Rd Sharma Xi 2018 Solutions for Class 12 Science Math Chapter 5 Trigonometric Functions are provided here with simple step-by-step explanations. These solutions for Trigonometric Functions are extremely popular among Class 12 Science students for Math Trigonometric Functions Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma Xi 2018 Book of Class 12 Science Math Chapter 5 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma Xi 2018 Solutions. All Rd Sharma Xi 2018 Solutions for class Class 12 Science Math are prepared by experts and are 100% accurate.

Question 1:

stion Prove the following identities (1-16)
sec4 x sec2 x = tan4 x + tan2 x

Question 2:

Prove the following identities (1-16)

Question 3:

Prove the following identities (1-16)

Question 4:

Prove the following identities (1-16)

Question 5:

Prove the following identities (1-16)

Question 6:

Prove the following identities (1-16)

Question 7:

Prove the following identities (1-16)

Question 8:

Prove the following identities (1-16)

Question 9:

Prove the following identities (1-16)

Question 10:

Prove the following identities (1-16)

Question 11:

Prove the following identities (1-16)

= RHS
Hence proved.

Question 12:

Prove the following identities (1-16)

= RHS

Hence proved.

Question 13:

Prove the following identities (1-17)

Hence proved.

Question 14:

Prove the following identities (1-16)

Question 15:

Prove the following identities (1-16)

Question 16:

Prove the following identities (1-16)

Question 17:

If , then prove that is also equal to a.

Disclaimer: There is some error in the given question.
The question should have been
Question: If , then prove that is also equal to a.
So, the solution is done accordingly.

Solution:

Hence proved.

Question 18:

If , then the values of tan x, sec x and cosec x

$\mathrm{tan}x=\frac{\mathrm{sin}x}{\mathrm{cos}x}=\frac{\frac{{a}^{2}-{b}^{2}}{{a}^{2}+{b}^{2}}}{\frac{2ab}{{a}^{2}+{b}^{2}}}=\frac{{a}^{2}-{b}^{2}}{2ab}\phantom{\rule{0ex}{0ex}}\mathrm{sec}x=\frac{1}{\mathrm{cos}x}=\frac{{a}^{2}+{b}^{2}}{2ab}\phantom{\rule{0ex}{0ex}}\mathrm{cosec}x=\frac{1}{\mathrm{sin}x}=\frac{{a}^{2}+{b}^{2}}{{a}^{2}-{b}^{2}}$

Question 19:

If , then find the values of $\sqrt{\frac{a+b}{a-b}}+\sqrt{\frac{a-b}{a+b}}$.

If show that .

Hence proved.

Question 21:

If , then prove that .

Question 22:

If prove that ${\left({m}^{2}+{n}^{2}\right)}^{2}=mn$.

Question 23:

If , then prove that , where ${m}^{2}\le 2$

Question 24:

If , then shown that $ab+a-b+1=0$.

Hence proved.

Prove the:

Question 26:

If , prove that

(i) $\frac{{T}_{3}-{T}_{5}}{{T}_{1}}=\frac{{T}_{5}-{T}_{7}}{{T}_{3}}$

(ii)

(iii)

(i) LHS:

RHS:
$\frac{{T}_{5}-{T}_{7}}{{T}_{3}}\phantom{\rule{0ex}{0ex}}=\frac{\left({\mathrm{sin}}^{5}x+{\mathrm{cos}}^{5}x\right)-\left({\mathrm{sin}}^{7}x+{\mathrm{cos}}^{7}x\right)}{{\mathrm{sin}}^{3}x+{\mathrm{cos}}^{3}x}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\frac{{\mathrm{sin}}^{5}x-\mathrm{si}{n}^{7}x+{\mathrm{cos}}^{5}x-{\mathrm{cos}}^{7}x}{{\mathrm{sin}}^{3}x+{\mathrm{cos}}^{3}x}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\frac{{\mathrm{sin}}^{5}x\left(1-{\mathrm{sin}}^{2}x\right)+{\mathrm{cos}}^{5}x\left(1-{\mathrm{cos}}^{2}x\right)}{{\mathrm{sin}}^{3}x+{\mathrm{cos}}^{3}x}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\frac{{\mathrm{sin}}^{5}x{\mathrm{cos}}^{2}x+{\mathrm{cos}}^{5}x{\mathrm{sin}}^{2}x}{{\mathrm{sin}}^{3}x+{\mathrm{cos}}^{3}x}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}={\mathrm{sin}}^{2}x.{\mathrm{cos}}^{2}x$

LHS = RHS

Hence proved.

(ii) LHS:

Hence proved.

(iii) LHS:

$6{T}_{10}-15{T}_{8}+10{T}_{6}-1\phantom{\rule{0ex}{0ex}}6\left({\mathrm{sin}}^{10}x+{\mathrm{cos}}^{10}x\right)-15\left({\mathrm{sin}}^{8}x+{\mathrm{cos}}^{8}x\right)+10\left({\mathrm{sin}}^{6}x+{\mathrm{cos}}^{6}x\right)-1\phantom{\rule{0ex}{0ex}}$

Question 1:

Find the values of the other five trigonometric functions in each of the following:

(i) x in quadrant III

(ii) x in quadrant II

(iii) x in quadrant III

(iv) x in quadrant I

Question 2:

If sin $x=\frac{12}{13}$ and x lies in the second quadrant, find the value of sec x + tan x.

Question 3:

If sin  find the value of 8 tan .

We have:

Question 4:

If sin x + cos x = 0 and x lies in the fourth quadrant, find sin x and cos x.

Question 5:

If  find the values of other five trigonometric functions and hence evaluate .

Question 1:

Find the values of the following trigonometric ratios:
(i) $\mathrm{sin}\frac{5\mathrm{\pi }}{3}$

(ii) sin 17π

(iii) $\mathrm{tan}\frac{11\mathrm{\pi }}{6}$

(iv) $\mathrm{cos}\left(-\frac{25\mathrm{\pi }}{4}\right)$

(v)

(vi) $\mathrm{sin}\frac{17\pi }{6}$

(vii) $\mathrm{cos}\frac{19\pi }{6}$

(viii) $\mathrm{sin}\left(-\frac{11\pi }{6}\right)$

(ix) $\mathrm{cosec}\left(-\frac{20\pi }{3}\right)$

(x) $\mathrm{tan}\left(-\frac{13\pi }{4}\right)$

(xi) $\mathrm{cos}\frac{19\pi }{4}$

(xii) $\mathrm{sin}\frac{41\pi }{4}$

(xiii) $\mathrm{cos}\frac{39\pi }{4}$

(xiv) $\mathrm{sin}\frac{151\pi }{6}$

Question 2:

Prove that:
(i) tan 225° cot 405° + tan 765° cot 675° = 0
(ii) $\mathrm{sin}\frac{8\mathrm{\pi }}{3}\mathrm{cos}\frac{23\mathrm{\pi }}{6}+\mathrm{cos}\frac{13\mathrm{\pi }}{3}\mathrm{sin}\frac{35\mathrm{\pi }}{6}=\frac{1}{2}$
(iii) cos 24° + cos 55° + cos 125° + cos 204° + cos 300° = $\frac{1}{2}$
(iv) tan (−225°) cot (−405°) −tan (−765°) cot (675°) = 0
(v) cos 570° sin 510° + sin (−330°) cos (−390°) = 0
(vi) $\mathrm{tan}\frac{11\mathrm{\pi }}{3}-2\mathrm{sin}\frac{4\mathrm{\pi }}{6}-\frac{3}{4}{\mathrm{cosec}}^{2}\frac{\mathrm{\pi }}{4}+4{\mathrm{cos}}^{2}\frac{17\mathrm{\pi }}{6}=\frac{3-4\sqrt{3}}{2}$
(vii) $3\mathrm{sin}\frac{\mathrm{\pi }}{6}\mathrm{sec}\frac{\mathrm{\pi }}{3}-4\mathrm{sin}\frac{5\mathrm{\pi }}{6}\mathrm{cot}\frac{\mathrm{\pi }}{4}=1$

Question 3:

Prove that

(i)

(ii)  $\frac{\mathrm{cosec}\left(90°+x\right)+\mathrm{cot}\left(450°+x\right)}{\mathrm{cosec}\left(90°-x\right)+\mathrm{tan}\left(180°-x\right)}+\frac{\mathrm{tan}\left(180°+x\right)+\mathrm{sec}\left(180°-x\right)}{\mathrm{tan}\left(360°+x\right)-\mathrm{sec}\left(-x\right)}=2$

(iii)

(iv)

(v)

Question 4:

Prove that: ${\mathrm{sin}}^{2}\frac{\mathrm{\pi }}{18}+{\mathrm{sin}}^{2}\frac{\mathrm{\pi }}{9}+{\mathrm{sin}}^{2}\frac{7\mathrm{\pi }}{18}+{\mathrm{sin}}^{2}\frac{4\mathrm{\pi }}{9}=2$

Question 5:

Prove that:
$\mathrm{sec}\left(\frac{3\mathrm{\pi }}{2}-x\right)\mathrm{sec}\left(x-\frac{5\mathrm{\pi }}{2}\right)+\mathrm{tan}\left(\frac{5\mathrm{\pi }}{2}+x\right)\mathrm{tan}\left(x-\frac{3\mathrm{\pi }}{2}\right)=-1.$

Question 6:

In a âˆ†ABC, prove that:
(i) cos (A + B) + cos C = 0
(ii) $\mathrm{cos}\left(\frac{A+B}{2}\right)=\mathrm{sin}\frac{C}{2}$
(iii) $\mathrm{tan}\frac{A+B}{2}=\mathrm{cot}\frac{C}{2}$

Question 7:

In a âˆ†A, B, C, D be the angles of a cyclic quadrilateral, taken in order, prove that
cos(180° − A) + cos (180° + B) + cos (180° + C) − sin (90° + D) = 0

Question 8:

Find x from the following equations:

$90°=\frac{\pi }{2}$

Question 9:

Prove that:
(i)

(ii) $\mathrm{sin}\frac{13\pi }{3}\mathrm{sin}\frac{8\pi }{3}+\mathrm{cos}\frac{2\pi }{3}\mathrm{sin}\frac{5\pi }{6}=\frac{1}{2}$

(iii)

(iv) $\mathrm{sin}\frac{10\pi }{3}\mathrm{cos}\frac{13\pi }{6}+\mathrm{cos}\frac{8\pi }{3}\mathrm{sin}\frac{5\pi }{6}=-1$

(v) $\mathrm{tan}\frac{5\mathrm{\pi }}{4}\mathrm{cot}\frac{9\mathrm{\pi }}{4}+\mathrm{tan}\frac{17\mathrm{\pi }}{4}\mathrm{cot}\frac{15\mathrm{\pi }}{4}=0$

Question 1:

Write the maximum and minimum values of cos (cos x).

Question 2:

Write the maximum and minimum values of sin (sin x).

Question 3:

Write the maximum value of sin (cos x).

Question 4:

If sin x = cos2x, then write the value of cos2x (1 + cos2x).

Question 5:

If sin x + cosec x = 2, then write the value of sinn x + cosecn x.

Question 6:

If sin x + sin2x = 1, then write the value of cos12x + 3 cos10x + 3 cos8x + cos6x.

Question 7:

If sin x + sin2x = 1, then write the value of cos8x + 2 cos6x + cos4x.

Question 8:

If sin θ1 + sin θ2 + sin θ3 = 3, then write the value of cos θ1+ cos θ2 + cos θ3.

Sine function can take the maximum value of 1.
If, $\mathrm{sin}{\theta }_{1}+\mathrm{sin}{\theta }_{2}+\mathrm{sin}{\theta }_{3}=3$, then we have:

sin${\theta }_{1}$ = 1

⇒ ${\theta }_{1}$=$\frac{\mathrm{\pi }}{2}$
Similarly, ${\theta }_{2}={\theta }_{3}=\frac{\mathrm{\pi }}{2}$

$⇒\mathrm{cos}{\theta }_{1}=\mathrm{cos}{\theta }_{2}=\mathrm{cos}{\theta }_{3}=0\phantom{\rule{0ex}{0ex}}⇒\mathrm{cos}{\theta }_{1}+\mathrm{cos}{\theta }_{2}+\mathrm{cos}{\theta }_{3}=0$

Question 9:

Write the value of sin 10° + sin 20° + sin 30° + ... + sin 360°.

Question 10:

A circular wire of radius 15 cm is cut and bent so as to lie along the circumference of a loop of radius 120 cm. Write the measure of the angle subtended by it at the centre of the loop.

Circumference of the circle of radius 15 cm:

Now, 94.2 cm will be the length of arc$\left(\mathrm{l}\right)$ for the circle with radius 120 cm.
We know:

45$°$ = $\frac{\mathrm{\pi }}{4}=\frac{22}{7×4}=0.785$ radians
Therefore, the angle subtended by it at the centre of the loop is 45$°$.

Question 11:

Write the value of 2 (sin6x + cos6 x) −3 (sin4 x + cos4x) + 1.

$2\left({\mathrm{sin}}^{6}x+{\mathrm{cos}}^{6}x\right)-3\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x\right)+1\phantom{\rule{0ex}{0ex}}=2\left({\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x\right)\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x-{\mathrm{sin}}^{2}x.c{\mathrm{os}}^{2}x\right)-3\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x\right)+1\phantom{\rule{0ex}{0ex}}=2.1\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x-{\mathrm{sin}}^{2}x.c{\mathrm{os}}^{2}x\right)-3\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x\right)+1\phantom{\rule{0ex}{0ex}}=2\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x\right)-2{\mathrm{sin}}^{2}x.c{\mathrm{os}}^{2}x-3\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x\right)+1\phantom{\rule{0ex}{0ex}}=-\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x\right)-2{\mathrm{sin}}^{2}x.c{\mathrm{os}}^{2}x+1\phantom{\rule{0ex}{0ex}}=-\left\{{\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x+2{\mathrm{sin}}^{2}x.c{\mathrm{os}}^{2}x\right\}+1\phantom{\rule{0ex}{0ex}}=-{\left({\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x\right)}^{2}+1\phantom{\rule{0ex}{0ex}}=-1+1\phantom{\rule{0ex}{0ex}}=0$

Question 12:

Write the value of cos 1° + cos 2° + cos 3° + ... + cos 180°.

Question 13:

If cot (α + β) = 0, then write the value of sin (α + 2β).

Question 14:

If tan A + cot A = 4, then write the value of tan4A + cot4A.

Question 15:

Write the least value of cos2x + sec2x.

We know:
cos x can take the minimum value of $-1$.

cos2x + sec2x

$=\frac{{\mathrm{cos}}^{4}x+1}{{\mathrm{cos}}^{2}x}\phantom{\rule{0ex}{0ex}}=\frac{{\left(-1\right)}^{4}+1}{{\left(-1\right)}^{2}}\phantom{\rule{0ex}{0ex}}=2$

Question 16:

If x = sin14x + cos20  x, then write the smallest interval in which the value of x lie.

If x = 0$°$, 90$°$, 180$°$, 270$°$, 360$°$, then

The smallest interval in which the value of x lie is $\left(0,1\right]$.

Question 17:

If 3 sin x + 5 cos x = 5, then write the value of 5 sin x − 3 cos x.

Question 1:

If tan x = $x-\frac{1}{4x}$, then sec x − tan x is equal to
(a) $-2x,\frac{1}{2x}$
(b) $-\frac{1}{2x},2x$
(c) 2x
(d) $2x,\frac{1}{2x}$

(a) $-2x,\frac{1}{2x}$

Question 2:

If sec $x=\mathrm{x}+\frac{1}{4\mathrm{x}}$, then sec x + tan x =
(a) $x,\frac{1}{x}$
(b) $2x,\frac{1}{2x}$
(c) $-2x,\frac{1}{2x}$
(d) $-\frac{1}{x},x$

(b) $2x,\frac{1}{2x}$

Question 3:

If  is equal to
(a) sec x − tan x
(b) sec x + tan x
(c) tan x − sec x
(d) none of these

(c) tan x − sec x

Question 4:

If π < x <2π, then  is equal to
(a) cosec x + cot x
(b) cosec x − cot x
(c) −cosec x + cot x
(d) −cosec x − cot x

(d) −cosec x − cot x

Question 5:

If $0, and if , then y is equal to
(a) $\mathrm{cot}\frac{x}{2}$

(b) $\mathrm{tan}\frac{x}{2}$

(c) $\mathrm{cot}\frac{x}{2}+\mathrm{tan}\frac{x}{2}$

(d) $\mathrm{cot}\frac{x}{2}-\mathrm{tan}\frac{x}{2}$

(b) $\mathrm{tan}\frac{x}{2}$

If  is equal to
(a) 2 sec x
(b) −2 sec x
(c) sec x
(d) −sec x

(b)  −2 sec x

Question 7:

If x = r sin θ cos Ï•, y = r sin θ sin Ï• and z = r cos θ, then x2 + y2 + z2 is independent of
(a) θ, Ï•
(b) r, θ
(c) r, Ï•
(d) r.

(a) θ, Ï•

We have:
x = r sin θ cos Ï•  ,  y = r sin θ sin Ï• and z = r cos θ,
x2 + y2 + z2

Question 8:

If tan x + sec x = $\sqrt{3}$, 0 < x < π, then x is equal to

(a) $\frac{5\mathrm{\pi }}{6}$

(b) $\frac{2\mathrm{\pi }}{3}$

(c) $\frac{\mathrm{\pi }}{6}$

(d) $\frac{\mathrm{\pi }}{3}$

(c) $\frac{\mathrm{\pi }}{6}$

Question 9:

If tan $x=-\frac{1}{\sqrt{5}}$ and θ lies in the IV quadrant, then the value of cos x is
(a) $\frac{\sqrt{5}}{\sqrt{6}}$

(b) $\frac{2}{\sqrt{6}}$

(c) $\frac{1}{2}$

(d) $\frac{1}{\sqrt{6}}$

(a) $\frac{\sqrt{5}}{\sqrt{6}}$

If  is equal to
(a)  1 − cot α
(b) 1 + cot α
(c) −1 + cot α
(d) −1 −cot α

(d) −1 −cot α

Question 11:

sin6A + cos6A + 3 sin2A cos2A =
(a) 0
(b) 1
(c) 2
(d) 3

(b) 1

Question 12:

If , then cos x is equal to
(a) $\frac{5}{3}$

(b) $\frac{3}{5}$

(c) $-\frac{3}{5}$

(d) $-\frac{5}{3}$

(b) $\frac{3}{5}$

Question 13:

If , then tan x =
(a) $\frac{21}{22}$

(b) $\frac{15}{16}$

(c) $\frac{44}{117}$

(d) $\frac{117}{44}$

(c) $\frac{44}{117}$

Question 14:

${\mathrm{sec}}^{2}x=\frac{4xy}{\left(x+y{\right)}^{2}}$ is true if and only if
(a) x + y ≠ 0
(b) x = y, x ≠ 0
(c) x = y
(d) x ≠0, y ≠ 0

(b) x = y, x ≠ 0

Question 15:

If x is an acute angle and , then the value of  is
(a) 3/4
(b) 1/2
(c) 2
(d) 5/4

(a) 3/4

Question 16:

The value of sin25° + sin210° + sin215° + ... + sin285° + sin290° is
(a) 7
(b) 8
(c) 9.5
(d) 10

(c) 9.5

Question 17:

sin2π/18 + sin2π/9 + sin2 7π/18 + sin2 4π/9 =
(a) 1
(b) 4
(c) 2
(d) 0

(c) 2

Question 18:

If tan A + cot A = 4, then tan4A + cot4A is equal to
(a) 110
(b) 191
(c) 80
(d) 194

(d) 194

Question 19:

If x sin 45° cos2 60° = , then x =
(a) 2
(b) 4
(c) 8
(d) 16

(c) 8

Question 20:

If A lies in second quadrant 3tanA + 4 = 0, then the value of 2cotA − 5cosA + sinA is equal to

(a) $-\frac{53}{10}$                           (b) $\frac{23}{10}$                           (c) $\frac{37}{10}$                           (d) $\frac{7}{10}$

It is given that $\frac{\mathrm{\pi }}{2}.

$3\mathrm{tan}A+4=0\phantom{\rule{0ex}{0ex}}⇒\mathrm{tan}A=-\frac{4}{3}\phantom{\rule{0ex}{0ex}}⇒\mathrm{cot}A=-\frac{3}{4}$

Now,

Also,

So,

$2\mathrm{cot}A-5\mathrm{cos}A+\mathrm{sin}A\phantom{\rule{0ex}{0ex}}=2×\left(-\frac{3}{4}\right)-5×\left(-\frac{3}{5}\right)+\frac{4}{5}\phantom{\rule{0ex}{0ex}}=-\frac{3}{2}+3+\frac{4}{5}\phantom{\rule{0ex}{0ex}}=\frac{-15+30+8}{10}\phantom{\rule{0ex}{0ex}}=\frac{23}{10}$

Hence, the correct answer is option B.

Question 21:

If , then tan x =
(a) $\frac{21}{22}$

(b) $\frac{15}{16}$

(c) $\frac{44}{117}$

(d) $\frac{117}{43}$

(c) $\frac{44}{117}$

Question 22:

If tan θ + sec θ =ex, then cos θ equals

(a) $\frac{{e}^{x}+{e}^{-x}}{2}$
(b) $\frac{2}{{e}^{x}+{e}^{-x}}$
(c) $\frac{{e}^{x}-{e}^{-x}}{2}$
(d) $\frac{{e}^{x}-{e}^{-x}}{{e}^{x}+{e}^{-x}}$

(b) $\frac{2}{{e}^{x}+{e}^{-x}}$

Question 23:

If sec x + tan x = k, cos x =

(a) $\frac{{k}^{2}+1}{2k}$

(b) $\frac{2k}{{k}^{2}+1}$

(c) $\frac{k}{{k}^{2}+1}$

(d) $\frac{k}{{k}^{2}-1}$

(b) $\frac{2k}{{k}^{2}+1}$

Question 24:

If $f\left(x\right)={\mathrm{cos}}^{2}x+{\mathrm{sec}}^{2}x$, then

(a) f(x) < 1                             (b) f(x) = 1                             (c) 1 < f(x) < 2                              (d) f(x) ≥ 2

Hence, the correct option is answer D.

Question 25:

Which of the following is incorrect?

(a)                 (b) cos x = 1                (c)                 (d) tan x = 20

(a) is correct as

(b) cos x = 1 is correct as

(c) is not correct as

(d) tan x = 20 is correct as tan x can take any real value.

Hence, the correct answer is option C.

Question 26:

The value of is

(a) $\frac{1}{\sqrt{2}}$                         (b) 0                          (c) 1                          (d) $-1$

Hence, the correct answer is option B.

Question 27:

The value of is

(a) 0                                  (b) 1                                  (c) $\frac{1}{2}$                                  (d) not defined

We know that, $\mathrm{tan}\left(90°-\theta \right)=\mathrm{cot}\theta$

So,

Hence, the correct answer is option B.

Question 28:

Which of the following is correct?

(a) $\mathrm{sin}1°>\mathrm{sin}1$                    (b) $\mathrm{sin}1°<\mathrm{sin}1$                    (c) $\mathrm{sin}1°=\mathrm{sin}1$                    (d) $\mathrm{sin}1°=\frac{\mathrm{\pi }}{180}\mathrm{sin}1$

We know that, 1 radian is approximately 57º.

Also, the value of sinx is always increasing for $0\le x\le 90°$ ( or sinx is an increasing function for $0\le x\le 90°$).

Now,

Hence, the correct answer is option B.

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