Straight Lines Test 43

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Straight Lines Test 43

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Weekly Quiz Competition

Question 1
1 / -0

The number of ways in which $$9$$ persons can be divided into three equal groups, is

A
$$1680$$

B
$$840$$

C
$$560$$

D
$$280$$

Solution
Question 2
1 / -0

How many integers are there such that $$2 \le n \le 100$$ and the highest common factor of $$n$$ and $$36$$ is $$1$$?

A
$$166$$

B
$$332$$

C
$$331$$

D
$$416$$

Solution

$$36.2^{2}.3^{2}$$
Since HCF (36,n) =1
Therefore, n sholud not be a multiple of 2 or 3.
$$ 2 \leq n\leq 100$$
Total numbers = T= 1000.
Number of numbers divisible by 2= $$N_{2}$$
Number of numbers divisible by 3= $$N_{3}$$
Number of numbers divisible by 6= $$N_{6}$$
Therefore, there are $$T-N_{2}-N_{3}+N_{6}$$ total integers
a=2
l=1000
d=2
We know, $$l=a+(N_{2}-1)d.$$
$$N_{2}=\frac{1000-2}{2}+1$$
$$N_{2}=\frac{998}{2}+1$$
$$N_{2}=499+1=500$$
Similarly,
$$N_{3}=\frac{999-3}{3}+1=333$$
$$N_{6}=\frac{999-6}{3}+1=166$$
Answer
= 999-500-333+166
= 332
Question 3
1 / -0

What is the next number in the series $$2,12,36,80,150?$$

A
$$194$$

B
$$210$$

C
$$252$$

D
$$258$$

Solution

$$12 - 2 = 10$$.
$$36 - 12 = 24$$.
$$80 - 36 = 44$$.
$$150 - 80 = 70$$.
Now,
$$24 - 10 = 14$$.
$$44 - 24 = 20$$.
$$70 - 44 = 26$$.
Then next difference of the difference will be $$26 + 6 = 32$$.
Hence answer is $$150 + 70 + 32 = 252$$.
Question 4
1 / -0

The number of permutations which can be formed out of the letters of the word "SERIES" three letters together, is:

A
120

B
60

C
42

D
none

Solution
Question 5
1 / -0

The coefficient of $$x^{18}$$ in the expansion of $$(1+x)(1-x)^{10}\{(1+x+x^2)^9\}$$ is?

A
$$84$$

B
$$126$$

C
$$-42$$

D
$$42$$

Solution

Coefficient of $$x^{18}$$ in $$(1+x)(1-x)^{10}(1+x+x^2)^9$$
$$=$$Coefficient of $$x^{18}$$ in $$(1-x)^2\{(1-x)(1+x+x^2)\}^9$$
$$=$$Coefficient of $$x^{18}$$ in $$(1-x^2)(1-x^3)^9$$
$$={^9C_6}=0=84$$.
Question 6
1 / -0

Area of the triangle with vertices $$(-2,2), (1,5)$$ and $$(6,-1)$$ is

A
$$15$$

B
$$\dfrac{3}{5}$$

C
$$\dfrac{29}{2}$$

D
$$\dfrac{33}{2}$$

Solution

Area of triangle having vertices $$(x_1,y_1), (x_2,y_2)$$ and $$(x_3,y_3)$$ is given by
Area $$ = \dfrac{1}{2} \times [ x_1 (y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)]$$

$$=\dfrac{1}{2} [-2(5+1) + 1(-1-2)+6(2-5)]$$

$$=\dfrac{1}{2}[-12-3-18]$$

$$=-\dfrac{33}{2}$$

$$\therefore$$ Area $$=\dfrac{33}{2}~sq.\, units$$
Question 7
1 / -0

The area of a triangle with vertices $$A(5,0),B(8,0)$$ and $$C(8,4)$$ in square units is

A
$$20$$

B
$$12$$

C
$$6$$

D
$$16$$

Solution

Area of Triangle using coordinates of A, B and C
Area $$ = \dfrac{1}{2} \times [ x_1 (y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)]$$--------(1)
Here in our case $$ A(5,0) = (x_1, y_1)$$
$$ A(8,0) = (x_2, y_2)$$
$$ A(8,4) = (x_3, y_3)$$

Substituting above values in Eq - 1, we get
Area $$ = 0.5 [ 5(0-4) + 8(4-0) + 8(0-0)]$$
Area $$ = 0.5 [ -20 + 32]$$
Area $$ = 0.5 \times (12)=6$$
$$ Area = 6 $$ square units.
Question 8
1 / -0

If $$\dfrac{1}{6!}+\dfrac{1}{7!}=\dfrac{x}{8!}$$, then $$x=?$$

A
$$32$$

B
$$48$$

C
$$56$$

D
$$64$$

Solution

$$\dfrac{1}{6!}+\dfrac{1}{7!}=\dfrac{x}{8!}\Rightarrow \dfrac{8\times 7}{8\times 7\times (6!)}+\dfrac{8}{8\times (7!)}=\dfrac{x}{8!}$$
$$\Rightarrow \dfrac{56}{8!}+\dfrac{8}{8!}=\dfrac{x}{8!}\Rightarrow x=56+8=64$$.
Question 9
1 / -0

The vertices of a $$\triangle ABC$$ are $$A(3,8),B(-4,2)$$ and $$C(5,-1)$$. The area of $$\triangle ABC$$ is

A
$$57$$ sq units

B
$$75$$ sq units

C
$$28\cfrac{1}{2}$$ sq units

D
$$37\cfrac{1}{2}$$ sq units

Solution

Here $$\left( { x }_{ 1 }=3,{ y }_{ 1 }=8 \right) ;\left( { x }_{ 2 }=-4,{ y }_{ 2 }=2 \right) ;\left( { x }_{ 3 }=5,{ y }_{ 3 }=-1 \right) $$
$$Ar\left( \triangle ABC \right) =\cfrac { 1 }{ 2 } \left[ { x }_{ 1 }\left( { y }_{ 2 }-{ y }_{ 3 } \right) +{ x }_{ 2 }\left( { y }_{ 3 }-{ y }_{ 1 } \right) +{ x }_{ 3 }\left( { y }_{ 1 }-{ y }_{ 2 } \right) \right] $$
$$=\cfrac { 1 }{ 2 } \left[ 3(2+1)-4(1-8)+5(8-2) \right] =\cfrac { 1 }{ 2 } (9+36+30)=\cfrac { 75 }{ 2 } =37\cfrac { 1 }{ 2 } $$
Question 10
1 / -0

The exponent of 3 in $$100!$$ is

A
$$12$$

B
$$24$$

C
$$48$$

D
$$96$$

Solution

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Re-Attempt Weekly Quiz Competition

Straight Lines Test 43

Result

Straight Lines Test 43

Score

-

Rank

-

SHARING IS CARING

Question 1

$$1680$$

$$840$$

$$560$$

$$280$$

Solution

Question 2

$$166$$

$$332$$

$$331$$

$$416$$

Solution

Question 3

$$194$$

$$210$$

$$252$$

$$258$$

Solution

Question 4

120

60

42

none

Solution

Question 5

$$84$$

$$126$$

$$-42$$

$$42$$

Solution

Question 6

$$15$$

$$\dfrac{3}{5}$$

$$\dfrac{29}{2}$$

$$\dfrac{33}{2}$$

Solution

Question 7

$$20$$

$$12$$

$$6$$

$$16$$

Solution

Question 8

$$32$$

$$48$$

$$56$$

$$64$$

Solution

Question 9

$$57$$ sq units

$$75$$ sq units

$$28\cfrac{1}{2}$$ sq units

$$37\cfrac{1}{2}$$ sq units

Solution

Question 10

$$12$$

$$24$$

$$48$$

$$96$$

Solution

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