Ncert Solutions Maths class 12th

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New answer posted•a year ago

Ncert Solutions Maths class 12th Class 12th

$\underset{n \to \infty}{l i m} (1 + \frac{1 + \frac{1}{2} + . . . . . . . . . . + \frac{1}{n}}{n^{2}}) i s e q u a l t o :$

3 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

$\underset{n \to \infty}{l i m} {(1 + \frac{1 + \frac{1}{2} + . . . . . . . + \frac{1}{n}}{n^{2}})}^{n}$ limit is in the form of $1^{\infty}$

$l = e x p (\underset{n \to \infty}{l i m} \frac{1 + \frac{1}{2} + \frac{1}{3} + . . . . . + \frac{1}{n}}{n^{2}})$

$0 \leq 1 + \frac{1}{2} + \frac{1}{3} + . . . . . + \frac{1}{n} \leq 1 + \frac{1}{\sqrt[]{2}} + \frac{1}{\sqrt[]{3}} + . . . . + \frac{1}{\sqrt[]{n}} \leq 2 \sqrt[]{n} - 1$

Taking limit $(n \to \infty)$

$l$ = exp (0) (from sandwich)

$l = 1$

Second Method :

1 + \frac{1}{2} + \frac{1}{3} + . . . . + \frac{1}{n} \geq l n (n + 1) . . . . . . . (i)

1 + \frac{1}{2} + \frac{1}{3} + . . . . . + \frac{1}{n} \leq 1 + \int_{1}^{2} \frac{1}{2} d x \leq 1 + l n n . . . . . . . . . . (i i)

From (i) & (ii)

$l n (n + 1) \leq 1 + \frac{1}{2} + \frac{1}{3} + . . . . + \frac{1}{n} \leq 1 + I n n, \forall n \in N, n \geq 2$ Views|Posted a year ago

Asked by Payal Gupta

Answered by

Payal Gupta | Contributor-Level 10

a year ago

$\vec{O} P = (2, - 1, 1)$

Normal vector to the plane

$= \vec{A} B \times A \vec{C} = | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & - 2 \\ 1 & 2 & - 1 \end{matrix} | = (3, - 1, 1) = \vec{n}$

Projection of $\vec{O} P o n \vec{n} i s | \frac{\vec{O} P . \vec{n}}{| \vec{n} |} |$

PN = $\frac{6 + 1 + 1}{\sqrt[]{11}} = \frac{8}{11}$

Projection of OP on plane = ON = $\sqrt[]{O P^{2} - P N^{2}} = \sqrt[]{6 - \frac{64}{11}} = \sqrt[]{\frac{2}{11}}$

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New answer posted•a year ago

Ncert Solutions Maths class 12th Class 12th

If the curves, $\frac{x^{2}}{a} + \frac{y^{2}}{b} = 1 a n d \frac{x^{2}}{c} + \frac{y^{2}}{d} = 1$ intersect each other at an angle of 90°, then which of the following relations is TRUE?

43 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

$\frac{x^{2}}{a} + \frac{y^{2}}{b} = 1, \frac{x^{2}}{c} - \frac{y^{2}}{(- d)} = 1$

Ellipse and Hyperbola are orthogonal so these will be confocal.

$\sqrt[]{a - b} = \sqrt[]{c + (- d)}$

a – b = c – d

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New answer posted•a year ago

Ncert Solutions Maths class 12th Class 12th

The integral $\int \frac{e^{3 l o g_{e} 2 x} + 5 e^{2 l o g_{e} 2 x}}{e^{4 l o g_{e} x} + 5 e^{3 l o g_{e} x} - 7 e^{2 l o g_{e} x}} d x, x > 0 i s e q u a l t o :$ (where c is constant of integration)

6 Views|Posted a year ago

Asked by Payal Gupta

Answered by

Payal Gupta | Contributor-Level 10

a year ago

$l = \int \frac{8 x^{3} + 20 x^{2}}{x^{4} + 5 x^{3} - 7 x^{2}} d x = 4 \int \frac{2 x + 5}{x^{2} + 5 x - 7} d x = 4 l n | x^{2} + 5 x - 7 | + c$

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New answer posted•a year ago

Ncert Solutions Maths class 12th Class 12th

If Rolle’s theorm holds for the function f(x) = $x^{3} - a x^{2} + b x - 4, x \in [1, 2]$  with  $f' (\frac{4}{3}) = 0,$ then ordered pair (a, b) is equal to:

2 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

$f (x) = x^{3} - a x^{2} + b x - 4$

f (1) = f (2)

->1 – a + b – 4 = 8 – 4a + 2b – 4

->3a – b = 7 . (i)

8a = 16 + 3b . (ii)

(i) and (ii) -> (a, b) = (5, 8)

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New answer posted•a year ago

Ncert Solutions Maths class 12th Class 12th

If a curve passes through the origin and the slope of the tangent to it at any point (x, y) is $\frac{x^{2} - 4 x + y + 8}{x - 2}$ , then this curve also passes through the point:

4 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

As per questions

$\frac{d y}{d x} = \frac{x^{2} - 4 x + y + 8}{x - 2}$

$\frac{d y}{d x} = \frac{{(x - 2)}^{2} + (y + 4)}{(x - 2)}$

$\frac{d y}{d x} = (x - 2) + \frac{y + 4}{x - 2}$ .(i)

Let $\frac{y + 4}{x - 2} = t$

(y + 4) = t(x – 2)

Putting in equation (i)

$(x - 2) \frac{d t}{d x} + t = (x - 2) + t$ &nb

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New answer posted•a year ago

Ncert Solutions Maths class 12th Class 12th

If In = $\int_{\frac{π}{4}}^{\frac{π}{2}} c o t^{n} x d x, t h e n :$

2 Views|Posted a year ago

Asked by Payal Gupta

Answered by

Payal Gupta | Contributor-Level 10

a year ago

$l_{n} = \int_{π / 4}^{π / 2} c o t^{n} x d x = \int_{π / 4}^{π / 2} c o t^{n - 2} x (c o s e c^{2} x - 1) d x$

$= {[\frac{- c o t^{n - 1} x}{n - 1}]}_{π / 4}^{π / 2} - l_{n - 2} = \frac{1}{n - 1} - l_{n - 2}$

$l_{n} + l_{n - 2} = \frac{1}{n - 1}$

$\begin{array}{l} n = 4 \Rightarrow l_{4} + l_{2} = \frac{1}{3} \\ n = 5 \Rightarrow l_{5} + l_{3} = \frac{1}{4} \\ n = 6 \Rightarrow l_{6} + l_{4} = \frac{1}{5} \end{array}} \Rightarrow \frac{1}{l_{2} + l_{4}}, \frac{1}{l_{3} + l_{5}}, \frac{1}{l_{4} + l_{6}}$ are in A.P.

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New answer posted•a year ago

Ncert Solutions Maths class 12th Class 12th

Let $l = \frac{2 b^{2}}{a} = \frac{2 (1)}{3}$ be a twice differentiable function such that $\Rightarrow 9 (e^{2} + l)$ $= 9 (\frac{10}{9} + \frac{2}{3})$ , for all $= 10 + 6$ . If $= 16$ , then the value of $= \frac{10!}{5! 3! 2!} = 2520$ is

2 Views|Posted a year ago

Asked by Payal Gupta

Answered by

Payal Gupta | Contributor-Level 10

a year ago

$f : R \to R$

$(s i n ? x c o s ? y) (f (2 x + 2 y) - f (2 x - 2 y)) = (c o s ? x$ Put $s i n ? y) (f (2 x + 2 y) + f (2 x - 2 y))$

$x, y \in R$ Put $f^{'} (0) = \frac{1}{2}$

$24 f^{''} (\frac{5 π}{3})$

$(s i n ? x c o s ? y) (f (2 x + 2 y) - f (2 x - 2 y)) = (c o s ? x s i n ? y)$

$(f (2 x + 2 y) + f (2 x - 2 y))$

$f (2 x + 2 y) (s i n ? (x - y)) = f (2 x - 2 y) s i n ? (x + y)$

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New answer posted•a year ago

Ncert Solutions Maths class 12th Class 12th

Let A be a matrix of order $3 \times 3$ and $| A | = 5$ . If $| 2 a d j (3 A a d j (2 A)) | = 2^{α} \cdot 3^{β} \cdot 5^{γ} α, β, γ \in N$ then $α + β + γ$ is equal to

10 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

: $? 2$ adj $(3 A$ adj(2A))|
${= 2}^{3} . ? 3 A$ adj(2A)| $|^{2}$

${= 2}^{3} \cdot {(3^{3})}^{2} \cdot | A |^{2} \cdot | a d j (2 A) |^{2}$ intersect the line $= 2^{3} \cdot 3^{6} \cdot | A |^{2} \cdot {(| 2 A |^{2})}^{2}$ at the point