Explore topic-wise InterviewSolutions in Current Affairs.

(D² - 2D + 5)y = e^2x sinx

For C.F, use m in place of D

m² - 2m + 5 = 0

 ⇒ m = \(\frac{2 ± \sqrt{4-20}}{2}\)

 = \(\frac{2±4c}{2}\)

= 1 ± 2c

\(\therefore\) C.F = e^x (C₁cos 2x + c₂ sin 2x)

P.I = \(\frac{1}{D^2 - 2D +5}\) e^2x sinx

= e^2x \(\frac{1}{(D+2)^2 -2(D+2) + 5} sinx\)

\(\bigg(\therefore \frac{1}{f(D) }e^{ax}V = e^{ax} \frac{1}{f(D+a)V}\bigg)\)

= e^2x \(\frac{1}{D^2+2D + 5} sinx\)

= e^2x \(\frac{1}{-1+2d+5}sinx\)

\(\bigg(\therefore \frac{1}{f(D^2)} sinax = \frac{1}{f(-a^2)} sinax\bigg)\)

 = e^2x \(\frac{1}{2D + 4}\) sinx

= \(\frac{e^{2x}}{2} \frac{D-2}{D^2 -4} sinx\)

 = \(\frac{e^{2x}}{2}\) \(\frac{(D-2)sinx}{-1-4}\)

= \(\frac{e^{2x}}{-10}(Dsinx - 2sinx)\)

= \(\frac{-e^{2x}}{10}(cos x -2sinx)\)

Complete solution y = e^x (c₁cos 2x + c₂ sin2x)

 = \(\frac{-e^{2x}}{10}\) (cosx -2sinx)

Equation of plane passing through intersection of the plane is (3x - y + 2z - 4) + λ (x + y + z + 2) = 0 

This plane pass through (2, 2, 1) 

3(2) - 2 + 2(1) -4 + λ(2 + 2 + 1 + 2) = 0 

2 + 7λ = 0 

λ = -2/7

Required equation of plane is 

(3x - y + 2z - 4) + (x + yz + 2) + 0 [x^ly by 7] 

21x - 7y + 14z - 28 - 2x - 2y - 2z - 4 = 0 

19x - 9y + 12z = 32

â = 1/√14 (2î + 3ĵ + k̂)

∴ Directions cosines are :

2/√14. 3/√14. 1/√14

\(\vec A=\hat i+2\hat j+2\hat k\)

Equation of  y-axis is \(\vec y=0\hat i+\hat j + 0\hat k\)

|\(\vec A\)| = \(\sqrt{1+4+4}=3\)

|\(\vec y\)| = \(\sqrt{0+1+0}=1\)

\(\vec A.\vec y=(\hat i+2\hat j+2\hat k).(0\hat i+\hat j+0\hat k)\)

⇒ |\(\vec A\)||\(\vec y\)| cos θ = 2, where θ is angle between \(\vec A\) and y - axis.

⇒ cos θ = \(\frac2{|\vec A||\vec y|}=\frac23\)

⇒ θ  = cos^-1(2/3) ≈ 48.2°

cos^-1 x = A Now 1/2≤ x ≤ 1 

cosA = x 

cos3A = 4cos³ A – 3cosA ⇒ 1/2 ≤ A < 1 

cos3A = 4x³ – 3x 

3A = cos^-1(4x³ – 3x) ⇒ 0 ≤ A ≤ π/3

3cos^-1x = cos^-1 [4x³ – 3x] ⇒ 0 ≤ 3A ≤ π

f(x) = cosx 

g(x) = 3x² 

gof(x) = g[f(x)] 

= g[cosx] 

= 3cos²x 

fog(x) = f[g(x)] 

= f[3x²] 

= cos3x² 

fog ≠ gof

(a) y = c e^{x^2} 

The correct answer is : (d) 1/3

The correct answer is : (b) y = -2/3

Correct answer is: (c) 50√2/π cm

2πr =100. So Diameter = 2r =100/π = diagonal of the square.

side√2 = diagonal of square = 100/ π 

\(\therefore\) side = 100/√2π = 50√2/π cm

Correct answer is: (b) 1

3^x+y = 243 = 3⁵

So x + y =5----------(1)

243^x-y = 3

(3⁵)^x-y = 3¹

So 5x -5y =1---------(2)

Since : \(\cfrac{a_1}{a_2}\neq\cfrac{b_1}{b_2}\), so unique solution

y = 4- x² intersects x- axis at A and B

∵ 4 - x² = 0 gives x = -2 or 2

Hence, (-2 ,0) & (2 ,0) are coordinates of A& B

Replace x by x + y in equation (i), we get 

y = 4- (x+ 4)²

⇒ 4 - (4 + 4)²  = 0 gives intersection points with x- axis

⇒ (x+4)²  4 = (+/- 2)² 

⇒ x + 4 = -2 or x+4 = 2

⇒ x = -6 or x = -2

Hence, (-2,0)  & (-6,0) are coordinates of B & C

∴ B = (-20) , A (2,0)  C & (-6,0)

∴ abscissa of c will be -6

C) What's wrong with it?

C) What did Mark begin to do?

1. or 

2. nor 

3. for 

4. so 

5. yet

1. can’t walk fast 

2. they were absent 

3. I did not pick 

4. I will not take you to the fair 

5. I will open the door 

6. the programme got over soon 

7. cancel the order 

8. not shrewd

D) Why did mother punish Michael?

1. Could you please tell me the way to the railway station?

2. Would you lend me your bike?

3. Could you exchange this book

4. Could you please open our classroom window?

Correct option is C) a small amount

Correct option is A) want / lend

1. on 

2. on 

3. for 

4. during 

5. at 

6. at

Correct option is E) 5,3,1,4,2

Correct option is E) quitted

Correct option is A) had come back

Correct option is A) is shown

Correct option is A) do you have

Answer is (c) RTI

Correct option is C) drastic

Correct option is (c) violent

Correct option is B) don’t

Correct option is C) could / have beaten

a. There were heaps of papers and writings and a burning candle on the table.

b. Newton was away for a short time. Then his pet dog suddenly jumped upon the table. This caused the lighted candle to fall on the papers and they caught fire.

c. iii) Diamond

d. i) kind and cool tempered

e. iii) He patted the dog.

a. The elephant never stops learning because he is always thinking.

b. i) If he cannot reach with his trunk some part of his body that itches, he picks up a long stick and gives himself a good stretch with that instead. 

ii) If he pulls up some grass, and it comes up by .the roots with a lump of earth, he will smack it against his foot, until the earth is shaken off.

c. iii) To rub that part of his body that itches

d. ii) To shake off the grass

e. iii) He either smacks it against the foot or washes it clean.

Two ways of creating awareness.

Door to door campaigning

Nukkaa Natak/ any other method.

Detailed Answer:

Two ways to bring awareness on "how to save water" are:

(i) By organising a demonstration of the practices in the households on how we can minimize the wastage of water.

(ii) By organising an awareness week every year with poster competitions on "how to save water.

Correct option is A) happy