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Q1. If h(x) = x^3 - 3x + 2, what is the critical point?
Solution:
h'(x) = 3x^2 - 3 = 0 gives x^2 = 1, so x = 1 and x = -1 are critical points.
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Q2. What is the sum of the roots of the quadratic equation f(x) = x^2 - 5x + 6?
Solution:
The sum of the roots is given by -b/a = 5/1 = 5.
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Q3. What is the inverse of the function f(x) = 3x + 1?
Solution:
To find the inverse, set y = 3x + 1, solve for x: x = (y - 1)/3.
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Q4. If \( y = \tan^{-1}(x) + \tan^{-1}(y) \), what is the value of \( y \) when \( x = 1 \)?
Solution:
When \( x = 1 \), \( y = \tan^{-1}(1) + \tan^{-1}(y) \) leads to \( y = \frac{\pi}{4} \).
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Q5. How many subsets does the set B = {a, b, c, d} have?
Solution:
The number of subsets of a set with n elements is 2^n. Here, n=4, so the number of subsets is 2^4 = 16.
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Q6. If f(x) = 3x - 4, find f(-2).
Solution:
f(-2) = 3(-2) - 4 = -6 - 4 = -10.
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Q7. The range of sin^(-1)(x) is:
Solution:
The range of sin^(-1)(x) is [-π/2, π/2].
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Q8. The function f(x) = x^2 - 4x + 4 can be expressed in which form?
Solution:
f(x) = (x - 2)^2 is the completed square form.
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Q9. If A = {x | x is a natural number less than 10} and B = {x | x is a prime number}, what is A ∩ B?
Solution:
The intersection A ∩ B includes all prime numbers less than 10, which are {2, 3, 5, 7}.
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Q10. If H = {x, y, z}, how many subsets of H have at least one element?
Solution:
The total number of subsets is 2^3 = 8. Subtracting the empty set gives 8 - 1 = 7.
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