure.
Now there are $$97$$ matchsticks
$$\implies 3+2(n-1)=97$$
$$\implies 3+2n-2=97$$
$$\implies 2n=96$$
$$\implies n=48$$
Hence, serial number of the figure having $$97$$ matchsticks is $$48$$.
Question 3
1 / -0

Ten points lie in a plane so that no three of them are collinear. The number of lines passing through exactly two of these points are dividing the plane into two regions each containing four of the remaining points is

A
1

B
5

C
10

D
Dependent on the configuration of points.

Solution

You can clearly see from the attached figure
Five such lines are possible joining $$AF,BG,CH,DI,EJ$$
Question 4
1 / -0

Area of the triangle formed by the points $$(0,0),(2,0)$$ and $$(0,2)$$ is

A
$$1$$ sq.units

B
$$2$$ sq.units

C
$$4$$ sq.units

D
$$8$$ sq.units

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,
$$=\dfrac{1}{2}\left|0(0-2)+2(2-0)+0(0-0)\right|$$

$$=\dfrac{1}{2}\left|0+4+0\right|$$

$$=\dfrac{1}{2}\times 4=2\ sq.\ units$$

Hence, this is the answer.
Question 5
1 / -0

Find the term independent of $$x$$ in $${ \left( \cfrac { 3 }{ 2 } { x }^{ 2 }-\cfrac { 1 }{ 3x } \right) }^{ 9 }$$.

A
$$\dfrac {6}{15}$$

B
$$\dfrac {7}{18}$$

C
$$\dfrac {7}{8}$$

D
$$\dfrac {4}{9}$$

Solution

We know that the general term in expansion of $$(a+b)^n$$ is given by $$^nC_{r}a^{n-r}b^{r}$$ where r ranges from $$0 $$ to $$ 9$$.
Here $$a= \cfrac{3}{2}x^2, \ b= -\cfrac{1}{3x} \ \And \ n=9$$
For any term to be independent of $$x$$, coefficient of $$x$$ in $$a^{n+1-r}b^{r-1}$$ should be zero.
$$\Rightarrow \ (x^2)^{9-r}(x^{-1})^{r}=x^0$$
$$\Rightarrow 18-2r-r=0$$
$$\Rightarrow r=6$$
Therefore, the term independent of $$x$$ is $$^9C_6 \left (\cfrac{3}{2}\right)^3 \left (-\cfrac{3}{3}\right)^6=\cfrac{7}{18}$$.
Question 6
1 / -0

If $$f(x)=\left| \log { \left| x \right| } \right| $$, then

A
$$f(x)$$ is continuous and differentiable for all $$x$$ in its domain

B
$$f(x)$$ is continuous for all $$x$$ in its domain but not differentiable at $$x=\pm 1$$

C
$$f(x)$$ is neither continuous nor differentiable at $$x=\pm 1$$

D
None of the above

Solution

It is evident from the graph of $$f(x)=\left| \log { \left| x \right| } \right| \quad $$ that $$f(x)$$ is everywhere continuous but not differentiable at $$x=\pm 1$$
Question 7
1 / -0

Three bells commenced to toll at the same time and tolled at intervals of $$20, 30, 40$$ seconds respectively. If they toll together at $$6$$ am, then which of the following is the time at which they can toll together

A
$$6:55$$am

B
$$6:56$$am

C
$$6:57$$am

D
$$6:59$$am

Solution
Question 8
1 / -0

Choose $$3, 4, 5$$ points other than vertices respectively on the sides $$AB, BC$$ and $$CA$$ of a $$\triangle ABC$$. The number of triangles that can be formed by using only these points as vertices, is

A
$$220$$

B
$$217$$

C
$$215$$

D
$$210$$

E
$$205$$

Solution

Required number of triangles that can be formed by using only given points as vertices
$$= ^{12}C_{3} - \left \{^{3}C_{3} + ^{4}C_{3} + ^{5}C_{3}\right \}$$
$$= \dfrac {12\times 11\times 10}{3\times 2\times 1} - \left \{1 + 4 + \dfrac {5\times 4}{2\times 1}\right \}$$
$$= 220 - (5 + 10)$$
$$= 220 - 15 = 205$$
Question 9
1 / -0

Find the $$4^{th}$$ term of $${ \left( 9x-\cfrac { 1 }{ 3\sqrt { x } } \right) }^{ 18 }$$.

A
$$16500$$

B
$$18564$$

C
$$16540$$

D
$$32600$$

Solution

We know that the $$r^{th}$$ term in expansion of $$(a+b)^n$$ is given by $$^nC_{r-1}a^{n+1-r}b^{r-1}$$
Here $$a=9x, \ n=18 \ \And \ b=-\cfrac{1}{3\sqrt{x}}$$
$$\therefore$$ the fourth term in expansion of $$\left (9x-\cfrac{1}{3\sqrt{x}}\right)^{18}$$ is $$^{18}C_{12} {x}^{6}\left (-\cfrac{1}{3\sqrt{x}}\right)^{12} =18564$$
Question 10
1 / -0

Directions For Questions

A trip of Saurashtra was organised by the school in a mini bus for 28 students and in a regular bus for 45 students. Each student contributed Rs 500. During the trip, the expenses were : food bill Rs 9,855, entry fees Rs 3,285, mini bus charge Rs 5,432, the regular bus charge Rs 8,730 and boarding charge Rs 6,132.

...view full instructions

What was the bus charge per student?

A
Rs 194

B
Rs 184

C
Rs 187

D
Rs 199

Solution

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Permutations and Combinations Test 28

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Permutations and Combinations Test 28

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Question 1

$$9000 < N < 10000$$

$$8000 < N < 9000$$

$$7000 < N < 8000$$

$$10000 < N < 12000$$

Solution

Question 2

Figure 32

Figure 95

Figure 49

Figure 48

Solution

Question 3

1

5

10

Dependent on the configuration of points.

Solution

Question 4

$$1$$ sq.units

$$2$$ sq.units

$$4$$ sq.units

$$8$$ sq.units

Solution

Question 5

$$\dfrac {6}{15}$$

$$\dfrac {7}{18}$$

$$\dfrac {7}{8}$$

$$\dfrac {4}{9}$$

Solution

Question 6

$$f(x)$$ is continuous and differentiable for all $$x$$ in its domain

$$f(x)$$ is continuous for all $$x$$ in its domain but not differentiable at $$x=\pm 1$$

$$f(x)$$ is neither continuous nor differentiable at $$x=\pm 1$$

None of the above

Solution

Question 7

$$6:55$$am

$$6:56$$am

$$6:57$$am

$$6:59$$am

Solution

Question 8

$$220$$

$$217$$

$$215$$

$$210$$

$$205$$

Solution

Question 9

$$16500$$

$$18564$$

$$16540$$

$$32600$$

Solution

Question 10

Rs 194

Rs 184

Rs 187

Rs 199

Solution

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