Limits – Trigonometric
\(\lim_{x\to 0}\frac{1 – \cos x}{x}\) is equal to:
(a) 0 (b) 1 (c) -1 (d) None of these
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✅ Solution
Answer: (a) 0
Using standard limit: \(\lim_{x\to 0}\frac{1 – \cos x}{x} = 0\)
Alternatively, using trigonometric identity:
\(1 – \cos x = 2\sin^2(\frac{x}{2})\)
\(\lim_{x\to 0}\frac{2\sin^2(\frac{x}{2})}{x} = \lim_{x\to 0}\frac{\sin(\frac{x}{2})}{\frac{x}{2}} \times \sin(\frac{x}{2}) = 1 \times 0 = 0\)
Differentiation – Logarithm
\(\frac{d}{dx}\left(\log x\right)\) is equal to:
(a) \(\frac{1}{x}\) (b) \(\frac{1}{x^2}\) (c) \(-\frac{1}{x}\) (d) None of these
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✅ Solution
Answer: (a) \(\frac{1}{x}\)
This is a standard derivative:
\(\frac{d}{dx}(\log x) = \frac{1}{x}\)
Note: This is for natural logarithm (base e). For logarithm with base a:
\(\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}\)
Sets – Power Set
A power set of a set ‘A’ is the collection of all subsets of A and is denoted by P(A). (Write True/False)
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✅ Solution
True
• Power set P(A) is the set of all subsets of A.
• If A has n elements, then P(A) has \(2^n\) elements.
• The empty set φ and A itself are always elements of P(A).
Ordered Pair
A pair of elements grouped together in a particular order is called _____ pair. (Fill in the blank)
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✅ Solution
Answer: Ordered
• An ordered pair (a, b) has a as the first element and b as the second element.
• Order matters: (a, b) ≠ (b, a) unless a = b.
• Used in Cartesian products, coordinates, and relations.
Limits – Trigonometric Standard Limit
\(\lim_{x\to 0}\frac{\tan x}{x}\) is equal to:
(a) 0 (b) 1 (c) -1 (d) None of these
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✅ Solution
Answer: (b) 1
Standard limit: \(\lim_{x\to 0}\frac{\tan x}{x} = 1\)
Proof: \(\frac{\tan x}{x} = \frac{\sin x}{x} \times \frac{1}{\cos x}\)
\(\lim_{x\to 0}\frac{\tan x}{x} = \lim_{x\to 0}\frac{\sin x}{x} \times \lim_{x\to 0}\frac{1}{\cos x} = 1 \times 1 = 1\)