Permutations and Combinations Test 45

Result

Permutations and Combinations Test 45

Score
-
out of -
Rank
-
out of -

TIME Taken - -

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Weekly Quiz Competition

Question 1
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 2
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 3
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 4
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 5
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 6
1 / -0

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

A
$$12$$

B
$$24$$

C
$$48$$

D
$$96$$

Solution
Question 7
1 / -0

If $$A$$ and $$B$$ are two points having co-ordinates $$(3,4)$$ and $$(5,-2)$$ respectively and $$P$$ is a point such that $$PA=PB$$ and area of triangle $$PAB=10$$ square units, then the co-ordinates of $$P$$ are

A
$$(7,4)$$ or $$(13,2)$$

B
$$(7,2)$$ or $$(1,0)$$

C
$$(2,7)$$ or $$(4,13)$$

D
$$None\ of\ these$$

Solution
Question 8
1 / -0

The next number in the pattern 62, 37, 12 ____ is

A
25

B
13

C
0

D
-13

Solution

Option (b) is correct; here, the given series has a common difference of +25.
$$\begin{array}{l}\Rightarrow-62+25=-37,-37+25=-12 \\\therefore-12+25=13\end{array}$$

Question 9

1 / -0

The letters of word 'ZENITH' are written in all positive ways. If all these words are written in the order of a dictionary, then the rank of the word 'ZENITH' is

$$716$$

$$692$$

$$698$$

$$616$$

Solution

The total number of words is $$6! = 720$$. Let us write the letters of word ZENITH alphabetically, i.e, EHINTZ.

For ZENITH word starts with	Word starting with	Number of words
$$Z$$	$$E$$	$$5!$$
	$$H$$	$$5!$$
	$$I$$	$$5!$$
	$$N$$	$$5!$$
	$$T$$	$$5!$$
ZEN	ZEH	$$3!$$
	ZEI	$$3!$$
ZENI	ZENH	$$2$$
ZENIT	ZENIH	$$1$$
	Total number of words before ZERNITH	$$615$$

Hence, there are $$615$$ words before ZENITH, so the rank of ZENITH is $$616$$.

Question 10
1 / -0

the area of a triangle with vertices $$ (-3, 0),(3,0) $$ and $$ (0, k) $$ is $$ 9 $$ sq units. then the value of $$ k $$ will be

A
$$ 9 $$

B
$$ 3 $$

C
$$ -9 $$

D
$$ 6 $$

Solution

We know that , area of a triangle with vertices $$ (a_1,y_1),(x_2,y_2) $$ and $$ (x_3,y_3) $$ is given by
$$ \Delta = \frac {1}{2} \left| \begin{matrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{matrix} \right| $$
$$ \therefore \Delta = \frac {1}{2} \left| \begin{matrix} -3 & 0 & 1 \\ 3 & 0 & 1 \\ 0 & k & 1 \end{matrix} \right| $$
Expanding along $$ R_1 $$
$$ 9 = \frac {1}{2} [ -3 ( -k) - 0 +1 ( 3k) ] $$
$$ \Rightarrow 18 = 3k + 3k = 6 k $$
$$ \therefore K = \frac {18 }{6} = 3 $$

User

Question Analysis

Correct -
Wrong -
Skipped -

My Perfomance

Score
-
out of -
Rank
-
out of -

Re-Attempt Weekly Quiz Competition

Permutations and Combinations Test 45

Result

Permutations and Combinations Test 45

Score

-

Rank

-

SHARING IS CARING

Question 1

$$84$$

$$126$$

$$-42$$

$$42$$

Solution

Question 2

$$20$$

$$12$$

$$6$$

$$16$$

Solution

Question 3

$$15$$

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

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

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

Solution

Question 4

$$32$$

$$48$$

$$56$$

$$64$$

Solution

Question 5

$$57$$ sq units

$$75$$ sq units

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

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

Solution

Question 6

$$12$$

$$24$$

$$48$$

$$96$$

Solution

Question 7

$$(7,4)$$ or $$(13,2)$$

$$(7,2)$$ or $$(1,0)$$

$$(2,7)$$ or $$(4,13)$$

$$None\ of\ these$$

Solution

Question 8

25

13

0

-13

Solution

Question 9

$$716$$

$$692$$

$$698$$

$$616$$

Solution

Question 10

$$ 9 $$

$$ 3 $$

$$ -9 $$

$$ 6 $$

Solution

User

Question Analysis

My Perfomance

Score

-

Rank

-

Results Announcement on: coming Monday

Stay Tuned: We’ll update you on your result via email or SMS.