# Past Papers Calculus 2

[OBJECTIVE]

Subject: Calculus-II

Time Allowed: 15 Minutes

Max Marks: 10

NOTE: Attempt this Paper on this Question Sheet only. Please encircle the correct option. Division of marks is given in front of each question. This Paper will be collected back after expiry of time limit mentioned above.

Part-I  Answer the following Questions, cutting and overwriting are not allowed. (10)

The sum of first six terms of the series 1+7+13+… is

12 b. 24

48 d. 96

2. H1,H2…,Hn are said to be n harmonic means between a and b if a,H1,H2…,Hn,b form

H.P b. A.P

G.P d. Harmonic series

If a and d are the first term and the common difference of the A.P respectively, then the nth term of corresponding H.P is

an = a+(n-1)d b. an = Iat(n-1)d

an = a/1+(n-1)d d. an = a/at{(n-1)d

The sum of an infinite geometric series exist only if the condition on the common ratio r is

-1< r <l

-1 ≤ r ≤

r< -1,r >1

r ≤ -1, r ≥ 1

The sum of the series 1+1/2+1/22+ … is

2 b. 2/3

4/3 d. 10/9

1,8,15,22,29,36,… is

G.P B. AP

Geometric series D. arithmetic series

7. The A.P whose nth term is 2n-1 is

1,3,6,… B. 2,3,5,…

1,3,5,… D. 5,3,1,…

The arithmetic mean between a and b is

a-b/2 B. b-a/2

a+b/2 D. +V(ab)

If an+2 + bn+2/a n+2 + b n+1 is the geometric mean between 8 and b, then n =

1 B. 1

-2 D. -0.5

The geometric mean between -2 and 8 are

±4 B. ±3

±2 D. ±1

[SUBJECTIVE]

Subject: Calculus-II

Time Allowed: 2 Hours 45 Minutes

Max Marks: 50

NOTE: ATTEMPT THIS (SUBJECTIVE) ON A SEPARATE ANSWER SHEET PROVIDED

Part-II  Give Short Answers, Each question carries equal marks. (20)

Part-III  Give Long Answers, Each question carries equal marks. (30)

Q#1. State and Prove the Euler’s Theorem.

Q#2. Investigate the behavior of the series:

Q#3.

(a) State and Proof any theorem regarding the differentiable implicit functions.

(b) Find the slope of the tangent to the hyperbola x2 = 4xy-3y2-9 at the point (2, -1).

[OBJECTIVE]

Subject: Mathematics B-III [Calculus (II)]

Time Allowed: 10 Minutes

Maximum Marks: 10

NOTE: Attempt this Paper on this Question Sheet only. Please encircle the correct option. Division of marks is given in front of each question. This Paper will be collected back after expiry of time limit mentioned above.

Part-I Encircle the right answer, cutting and overwriting is not allowed. (10)

[SUBJECTIVE]

Subject: Mathematics B-III [Calculus (II)]

Time Allowed: 2 Hours 45 Minutes

Maximum Marks: 50

NOTE: ATTEMPT THIS (SUBJECTIVE) ON THE SEPARATE ANSWER SHEET PROVIDED.

Q#1: Define triple integral with an example.

Q#2: Define Alternating series test and give an example.

Q#3: Find equation of tangent plane at any point P(x1,y1,z1) of elliptic parabola z = a2 + 4y2.

Q#4: Work out the critical points of f(x,y) = 2x2 — 4x + xy2 – 1.

Q#5: Investigate the behavior of the series

Prof.Fazal Rehman Shamil (Available for Professional Discussions)
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