Straight Lines Test 27

Result

Straight Lines Test 27

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Weekly Quiz Competition

Question 1
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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 2
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An alphabet contains a $$A^{'s}$$ and b $$ B^{'s}$$ . (In all a+b letters ). The number of words each containing all the $$ A^{'s}$$ and any number of $$ B^{'s}$$, is

A
$$^{a+b}C_b$$

B
$$^{a+b+1}C_a$$

C
$$^{a+b+1}C_b$$

D
none of these

Solution
Question 3
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 4
1 / -0

The maximum number of intersection points of n circles and n straight lines , among themselves is 80.The value of n is

A
$$7$$

B
$$6$$

C
$$5$$

D
$$8$$

Solution
Question 5
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 6
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 7
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If $$\alpha ,\beta , \gamma $$ are three consecutive integers. If these integers are raised to first, second and third positive powers respectively, and added then they form a perfect square, the square root of which is equal to the sum of these integers. Also, $$\alpha < \beta < \gamma $$. Then, $$\gamma$$ is equals to:

A
$$3$$

B
$$14$$

C
$$5$$

D
$$11$$

Solution

Let the numbers be $$n-1,n,n+1$$
As per the given information, we have
$$(n-1)+{ n }^{ 2 }+{ (n+1) }^{ 3 }={ (n-1+n+n+1) }^{ 2 }$$
$$\Rightarrow { n }^{ 3 }-5{ n }^{ 2 }+4n=0$$
$$\Rightarrow n=0,1,4$$
For $$n=0,$$ the numbers are: $$-1,0,1$$ this is out as all the numbers should be positive
For $$n=1,$$ the numbers are: $$0,1,2$$ this is out as all the numbers should be positive (‘0’ can’t be taken as positive)
For $$n=4,$$ the numbers are: $$3,4,5$$
We have got largest number which is $$5$$
Hence, the value of $$\gamma$$ is $$5$$.
Question 8
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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 9
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Find the missing number in the circle:

A
$$66$$

B
$$72$$

C
$$71$$

D
$$78$$

Solution

$$3\times 2 - 1, 5\times 2 - 1, 9\times 2 - 1, 17\times 2 - 1 + 39 = 72$$.

$$\therefore$$ The solution is $$72$$.
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