Class 12th

The graph of function f(x) = x⁵/20 - x⁴/12 + 5 has :

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Contributor-Level 10

f (x) = x? /20 - x? /12 + 5
f' (x) = x? /4 - x³/3 = x³ (x/4 - 1/3)
Local maxima at 0, Local minima at 4/3
f' (x) = x³ - x² = x² (x-1)
x = 1 point of inflection

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2 months ago

Suppose y is a function of x that satisfies dy/dx = √(1-y²)/x² and y=0 at x = 2/π, then y(3/π) is equal to:

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Contributor-Level 10

dy/√ (1-y²) = dx/x²
sin? ¹ (y) = -1/x + c ⇒ c = π/2
sin? ¹ (y) = -π/3 + π/2 = π/6
y = 1/2

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2 months ago

If $\int_{- a}^{a} (| x | + | x - 2 |) d x = 22, (a > 2) a n d [x]$ denotes the greatest integer $\leq x,$ then $\int_{a}^{- a} (x + [x]) d x$ is equal to__________.

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Contributor-Level 10

$\int_{- a}^{a} (| x | + | x - 2 |) d x = 22, a > 2$

$\int_{- a}^{0} (- 2 x + 2) d x + \int_{0}^{2} (x - x + 2) d x + \int_{2}^{a} (2 x - 2) d x = 22$

$\Rightarrow 2 a^{2} + 2 = 20 \Rightarrow a^{2} = 9 \Rightarrow a = 3$

$∴ \int_{3}^{- 3} (x + [x]) d x = - \int_{- 3}^{3} (2 x - {x}) d x = - \int_{- 3}^{3} 2 x d x + 6 {\int_{0}^{1} x d x = 6 . \frac{x^{2}}{2} |}_{0}^{1} = 3$

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2 months ago

Class 12th Physics

The diagram shows three concentric conducting spherical shells having radii $R, 2 R$ and $3 R$ . All three shells are given some charges. The initial potential of each shell is as mentioned in the figure. If the inner most shell is earthed then the final charge present on its outer surface would be equal to:

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Raj Pandey

Contributor-Level 9

$V_{1} = \frac{k Q_{3}}{3 R} + \frac{k Q_{2}}{2 R} + \frac{k Q_{1}}{R} = 30$

$V_{2} = \frac{k Q_{3}}{3 R} + \frac{k Q_{2}}{2 R} + \frac{k Q_{1}}{2 R} = 15$

$V_{3} = \frac{k Q_{3}}{3 R} + \frac{k Q_{2}}{3 R} + \frac{k Q_{1}}{3 R} = 10$

Eq. (1) $&$ (2) $\to \frac{k Q_{1}}{R} = 30$

Eq. (2) $&$ (3) $\to \frac{k Q_{2}}{6 R} + \frac{k Q_{1}}{6 R} = 5 \Rightarrow \frac{k Q_{2}}{R} = - \frac{k Q_{1}}{R} + 30 = 0$

By (3) $\to \frac{k Q_{3}}{R} = 30 - \frac{k Q_{2}}{R} - \frac{k Q_{1}}{R} = 0$

Potential of innermost shell (earthed) $= \frac{k Q_{1}^{'}}{R} + \frac{k Q_{2}}{2 R} + \frac{k Q_{3}}{3 R} = 0$

$\Rightarrow \frac{k Q_{1}^{'}}{R} = 0 ∴ Q_{1}^{'} = 0$

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2 months ago

Let P = $[\begin{matrix} 3 & - 1 & - 2 \\ 2 & 0 & α \\ 3 & - 5 & 0 \end{matrix}] w h e r e α \in R .$ Suppose Q = [q_ij] is a matrix satisfying PQ = kI₃ for some non-zero k $\in$ R. If $q_{23} = \frac{- k}{8} a n d | Q | = \frac{k^{2}}{2}, t h e n α^{2} + k^{2}$ is equal to_________.

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Contributor-Level 10

$P = [\begin{matrix} 3 & - 1 & - 2 \\ 2 & 0 & α \\ 3 & - 5 & 0 \end{matrix}] a n d Q = [q_{i j}] \Rightarrow P Q = k l_{3}$

$q_{23} = - \frac{k}{8} a n d | Q | = \frac{k^{2}}{2}$

$P Q = k l_{3} \Rightarrow P^{- 1} = \frac{Q}{k} = {(\begin{matrix} 3 & - 1 & - 2 \\ 2 & 0 & α \\ 3 & - 5 & 0 \end{matrix})}^{- 1}$

$| p | | Q | = (k l_{3}) \Rightarrow 8 . \frac{k^{2}}{2} = k^{3}$

$k \neq 0 \Rightarrow k = 4$

$∴ α^{2} + k^{2} = 1 + 16 = 17 .$

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2 months ago

Class 12th Biomolecules

Which is a nucleic acid, DNA or RNA?

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Vikash Kumar Vishwakarma

Contributor-Level 7

DNA and RNA both are the types of nucleic acids.

DNA (Deoxyribonucleic acid) store the genetic information.
RNA (Ribonucleic acid) helps in protein synthesis. It carry and translates the genetic codes.

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2 months ago

Class 12th Vector Algebra

Let three vectors $\vec{a}, \vec{b} a n d \vec{c}$ be such that $\vec{c}$ is coplanar with $\vec{a} a n d \vec{b}, \vec{a} . \vec{c} = 7 a n d \vec{b}$ is perpendicular to $\vec{c}, w h e r e \vec{a} = - \hat{i} + \hat{j} + \hat{k} a n d \hat{b} = 2 \hat{i} + \hat{k},$ then the value of 2 ${| \vec{a} + \vec{b} + \vec{c} |}^{2}$ is____________.

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Class 12th Complex Numbers and Quadratic Equations

Contributor-Level 10

$\vec{c} = α \vec{a} + β \vec{b} . . . . . (i)$

$\vec{a} . \vec{c} = 7 \vec{b} . \vec{c} = 0$

$\vec{a} = - \hat{i} + \hat{j} + \hat{k} \Rightarrow | \vec{a} | = \sqrt[]{3}$

$\vec{b} = 2 \hat{i} + \hat{k} \Rightarrow | \vec{b} | = \sqrt[]{5} \vec{a} . \vec{b} = - 2 + 1 = - 1$

From $(i) \vec{a} . \vec{c} = α {| \vec{a} |}^{2} - β$

$3 α - β = 7 . . . . . . . . . . (i i)$

$\bar{b} . \bar{c} = α \bar{b} . \bar{a} + β {| \vec{b} |}^{2} \Rightarrow - α + 5 β = 0 . . . . . . . (i i i)$

Solving $α = \frac{5}{2} a n d β = \frac{1}{2}$

$\Rightarrow 2 {| \vec{a} + \vec{b} + \vec{c} |}^{2} = 75$

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2 months ago

A quadratic function f(x) satisfies f(x) ≥ 0 for all real x. If f(2) = 0 and f(4) = 12, then the value of f(6) is :

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Contributor-Level 10

f (x) = λ (x-2)²
⇒ 12 = λ (2)² ⇒ λ = 3
f (x) = 3 (x-2)² f (6) = 3 * 4² = 48

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2 months ago

The area bounded by the curve y=(x-1)(x-2)(x-3), x-axis and ordinates x=0, x=3 is :

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Contributor-Level 10

∫ (from 0 to 3) (x-1) (x-2) (x-3) dx = ∫ (from 0 to 3) (x³ - 6x² + 11x - 6) dx

A = ∫ (from 0 to 1) (x³ - 6x² + 11x - 6) dx - ∫ (from 1 to 2) (x³ - 6x² + 11x - 6) dx + ∫ (from 2 to 3) (x³ - 6x² + 11x - 6) dx
A = 11/4

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2 months ago

Let B_i (i = 1, 2, 3) be three independent events in a sample space. The probability that only B₁ occurs is a, only B₂ occurs is b and B₃ occurs is Let p be the probability that none of the events B₁ occurs and these 4 probabilities satisfy the equations (a - 2b) = ab and (b - 3 p = (All the probabilities are assumed to lie in the integral (0, 1)). The is equal to_________.

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