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ఇంటర్మీడియట్ – రెండవ సంవత్సరం
ద్వితీయ భాష – తెలుగు : భాగం-II
మాదిరి ప్రశ్నపత్రం – వివరణ

ఇంటరు రెండవ సంవత్సరము తెలుగు పేపరులో మొత్తం 16 ప్రశ్నలు ఉంటాయి. మూడు గంటల సమయంలో జవాబులు రాయాలి.
మొత్తం మార్కులు = 100

సూచనలు :
1. ప్రశ్నపత్రం ప్రకారం వరుసక్రమంలో సమాధానాలు రాయాలి.
2. ఒక్క మార్కు ప్రశ్నల జవాబులను కేటాయించిన ప్రశ్న కింద వరుసక్రమంలో రాయాలి.

I. పద్య భాగం నుండి రెండు పద్యాలు ఇస్తారు. అందులో ఒకదానికి ప్రతిపదార్థ తాత్పర్యాలను రాయాలి. (1 × 8 = 8 మార్కులు)

II. పద్య భాగం నుండి రెండు వ్యాసరూప సమాధాన ప్రశ్నలు ఇస్తారు. అందులో ఒక్క ప్రశ్నకు 20 పంక్తులలో సమాధానం రాయాలి. (1 × 6 = 6 మార్కులు)

III. గద్య భాగం నుండి రెండు వ్యాసరూప సమాధాన ప్రశ్నలు ఇస్తారు. అందులో ఒక్క ప్రశ్నకు 20 పంక్తులలో సమాధానం రాయాలి. (1 × 6 = 6 మార్కులు)

IV. ‘యాత్రారచన’ ఉపవాచకం నుండి నాలుగు ప్రశ్నలు ఇస్తారు. అందులో రెండు ప్రశ్నలకు 15 పంక్తులలో సమాధానాలు రాయాలి. (2 × 4 = 8 మార్కులు)

V. పద్య భాగం నుండి నాలుగు సందర్భ సహిత వ్యాఖ్యలు ఇస్తారు. అందులో రెండింటికి జవాబులు రాయాలి. (2 × 3 = 6 మార్కులు)

VI. ‘యాత్రారచన’ నుండి నాలుగు సందర్భసహిత వ్యాఖ్యలు ఇస్తారు. అందులో రెండింటికి సమాధానాలు రాయాలి. (2 × 3 = 6 మార్కులు)

VII. పద్య భాగంపై నాలుగు సంగ్రహ సమాధాన ప్రశ్నలు ఇస్తారు. అందులో రెండు ప్రశ్నలకు సంగ్రహంగా సమాధానాలు రాయాలి. (2 × 2 = 4 మార్కులు)

VIII. గద్య భాగంపై నాలుగు సంగ్రహ సమాధాన ప్రశ్నలు ఇస్తారు. అందులో రెండు ప్రశ్నలకు సంగ్రహంగా సమాధానాలు రాయాలి. (2 × 2 = 4 మార్కులు)

IX. పద్య భాగం నుండి ఒక వాక్యంలో సమాధానాలు రాయవలసిన ఎనిమిది ప్రశ్నలు ఇస్తారు. అందులో ఆరింటికి జవాబులు రాయాలి. (6 × 1 = 6 మార్కులు)

X. గద్య భాగం నుండి ఒక వాక్యంలో సమాధానాలు రాయవలసిన ఎనిమిది ప్రశ్నలు ఇస్తారు. అందులో ఆరింటికి జవాబులు రాయాలి. (6 × 1 = 6 మార్కులు)

XI. ఛందస్సు : మూడు పద్యములు ఇస్తారు. దానిలో ఒక పద్యానికి లక్షణాలు తెలిపి, ఉదాహరణతో సమన్వయించాలి. (1 × 6 = 6 మార్కులు)

XII. ఛందస్సుపై ఎనిమిది ఏకవాక్య సమాధాన ప్రశ్నలు ఇస్తారు. అందులో ఆరింటికి జవాబులు రాయాలి. (6 × 1 = 6 మార్కులు)

XIII. అలంకారములు : మూడు అలంకారాలు ఇస్తారు. అందులో ఒకదానికి లక్షణాలు తెలిపి ఉదాహరణతో సమన్వయించాలి. (1 × 6 = 6 మార్కులు)

XIV. అలంకారాలపై ఎనిమిది ఏకవాక్య సమాధాన ప్రశ్నలు ఇస్తారు. అందులో ఆరింటికి జవాబులు రాయాలి. (6 × 1 = 6 మార్కులు)

XV. సంక్షిప్తీకరణ : ఇచ్చిన విషయాన్ని 1/3 వంతుకు సంక్షిప్తం చేసి రాయాలి. (1 × 6 = 6 మార్కులు)

XVI. (అ) ఇచ్చిన పదాలు ఆధారంగా చేసుకుని సంభాషణ రాయాలి. (1 × 5 = 5 మార్కులు)
(ఆ) భాషాభాగాలపై ఐదు ఏకవాక్య సమాధాన ప్రశ్నలు ఇస్తారు. అన్నింటికీ జవాబులు రాయాలి. (1 × 8 = 8 మార్కులు)

మొత్తం = 100 మార్కులు

TS Inter 2nd Year Study Material

TS Inter 2nd Year Maths 2B Differential Equations Important Questions

Students must practice these TS Inter 2nd Year Maths 2B Important Questions Chapter 8 Differential Equations to help strengthen their preparations for exams.

TS Inter 2nd Year Maths 2B Differential Equations Important Questions

Very Short Answer Type Questions

Question 1.
Find the order and degree of \(\frac{d y}{d x}=\frac{x^{1 / 2}}{y^{1 / 2}\left(1+x^{1 / 2}\right)}\)
Solution:
Order is 1 and Degree is ‘1’
Since there is first order derivative with highest degree is ‘1’.

Question 2.
Find the degree and order of the differential equation \(\frac{d^2 y}{d x^2}=\left[1+\left(\frac{d y}{d x}\right)^2\right]^{5 / 3}\)
Solution:
The equation can be written as \(\left(\frac{d^2 y}{d x^2}\right)^3=\left[1+\left(\frac{d y}{d x}\right)^2\right]^5\)
The order is 2 and degree is ‘3’

TS Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 3.
Find the order and degree of the equation
\(1+\left(\frac{d^2 y}{d x^2}\right)^2=\left[2+\left(\frac{d y}{d x}\right)^2\right]^{3 / 2}\)
Solution:
The equation can be expressible as
\(\left[1+\left(\frac{d^2 y}{d x^2}\right)^2\right]^2=\left[2+\left(\frac{d y}{d x}\right)^2\right]^3\)
Order is 2 and degree is 4.

Question 4.
Find the order and degree of \(\frac{d^2 y}{d x^2}+2 \frac{d y}{d x}+y=\log \left(\frac{d y}{d x}\right)\)
Solution:
Order is 2and degree is not defined since the equation cannot be expressed as a polynomial equation In the derivatives.

Question 5.
Find the order and degree of \(\left[\left(\frac{d y}{d x}\right)^{\frac{1}{2}}+\left(\frac{d^2 y}{d x^2}\right)^{\frac{1}{3}}\right]^{\frac{1}{4}}=0\)
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions1

Question 6.
Find the order and degree of = \(\frac{d^2 y}{d x^2}=-p^2 y\)
Solution:
Equation is a polynomial equation in \(\frac{d^2 y}{d x^2}\)
So degree is ‘1′ and order is ‘2’.

TS Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 7.
Find the order and degree of \(\left(\frac{d^3 y}{d x^3}\right)^2-3\left(\frac{d y}{d x}\right)^2-e^x=4\)
\(\left(\frac{d^3 y}{d x^3}\right)^2-3\left(\frac{d y}{d x}\right)^2-e^x=4\)
Solution:
The equation is a polynomial equation in and \(\frac{d y}{d x}\) \(\frac{\mathrm{d}^3 \mathrm{y}}{\mathrm{dx}^3}\)
∴ Order is 3 and degree is 2.

Question 8.
Find the order and degree of \(x^{\frac{1}{2}}\left(\frac{d^2 y}{d x^2}\right)^{\frac{1}{3}}+x \frac{d y}{d x}+y=0\)
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions2

Question 9.
Find the order and degree of \(\left[\frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^3\right]^{\frac{6}{5}}=6 y\)
Solution:
The given equation can be written as
\(\frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^3=(6 y)^{5 / 6}\)
order is ‘2’ and degree is ‘1’.

Question 10.
Find the order of the differential equation corresponding to y = Aex + Be3x + Ce5x (A, B, C are parameters) is a solution.
Solution:
Since there are 3 constants in
y = Aex + Be3x + Ce5x we can have a differential equation of third order by eliminating A,B,C.
∴ Order of the differential equation is ‘3’.

Question 11.
Form the differential equation to y = cx – 2c2 where c is a parameter.
Solution:
Given y = cx-2c2 ………….. (1)
we have y1=c ……………….. (2)
∴From(1)
y=xy1 – 2y21 ………………….. (3)
∴ This is a differential equation corresponding to (1).

TS Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 12.
Form the differential equation corresponding to y = A cos 3x+ B sin 3x where A and B are parameters.
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions3

Question 13.
Express the following differential equations in the form f(x) dx + g(y) dy = 0
(i) \( \frac{d y}{d x}=\frac{1+y^2}{1+x^2}\)
Solution:
\(\frac{d x}{1+x^2}-\frac{d y}{1+y^2}=0\)

(ii) \(y-x \frac{d y}{d x}=a\left(y^2+\frac{d y}{d x}\right)\)
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions4

(iii) \(\frac{d y}{d x}=e^{x-y}+x^2 e^{-y}\)
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions5

(iv) \(\frac{d y}{d x}+x^2=x^2 e^{3 y}\)
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions6

Question 14.
Find the general solution of x + y \(\frac{dy}{dx}\) = 0.
Solution:
The given equation can be written as
x dx + y dy = 0
∴ ∫ xdx+∫ ydy = c
⇒ x2 + y2 = 2c

Question 15.
Find the general solution of \(\frac{d y}{d x}=e^{x+y}\)
Solution:
The given equation can be written as \(\frac{d y}{d x}=e^x \cdot e^y\)
writing in variable separable form ex dx = e-y dy = 0
∴ ex + e-y = c is the required solution.

TS Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 16.
Find the degree of the following homogeneous functions.

(i) f(x, y) = 4x2y + 2xy2
Solution:
Given f(x, y) = 4x2y+2xy2
we have f(kx, ky) = 4k2x2ky + 2kxk2y2
⇒ 4k3x2y + 2k3xy2
⇒ k3(4x2y + y2)
⇒ k3 f(x, y) ∀ k
and f(x, y), x3 Φ \(\left(\frac{\mathrm{y}}{\mathrm{x}}\right)\) and hence f(x, y) is a homogeneous function of degree ‘3’.

(ii) g(x,y)=xy1/2+yx1/2
Solution:
Given g(x, y) =xy1/2+ yx1/2
g(kx, ky) = kx(ky)1/2 + (ky)(kx)1/2
⇒ k3/2 (xyk1/2 + yx1/2)
⇒ k3/2 g(x, y)
∴ g(x, y) is a homogeneous function of degree ‘3’.

(iii) \(h(x, y)=\frac{x^2+y^2}{x^3+y^3}\)
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions8

∴ h(x, y) is a homogeneous function of degree – 1.

(iv) Show that f(xy) = I +ex/y is a homogeneous function of x and y.
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions9

(v) f(x,y) = x \(\sqrt{\mathbf{x}^2+y^2}-y^2\) is a homogeneous function of x and y.
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions31
∴f(x, y) is a homogeneous function of degree ‘1’.

(vi) f(x,y) = x – y log y + y log x
Solution:
Givenf(x, y) =x-ylogy+ylogx
∴ f(kx, ky) – kx – ky log (ky) + ky log(kx)
= k[x-y log(ky) + ylog(kx)]
= k[x- y(logk+logy) +y(logk+logx)]
= k[x – y log y + y log x]
= k f(x, y)
∴ f(x, y) is a homogeneous function of degree ‘F.

Question 17.
Express (1+ex/y) dx + ex/y \(\left(1-\frac{x}{y}\right)\) dy = 0 in the form \(\frac{\mathbf{d x}}{\mathbf{d y}}=F\left(\frac{x}{y}\right)\)
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions10

Question 18.
Express \(\left(x \sqrt{x^2+y^2}-y^2\right)\) dx+xy dx = 0 in the form \(\frac{\mathbf{d y}}{\mathbf{d x}}=F\left(\frac{x}{y}\right)\)
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions12

Question 19.
Express \(\frac{d y}{d x}=\frac{y}{x+y e^{-\frac{2 x}{y}}}\) in the form \(\frac{d x}{d y}=F\left(\frac{x}{y}\right)\)
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions13

Question 20.
Transform x logx \(\frac{d y}{d x}\) y into linear form.
Solution:
Dividing both sides by x log x we get
TS Inter 2nd Year Maths 2B Differential Equations Important Questions14

TS Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 21.
Transform \(\left(x+2 y^3\right) \frac{d y}{d x}=y\) into linear form
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions15

Question 22.
Find I.F. of the following differential equations by converting them into linear form.

(i) cosx\(\frac{d y}{d x}\)+y sinx=tanx
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions16

(ii) (2y -10y3) \(\frac{d y}{d x}\) + y = 0
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions17

Short Answer Type Questions

Question 1.
Find the order of the differential equation corresponding to y = c( x- c)2 where c is an arbitrary constant
Solution:
Given y = c(x – e)2; eliminate ‘c’ and form the differential equation.
TS Inter 2nd Year Maths 2B Differential Equations Important Questions18
TS Inter 2nd Year Maths 2B Differential Equations Important Questions19

Question 2.
Form the differential equation corresponding to the family of circles of radius ‘r’ given by (x-a)2+(y-b)2=r2 where a and b are parameters.
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions20

TS Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 3.
Form the differential equation corresponding to the family of circles passing through the origin and having centres on Y- axis.
Solution:
The equation of family of circles passing through the origin and having centres on Y-axis is
x2+y2-2fy=0 ……………….. (1)
Differentiating w.r.t x, we get
TS Inter 2nd Year Maths 2B Differential Equations Important Questions21

Question 4.
Solve \(y^2-x \frac{d y}{d x}=a\left(y+\frac{d y}{d x}\right)\)
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions22
TS Inter 2nd Year Maths 2B Differential Equations Important Questions23

Question 5.
Solve \(\frac{d y}{d x}=\frac{y^2+2 y}{x-1}\)
Solution:
The equation can be written as
TS Inter 2nd Year Maths 2B Differential Equations Important Questions24

Question 6.
Solve \(\frac{d y}{d x}=\frac{x(2 \log x+1)}{\sin y+y \cos y}\)
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions25

TS Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 7.
Find the equation of the curve whose slope at any point (x, y) is \(\frac{y}{x^2}\) and which satisfy the condition y = 1 when x =3.
Solution:
We have the slope at any point x, y) on the
TS Inter 2nd Year Maths 2B Differential Equations Important Questions26

Question 8.
Solve y (1+x)dx+x(1+y) dy = 0
Solution:
The given equation can be expressed as
TS Inter 2nd Year Maths 2B Differential Equations Important Questions28
logx+x+logy+y=c
x + y + log (xy) = c which is the required solution.

Question 9.
Solve \(\frac{d y}{d x}\) = sin(x + y) +cos(x + y)
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions29

Question 10.
Solve that (x – y)2  \(\frac{d y}{d x}=a^2\)
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 90

Question 11.
Solve \(\frac{d y}{d x}=\frac{x-2 y+1}{2 x-4 y}\)
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions32

Question 12.
Solve \(\frac{d y}{d x}=\sqrt{y-x}\)
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions33

TS Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 13.
Solve \(\frac{d y}{d x}\) +1 = ex+y
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions34

Question 14.
Solve \(\frac{d y}{d x}\) = (3x + y + 4)2
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions35

Question 15.
Solve \(\frac{d y}{d x}\) – x tan(y-x)= 1
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions37

Question 16.
Solve \(\frac{d y}{d x}=\frac{y^2-2 x y}{x^2-x y}\)
Solution:
The given equation ¡s a homogeneous equation of degree ‘2’.
TS Inter 2nd Year Maths 2B Differential Equations Important Questions38
TS Inter 2nd Year Maths 2B Differential Equations Important Questions39
which is athe general solution of the given equation.

TS Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 17.
Solve(x2+y2)dx=Zxydy
Solution:
The given equation can be written as
TS Inter 2nd Year Maths 2B Differential Equations Important Questions41

Question 18.
Solve xy2dy – (x3+y)dx=0
Solution:
The given equation can be written as \(\frac{d y}{d x}=\frac{x^3+y^3}{x y^2}\) which is a homogeneous equation.
TS Inter 2nd Year Maths 2B Differential Equations Important Questions42
which is the general solution of the given equation.

Question 19.
Solve \(\frac{d y}{d x}=\frac{x^2+y^2}{2 x^2}\)
Solution:
The given equation \(\frac{d y}{d x}=\frac{x^2+y^2}{2 x^2}\) homogeneous equation.
TS Inter 2nd Year Maths 2B Differential Equations Important Questions44
which is the general solution of the given equation.

Question 20.
Give the solution of x sin2 \(\left(\frac{y}{x}\right)\) dx = y dx – x dy which passes through the point \(\left(1, \frac{\pi}{4}\right)\)
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions45
is the required particular solution of the given equation.

Question 21.
Solve(x3-3xy2)dx+(3x2y-y3)dy=0
Solution:
The given equation can be written as
TS Inter 2nd Year Maths 2B Differential Equations Important Questions46
TS Inter 2nd Year Maths 2B Differential Equations Important Questions47
TS Inter 2nd Year Maths 2B Differential Equations Important Questions48

TS Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 22.
Solve the equation \(\frac{d y}{d x}=\frac{3 x-y+7}{x-7 y-3}\)
Solution:
Here a=3, b =-1,c = 7
a’=1, b’=-7,c’ = -3
and b =- a’. Hence that solution can be obtained by grouping.
∴ From the given equation
3xdx – ydx+7dx = xdy-7ydy – 3dy
= (xdy+ydx) – 7ydy – 7dx – 3xdx – 3dy = 0
= ∫d(xy) -∫7ydy – 7∫dx – 3∫xdx – 3∫dy = 0
= xy – 7\(\frac{y^2}{2}\) – 7x -3 \(\frac{x^2}{2}\) -3y =c
⇒ 2xy – 7y2-14x-3x2– 6y=2c
⇒ 2xy – 7y2 – 14x-3x2 – 6y= c’ where C – 2c
Is the required solution.

Question 23.
Solve (1+x2) \(\frac{\mathrm{dy}}{\mathbf{d x}}\) +2xy = 4x2
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions49

Question 24.
Solve sin 2 x \(\frac{d y}{d x}\) +y = cot x
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions50

Question 25.
Find the solution of the equation x(x – 2) \(\frac{d y}{d x}\) (x – 1)y=x3(x-2) which sotisfies the condition that y=9 where x=3.
Solution:
The equation can be written as
TS Inter 2nd Year Maths 2B Differential Equations Important Questions51
TS Inter 2nd Year Maths 2B Differential Equations Important Questions52

Question 26.
Solve (1+y2)dx = (tan-1 y-x)dy
Solution:
The given equation can be written as
TS Inter 2nd Year Maths 2B Differential Equations Important Questions53
Long Answer Type Questions

Question 1.
Solve \(\sqrt{1+x^2} \sqrt{1+y^2}\)dx + xy dy =0.
Solution:
The given equation can he written as
TS Inter 2nd Year Maths 2B Differential Equations Important Questions54
TS Inter 2nd Year Maths 2B Differential Equations Important Questions55
Is the solution of the given differential equation.

TS Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 2.
Solve x sec \(\left(\frac{\mathbf{y}}{\mathbf{x}}\right)\) (y dx+xdy)=y cosec \(\left(\frac{\mathbf{y}}{\mathbf{x}}\right)\)
Solution:
The given equation can be written as
TS Inter 2nd Year Maths 2B Differential Equations Important Questions56
TS Inter 2nd Year Maths 2B Differential Equations Important Questions57
TS Inter 2nd Year Maths 2B Differential Equations Important Questions58
TS Inter 2nd Year Maths 2B Differential Equations Important Questions59
which is the general solution of the given equation.

TS Inter 2nd Year Maths 2B Differential Equations Important Questions

Question 3.
Solve (2x+y+3)dx=(2y+x+1)dy
Solution:
TS Inter 2nd Year Maths 2B Differential Equations Important Questions60
TS Inter 2nd Year Maths 2B Differential Equations Important Questions61
TS Inter 2nd Year Maths 2B Differential Equations Important Questions62

 

TS Inter 2nd Year Maths 2B Definite Integrals Important Questions

Students must practice these TS Inter 2nd Year Maths 2B Important Questions Chapter 7 Definite Integrals to help strengthen their preparations for exams.

TS Inter 2nd Year Maths 2B Definite Integrals Important Questions

Very Short Answer Type Questions

Question 1.
Evaluate \(\int_1^2 x^5\) dx
Solution:
\(\int_1^2 x^5 d x=\left[\frac{x^6}{6}\right]_1^2=\frac{2^6}{6}-\frac{1}{6}=\frac{64}{6}-\frac{1}{6}=\frac{63}{6}=\frac{21}{2}\)

Question 2.
Evaluate \(\int_0^\pi \) sinx dx
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 1

TS Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 3.
Evaluate \(\int_0^a \frac{d x}{x^2+a^2}\)
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 2

Question 4.
Evaluate \(\int_1^4 x \sqrt{x^2-1}\) dx
Solution:
Let x2 – 1 – t ⇒ 2x dx dt then
Upper limit when x = 4 is t = 15.
Lower Limit when x = 1 is t = 0.
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 3

Question 5.
Evaluate \(\int_0^2 \sqrt{4-x^2}\) dx
Solution:
Let x= 2 sin θ = dx – 2cosθ dθ then
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 4

Question 6.
Show that \(\int_0^{\frac{\pi}{2}} \sin ^n x d x=\int_0^{\frac{\pi}{2}} \cos ^n x dx\)
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 6

TS Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 7.
Evaluate \(\int_0^{\frac{\pi}{2}} \frac{\cos ^{\frac{5}{2}} x}{\sin ^{\frac{5}{2}} x+\cos ^{\frac{5}{2}} x}\) dx
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 5

Question 8.
Evaluate \(\int_0^{\frac{\pi}{2}} \) x sin x dx
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 7

Question 9.
Evaluate
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 8
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 9

TS Inter 2nd Year Maths 2B Definite Integrals Important Questions

(iii) \(\int_0^{\frac{\pi}{2}} \sin ^6 x \cos ^4 x dx\)
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 10

Question 10.
Find \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^2 x \cos ^4 x d x\)
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 11

Short Answer Type Questions

Question 1.
Find \(\int_0^2\left(x^2+1\right) dx\) as the limit of a sum
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 12

TS Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 2.
Evaluate \(\int_0^2 e^x dx\) as the limit of a sum.
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 13
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 14

Question 3.
Lets define f : [0,1]→ R by
f(x) = 1 if x is rational
= 0 if x is irrational
then show that f is nor R Integrable over [0, 1].
Solution:
Let P = (x0, x1,…., xn] be a partition of [0, 1].
Since between any two real numbers there exists rational and irrational numbers and
let ti, si ∈ [Xi -i xj] be the rational and irrational numbers.
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 15

Question 4.
Evalute \(\int_0^{16} \frac{x^{\frac{1}{4}}}{1+x^{\frac{1}{2}}}\) dx
Solution:
Let x = t4 then dx – 4t3 dt
Upper limit when x = 16 is t = 2.
and Lower limit when x = 0 is t = 0.
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 16

TS Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 5.
Evaluate \(\int_{-\frac{\pi}{2}}^\pi \sin\) |x| dx
Solution:
We have sin |x| = sin(-x) if x < 0
= sinx if x ≥ 0
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 17

Question 6.
Evaluate by using the method of finding definite integral as the limit of a sum.
\(\lim _{n \rightarrow \infty} \sum_{i=1}^n \frac{1}{n}\left(\frac{n-1}{n+1}\right)\)
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 18

Question 7.
Evaluate \(\lim _{n \rightarrow \infty} \frac{2^k+4^k+6^k+\ldots+(2 n)^k}{n^{k+1}}\) using the method of finding definite integral as the limit of a sum.
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 19

TS Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 8.
Evaluate \(\lim _{n \rightarrow \infty}\left[\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right) \ldots\left(1+\frac{n}{n}\right)\right]^{\frac{1}{n}}\)
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 20
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 21

Question 9.
Obtain Reduction formula for \(\int_0^{\frac{\pi}{2}} \sin ^n x d x\) and hence find
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 22
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 23
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 24
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 25
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 26

TS Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 10.
Evaluate \(\int_0^a \sqrt{a^2-x^2} dx\)
Solution:
Let x = a sinθ then dx = a cosθ dθ
Upper limit when x = a is θ = \(\frac{\pi}{2}\)
and Lower limit when x = 0 is θ = 0
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 27

Question 11.
Find \(\int_{-a}^a x^2\left(a^2-x^2\right)^{3 / 2} dx\)
Solution:
Since f(x) = x2 (a2 – x2)3/2 is an even function and f(- x) = f(x) we have
\(\int_{-a}^a x^2\left(a^2-x^2\right)^{3 / 2} d x=2 \int_0^a x^2\left(a^2-x^2\right)^{3 / 2} d x\)
Let x = a sin θ then dx = a cos θ dθ
∴ Upper limit when x = a is θ = \(\frac{\pi}{2}\)
Lower limit when x = 0 is θ = 0
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 28

Question 12.
Find \(\int_0^1 x^{3 / 2} \sqrt{1-x} dx\)
Solution:
Let x = sin2θ then dx = 2 sinθ cosθ dθ
Upper limit when x = 1 is θ = \(\frac{\pi}{2}\)
Lower Limit when x = θ is θ = 0.
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 29
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 30

TS Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 13.
Find the area under the curve f(x) = sin x in (0, 2π).
Solution:
Consider the graph of the function f(x) = sinx in [0, 2π];
we have sin x ≥ 0 ∀ x ∈ [0,π] and sin x≤0∀x∈[π,2π].
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 31

Question 14.
Find the area under the curve f(x) = cos x in [0, 2π].
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 32
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 33

Question 15.
Find the bounded by the y = x2 parabola the X- axis and the lines x = – 1, x = 2.
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 34

Question 16.
Find the area cut off between the line y = 0 and the parabola y = x2– 4x + 3.
Solution:
The point of intersection of y – 0 and y = x2 – 4x + 3 is given by x2 – 4x + 3 = 0
= (x – 3)(x-1) = 0 = x = 1 or 3
y=x2– 4x + 3 ⇒ y+1 =  x2– 4x + 4 (x-2)2
Hence the equation represents a parabola
with vertex (2, -1) lies in IV quadrant.
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 34

TS Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 17.
Find the area bounded by the curves y = sin x and y = cos x between any two consecutive points of intersection.
Solution:
The given curves y = sin x and y = cosx and
tan x = 1 ⇒ x = \(\frac{\pi}{4}\)
∴ x = \(\frac{\pi}{4}\) and x = \(\frac{5 \pi}{4}\) are the two consecutive points of intersection.
Taking f(x) = sin x and g(x) cos x over \(\left[\frac{\pi}{4}, \frac{5 \pi}{4}\right]\) we have
f(x)> g(x) ∀ x ∈\(\left[\frac{\pi}{4}, \frac{5 \pi}{4}\right]\).
Hence the area bounded by y = sin x, y = cos x and the two points of intersection is
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 35

Question 18.
Find the area of one of the curvilinear rectangles bounded by y = sin x, y cos x and X-axis.
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 36
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 37

Question 19.
Find the area of the right angled triangle with base b and altitude ‘h’ using the fundamental theorem of integral calculus.
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 38

TS Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 20.
Find the area bounded between the curves y2 – 1 = 2x and x = 0.
Solution:
The given curves are
y2-1-2x-2(x-0) ……………. (1)
= (y-0)2 2(x)+1=2 \(\left[\mathrm{x}+\frac{1}{2}\right]\)
(1) represents parabola with vertex \(\left(-\frac{1}{2}, 0\right)\)
Solving (1) and x = 0 we get
y2 -1 = 0 ⇒ y = ±1
∴ The points of intersection are (0, 1), (0, -1).
The parabola meets the X- axis and y = 1 and y = – 1 and the curve is symmetric with respect to X – axis
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 39

Question 21.
Find the area enclosed by the curves y = 3x and y = 6x-x2.
Solution:
Given curves are y3x and y=6x – x2
Solving 6x – x2 = 3x = 3x – x2 = 0
= x(3- x)=0 =x=0 or x=3
Taking f(x) = 3x and g(x) = 6x – x2
then g(x) ≥ 1(x) in [0, 3] and area enclosed between the line y = 3x and the parabola y = 6x-x2 is
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 40

Long Answer Type Questions

Question 1.
Show that \(\int_0^{\frac{\pi}{2}} \frac{x}{\sin x+\cos x}\) dx =\(\frac{\pi}{2 \sqrt{2}} \log (\sqrt{2}+1)\)
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 41
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 42
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 43
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 44

TS Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 2.
Evaluate \(\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\) dx
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 45
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 46

Question 3.
Evaluate \(\int_{-a}^a\left(x^2+\sqrt{a^2-x^2}\right) dx\)
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 47
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 48

TS Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 4.
Evaluate \(\int_0^\pi \frac{x \sin x}{1+\sin x}\) dx
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 49
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 50
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 51

Question 5.
Find \(\int_0^\pi \mathbf{x}\) sin7 x cos 6 x dx.
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 52
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 53

TS Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 6.
Find the area enclosed between y=x2-5x and y=4-2x.
Solution:
The graphs of curves are shown below.
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 54
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 55

TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 56

Question 7.
Find the area bounded between the curves y = x2, y = \(\sqrt{\mathbf{x}} \)
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 57
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 58

TS Inter 2nd Year Maths 2B Definite Integrals Important Questions

Question 8.
Find the area bounded between the curves y2=4ax, x2= 4by(a>0,b>0).
Solution:
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 59
TS Inter 2nd Year Maths 2B Definite Integrals Important Questions 60

TS Inter 2nd Year Maths 2B Integration Important Questions

Students must practice these TS Inter 2nd Year Maths 2B Important Questions Chapter 6 Integration to help strengthen their preparations for exams.

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 1.
Find ∫2x7 dx on R.
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 1

Question 2.
Evaluate ∫cot2xdx on l⊂R {nπ:n∈Z)
Solution:
∫cot2xdx =∫(cosec2x – 1)dx
= ∫ cosec2 x – ∫dx = – dx = – cotx – x + c

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 3.
Evaluate \(\int\left(\frac{x^6-1}{1+x^2}\right)\) dx for x ∈ R
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 2

Question 4.
Find ∫(1 – x)(4 -3x) (3 + 2x) dx ; x ∈R.
Solution:
(1- x)(4 – 3x)(3+2x) = 6x3 – 5x2 – 13x + 12
∴ ∫ (1 – x)(4 – 3x)(3+2x)dx
=∫(6x3– 5x2_13x+ 12)dx
TS Inter 2nd Year Maths 2B Integration Important Questions 3

Question 5.
Evaluate \(\int\left(x+\frac{1}{x}\right)^3 d x, x>0\)
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 4

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 6.
Find \(\int \sqrt{1+\sin 2 x}\) dx on R.
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 5

Question 7.
Find \(\int \frac{6 x}{3 x^2-2}\) dx on any interval I ⊂ R \(\left\{-\sqrt{\frac{2}{3}}, \sqrt{\frac{2}{3}}\right\}\)
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 6

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 8.
\(\int \frac{\left(\sin ^{-1} x\right)^2}{\sqrt{1-x^2}}\) dx on R
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 7
Question 9.
Evaluate \(\int \frac{1}{1+(2 x+1)^2}\) dx on R.
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 8

Question 10.
Evaluate \(\int \frac{x^5}{1+x^{12}}\) dx on R
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 9

Question 11.
∫ cos3 sinx dx on R.
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 11

Question 12.
Find \(\int\left(1-\frac{1}{x^2}\right) e^{\left(x+\frac{1}{x}\right)}\) dx on I where I = (0,∞)
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 12

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 13.
Evaluate \(\int \frac{1}{\sqrt{\sin ^{-1} x} \sqrt{1-x^2}}\)
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 13

Question 14.
Evaluate \(\int \frac{\sin ^4 x}{\cos ^6 x} d x \), x ∈ I ⊂ R – {(2n+1) \(\frac{\pi}{2}\) …………… n∈Z}
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 18

Question 15.
Evaluate ∫ sin2 x dx on R.
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 19

Question 16.
Find \(\int \frac{x^2}{\sqrt{x+5}}\) on (-5, ∞)
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 20

Question 17.
Find  \(\int \frac{x}{\sqrt{1-x}}\) dx, x∈1=(0,1)
Solution:
Let 1 – x = t2 over (0, 1)
then – dx = 2t dt and x = 1 – t2
TS Inter 2nd Year Maths 2B Integration Important Questions 27

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 18.
Evaluate \(\int \frac{d x}{(x+5) \sqrt{x+4}}\) on (-4,∞)
Solution:
Let x + 4 – t2 then dx – 2t dt
defined over (- 4, ∞)
TS Inter 2nd Year Maths 2B Integration Important Questions 28

Question 19.
Evaluate \(\int \frac{d x}{\sqrt{4-9 x^2}} \text { on } I=\left(-\frac{2}{3}, \frac{2}{3}\right)\)
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 30

Question 20.
\(\int \frac{1}{a^2-x^2}\) dx for x E I = (- a, a).
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 32

Question 21.
Evaluate \(\int \frac{1}{1+4 x^2}\) dx on R.
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 33

Question 22.
Evaluate \(\int \sqrt{4 x^2+9}\) dx on R.
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 34

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 23.
Evaluate \(\int \frac{1}{\sqrt{4-x^2}}\) dx on (-2,2)
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 35

Question 24.
Evaluate \(\int \sqrt{9 x^2-25} d x \text { on }\left(\frac{5}{3}, \infty\right)\)
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 36

Question 25.
Evaluate \(\int \sqrt{16-25 x^2} d x \text { on }\left(-\frac{4}{5}, \frac{4}{5}\right)\)
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 37

Question 26.
Find \(e^x \frac{(1+x)}{(2+x)^2}\) dx on l ⊂ R – {-2}
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 39

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 27.
Evaluate \(\int \frac{d x}{\sqrt{x^2+2 x+10}}\)
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 40

Question 28.
Evaluate \(\int \frac{d x}{\sqrt{1+x-x^2}}\)
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 41

Short Answer Type Questions

Question 1.
Evaluale \(\int\left(\frac{2 x^3-3 x+5}{2 x^2}\right)\) dx for x>0 and verify the result by differentiation.
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 42
This is the given expression and the result is correct.

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 2.
Evaluate \(\int \frac{1}{a \sin x+b \cos x}\) dx where a, b ∈ R and a2 + b 2 ≠ 0 on R
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 43

Question 3.
Evaluate ∫ x sin-1 x dx on (-1, 1).
Solution:
We use integration by parts by suitably
choosing y x and u = sin-1 x so that
TS Inter 2nd Year Maths 2B Integration Important Questions 44
TS Inter 2nd Year Maths 2B Integration Important Questions 45

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 4.
Evaluate ∫ x2cosx dx dx
solution:
Use integration by parts by choosing u x2 and y = cos x, we get
∫ x2 cos x dx dx = x2 ∫cos x dx
\(-\int\left[\frac{d}{d x}\left(x^2\right) \int \cos x d x\right] dx\)
⇒ x2 sin x – f 2x sin x dx
⇒ x2 sinx-[2x(- cosx) – ∫2(- cosx)dx]
⇒ x2 sinx+ 2xcosx – 2sinx+c
⇒ (x2 -2) sinx + 2xcosx+ c
(again using integration by parts on ∫ 2x sin x dx)

Question 5.
Evaluate ∫ ex sinx dx on R.
Solution:
Let I = ∫ ex sinx dx. Then using integration by parts by taking u = ex v = sin x we get
TS Inter 2nd Year Maths 2B Integration Important Questions 46

Question 6.
Find ∫ eax cos(bx +c) dx on R, where a,b,c are real numbers and b ≠ 0.
Solution:
Let I =∫ eax cos(bx +c) dx
using integration by parts by suitably choosing eax = u and cos (bx + c) = v, we get
TS Inter 2nd Year Maths 2B Integration Important Questions 47
TS Inter 2nd Year Maths 2B Integration Important Questions 48

Question 7.
Evaluate \(\int \tan ^{-1} \sqrt{\frac{1-x}{1+x}}\) dx on (-1,1)
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 49

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 8.
Evaluate \(\int e^x\left(\frac{1-\sin x}{1-\cos x}\right)\) dx on I⊂R {2nπ : n ∈Z}.
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 50
Question 9.
\(\int \frac{d x}{(x+5) \sqrt{x+4}}\)
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 51

Question 10
Evaluate \(\int \frac{d x}{5+4 \cos x}\)
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 52

Question 11.
\(\int \frac{d x}{3 \cos x+4 \sin x+6}\)
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 53
TS Inter 2nd Year Maths 2B Integration Important Questions 54

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 12.
Find \(\int \frac{d x}{d+e \tan x}\)
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 55
TS Inter 2nd Year Maths 2B Integration Important Questions 56

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 13.
Evaluate \(\int\left(\frac{\cos x+3 \sin x+7}{\cos x+\sin x+1}\right) d x\)
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 57
TS Inter 2nd Year Maths 2B Integration Important Questions 58

Question 14.
Find \(\int \frac{x^3-2 x+3}{x^2+x-2}\) dx
Solution:
Integrand is a rational function in which the degree of the numerator Is greater than the denominator. Hence using synthetic division.
TS Inter 2nd Year Maths 2B Integration Important Questions 59
TS Inter 2nd Year Maths 2B Integration Important Questions 60

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 15.
Find \(\int \frac{2 x^2-5 x+1}{x^2\left(x^2-1\right)}\) dx
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 60

Question 16.
Find \(\int \frac{3 x-5}{x\left(x^2+2 x+4\right)}\) dx
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 61
TS Inter 2nd Year Maths 2B Integration Important Questions 62

Long Answer Type Questions

Question 1.
Evaluate \(\int \tan ^{-1}\left(\frac{2 x}{1-x^2}\right)\) dx on l ⊂ R {-1,1}
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 65
TS Inter 2nd Year Maths 2B Integration Important Questions 66

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 2.
Find \(\int \frac{x^2 e^{m \sin ^{-1} x}}{\sqrt{1-x^2}}\) dx on (-1,1) where m is a real number.
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 67
TS Inter 2nd Year Maths 2B Integration Important Questions 68
TS Inter 2nd Year Maths 2B Integration Important Questions 69
TS Inter 2nd Year Maths 2B Integration Important Questions 70

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 3.
Evaluate \(\int \frac{x+1}{x^2+3 x+12}\) dx
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 71
TS Inter 2nd Year Maths 2B Integration Important Questions 72

Question 4.
Evaluate \(\int(3 x-2) \sqrt{2 x^2-x+1}\) dx
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 73
TS Inter 2nd Year Maths 2B Integration Important Questions 74

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 5.
Evaluate \(\int \frac{2 x+5}{\sqrt{x^2-2 x+10}}\) dx
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 76

Question 6.
Evaluate \(\int \frac{2 x+1}{x\left(x^2+4\right)^2}\) dx
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 77
TS Inter 2nd Year Maths 2B Integration Important Questions 78
TS Inter 2nd Year Maths 2B Integration Important Questions 79
TS Inter 2nd Year Maths 2B Integration Important Questions 80

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 7.
Find reduction formula for ∫ xn eax dx, n being a positive integer and hence evaluate dx
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 81
TS Inter 2nd Year Maths 2B Integration Important Questions 82

Question 8.
Obtain reduction formula for ∫ sinn x dx for an integer n ≥ 2 and hence obtain ∫ sin4 xdx.
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 83

TS Inter 2nd Year Maths 2B Integration Important Questions 84

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 9.
Obtain reduction formula for ∫ sinm x cosn x dx for a positive integer m and integer n≥ 2.
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 85
TS Inter 2nd Year Maths 2B Integration Important Questions 86

Question 10.
Obtain reduction formula for ∫ tann x dx for an integar n ≥ 2 and hence find ∫ tan6 x dx.
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 87
TS Inter 2nd Year Maths 2B Integration Important Questions 88

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 11.
Obtain reduction formula for ∫ secn x dx for n ≥ 2 and hence evaluate ∫ sec5 xdx
Solution:
TS Inter 2nd Year Maths 2B Integration Important Questions 89
TS Inter 2nd Year Maths 2B Integration Important Questions 90

TS Inter 2nd Year Maths 2B Hyperbola Important Questions

Students must practice these TS Inter 2nd Year Maths 2B Important Questions Chapter 5 Hyperbola to help strengthen their preparations for exams.

TS Inter 2nd Year Maths 2B Hyperbola Important Questions

Short Answer Type Questions

Question 1.
If e, e1 are the eccentricities of a hyperbola and its conjugate hyperbola, prove that \(\frac{1}{e^2}+\frac{1}{e_1^2}=1\)
Solution:
Let e, e1 be the eccentricities of hyperbola
TS Inter 2nd Year Maths 2B Hyperbola Important Questions 1

TS Inter 2nd Year Maths 2B Hyperbola Important Questions

Question 2.
Find the equations of the tangents to the hyperbola 3x2 – 4y2 = 12 which are (I) parallel and (H) perpendicular to the line y = x – 7.
Solution:
Equation of given hyperbola is \(\frac{x^2}{4}-\frac{y^2}{3}=1\)
So that a2 = 4, b2 = 3 and equation to the given line y = x  – 7 and slope is ‘1’.

(i) Slope of the tangents which are parallel to the given line is ‘1’.
∴ Equation of tangents are
y = mx ± \(\sqrt{a^2 m^2-b^2}\)
⇒ y=x± \(\sqrt{4-3}\) and
⇒ y = x ± 1

(ii) Slope of the tangent which are perpendicular to the given line is – 1.
∴ Equations of tangents which are perpendicular to the given line are
y = (-1) x ± \(\sqrt{4(-1)^2-3}\)
x+y = ±1

Question 3.
A circle on the rectangular hyperbola xy = 1. In the points (Xr Yr)’ (r = 1, 2, 3, 4) Prove that x1 x2 x3 x4 = y1 y2 y3 y4 = 1.
Solution:
Let the circle be x2 + y2 = a2.
Since \((\mathrm{t}, \frac{1}{t})\)(t ≠ 0) lies on xy= 1, the points of intersection of the circle and the hyperbola are given by
\(t^2+\frac{1}{t^2}=a^2\)
TS Inter 2nd Year Maths 2B Hyperbola Important Questions 2

Long Answer Type Questions

Question 1.
Find the centre, eccentricity, foci, directrices and the length of latus rectum of the following hyperbolas.
4x2 – 9y2 – 8x -32 = 0
Solution:
Given equation is 4x2 – 9y2 –  8x – 32 = 0
⇒ 4x2 – 8x -9y2 = 32
⇒ 4(x -2x)-9y=32
⇒ 4(x2-2x+ 1) – 9y2 = 32 + 4 = 36
⇒ 4 (x-1)2 – 9y2 = 36
\(\frac{(x-1)^2}{9}-\frac{(y-0)^2}{4}=1\)
∴ Centre of the hyperbola = (1, 0)
The semi-transverse axis a = 3, and the semiconjugate axis b = 2.
∴ \(e=\sqrt{\frac{a^2+b^2}{a^2}}=\sqrt{\frac{9+4}{9}}=\sqrt{\frac{13}{3}}\)
Coordinates of foci (h ± ae, k)
TS Inter 2nd Year Maths 2B Hyperbola Important Questions 3

TS Inter 2nd Year Maths 2B Hyperbola Important Questions

(ii) 4(y+3)2 -9(x-2)2= 1
Solution:
The equation can be written as
TS Inter 2nd Year Maths 2B Hyperbola Important Questions 4

Question 2.
(i) If t be line lx+ my+n= 0 is a tangent to the hyperbola \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) then show that
a2l2 – b2m2 = n2

(ii) If the lx + my = t is a normal to the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) then show that
\(\frac{a^2}{l^2}-\frac{b^2}{m^2}=\left(a^2+b^2\right)^2\)
Solution:
(i) Let the line lx + my + n = 0 ……………….. (1) is a tangent to the hyperbola S = \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) at P(θ).
Then the equation of tangent at P(θ) is
\(\frac{x}{a}\) secθ – \(\frac{x}{b}\) tanθ – 1 = θ ………….. (2)
Since (1) and (2) represent the same line,
TS Inter 2nd Year Maths 2B Hyperbola Important Questions 5
TS Inter 2nd Year Maths 2B Hyperbola Important Questions 6

TS Inter 2nd Year Maths 2B Hyperbola Important Questions

Question 3.
Prove that the point of intersection of two per perpendicular tangents to the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) lies on the circle x +y =a – b.
Solution:
Let P (x1, y1) be a point of intersection of two perpendicular tangents to the hyperbola
The equation of any tangent to S = 0 is of the form
\(S \equiv \frac{x^2}{a^2}-\frac{y^2}{b^2}-1=0\)
The equation of any tangent to S = 0 is of the form
TS Inter 2nd Year Maths 2B Hyperbola Important Questions 7
This is a quadratic equation ¡n ‘m’ which has two roots m1, m2 (say) which corresponds to slopes of tangents.
TS Inter 2nd Year Maths 2B Hyperbola Important Questions 8

Question 4.
If your points be taken on a rectangular hyperbola such that the chords joining any two points is perpendicular to the chord Joining the other two, and if α , β , γ and δ be the inclinations to either asymptote of the straight lines joining these points to the centre, prove that tanα, tanβ, tanγ , tanδ = 1
Solution:
Let the equation of rectangular hyperbola be x2 – y2 = a2. By rotating the X – axis and Y – axis about the orgin through an angle \(\frac{\pi}{4}\) in the clockwise direction the equation x2 – y2 = a2 will be transformed to xy = C2.
TS Inter 2nd Year Maths 2B Hyperbola Important Questions 9
Since \(\overline{\mathrm{AB}}\) is perpendicular \(\overline{\mathrm{CD}}\) we have
\(\left(-\frac{1}{t_1 t_2}\right)\left(-\frac{1}{t_3 t_4}\right)=-1\)
⇒ t1 t1 t1 t1 = – 1 ………………. (1)
We have the coordinate axis as the a asymptotes of the curves.
If \(\overline{\mathrm{OA}}, \overline{\mathrm{OB}}, \overline{\mathrm{OC}}, \overline{\mathrm{OD}}\) make angles α, β, γ and δ with positive direction of X-axis then tanα, tanβ, tanγ, and tanδ are the slopes.

TS Inter 2nd Year Maths 2B Hyperbola Important Questions 10
If \(\overline{\mathrm{OA}}, \overline{\mathrm{OB}}, \overline{\mathrm{OC}}, \overline{\mathrm{OD}}\) make angles α, β, γ and δ with the other asymptote the Y – axis then cot α, cot β, cot γ, cot δ are the respective slopes.
So that cot α cot β cot γ cot δ = tan α tan β tan γ tan δ = I

TS Inter 2nd Year Maths 2B Ellipse Important Questions

Students must practice these TS Inter 2nd Year Maths 2B Important Questions Chapter 4 Ellipse to help strengthen their preparations for exams.

TS Inter 2nd Year Maths 2B Ellipse Important Questions

Very Short Answer Type Questions

Question 1.
If the length of the latus rectum is equal to half of its minor axis of an ellipse in the standard form, then find the eccentricity of the ellipse.
Solution:
Let \(\frac{x^2}{a^2}+\frac{y^2}{b^2}\) + 1(a > b) be the ellipse in its standard form.
Given that the length of the latus rectum = \(\frac{1}{2}\) (minor axis)
TS Inter 2nd Year Maths 2B Ellipse Important Questions 1

TS Inter 2nd Year Maths 2B Ellipse Important Questions

Question 2.
The orbit of the Earth is an ellipse with eccentricity with the \(\frac{1}{60}\) Sun at one of its foci, the major axis being approximately 186 x 106 miles in length. Find the shortest and longest distance of the Earth from the Sun.
Solution:
Let the earths orbit be an ellipse given by
\(\frac{x^2}{a^2}+\frac{y^2}{b^2}\) + 1(a > b)
Since the major axis is 186 x 106 miles
we have 2a = 186 x 106 ⇒ a = 93 x 106 miles
If e is the eccentricity of ellipse then e = \(\frac{1}{60}\)
The longest and shortest distances of the earth from the sun are respectively a + ae and a – ae.
Here the longest distance of earth from the sun = a +ae \(\left(1+\frac{1}{60}\right)\)
= 9445 x 104 miles and shortest distance of earth from the sun = a – ae
= 93 x 10 \(\left(1-\frac{1}{60}\right)\)
= 9145 x 10 miles

Short Answer Type Questions

Question 1.
Find the eccentricity, coordinates of foci, length of latus rectum and equations of directrices of the following ellipse
9x2 + 16y2– 36x + 32y – 92 = 0
Solution:
Given equation of ellipse is
9x2 + 16y2– 36x + 32y – 92 = 0
which can be written as
TS Inter 2nd Year Maths 2B Ellipse Important Questions 2

TS Inter 2nd Year Maths 2B Ellipse Important Questions

(ii) 3x+ y2 – 6x -2y -5 = 0
Solution:
Given equation can be written as
3x2 – 6x + y2 – 2y = 5
⇒ 3(x2– 2x) + (y2– 2y) = 5
⇒ 3(x2 -2x + 1)+(y2-2y+1) = 5 + 4
⇒ 3(x – 1)2 + (y – 1)2 = 9
⇒ \(\frac{(x-1)^2}{3}+\frac{(y-1)^2}{9}=1\)
which is of the form
\(\frac{(\mathrm{x}-\mathrm{h})^2}{\mathrm{a}^2}+\frac{(\mathrm{y}-\mathrm{k})^2}{\mathrm{~b}^2}=1\)
which a2 = 3 and b2 = 9 and a < b.
Also (h, k) = (1, 1); eccentrIcity
TS Inter 2nd Year Maths 2B Ellipse Important Questions 3

Question 2.
Find the equation of the ellipse referred to its major and minor axes as the coordinate axes X, Y – respectively with latus rectum of length 4 and distance between foci \(4 \sqrt{2}\).
Solution:
Let the equation of ellipse be
TS Inter 2nd Year Maths 2B Ellipse Important Questions 4

TS Inter 2nd Year Maths 2B Ellipse Important Questions

Question 3.
C is the centre, A A’ and B B’ are major and minor axis of ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\).If
PN is the ordinate of a point P on the ellipse then show that \(\frac{(\mathrm{PN})^2}{(\mathrm{AN})(\mathrm{AN})}+\frac{(\mathrm{BC})^2}{(\mathrm{CA})^2} \)
Solution:
Let P(θ) = (a cos θ, b sin θ) be any point on the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)
TS Inter 2nd Year Maths 2B Ellipse Important Questions 6

Question 4.
S and T are the foci of an ellipse and B is one end of the minor axis. If STB is an equilateral triangle, then find the
eccentricity of the ellipse.
Solution:
Let \(\frac{x^2}{a^2}+\frac{y^2}{b^2}\) = 1 :(a>b) be an ellipse whose foci are S and T. B is the end of minor axis such that STB is an equilateral triangle.
than SB = ST = SB. Also S = (ae, 0).
T = (- ae. 0) and B = (0, b).
TS Inter 2nd Year Maths 2B Ellipse Important Questions 7

Question 5.
Show that among the points on the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}\)=1,(a>b) ;(-a, 0) is the farthest point and (a, 0) is the nearest point form the focus (ae, 0).
Solution:
Let P(x, y) be any point on the ellipse so that < x < a and S = (ae, 0) is the focus. Since (x. y) is on the ellipse.
TS Inter 2nd Year Maths 2B Ellipse Important Questions 8
Maximum value of SP is a + ae when P(-a.0)
and Minimum value of SP is a – ae when P (a. 0).
The nearest point is (a, 0) and the farthest point is (-a, 0).

TS Inter 2nd Year Maths 2B Ellipse Important Questions

Question 6.
Find the equation of tangent and normal to the ellipse 9x2 + 16y2 = 144 at the end of latus rectum in the first quadrant.
Solution:
Given equation of ellipse is 9x2 + 16y2 = 144
TS Inter 2nd Year Maths 2B Ellipse Important Questions 9
TS Inter 2nd Year Maths 2B Ellipse Important Questions 10

Question 7.
If a tangent to the ellipse = \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)\) meets its major axis and minor axis at M and N respectively then prove that \(\frac{a^2}{(\mathrm{CM})^2}+\frac{b^2}{(\mathrm{CN})^2}=1\) where C is the centre of the ellipse.
Solution:
TS Inter 2nd Year Maths 2B Ellipse Important Questions 11

TS Inter 2nd Year Maths 2B Ellipse Important Questions

Question 8.
Find the condition for the line,
(i) lx + my + n = 0 to be a tangent to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)
(ii) lx + my + n = 0 to be a normal to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)
Solution:
Let lx + my + n = 0 be a tangent at
P (θ) (a cos θ . b sin θ) on the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)
TS Inter 2nd Year Maths 2B Ellipse Important Questions 12

(ii) Let lx + myn = 0 be a normal to the ………………  (2)
Ellipse at the point P (θ). Then equation of normal at ‘θ’ is
TS Inter 2nd Year Maths 2B Ellipse Important Questions 14

TS Inter 2nd Year Maths 2B Ellipse Important Questions

Question 9.
If PN is the ordinate of a point P on the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) and the tangent at P
meets the x-axis at T then show that (CN) (CT) = a2 where C is the centre of ellipse.
Solution:
Let P (θ) = P (a cos θ, b sin θ) be a point on the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) .
Then the equation of the tangent at p (θ) is.
TS Inter 2nd Year Maths 2B Ellipse Important Questions 15

Question 10.
Show that the points of intersection of the perpendicular tangents to any ellipse lie on the circle.
Solution:
Let the equation of ellispe be \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) (a>b)
Any tangent to the above ellipse is of the form y = mx ± \(\pm \sqrt{a^2 m^2+b^2}\)
Let the perpendicular tangents intersect at
TS Inter 2nd Year Maths 2B Ellipse Important Questions 16
This being a quadratic is ‘rn has two roots m1 and m2 which corresponds to the slopes of tangents drawn from P to ellipse then
TS Inter 2nd Year Maths 2B Ellipse Important Questions 17
(∵ Product of slopes = – 1 for perpendicular tangents)
⇒ x12+y12 = a2 +b2
∴ Locus of (x1, y1) is x2 + y2 = a2+ b2 which is a circle.

TS Inter 2nd Year Maths 2B Ellipse Important Questions

Long Answer Type Questions

Question 1.
If θ1, θ2 are the eccentric angles of the extremeties of a focal chord (other than the verticles) of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) (a> b) and e is its eccentricity. Then show that
TS Inter 2nd Year Maths 2B Ellipse Important Questions 18
Solution:
TS Inter 2nd Year Maths 2B Ellipse Important Questions 19

Let P(θ1), Q(θ2) be the two extremeties of a focal chord of the ellipse
\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)\)
∴ P = (acos θ1, b sin θ1),(θ1 ≠ 0)
Q = (a cos θ2, b sin θ2), (θ2 ≠ π)
and focus S = (ae, 0). Now PQ is a focal chord and hence P, S. Q are collinear.
∴ Slope of \(\overline{\mathrm{PS}}\) = slope of \(\overline{\mathrm{SQ}}\)
\(\frac{b \sin \theta_1}{a\left(\cos \theta_1-\mathrm{e}\right)}=\frac{\mathrm{b} \sin \theta_2}{\mathrm{a}\left(\cos \theta_2-\mathrm{e}\right)}\)
TS Inter 2nd Year Maths 2B Ellipse Important Questions 20

Question 2.
If the normal at one end of a latus rectum of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) passes through one end of the minor axis, then show that e4 + e2 = 1 (e is the eccentricity of the ellipse)
Solution:
Let L be the one end of the latus rectum of \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\). Then the coordinates of
\(L=\left(a e, \frac{b^2}{a}\right)\)
∴Equation of normal at L is
TS Inter 2nd Year Maths 2B Ellipse Important Questions 22

TS Inter 2nd Year Maths 2B Ellipse Important Questions

Question 3.
If a circle is concentric with the ellipse, find the inclination of their common tangent to the major axis of the ellipse.
Solution:
Let the circle x2 + y2 = r2 and the ellipse be \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) with a> b.
The major axis of ellipse is X – axis.
If r< b <a, then the circle lies completely in the ellipse making no common tangents.
If b < a < r (ellipse lies completely in circle) no common tangent is passive.

Case (i) : If b <r <a
TS Inter 2nd Year Maths 2B Ellipse Important Questions 23
Let one of the common tangent make angle θ with positive X- axis and suppose the equation of tangent to the circle be x Cos α + y sin α = r where a is the angle made by the radius of circle with positive X – axis.
∴ \(\theta=\frac{\pi}{2}+\alpha \text { (or) } \theta=\alpha-\frac{\pi}{2}\)
Since x cos α + y sina r touches the ellipse also, we have a2 cos2a + b2 sin2 = r2

TS Inter 2nd Year Maths 2B Ellipse Important Questions 24

Case (ii): When r = a the circle touches the ellipse at the ends of major axis of the ellipse so that the common tangents are x = ± a and θ = \(\frac{\pi}{2}\)

TS Inter 2nd Year Maths 2B Ellipse Important Questions

Case (iii): When r = b, the circle touches the ellipse at the ends of minor axis of ellipse so that common tangents
y = ± b making θ = 0.

TS Inter 2nd Year Maths 2B Parabola Important Questions

Students must practice these TS Inter 2nd Year Maths 2B Important Questions Chapter 3 Parabola to help strengthen their preparations for exams.

TS Inter 2nd Year Maths 2B Parabola Important Questions

Very Short Answer Type Questions

Question 1.
Find the coordinates of the vertex and focus and the equations of the directrix and axis of the following parabolas.
(i) y2 = 16x
Comparing with y2 = 4ax, we have 4a = 16
a = 4
Vertex = (0, 0)
Focus = (a, 0) = (4, 0)
Equation of clirectrix is x + a = 0 ⇒ x + 4 = 0
Equation of axis is y = 0.

TS Inter 2nd Year Maths 2B Parabola Important Questions

(ii) x2 =-4y
Comparing with x2 = – 4ay, we have 4a = 4 ⇒ a = I
∴ Vertex = (0. 0)
Focus =(0,-a)=(0,-1)
Equation of directrix is y-a=0=y-1 =0
Equation of axis is x – 0.

(iii) 3x- 9x + 5y – 2 = 0
Given equation is 3x2 – 9x – 5y +2
= 3(x2 – 3x) =- \(5\left(\mathrm{y}-\frac{2}{5}\right)\)
TS Inter 2nd Year Maths 2B Parabola Important Questions 1
(i) Vertex (h, k) \(\left(\frac{3}{2}, \frac{7}{4}\right)\)
(ii) Focus=(h,k,-a) = \(\left(\frac{3}{2}, \frac{7}{4}-\frac{5}{12}\right)=\left(\frac{3}{2}, \frac{4}{3}\right)\)
(iii) Equation of directrix is y – k = 0
⇒ 6y – 13= 0
(iv) Equation of axis x – h = 0
= x = \(\frac{3}{2}\) – 0 ⇒ 2x – 3 = 0

TS Inter 2nd Year Maths 2B Parabola Important Questions

(iv) y2 – x + 4y + 5 = 0
The given equation can be written as
y2 + 4y – x – 5
9 + 4y + 4 – x – 1
(y+2)2 – (x-1)
Comparing with (y – k)2 – 4a (x – h)
we have h=1, k = -2, a= \(\frac{1}{4}\)
(i) Vertex = (h, k) – (1, -2)
(ii) Focus = (h+a,k) = \(\left(\frac{5}{4},-2\right)\)
(iii) Directrix is x – h + a = 0 ⇒ 4x – 3 = 0
(Iv) Equation of axis is y-k=0 ⇒ y+2 = 0

Question 2.
Find the equation of parabola whose vertex is (3-2) and focus is (3,1)
Solution:
The abscissae of vertex and focus is ‘3’.
Hence the axis of the parabola is x -3, a line parallel to y-axis, focus is above the vertex.
a = SA = \(\sqrt{(3-3)^2+(1+2)^2}=3\)
(∵ A=(3,-2) S= (3,3))
∴ Equation of the parabola
(x-h)2 =4a (y-k)
(x-3)2– 12(y+2)
(∵ A = (h,k) =3,-2 and S (h, k-a)]

Question 3.
Find the coordinates of the points on the parabola y2 = 2x whose focal distance is \(\frac{5}{2}\)
Solution:
Let P(x1, y1) be any point on the parabola
y2 = 2x whose focal distance is \(\frac{5}{2}\) then
\(y_1^2=2 x_1\) and \( x_1+a=\frac{5}{2} \Rightarrow \frac{5}{2}-\frac{1}{2}\)
Where 4a=2 ⇒ a = \(\frac{1}{2}\)
∴ x1= 2, and since y12 = 2x1 we have
y12 = 4 = y1 = ± 2
∴ The required points are (2, 2), (2, -2).

Short Answer Type Questions

Question 1.
Find the equation of the parabola passing through the points (- 1, 2), (1, – 1) and (2, 1) and having its axis parallel to the x-axis.
Solution:
Since the axis is parallel to x – axis the equation of the parabola is in the form of
x – ly2+my+n ……………. (1)
Given that the parabola passes through the points (-1, 2), (1, -1) and (2, 1) we have
– 1 = 14l+2m+n ……………………. (2)
1= l – m+n ……………….. (3)
2=l+m+n …………………….. (4)
From (2) and (3), -2 = 3l + 3m
TS Inter 2nd Year Maths 2B Parabola Important Questions 2
∴  Equation of required parabola is
\(x=-\frac{7}{6} y^2+\frac{1}{2} y+\frac{8}{3}\)

TS Inter 2nd Year Maths 2B Parabola Important Questions

Question 2.
A double ordinate of the curve y2 = 4ax is of length 8a. Prove that the lines from the vertex to its ends are at right angles.
Solution:
TS Inter 2nd Year Maths 2B Parabola Important Questions 3

Question 3.
Find the condition for the straight line lx + my + n=0 to be a tangent to the parabola y2 = 4ax and find the coordinates
of the point of contact.
Solution:
Let the line ix + my + n = 0 …………….. (1) touches
y2 = 4ax …………….. (2) at the point (x1, y1).
Then the equation of tangent at (x1, y1) to the parabola
y2=4ax is yy1– 2ax – 2ax1=0 …………. (2)
Since (1) and (2) represent the same line, coefficients are proportional.
TS Inter 2nd Year Maths 2B Parabola Important Questions 4

TS Inter 2nd Year Maths 2B Parabola Important Questions

Question 4.
Show that the straight line 7x+6y=13 is a tangent to the parabola y2– 7x – 8y + 14 = 0 and find the point of contact
Solution:
Given line is 7x + 6y – 13 = 0 ……………….. (1)
and parabola is y2 – 7x – 8y+ 14 = 0 ………………..(2)
From (1),
TS Inter 2nd Year Maths 2B Parabola Important Questions 5
Point of contact is (1, 1) and this point satisfies (1) and (2). Hence (1) is tangent to the parabola (2).

Question 5.
From an external point P, tangents are drawn to the parabola y2 = 4ax and these tangents make angles θ1, θ2 with its axis such that tan θ1+ tan θ2 is a constant b. Then show that P lies on the line y = bx.
Solution:
Let P(x1, y1) be the external point.
Any tangent to the parabola is of the form
y=mx+ \(\frac{\mathrm{a}}{\mathrm{m}}\) =mx1– my1+a = 0
Let the roots of (1) be 4 m1, m2 ………………  (1)
Then m1 + m2 = \(\frac{y_1}{x_1}\) and m1 + m2 = \(\frac{\mathrm{a}}{\mathrm{x}_1}\)
The tangents make angles θ1 and θ2 with the axis such that m1 = tan θ1 and m2= tan θ2
Given tan θ1 + tan θ2 = b
⇒ m1 + m2 = b
\(\frac{\mathrm{y}_1}{\mathrm{x}_1}\) = b ⇒ y1 = bx1
∴ Locus of (x1, y2) is y = bx.

TS Inter 2nd Year Maths 2B Parabola Important Questions

Question 6.
Show that the common tangents to the parabola y2 = 4ax and x2 = 4by is xa1/3 + yb1/3 + xa2/3 yb2/3
Solution:
The equations of the parabolas are
y2 = 4ax ………………….. (1)
and x2 = 4by ………………… (2)
Equation of any tangent to (1) is of the form
y=mx+\(\frac{\mathrm{a}}{\mathrm{m}}\) ………….. (3)
lf the line (3) is tangent to (2) also the point of intersection of (2) and (3) coincide.
From (3) and (2)
x2 = 4b \(\left(\mathrm{mx}+\frac{\mathrm{a}}{\mathrm{m}}\right)\)
⇒ mx2 – 4bm2x – 4ab = 0
which should have equal roots.
∴ Therefore the discriminant is zero.
TS Inter 2nd Year Maths 2B Parabola Important Questions 6

Long Answer Type Questions

Question 1.
Find the equation of a parabola in his standard form.
Solution:
TS Inter 2nd Year Maths 2B Parabola Important Questions 7

Let S be the focus, l be the directrix as shown in the figure. Let Z be the projection of S on I and A be the mid point of \(\overline{\mathrm{SZ}}\). A lies on the parabola since SA = AZ. A is the vertex of the parabola. YY’ is parallel to the clirectrix ‘l’.
Let S=(a,0), (a>0) then Z =(-a, 0) and the equation of the directrix is x + a = 0.
Let P(x, y) be a point on the parabola and PM is the perpendicular distance from P to x + a = 0.
Then by definition \(\frac{\mathrm{SP}}{\mathrm{PM}}\) = e = 1
= SP2 = PM2
= (x- a)2 +(y-0)2 = (x+a)2
= y2 =(x+a)2-(x-a)2 = 4ax
Conversely also if P(x, y) is a point such that y2 = 4ax then SP = PM.
∴ P(x, y) lies on the locus. Equation of parabola in standard form is y2 = 4ax.

Question 2.
(i) If the coordinates of the ends of a focal chord of the parabola y2 = 4ax are (x1, y1) and (x2, y2) then prove that
x1x2 = a2 and y1y2 = – 4a2.
(ii) For a focal chord PQ of the parabola y2 = 4ax if SP =l and SQ = l’ then prove that \(\frac{1}{l}+\frac{1}{l}=\frac{1}{a}\)
Solution:
(i) Let P(x1, y1) = (at12, 2at1)
and Q(x2, y2) = (at22, 2at2)
be two end points of a focal chord and S(a, 0) be the focus of the parabola y2 = 4ax. PSQ is a focal chord P, S, Q are collinear.
TS Inter 2nd Year Maths 2B Parabola Important Questions 8

(ii) Let P(at21, 2at1) and Q (at22, 2at2) be the extremities of a focal chord of parabola such that t1,t2 = – 1.
TS Inter 2nd Year Maths 2B Parabola Important Questions 9

TS Inter 2nd Year Maths 2B Parabola Important Questions

Question 3.
If Q is the foot of the perpendicular from a points P on the parabola y2 = 8(x -3) to its directrix. S is the focus of parabola and if SPQ is an equilateral triangle then find the length of side of triangle.
Solution:
Given equation of parabola is
(y – 0)2 = 8(x – 3)
which is of the form (y – k)2 = 4a (x – h)
where 4a = 8 ⇒ a = 2
∴ Vertex = (h, k) = (3,0)
and focus = (h + a, k) = (3 +2, 0) =  (5, 0)
TS Inter 2nd Year Maths 2B Parabola Important Questions 10
Since PQS is an equilateral triangle.
\(\angle \mathrm{SQP}=60^{\circ} \Rightarrow \angle \mathrm{SQZ}=30^{\circ}\)
Also in ΔSZQ , we have sin 30° = \(\frac{\mathrm{SZ}}{\mathrm{SQ}}\)
TS Inter 2nd Year Maths 2B Parabola Important Questions 11
Hence length of each side of the triangle is 8

Question 4.
The cable of a uniformly loaded suspension bridge hangs In the form of a parabola. The roadway which is horizontal and 72mt. long is supported by vertical wires attached to the cable. The longest being 30 mts. and the shortest being 6 mts. Find the length of the supporting wire attached to the roadway 18 mts. from the middle.
Solution:
TS Inter 2nd Year Maths 2B Parabola Important Questions 12

Let AOB be the cable [O is the lowest point and A, B are highest points). Let PRQ be the suspension bridge suspended with PR – RQ = 36 mts.
PA = QB = 30 mts (longest wire)
OR = 6 mts (shortest wire)
∴ PR = RQ = 36 mts. We take O as origin of coordinates at O, X-axis along the tangent \(\overline{\mathrm{RO}}\). So the equation of the cash is x2 = 4ay for some a> 0.
∴ B(36,24) and B is a point on x2=4ay
We have (36)2 = 4a(24)
⇒ 4a = \(\frac{36 \times 36}{24}\) = 54 mts
If RS = 18 mts and SC is the vertical through S meeting the cable at ‘C’ and the X- axis at D.
Then SC is the length of the required wire.
Let SC = l mts then DC = L – 6 mts.
∴ C = (18, 1 – 6) which lies on x2 = 4ay
⇒ (18)2 = 4a (l – 6)
⇒ (18)2= 54 (l -6)
⇒ l – 6 = \(\frac{18 \times 18}{54}\) = 6
⇒ l = 12.

Question 5.
Prove that the normal chord at the point other than origin whose ordinate is equal to Its abscissa subtends a right angle at the focus
Solution:
Let the equation of the parabola
y2 = 4ax ……………. (1)
and P(at2, 2at) be any point
Ordinate – abscissa
= 2at = at2 = t=0 or t = 2
But t ≠ 0, Hence the point (4a, 4a) at which normal is
⇒  y + 2x = 2a(2) + a(2)3
⇒  y + 2x = 4a + 8a
⇒  y=12a -2x …………….. (2)
∴ From (1) we get
(12a – 2x)2 – 4ax
⇒ 4x2 – 48ax + 144a2 = 4ax
⇒ 4x2 – 52ax+ 144a2= 0
⇒ x2 – 13ax + 36a2 = 0
⇒ (x – 4a) (x – 9a) – 0
⇒ x = 4a, 9a
Correspondingly we get y = 12a – 8a = 4a and y = 12a – 18a= – 6a
Hence the other points of intersection of normal at P(4a, 4a) is Q(9a – 6a).
We have S = (a, 0)
TS Inter 2nd Year Maths 2B Parabola Important Questions 13

TS Inter 2nd Year Maths 2B Parabola Important Questions

Question 6.
Prove that the area of the triangle formed by the tangents at (x1, y1), (x2, y2) and (x3,y3) to the parabola y2 = 4ax
\(\frac{1}{16 a}\left|\left(y_1-y_2\right)\left(y_2-y_3\right)\left(y_3-y_1\right)\right|\)  sq. unit
Solution:
Let A (x1, y1) = (at22, 2at1)
B (x2, y2) =  (at22, 2at2)
C (x3, y3) – (at32, 2at3) be the three points on the parabola y2=4ax.
The equations of tangents at A, B, C are
TS Inter 2nd Year Maths 2B Parabola Important Questions 14
∴ Point of intersection of tangents at A and B is P(at1t2, a(t1 + t2)).
Similarly the points of intersection of tangents at B and C as well as A & C are
TS Inter 2nd Year Maths 2B Parabola Important Questions 15

Question 7.
Prove that the two parabolas y2 = 4ax and x2 = 4by intersect other than the origin) at tan-1
\(\left(\frac{3 a^{1 / 3} b^{1 / 3}}{2\left(a^{2 / 3}+b^{2 / 3}\right)}\right)\)
Solution:
Take a>0, b>0
and y2=4ax ………………….. (1)
and x2 = 4by are ……………. (2)
the given parabolas.
Solving (1) and (2) we get the point of intersection other than the origin.
TS Inter 2nd Year Maths 2B Parabola Important Questions 16
TS Inter 2nd Year Maths 2B Parabola Important Questions 17
TS Inter 2nd Year Maths 2B Parabola Important Questions 18

TS Inter 2nd Year Maths 2B Parabola Important Questions

Question 8.
Prove that the orthocentre of the triangle formed by any three tangents to a parabola lies on the directrix of the parabola.
Solution:
Let y2 = 4ax be the parabota and
A = (at12, 2at1), B = (at22, 2at2),
C = (at32, 2at3) be any three points on It.
If P, Q, R are the points of intersection of tangents at A and B, B and C, C and A then
P = [at1t2, a(t1 + t2)] Q = [at2t3, a(t2 + t3)] R = [at1t3, a(t1 + t3)]
Consider the ΔPQR then equation \(\overline{\mathrm{QR}}\) of (Tangent at C) is
x – yt3 + at32 = 0
∴ Altitude through P of Δ PQR is
t3x + y = at1t2t3 + a(t1 + t2) …………….. (1)
∴ Slope = \(\frac{1}{t_3}\) and equation is
y – a(t1+ t2) = – t3 [x – at1t2]
y+xt3 – at1t2t3+a(t1 +t2)
Similarly the altitude through Q is
t1x + y – at1t2t3 + a(t2+ t) …………… (2)
Solving (1) and (2)
x(t3 – t1) = a(t1– t)
⇒ x = – a
Hence the orthocentre of ΔPQR with x coordinate as – ‘a’ lies on the directrix of the parabola.

TS Inter 2nd Year Maths 2B System of Circles Important Questions

Students must practice these TS Inter 2nd Year Maths 2B Important Questions Chapter 2 System of Circles to help strengthen their preparations for exams.

TS Inter 2nd Year Maths 2B System of Circles Important Questions

Very Short Answer Type Question

Question 1.
Find the angle between the circles x2+y2+4x-14y+28=0
x2+y2+4x – 5 = 0
Solution:
Comparing with general equation
x2+y2+2gx+2fy+c=0, we have
g=2, 1=-7, c=28, C1=(-2,7)
g’=2, f’=0, c=-5, C2=(-2,0)
Let θ be the angle between circles (1) and (2) then
TS Inter 2nd Year Maths 2B System of Circles Important Questions 1

TS Inter 2nd Year Maths 2B System of Circles Important Questions

Question 2.
If the angle between the circles x2+y2-12x- 6y+41=0 ………………..(1) x2+y2+kx+6y-59=0 …………… (2) is 45° find k.
Solution:
TS Inter 2nd Year Maths 2B System of Circles Important Questions 2
TS Inter 2nd Year Maths 2B System of Circles Important Questions 3
⇒ k2 + 272. 18k2
⇒ 17k2 = 272
⇒ k2=16
k = ± 4

Short Answer Type Questions

Question 1.
find the equation of the circle passing through the points of the intersection of the circles
x2+y2– 8x-6y+21=0 …………… (1)
x2+y2– 2x-15=0 …………. (2) and (1, 2)
Solution:
The equation of circle passing through the point of intersection o! circles S = 0, S’ = 0 is S+ λS’+ 0 where λ is a parameter.
∴(x2+y2-8x-6y+21) +λ(x2+y2-2x-15)=0 ……………….. (1)
If this passes through (1, 2) then
(1+4-8-12+21)+λ(1+4-2-15)=0
= 6-12λ=0 ⇒ λ= \(\frac{1}{2}\)
Hence the equation of the required circle
is (x2+y2-8x-6y+ 21) + \(\frac{1}{2}\) (x2+y2-2x-15) = 0
= 3(x2+y2)-18x-12y+ 27 = 0

TS Inter 2nd Year Maths 2B System of Circles Important Questions

Question 2.
Find the equation and length of the common chord of the circles
S ≡ x2 +y2+ 3x + 5y+ 4 = 0 ……………….. (1)
and S ≡ x2 +y2+ 5x+ 3y + 4 = 0  …………………. (2)
Solution:
The common chord of two intersecting circles is the radical axis given by S – S’ = 0.
⇒  2x + 2y = 0
⇒ x-y=0 ……………… (3)
Centre of cricle (1) is \(\mathrm{C}_1=\left(-\frac{3}{2},-\frac{5}{2}\right)\) and
TS Inter 2nd Year Maths 2B System of Circles Important Questions 4
∴ AD = 2
∴ Length of the common chord AB = 2(AD) = 2(2) 4

Question 3.
Find the equation of the circle whose diameter is the common chord of the circles
S = x2 +y2+2x+3y+ 1=0 …………………. (1)
and S’= x2 +y2+4x+3y+2=0 …………………. (2)
Solution:
The common chord is the radical axis of (1) and (2) given by S – S’ = 0.
⇒ 2x-1 =0
⇒  2x+1=0 ………………. (3)
The equation of any circle passing through the point of intersection of (1) and (3) is
S + λL = 0.
(x2 +y2+2x+3y+1)+λ(2x+1) = 0
x2 +y2+2(λ+ 1)x+3y+(1 +λ) = 0 ……………….. (4)
Centre of this circle = [- (λ + 1), \(-\frac{3}{2}\)]
For the circle (4), 2x + 1 -0 is one chord. This will be the diameter of the circle (4).
If the centre of (4) lies on (3) then
– 2(λ + 1) + 1 = 0
⇒ λ=-\(-\frac{3}{2}\)
∴ The equation of circle whose diameter is the common chord of (1) and (2) is
x2 +y2 + 2x + 3y + 1) – (2x + 1) = 0
2(x2 +y2)+2x+6y+1=0

TS Inter 2nd Year Maths 2B System of Circles Important Questions

Question 4.
Find the equation of a circle which cuts each of the following circles orthogonally
S’ = x2+y2+3x+2y+1=0 ……….. (1)
S”=x2+y2-x+6y+5=0 ………… (2)
S”’=x2+y+5x-6y+15=0 …………… (3)
Solution:
Radical axis of circles (1) and (2) is S’ -S”= 0.
⇒ 4x-4y-4=0
⇒ x-y-1=0   ………………… (4)
Radical axis of circles (2) and (3) is S’’ -S”’= 0.
⇒ -6x+ 14y-10 =0
⇒ 3x-7y+5= 0 ……………. (5)
Solving (4) and (5) we get
TS Inter 2nd Year Maths 2B System of Circles Important Questions 5
∴ Radical centre – (3, 2)
Also the length of the tangent from (3, 2) to
S’=\(\sqrt{9+4+9+4+1}=\sqrt{27}\)
Hence equation of circle which cuts orthogonally each of the given circles is obtained by taking radical centre as centre and length of the tangent as radius.
∴ Equation of the required circle is
(x-3)2 +(y-2)2= 27
⇒ x2+y2-6x-4y-14=0.

Long Answer Type Questions

Question 1.
Find the equation of the circle which passes through (1, 1) and cuts orthogonally each of the circles
x2+y2-8x-2y+ 16=0 …………. (1)
and x2+y2-4x-4y-1=0 …………. (2)
Solution:
Let the equation of the required circle be
x2+y2+2gx+2fy+c = 0 ………….. (3)
1f this passes through (1, 1) then
1+1 + 2g. 2f + c = 0
⇒ 2g+2f+c- 2 ………….. (4)
If (3) is orthogonal to (1) then
2g(-4) + 2f(-1) = c + 16
=-8g-21=c+16 …………………. (5)
11(3) is orthogonal to (2) then
2g(-2) + 2f(-2) =c-1
= – 4g – 41=c – 1 ………….. (6)
From (5) and (6) we have
– 4g + 21=17
⇒ 4g+2f=-17 …………….. (7)
From (4) and (5) we have
– 6g=+ 14=g= – \(\frac{7}{3}\)
∴ From (7)
TS Inter 2nd Year Maths 2B System of Circles Important Questions 6

TS Inter 2nd Year Maths 2B System of Circles Important Questions

Question 2.
Find the equation of circle which is orthogonal to each of the following three circles x2+y2+ 2x + 17y+ 4=0
x2+y2+7x+6y+ 11=0 and x2+y2-x+22y+3=0
Solution:
Denote the given circles by
S=x2+y2+2x+ 17y+4 = 0 ……………….. (1)
S’=x2+y2+7x+6y+11 = 0 ……………… (2)
and S”=x2+y2 x+22y+3 = 0 …………………. (3)
The radical axis of S = 0, S’ = 0 is s – S = 0.
⇒ – 5x+11y – 7=0
⇒ 5x – 11y-7=0 ……………….. (4)
The radical axis of S’ = 0. S” = 0 is S’ = S’’ = 0.
⇒ 8x-16y+8=0
⇒ x-2y+1=0 …………… (5)
Solving equations (4) and (5), we get the coordinates of radical centre.
TS Inter 2nd Year Maths 2B System of Circles Important Questions 7
⇒ x = 3, y = 2 ∴ C (3, 2) is the radical centre.
Length of the tangent from C(3, 2) to the circles = 0 is \(\sqrt{9+4+6+34+4}=\sqrt{57}\)
The equation of the circle which cuts or orthogonally each of the three circles if obtained by considering the equation of circle with radical centre (3,2) as centre and length of the tangent \(\sqrt{57}\) as the radius.
∴ Equation of the required circle is
(x-3)2 + (y – 2)2 = 57
= x2 + y2-6x-4y-44 = 0

Question 3.
If the straight line represented by x cos α + y sin α = p intersects the code x2 + y2 = a2 at the points A and B then show that the equation of circle with \(\overline{\mathbf{A B}}\) as diameter is (x2+y2-a2)-2p(x cosα+ysinα – p)=0.
Solution:
Given x cos α + y sin α p …….. (1) intersects the circle
x2 + y2 = a2 ……….. (2) at points A and B.
So the equation of circle passing through the point of intersection of (1) and (2) is
x2 + y2 = a2+λ(xcos α + ysinα-p) = 0 …………. (3)
Where λ is a parameter.
∴ Centre of (3) is  \(\left(\frac{-\lambda \cos \alpha}{2}, \frac{-\lambda \sin \alpha}{2}\right)\)
If the circle given by (3) has \(\overline{\mathrm{AB}}\) as diameter then the centre of it must lie on (1). Then
TS Inter 2nd Year Maths 2B System of Circles Important Questions 8
Hence the equation of the required circle is (x2 + y2– a2) – 2p(xcosα+ysinα – p)=0.

TS Inter 2nd Year Maths 2B System of Circles Important Questions

Question 4.
Show that the circles
S=x2+y2-2x-4y-20 =0 …………….. (1)
and S’=x2+y2+6x – 2y- 90=0 …………… (2)
touch each other Internally. Find their point of contact and the equation of common tangent
Solution:
Let C1 C2 be the centres of circles (1) and (2) and r1 r2 be the radii of circles (1) and (2).
Then C1 = (1, 2) and r1 = \(\sqrt{1+4+20}\) = 5
TS Inter 2nd Year Maths 2B System of Circles Important Questions 9
Since C1 C2 – |r1 r2|, the two circles touch
Internally. The point of contact P divides C3 C4 externally In the ratio of the radii of circles 5: 10 1: 2.
TS Inter 2nd Year Maths 2B System of Circles Important Questions 10
Since the two circles touch internally the common tangent at the point of contact is only the radical axis of
S = 0 and S’- 0 given by S – S’ – 0.
= – 8x – 6y + 70 = 0
= 4x + 3y – 35 = 0