5 Marks Questions

Question 15 Marks

A housewife buys the following articles from departmental store sugar 5 kg at rate of ₹ $55 / \mathrm{kg}$ and GST $5 \%$, butter 500 g at rate of ₹ $600 / \mathrm{kg}$, GST $12 \%$, shampoo 200 ml bottle at rate of ₹ $110 / \mathrm{bottle}$, GST $18 \%$, hair colour 2 packets at rate of 160 per packet GST 18%. Prepare the bill for the above purchases.

Answer

Bill for housewife

Article	Weight/Number of articles	Cost/article (in ₹)	Amount (in ₹)	Total amount (in ₹)
Sugar	5 Kg	₹ 55 per kg GST 5%	275.0013.75	288.75
Butter	500 g	₹ 600/kg GST 12%	300.0036.00	336.00
Butter	1 bottle	₹ 110/bottle CST 18%	110.0019.80	129.80
Hair colour	2 packers	₹ 160/packers GST 18%	320.0057.60	377,60
Total bill				1132.15

Amount of bill = ₹ 1132.15.

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Question 25 Marks

Compute mean deviation from mean of the following distribution:

Marks	10-20	20-30	30-40	40-50	50-60	60-70	70-80	80-90
No. of students	8	10	15	25	20	10	9	5

Answer

For Computation of mean deviation from the mean: we prepare the following table.

$\mathrm{N}=\sum_{i=1}^{8} \mathrm{f}_{\mathrm{i}}=110$ and $\sum_{i=1}^{8} \mathrm{f}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}=5390$
$\mathrm{X}=\frac{\sum_{i=1}^{8} f_{i} x_{i}}{N}$
$=\frac{5390}{110}=49$
Mean deviation $=\frac{\sum_{i=1}^{8} f_{i}\left|x_{i}-X\right|}{N}$
$=\frac{1644}{110}$
= 14.945
$\approx 14.95$

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Question 35 Marks

Find the Spearman's rank correlation between marks in Mathematics and Statistics obtained by 10 students:

Maths in Mathematics	80	38	95	30	74	84	91	60	66	40
Marks in Statistics	85	50	92	58	70	65	88	56	52	46

Answer

We construct the following table:

Marks in Mathematics	Rank R₁	Marks in Statistics	Rank R₂	Diff. d = R₁ - R₂	d²
80	4	85	3	1	1
38	9	50	9	0	0
95	1	92	1	0	0
30	10	58	6	4	16
74	5	70	4	1	1
84	3	65	5	-2	4
91	2	88	2	0	0
60	7	56	7	0	0
66	6	52	8	-2	4
40	8	46	10	-2	4
					30

$\therefore$ Spearman's rank correlation $=\mathrm{r}=1-\frac{6 \Sigma d^{2}}{n\left(n^{2}-1\right)}=1-\frac{180}{10 \times 99}=+\frac{9}{11}$
$\therefore r=+0.82$

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Question 45 Marks

Evaluate $\lim _{x \rightarrow 2} \frac{x^{3}+4 x-16}{2 x^{3}-3 x-10}$

Answer

$\lim _{x \rightarrow 2} \frac{x^{3}+4 x-16}{2 x^{3}-3 x-10}$
$=\lim _{x \rightarrow 2} \frac{(x-2)\left(x^{2}+2 x+8\right)}{(x-2)\left(2 x^{2}+4 x+5\right)}$
$=\lim _{x \rightarrow 2} \frac{x^{2}+2 x+8}{2 x^{2}+4 x+5}=\frac{4+4+8}{8+8+5}=\frac{16}{21}$

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Question 55 Marks

If ${ }^{n} P_{r}=336,{ }^{n} C_{r}=56$. Find $n$ and $r$ and hence find ${ }^{n-1} C_{r-1}$.

Answer

${ }^{n} P_{r}=336,{ }^{n} C_{r}=56$
We know ${ }^{\mathrm{n}} \mathrm{P}_{\mathrm{r}}=\mathrm{r}$! ${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}} \Rightarrow 336=56 \mathrm{r}$!
$\Rightarrow r!=6=3!\Rightarrow r=3$
Consider ${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}=56 \Rightarrow{ }^{\mathrm{n}} \mathrm{C}_{3}=56$
$\Rightarrow \frac{n(n-1)(n-2)}{3!}=56$
$\Rightarrow \mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)=56 \times 6=8 \times 7 \times 6$
$\Rightarrow \mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)=8(8-1)(8-2) \Rightarrow \mathrm{n}=8$
Hence $n=8, r=3$
$\therefore{ }^{\mathrm{n}-1} \mathrm{C}_{\mathrm{r}-1}={ }^{7} \mathrm{C}_{2}=\frac{7 \times 6}{2}=21$.

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Question 65 Marks

i. How many different words can be formed with the letters of the word HARYANA?
ii. How many of these begin with H and end with N?
iii. In how many of these H and N are together?

Answer

i. There are 7 letters in the word 'HARYANA' of which 3 are A's and remaining all are each of its own kind.
So, total number of words $=\frac{7!}{3!1!1!1!1!}=\frac{7!}{3!}=840$
ii. After fixing H in first place and N in last place, we have 5 letters out of which three are alike i.e. A's and remaining all are each of its own kind.
So, total number of words $=\frac{5!}{3!}=20$
iii. Considering H and N together we have $7-2+1=6$ letters out of which three are alike i.e A's and others are each of its own kind. These six letters can be arranged in $\frac{6!}{3!}$ ways. But H and N can be arranged amongst themselves in $2!$ ways.
Hence, the requisite number of words $=\frac{6!}{3!} \times 2!=120 \times 2=240$

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Applied Maths 5 Marks Questions

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