alleryStart learning →If $f(x) = kx^2(1 - x)$ for $0 < x < 1$ is the p.d.f. of $X$, the value of $k$ is:
- 10
- 12
- 6
- 8
Show answer and solution
Correct answer: B. 12
$\int_0^1 k(x^2 - x^3)\,dx = k[x^3/3 - x^4/4]_0^1 = k(1/3 - 1/4) = k/12 = 1$ ⇒ $k = 12$.
Indian Navy 10+2 B.Tech Cadet Limits and Continuity — practice questions
Explore 53 free Limits and Continuity questions in Indian Navy 10+2 B.Tech Cadet Mathematics. Open each question or exercise to read its content, reference answers and available explanation.
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Showing questions 1–30 of 53.
If $f(x) = kx^2(1 - x)$ for $0 < x < 1$ is the p.d.f. of $X$, the value of $k$ is:Read the answer and worked solution →Find dy/dx by implicit differentiation for x² + y² = 25:Read the answer and worked solution →lim_(x→0) [(1 + x)^(1/x)] =Read the answer and worked solution →What does the sign of f ′(x) tell you?Read the answer and worked solution →Complete the definition of a critical pointRead the answer and worked solution →Order the first derivative testRead the answer and worked solution →Match each sign change to its verdictRead the answer and worked solution →The second derivative testRead the answer and worked solution →True or false: maxima and minimaRead the answer and worked solution →Order the working rule for absolute maxima and minimaRead the answer and worked solution →Tap every point you must test for an absolute maximumRead the answer and worked solution →Local or absolute?Read the answer and worked solution →Complete the rate of change statementRead the answer and worked solution →The chain rule for related ratesRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →True or false: increasing and decreasingRead the answer and worked solution →Flip and pair: symbol with meaningRead the answer and worked solution →Match each situation to the right toolRead the answer and worked solution →Tap every statement true of a local maximum at cRead the answer and worked solution →Reading the tests backRead the answer and worked solution →Which test is being described?Read the answer and worked solution →Assertion and reason: rates of changeRead the answer and worked solution →Spot the odd oneRead the answer and worked solution →Complete the second derivative testRead the answer and worked solution →True or false: scope of this chapterRead the answer and worked solution →Match each kind of matrix to its conditionRead the answer and worked solution →Sort each matrix by its shapeRead the answer and worked solution →
Worked practice questions
Find dy/dx by implicit differentiation for x² + y² = 25:
- x/y
- -x/y
- 2x
- y/x
Show answer and solution
Correct answer: B. -x/y
**Differentiate both sides:** 2x + 2y(dy/dx) = 0. So dy/dx = -x/y.
(This is the slope of a circle of radius 5; tangent perpendicular to radius.)
lim_(x→0) [(1 + x)^(1/x)] =
- Infinity
- 0
- e
- 1
Show answer and solution
Correct answer: C. e
**Take log:** y = (1+x)^(1/x), ln y = (1/x) ln(1+x).
As x → 0, ln(1+x) ≈ x, so ln y → x/x = 1. Hence y → e.
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