Section A (1 Mark each)
Q1. (b) median
The intersection of the 'less than' and 'more than' ogives corresponds to the median on the x-axis.
Q2. (b) 25
Max frequency is 20. Modal class: 15-20. Lower limit = 15.
Total $N=66$. $N/2 = 33$. CF: 10, 25, 37... Median class: 10-15. Lower limit = 10.
Sum = $15 + 10 = 25$.
Q3. (b) 8
Empirical formula: 3 Median = Mode + 2 Mean.
$3 \text{ Median} = 8 + 2(8) = 24 \Rightarrow \text{Median} = 8$.
Q4. (b) Both A and R are true but R is not the correct explanation of A.
Assertion is the definition of median for even $n$. Reason is the general property of median. Reason doesn't explain the specific formula for even $n$.
Section B (2 Marks each)
Q5. New Mean.
If each observation is increased by $k$, the mean increases by $k$.
New Mean = $18 + 2 = 20$.
Q6. Class marks.
For 10-25: $\frac{10+25}{2} = \frac{35}{2} = 17.5$.
For 35-55: $\frac{35+55}{2} = \frac{90}{2} = 45$.
Q7. Mean calculation.
$\bar{x} = a + h (\frac{\sum f_i u_i}{\sum f_i}) = 55 + 10 (\frac{-30}{100})$
$= 55 + 10 (-0.3) = 55 - 3 = 52$.
Section C (3 Marks each)
Q8. Missing frequencies $f_1, f_2$. Mean = 53.
Total freq: $15 + f_1 + 21 + f_2 + 17 = 100 \Rightarrow f_1 + f_2 = 47$.
$x_i$: 10, 30, 50, 70, 90.
$\sum f_i x_i = 150 + 30f_1 + 1050 + 70f_2 + 1530 = 2730 + 30f_1 + 70f_2$.
Mean $53 = \frac{2730 + 30f_1 + 70f_2}{100} \Rightarrow 5300 = 2730 + 30f_1 + 70f_2$.
$30f_1 + 70f_2 = 2570 \Rightarrow 3f_1 + 7f_2 = 257$.
Solve $f_1 + f_2 = 47$ and $3f_1 + 7f_2 = 257$.
$3(47 - f_2) + 7f_2 = 257 \Rightarrow 141 - 3f_2 + 7f_2 = 257 \Rightarrow 4f_2 = 116
\Rightarrow f_2 = 29$.
$f_1 = 47 - 29 = 18$.
Q9. Mode.
Max freq is 20. Modal class 30-40.
$l=30, f_1=20, f_0=10, f_2=8, h=10$.
Mode = $30 + \frac{20-10}{40-10-8} \times 10 = 30 + \frac{10}{22} \times 10$
$= 30 + \frac{100}{22} = 30 + 4.54 = 34.54$.
Section D (5 Marks)
Q10. Median = 28.5. Total $N=60$.
$5+x+20+15+y+5 = 60 \Rightarrow x+y = 15$.
Median 28.5 lies in 20-30. $l=20, h=10, f=20, cf=5+x$.
$28.5 = 20 + \frac{30 - (5+x)}{20} \times 10$
$8.5 = \frac{25-x}{2} \Rightarrow 17 = 25-x \Rightarrow x = 8$.
$y = 15 - 8 = 7$.
Section E (Case Study - 4 Marks)
(i) Max frequency is 14. Modal Class is 4-6.
(ii) Mean: $x_i$: 1, 3, 5, 7, 9. $f_i$: 4, 12, 14, 12, 8.
$f_i x_i$: 4, 36, 70, 84, 72. Sum = 266.
Mean = $266/50 = 5.32$ units.
(iii) Median: $N=50, N/2=25$. CF: 4, 16, 30... Median class 4-6.
$l=4, cf=16, f=14, h=2$.
Median = $4 + \frac{25-16}{14} \times 2 = 4 + \frac{9}{7} = 4 + 1.28 = 5.28$ units.