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We have sum of p terms i. e.
S_p = p/2 (2a+(n-1)d)=0
or d=-2a/(n-1)d
also
Sum of next q terms means (Sum of p+q terms) - ( sum of p terms)

or sum of p+q terms because sum of p terms is 0

=> p+q/2 ( 2a +(n-1)d)
=> p+q/2 (2a + (n-1)(-2a/(n-1)d)
or
=> -a (p+q)q/p-1

Therefore, sum of the next q terms is given by

S_p+q-S_p = -a (p+q)q/p-1

We have to find LCM and HCF, We can write

8 =1 x 2 x 2 x 2
9 =1 x 3 x 3
25=1 x 5 x 5
HCF =1
LCM = 1 x 2 x 2 x 2 x 3 x 3 x 5 x 5 = 1800

\(\frac{3x^3 - x^2 + x - 1}{3x^3 - x^2 - x + 1}\) \(= \frac{4x^3 + 7x^2 + x + 1}{4x^3 + 7x^2 - x - 1}\)

\(\Rightarrow\) \((3x^3 - x^2 + x - 1)\) \((4x^3 + 7x^2 - x - 1)\) \(= (4x^3 + 7x^2 + x + 1)\) \((3x^3 - x^2 - x + 1)\)

\(\Rightarrow\) \(12x^6 + 17x^5 - 6x^4\) \(+\, x^3 - 7x^2 + 1\) \(= 12x^6 + 17x^5 - 8x^4\) \(-\, x^3 + 5x^2 + 1\)

\(\Rightarrow\) \(8x^4 - 6x^4 + x^3 + x^3\) \(-\, 7x^2 - 5x^2 + 1 - 1 = 0\)

\(\Rightarrow\) \(2x^4 + 2x^3 - 12x^2 = 0\)

\(\Rightarrow\) \(x^4 + x^3 - 6x^2 = 0\)

\(\therefore\) sum of roots \(= \frac{-\text{Coefficient of}\,x^3}{\text{Coefficient of}\,x^4}\)

\(= \frac{-1}{1} = -1.\)

Answer is (b) Both the statement are true, but statement II is not the correct explanation of Statement I. 

Answer is 

I. (b) (e - 1)

II. (c) 1

III. (d) sin 2x

IV. (a) x(log x - 1) + x

Given differential equation is \(\frac{ydx - xdy}{y} = 0\)

⇒ dx - \(\frac{x}{y}\)dy = 0

⇒ dx = \(\frac{x}{y}\)dy ⇒ \(\frac{1}{x}\)dx = \(\frac{1}{y}\)dy

\(\int \) \(\frac{1}{x}\)dx = \(\frac{1}{y}\)dy (By integrating both sides of above equation)

⇒ log x = log y + log C, where log is an integral constant.

⇒ log x = log Cy ( \(\because\) log a + log b = log ab) 

⇒ x = Cy, which is general solution of given differential equation. 

(By taking anti-log both sides of above equation. )

Hence, the general solution of the differential equation \(\frac{y dx - x dx}{y} = 0\)is x = Cy.

Let one number be x. Then other number is (16 - x) 

Let S(x) = x³ +(16 - x)³ 

S¹ (x) = 3x² + 3(16 - x)² (-1) 

S¹¹(x) = 6x + 6(16 - x)¹(+1) 

S¹(x) = 0 gives, 3x² - 3 (16 - x)² =0 

⇒ x² - (16 - x)² = 0 

⇒ x² = (16 - x)² 

⇒ x² - 256 + x² - 32x 

⇒ 32x = 256 

x = 256/32

x = 8 

Also, S¹¹ (x) = 6(8) + 6(16 - 8) = 48 + 6(8)

= 48 + 48 = 96 > 0 

∴ By second derivative test, x = 8 is the point of local minima of s. 

Hence sum of cubes of numbers is minimum when the numbers are 8 and 16 - 8 = 8

Total number of cards in the deck = 52.

Total number of black suited 10 cards = 2. 

∴ Probability of drawing a 10 of a black suit 

= \(\frac{total\,number\,of\,black\,suited\,10\,cards}{total\,number\,of\,cards\,in\,the\,deck}\)

= \(\frac{2}{52}\) 

= \(\frac{1}{26}\).

Answer is (d) (1,0,0)

The correct option is: (c) 2πrh 

The following is not quadratic equation x² +1/x² = 2

Since, the number of arbitrary constant in the general solution and particular solution of n^th order differential equation are n and 0, respectively. 

Therefore, the number of arbitrary constant in the general solution and particular solution of fourth order and third order differential equation are 4, 0 and 3, 0, respectively.

Correct option is  B) in / at

Correct option is  B) in

Correct option is B) who

Correct option is  B) In / in

The value of a₁₁A₂₁ + a₁₂A₂₂ + a₁₃A₂₃ is 0

Equation of any plane through the intersection of the given planes, is of the form

3x - y + 2z - 4 + λ (x + y - z - 2) = 0 

(3 + λ)x + (λ - 1)y + (2 - λ) z - (4 + 2λ) = 0 

By Data it is passes through (2, 2, 1) 

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

⇒ 6 + 2λ + 2λ - 2 + 2 - λ - 4 - 2λ = 0 

⇒ 2 + λ = 0 

λ = - 2 

Thus the required equation is 

(3 - 2)x + (-2 -1 )y + (2 - 2)1 - (4 + 2(-2)) = 0 

x - 3y + (0)(1) - (4 - 4) = 0 

x - 3y = 0

→ R – {(xy): 2 divides (x - y)} 

→ R is Reflexive as 2 divides (a - a) ∀ a ∈ z. 

If (a b) ∈ R then 2 divides a - b. 

2 divides b - a Hence (ba) ∈ R 

⇒R is Symmetric. 

(ab) ∈ R and (be) ∈ R 

⇒ a - b and b-c are divisible by 2 

Now a - c = (a - b) + (b - c) = is even 

∴ a - c is divisible by 2. 

⇒ R is Transitive

R is Reflexive, Transitive, Symmetric 

∴ R is an Equivalence Relation in Z.

e^x(1 − cot x) + C

Laurent series is a power series that contains negative terms.

The method of Laurent series expansions is an important tool in complex analysis.

Laurent series allows us to work around the singularities of a complex function.

P.I. = \(\frac{1}{D^2+4}\) sin2x

= \(\frac{1}{D^2+4}\) \(\Big(\frac{1-cos2x}{2}\Big)\) 

= \(\frac{1}{2}\)   \(\frac{1}{D^2+4}\) e^x -  \(\frac{1}{2}\) x  \(\frac{1}{D^2+4}\) cos2x

= \(\frac{1}{8}-\frac{1}{2}\times \frac{x}{2x^2}\) sin2x

= \(\frac{1}{8}-\frac{x}{8}\) sin2x

= \(\frac{1}{8}\) (1-sin2x)

The higher the amount of deposit for a longer period in the bank the higher is the Rate of interest.

A cheque is a bill of exchange drawn on any bank on demand.

b. 14 days

A minimum of 14 days  notice is generally required for an Extra Ordinary General Meeting.

c. Open cheque

Open is a cheque that is not crossed on the left corner and payable at the counter of the drawee bank.

It Is a small handy book issued by the bank to a customer to record all the dealings between them.

The Secretary acts as a Liaison Officer Officer between the employer and the employee.

a. Cash Credit

Cash Credit is also called Bank Overdraft.

The second vote cast by the Chairman in order to break a tie.

Correct answer is b. Ruhaan’s statement is wrong, Rohini’s statement is right 

No Seconder is required if the motion in a meeting is moved by the Chairperson.

Liabilities are obligations of the company; they are amounts owed to creditors for a past transaction and they usually have the word "payable" in their account title. 1

(d) Under the head Reserves and Surplus as a negative item

Office cost is the summation of factory cost and Office and administrative overheads.