📚 JKBOSE Class 11th
MATHEMATICS DAWN GUESS PAPER – Section B
(2 Mark Questions – Q.5 to Q.12)
📋 Section B Details
Sets – Difference of Sets
If \(X = \{a, b, c, d\}\) and \(Y = \{f, b, d, g\}\), find \(X – Y\) and \(Y – X\).
Show Answer (2 Marks)
✅ Solution
Given:
\(X = \{a, b, c, d\}\)
\(Y = \{f, b, d, g\}\)
Step 1: Find \(X – Y\)
\(X – Y\) = Elements in X but not in Y
Common elements: b, d
\(X – Y = \{a, c\}\)
Step 2: Find \(Y – X\)
\(Y – X\) = Elements in Y but not in X
Common elements: b, d
\(Y – X = \{f, g\}\)
Answers:
\(X – Y = \{a, c\}\)
\(Y – X = \{f, g\}\)
Trigonometry – Radian to Degree Conversion
Convert \(\frac{7\pi}{6}\) radians into degree measure.
Show Answer (2 Marks)
✅ Solution
Step 1: Use conversion formula
\(180^\circ = \pi\) radians
\(1\) radian = \(\frac{180}{\pi}\) degrees
Step 2: Convert \(\frac{7\pi}{6}\) radians
\(\frac{7\pi}{6}\) radians = \(\frac{7\pi}{6} \times \frac{180}{\pi}\) degrees
= \(\frac{7 \times 180}{6}\)
= \(\frac{1260}{6} = 210^\circ\)
Answer: \(210^\circ\)
Complex Numbers – Cube of Complex Number
Express \((5 – 3i)^3\) in the form of \(a + ib\).
Show Answer (2 Marks)
✅ Solution
Step 1: Expand using binomial theorem or direct multiplication
\((5 – 3i)^3 = (5 – 3i)(5 – 3i)(5 – 3i)\)
First multiply two terms:
\((5 – 3i)^2 = 25 – 30i + 9i^2\)
= \(25 – 30i – 9\) (since \(i^2 = -1\))
= \(16 – 30i\)
Step 2: Multiply with third term
\((16 – 30i)(5 – 3i) = 16(5 – 3i) – 30i(5 – 3i)\)
= \(80 – 48i – 150i + 90i^2\)
= \(80 – 198i + 90(-1)\)
= \(80 – 198i – 90\)
= \(-10 – 198i\)
Answer: \(-10 – 198i\)
Relations – Domain and Range
Let \(A = \{1, 2, 3, 4, 5, 6\}\), define a relation \(R\) from \(A\) to \(A\) by \(R = \{(x, y): y = x + 1\}\). Write down its range and domain.
Show Answer (2 Marks)
✅ Solution
Given: \(A = \{1, 2, 3, 4, 5, 6\}\)
\(R = \{(x, y): y = x + 1\}\)
Step 1: List all ordered pairs in R
For \(x \in A\), \(y = x + 1\) must also be in A
• When \(x = 1\), \(y = 2\) ⇒ (1, 2)
• When \(x = 2\), \(y = 3\) ⇒ (2, 3)
• When \(x = 3\), \(y = 4\) ⇒ (3, 4)
• When \(x = 4\), \(y = 5\) ⇒ (4, 5)
• When \(x = 5\), \(y = 6\) ⇒ (5, 6)
• When \(x = 6\), \(y = 7\) (not in A, so not included)
∴ \(R = \{(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)\}\)
Step 2: Find domain and range
Domain = Set of all x-coordinates = \(\{1, 2, 3, 4, 5\}\)
Range = Set of all y-coordinates = \(\{2, 3, 4, 5, 6\}\)
Answers:
Domain = \(\{1, 2, 3, 4, 5\}\)
Range = \(\{2, 3, 4, 5, 6\}\)
Limits – Trigonometric Limit
Evaluate: \(\lim_{x \to 0} \frac{1 – \cos x}{x}\)
Show Answer (2 Marks)
✅ Solution
Step 1: Use trigonometric identity
\(1 – \cos x = 2\sin^2\left(\frac{x}{2}\right)\)
Step 2: Rewrite the limit
\(\lim_{x \to 0} \frac{1 – \cos x}{x} = \lim_{x \to 0} \frac{2\sin^2\left(\frac{x}{2}\right)}{x}\)
= \(\lim_{x \to 0} \frac{\sin\left(\frac{x}{2}\right)}{\frac{x}{2}} \times \sin\left(\frac{x}{2}\right)\)
Step 3: Evaluate using standard limit
We know: \(\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1\)
Let \(\theta = \frac{x}{2}\)
\(\lim_{x \to 0} \frac{\sin\left(\frac{x}{2}\right)}{\frac{x}{2}} = 1\)
and \(\lim_{x \to 0} \sin\left(\frac{x}{2}\right) = 0\)
∴ \(\lim_{x \to 0} \frac{1 – \cos x}{x} = 1 \times 0 = 0\)
Answer: 0
Probability – Sample Space
A box contains 1 red and 3 identical white balls. Two balls are drawn at random in succession without replacement. Write the sample space for it.
Show Answer (2 Marks)
✅ Solution
Given:
• 1 red ball (R)
• 3 identical white balls (W₁, W₂, W₃) – but since identical, we can denote as W
Step 1: Since balls are drawn without replacement
Total balls = 4 (1R + 3W)
Two balls drawn without replacement means order matters if we consider them as distinct draws.
Step 2: List all possible outcomes
Possible outcomes when drawing 2 balls without replacement:
1. First red, then white: (R, W)
2. First white, then red: (W, R)
3. First white, then another white: (W, W)
But since the 3 white balls are identical, we don’t need to distinguish between them.
Sample Space:
S = {(R, W), (W, R), (W, W)}
Note: If considering white balls as distinct (W₁, W₂, W₃), the sample space would be larger with more combinations.