This document is a question paper for an Applied Mathematics exam covering various topics in linear algebra, vector calculus, differential equations, and multivariable calculus. It contains 7 main questions with multiple parts each, testing skills like matrix operations, vector/multivariable calculus concepts, solving differential equations, and applying calculus theorems. Students have 3 hours to complete as many questions and parts as possible from the paper.

1. A p p l i e d M a t h e m a t i c s – I
Q u e s t i o n P a p e r ( A p r i l – 2 0 1 4 ) [ R e v i s e d C o u r s e ]
1 | Page
M u m b a i B . S c . I T S t u d y
F a c e b o o k | T w i t t e r | I n s t a g r a m | G o o g l e + | Y o u T u b e
– Kamal T.
Time: 3 Hours Total Marks: 100
N.B.: (1) All Questions from 1 to 7 are Compulsory.
Q.1 Solve: (10 Marks)
(A) Given that 𝐴 = [
1 −1
2 3
]. Find 𝐴2
− 4𝐴 + 5𝑖 (5)
(B) Solve: (2𝑥3
+ 3𝑦)𝑑𝑥 + (3𝑥 + 𝑦)𝑑𝑦 = 0 (5)
Q.2 Solve: (15 Marks)
(A)
Define Unitary Matrix. Show that A is Unitary where 𝐴 = [
1(1+0)
2
−
1(1−0)
2
1(1+0)
2
1(1−0)
2
]
(10)
(B)
Determine the rank of the matrix 𝐴 = [
1 2 3
2 4 6
−3 −6 −9
]
(5)
OR
(C)
Find the inverse of the matrix using adjoint method 𝐵 = [
2 1 3
3 1 2
1 2 3
]
(10)
(D) Show that the matrix is Skew-Hermitian 𝐵 = [
3𝑖 2 ÷ 𝑖
−2 + 𝑖 −𝑖
] (5)
Q.3 Solve: (15 Marks)
(A) Is the vector Linearly dependent? If, so, find the relation between them
(3, 2, 7) (2, 4, 1) (1, −2, 6)
(10)
(B) Find Eigen Value and Eigen Vector of the matrix 𝐴 = [
1 −2
−5 4
] (5)
OR
(C)
Verify Cayley Hamilton theorem for [𝐴 =
1 2 2
2 1 2
2 2 1
]
(10)
(D)
Find Eigen Value and Eigen vector of the matrix 𝐴 = [
1 −6 −4
0 4 2
0 −6 −3
]
(5)
Q.4 Attempt Any Three From The Following: (15 Marks)
(A) If 𝐹 = (𝑥 + 𝑦 + 1)𝐼 + 𝐽 − (𝑥 + 𝑦) 𝐾, that show that F.curl F =- 0 (5)
(B) If A is a constant vector and 𝑅 = 𝑥𝑖 = 𝑦𝐽 + 2𝐾. Prove that, than
(i) 𝑔𝑟𝑎𝑛𝑑(𝐴. 𝑅) = 𝐴
(ii) 𝑑𝑖𝑣(𝐴 × 𝑅) = 0
(5)
(C) Show that the vector is solenoidal (𝑥 + 3𝑦) + (𝑦 − 3𝑧) + (𝑥 − 2𝑧)𝐾 (5)
(D) Prove that 𝑐𝑢𝑟𝑙 [(𝐴. 𝑅)𝑅] = 𝐴 ∗ 𝑅 where A is constant vector. (5)

2. A p p l i e d M a t h e m a t i c s – I
Q u e s t i o n P a p e r ( A p r i l – 2 0 1 4 ) [ R e v i s e d C o u r s e ]
2 | Page
M u m b a i B . S c . I T S t u d y
F a c e b o o k | T w i t t e r | I n s t a g r a m | G o o g l e + | Y o u T u b e
– Kamal T.
Q.5 Attempt Any Three From The Following: (15 Marks)
(A) Solve: (2𝑥 − 𝑦 + 1)𝑑𝑥 + (2𝑦 − 𝑥 − 1)𝑑𝑦 = 0 (5)
(B) Solve: 𝑦2
+ 𝑥2 𝑑𝑦
𝑑𝑥
= 𝑥𝑦
𝑑𝑦
𝑑𝑥
(5)
(C) Solve: (𝑥2
+ 𝑦2)𝑑𝑥 − 2𝑥𝑦𝑑𝑦 = 0 (5)
(D) If the population of town doubles in 20 years, in how many years will it triple under
the assumption that the rate of increase is proportional to the number of inhabitants?
(5)
Q.6 Attempt Any Three From The Following: (15 Marks)
(A) Verify Rolle’s Theorem for 𝑓(𝑥) = (𝑥2
− 1)(𝑥 + 28) in [4, 7] if possible. (5)
(B) Find 𝑛 𝑡ℎ
derivatives of the 𝑦 = 𝑐𝑜𝑠ℎ
𝑥. (5)
(C) Verify Lagranges Mean Value Theorem for 𝑓(𝑥) = 2𝑥 − 𝑥2
in [0, 1] (5)
(D) Verify Cauchy MVT for 𝑓(𝑥) = 𝑥3
= 4𝑥 and 𝑔(𝑥) = 𝑥2
+ 1 in [0, 1] (5)
Q.7 Attempt Any Three From The Following: (15 Marks)
(A) Find the maximum and minimum values of 𝑧 = 𝑓(𝑥, 𝑦) = 𝑥3
+ 𝑦3
− 3𝑎𝑥𝑦 (5)
(B) Divide 640 into three parts such that the sum of their products taken two at a time is
maximum.
(5)
(C) Find the percentage error in the area of an ellipse when an error of 1.5% and 2% is
made in measuring its major axis and minor axis respectively.
(5)
(D) If 𝑢 = log(𝑥3
+ 𝑥𝑦 − 𝑦2) show that 𝑥
𝑑𝑦
𝑑𝑥
+ 𝑦
𝑑𝑥
𝑑𝑦
= 2 (5)