📝 SECTION B
Very Short Answer (VSA) · 5 × 2 = 10 marks
🔢 Sets (VSA)
If P = {1, 2, 3, 4}, Q = {3, 4, 5, 6}, R = {5, 6, 7, 8} and S = {7, 8, 9, 10}. Find:
(a) \(P \cup Q\)
(b) \(P \cup Q \cup R\)
OR
Which of the given pairs of sets are disjoint?
(i) A = {1, 2, 3, 4} and B = {x: x is a natural number and 4 ≤ x ≤ 6}
(ii) A = {x: x is an even integer} and B = {x: x is an odd integer}
🔍 VIEW ANSWER
✅ STEP-BY-STEP SOLUTION:
Main Part:
(a) \(P \cup Q\) \(= \{1,2,3,4\} \cup \{3,4,5,6\}\)\( = \{1,2,3,4,5,6\}\)
(b) \(P \cup Q \cup R\)\( = \{1,2,3,4,5,6\} \cup \{5,6,7,8\}\)\( = \{1,2,3,4,5,6,7,8\}\)
OR Part (Disjoint sets):
(i) A = {1,2,3,4}, B = {4,5,6} (since natural numbers 4≤x≤6). They have common element 4 → not disjoint.
(ii) Even integers = {…,-4,-2,0,2,4,…}, Odd integers = {…,-3,-1,1,3,…}. No common element → disjoint.
✅ Final answers: (a) {1,2,3,4,5,6} (b) {1,2,3,4,5,6,7,8} | OR: (i) not disjoint, (ii) disjoint
📐 Trigonometry (Proof)
Prove that: \(\frac{\sin x – \sin y}{\cos x + \cos y} = \tan\frac{x – y}{2}\)
🔍 VIEW PROOF
✅ STEP-BY-STEP PROOF:
Step 1: Use sum-to-product identities:
\(\sin x – \sin y\) \(= 2 \cos\frac{x+y}{2} \sin\frac{x-y}{2}\)
\(\cos x + \cos y\) \(= 2 \cos\frac{x+y}{2} \cos\frac{x-y}{2}\)
Step 2: Substitute into LHS:
\(\frac{\sin x – \sin y}{\cos x + \cos y}\) \(= \frac{2 \cos\frac{x+y}{2} \sin\frac{x-y}{2}}{2 \cos\frac{x+y}{2} \cos\frac{x-y}{2}}\)
Step 3: Cancel common factor \(2 \cos\frac{x+y}{2}\) (provided \(\cos\frac{x+y}{2} \neq 0\)):
\(= \frac{\sin\frac{x-y}{2}}{\cos\frac{x-y}{2}} = \tan\frac{x-y}{2}\)
✅ Hence proved: LHS = RHS.
🧮 Quadratic Equations
Solve the quadratic equation: \(x^{2} + 3x + 9 = 0\)
OR
Solve: \(-x^{2} + x + 2 = 0\)
🔍 VIEW SOLUTION
✅ STEP-BY-STEP SOLUTIONS:
Main: \(x^{2} + 3x + 9 = 0\)
Using quadratic formula \(x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}\) with a=1, b=3, c=9:
Discriminant \(D = 9 – 36 = -27 = 27i^2\)
\(\sqrt{D} = \sqrt{27}i = 3\sqrt{3}i\)
\(x = \frac{-3 \pm 3\sqrt{3}i}{2}\)
Thus roots are \(\frac{-3 + 3\sqrt{3}i}{2}\) and \(\frac{-3 – 3\sqrt{3}i}{2}\)
OR: \(-x^{2} + x + 2 = 0\) (Multiply by -1)
\(x^{2} – x – 2 = 0\)
Factor: \((x – 2)(x + 1) = 0\)
Thus \(x = 2\) or \(x = -1\)
✅ Main: \(\frac{-3 \pm 3\sqrt{3}i}{2}\) | OR: x = 2, -1
📈 Arithmetic Progression
Insert five numbers between 8 and 26 so that the resulting series is an A.P.
🔍 VIEW SOLUTION
✅ STEP-BY-STEP SOLUTION:
Let the AP be: 8, A₁, A₂, A₃, A₄, A₅, 26
Here first term a = 8, seventh term a₇ = 26.
Using formula aₙ = a + (n-1)d:
a₇ = 8 + 6d = 26
⇒ 6d = 18 ⇒ d = 3
Thus the five numbers are:
A₁ = 8 + 3 = 11
A₂ = 11 + 3 = 14
A₃ = 14 + 3 = 17
A₄ = 17 + 3 = 20
A₅ = 20 + 3 = 23
✅ The five numbers are: 11, 14, 17, 20, 23
Check: 8,11,14,17,20,23,26 forms AP with common difference 3.
📊 Mean Deviation
Find the mean deviation about the mean for the following data:
| \(x_i\) | 10 | 30 | 50 | 70 | 90 |
|---|---|---|---|---|---|
| \(f_i\) | 4 | 24 | 28 | 16 | 8 |
🔍 VIEW SOLUTION
✅ STEP-BY-STEP CALCULATION:
Step 1: Calculate mean (\(\bar{x}\))
| \(x_i\) | \(f_i\) | \(f_i x_i\) |
|---|---|---|
| 10 | 4 | 40 |
| 30 | 24 | 720 |
| 50 | 28 | 1400 |
| 70 | 16 | 1120 |
| 90 | 8 | 720 |
| Total | \(\sum f_i = 80\) | \(\sum f_i x_i\) \(= 4000\) |
\(\bar{x} = \frac{4000}{80} = 50\)
Step 2: Calculate \(|x_i – \bar{x}|\) and \(f_i|x_i – \bar{x}|\)
| \(x_i\) | \(f_i\) | \(|x_i – 50|\) | \(f_i|x_i – 50|\) |
|---|---|---|---|
| 10 | 4 | 40 | 160 |
| 30 | 24 | 20 | 480 |
| 50 | 28 | 0 | 0 |
| 70 | 16 | 20 | 320 |
| 90 | 8 | 40 | 320 |
| Total | \(\sum f_i|x_i – \bar{x}|\) \(= 1280\) |
Step 3: Mean deviation about mean
M.D. = \(\frac{\sum f_i|x_i – \bar{x}|}{\sum f_i} = \frac{1280}{80} = 16\)
✅ Mean deviation about mean = 16