Class notes BT 102

SAGAR INSTITUTE OF SCIENCE & TECHNOLOGY(SISTec)
GANDHINAGAR, BHOPAL

FIRST YEAR CELL

BRANCH ALL QUESTION BANK
SESSION 2024-25


SUBJECT:ENGINEERINGMATHEMATICS-1SUBJECT CODE:BT102




UNIT - 5

BLOOM’S COURSE
S.NO QUESTION TAXONO OUTCOM
MYLEVEL E
1 Find the normal form of the matrix and hence find its rank and 1 [REMEMBER] CO5
nullity:
2 3 4 5
3 4 5 6
[ ]
4 5 6 7
9 10 11 12
2 Find the rank & nullity of the matrix 1 [REMEMBER] CO5
2 3 −1 −1
1 −1 −2 −4
[ ]
3 1 3 −2
6 3 0 −7
3 Test the consistency of the following equations and solve: 2 [ANALYZE] CO5
3 x + 3y + 2z = 1, x + 2y = 4, 10y + 3z = −2, 2x − 3y − z = 5.
4 Test for consistency and solve it : 2 [ANALYZE] CO5
5x + 3y + 7z = 4, 3x + 26y + 2z = 9, 7x + 2y + 10z = 5.
5 Investigate the value of  and  so that the equations: 2 [ANALYZE] CO5
2x + 3y + 5z = 9, 7 x + 3y – 2z = 8, 2𝑥 + 3𝑦 + 𝑧 = 𝜇
have (a) No solution (b) A unique solution (c) An infinite no. of
solutions.
6 For what values of k the equations: x + y + z = 1, x + 2y + 4z =k, x 1 [REMEMBER] CO5
+ 4y + 10z =𝑘 2 have a solution and solve completely each case.
7 Find the Eigen values and Eigen vectors of the following matrix 1 [REMEMBER] CO5

, 8 −6 2
[−6 7 −4].
2 −4 3
8 Find the Characteristic roots and characteristic vectors of the matrix 1 [REMEMBER] CO5
2 1 1
[1 2 1]
0 0 1
9 Find the Characteristic equation and VerifyCayley Hamilton 1 [REMEMBER] CO5
Theoremfor the matrix
4 3 1
A=[ 2 1 −2] and Hence compute A-1.
1 2 1
10 Show that the following matrix satisfies Cayley-Hamilton Theorem and 1 [REMEMBER] CO5
5 4
hence obtain the inverse of the given matrix [ ]
1 2
11 Diagonalise the matrix 1 [REMEMBER] CO5
2 −2 3
A=[1 1 1]
1 3 −1
12 1 [REMEMBER] CO5
6 −2 2
If 𝐴 = [ −2 3 −1]then find a matrix P that diagonalizes A, and
2 −1 3
determine 𝑃 −1 𝐴𝑃.
UNIT-1

1 1 3 (Apply) CO1
Verify Lagrange’s Mean value theorem for the 𝑓(𝑥) = 𝑙𝑜𝑔𝑥 𝑖𝑛 [2 , 2]

2 Verify Lagrange’s mean value theorem for the function 𝑓(𝑥) = 3 (Apply) CO1
𝑥−1
𝑖𝑛 (4, 5).
𝑥−3
3 Verify Rolle’s theorem for 𝑓(𝑥) = 𝑥 3 − 4. 3 (Apply) CO1

4 𝑠𝑖𝑛𝑥 3 (Apply) CO1
Check Rolle’s theorem for 𝑓(𝑥) = 𝑖𝑛 (0, 𝜋).
𝑒𝑥

5 Verify Cauchy’s mean value theorem for 𝑓(𝑥) = 𝑠𝑖𝑛𝑥, 𝑔(𝑥) = 3 (Apply) CO1
𝑐𝑜𝑠𝑥 𝑖𝑛 [−𝜋/2,0].
−1 𝑥
6 Expand 𝑒 𝑎𝑠𝑖𝑛 in ascending power of 𝑥 3 (Apply) CO1

7 Expand log 𝑥 in powers of (𝑥 − 1)and hence evaluate log 1.1 . 3 (Apply) CO1

8 𝜋 3 (Apply) CO1
Expand tan (𝑥 + 4 ) as far as the term 𝑥 4 and evaluate tan (46.5°) to
four significant digits.
9 Calculate the approximated values of Cos 640 3 (Apply) CO1

10 Calculate the approximated value of √10 3 (Apply) CO1

11 Apply Maclaurin’s theorem to prove that 3 (Apply) CO1
𝑥2 𝑥4 𝑥6
𝑙𝑜𝑔𝑠𝑒𝑐𝑥 = + 12 + 45 + ⋯
2

,12 If u= log (𝑥 3 + 𝑦 3 + 𝑧 3 − 3𝑥𝑦𝑧) then prove that 3 (Apply) CO1
∂u ∂u ∂u 3 ∂ ∂ ∂ 2 −9
(i) ∂x + ∂y + ∂z = x+y+z (ii) (∂x + ∂y + ) 𝑢 = (𝑥+𝑦+𝑧)2
∂z

13 𝑦 𝜕𝑢 𝜕𝑢 3 (Apply) CO1
If 𝑢 = 𝑓 (𝑥 ) 𝑡ℎ𝑒𝑛 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝑥 𝜕𝑥 + 𝑦 𝜕𝑦 = 0

14 𝑥 2 +𝑦 2 3 (Apply) CO1
If 𝑢 = 𝑡𝑎𝑛−1 𝑡ℎ𝑒𝑛 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡
𝑥+𝑦
𝜕𝑢 𝜕𝑢
(𝑖) 𝑥 +𝑦 = 𝑠𝑖𝑛2𝑢
𝜕𝑥 𝜕𝑦
𝜕 2𝑢 𝜕 2𝑧 𝜕 2𝑢
(𝑖𝑖)𝑥 2 2 + 2𝑥𝑦 + 𝑦 2 2 = 𝑠𝑖𝑛4𝑢 − 𝑠𝑖𝑛2𝑢
𝜕𝑥 𝜕𝑥𝜕𝑦 𝜕𝑦

15 If 𝑧(𝑥 + 𝑦) = 𝑥 2 + 𝑦 2 𝑡ℎ𝑒𝑛 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 3 (Apply) CO1
∂z ∂z ∂z ∂z
( − )² = 4(1 − − )
∂x ∂y ∂x ∂y
16 Find the Maxima or Minima of the function 𝑢 = 𝑥 3 + 𝑦 3 − 3𝑎𝑥𝑦. 3 (Apply) CO1

17 1 1 3 (Apply) CO1
Find the Maxima or Minima of the function 𝑢 = 𝑥𝑦 + 𝑥 + 𝑦.

18 Find the Maxima or Minima of the function 3 (Apply) CO1
𝑢 = 𝑠𝑖𝑛𝑥 + 𝑠𝑖𝑛𝑦 + 𝑠𝑖𝑛(𝑥 + 𝑦)
19 Find the Maxima or Minima of the function 3 (Apply) CO1
𝑢 = 𝑥 3 + 𝑦 3 + 120𝑥𝑦 − 63(𝑥 + 𝑦).
20 Find the Taylor’s series for 𝑒 𝑥 𝑠𝑖𝑛𝑦 in powers of x and y, x=y=0. 3 (Apply) CO1

UNIT_-2

𝜋
1 3 [APPLY] CO2
Apply Ab initio method to Evaluate ∫02 𝑠𝑖𝑛𝑥𝑑𝑥 .

2 𝑏 3 [APPLY] CO2
Apply Ab initio method to Evaluate ∫𝑎 𝑒 𝑥 𝑑𝑥

3 3 3 [APPLY] CO2
Apply Ab initio method to Evaluate ∫1 (𝑥 2 + 𝑥)𝑑𝑥

4 Apply Ab initio method to Evaluate 3 [APPLY] CO2
1
1 22 32 𝑛2 𝑛
lim [(1 + ) (1 + ) (1 + ) … (1 + )]
𝑛→∞ 𝑛2 𝑛2 𝑛2 𝑛2
1 3 [APPLY]
5 𝑛! 𝑛 CO2
Apply Ab initio method to Evaluate lim { }
𝑛→∞ 𝑛𝑛
6 Apply Ab initio method to Evaluate the limit 3 [APPLY] CO2
1
when 𝑛 → ∞ 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑒𝑟𝑖𝑒𝑠 ∑𝑛−1
𝑟=1 √𝑛2 2 −𝑟

7 Apply definition of Beta and Gamma function and State and prove 3 [APPLY] CO2
the relationship between Beta and Gamma function.

, 8 Apply definition of Beta and Gamma function and Prove that 3 [APPLY] CO2
1 √π
Γ(m) Γ (m + ) = 2m−1 Γ(2m) where m > 0
2 2

9 Apply definition of Beta and Gamma function and 3 [APPLY] CO2
∞ 𝑥𝑐 Γ(c+1)
Prove that ∫0 𝑐 𝑥 𝑑𝑥 = (log 𝑐)𝑐+1 , 𝑐 > 1

10 Apply definition of Beta and Gamma function 3 [APPLY] CO2
5
∞
toEvaluate∫0 𝑒 −4𝑥 . 𝑥 𝑑𝑥
2

11 3 [APPLY] CO2
Apply the concept of double integral to Evaluate ∬𝐷 𝑥 2 𝑦 2 𝑑𝑥𝑑𝑦 ,
where D is the region bounded by
𝑥 = 0, 𝑦 = 0 𝑎𝑛𝑑 𝑥 2 + 𝑦 2 = 1, 𝑥 > 0, 𝑦 > 0.
12 3 [APPLY] CO2
Apply the concept of double integral to Evaluate∬𝑅 𝑒 2𝑥+3𝑦 𝑑𝑥𝑑𝑦
over the triangle bounded by x=0,y=0 and x+y=1.
13 Apply the concept of double integral and Evaluate the following 3 [APPLY] CO2
1 √2−𝑥 2 𝑥𝑑𝑦𝑑𝑥
integral by changing the order of integration∫0 ∫𝑥
√𝑥 2 +𝑦 2
14 Apply the triple integration to find the volume of the sphere 3 [APPLY] CO2
𝑥 2 + 𝑦 2 + 𝑧 2 = 𝑎2
MODULE3:


Find the Fourier series to represent the function f(x) = x 2 , 3 [APPLY] CO3
1 1 l4
−l < x < l, hence show that ∑∞
n=1 n4 = 90

2 Find Fourier series of the function (x − x 2 ) in the interval (−π, π). 3 [APPLY] CO3
π2 1 1 1 1
Also deduce that = 12 – 22 + 32 – 42 ………
12

3 Expand f(x) = x. sinx, 0 < 𝑥 < 2π in a Fourier series. 3 [APPLY] CO3


4 Find the half range sine series of the function πx − x 2 , in 3 [APPLY] CO3
0 < 𝑥 < π.
5 1. Expess f ( x) = x as a half range sine series and half range 3 [APPLY] CO3
2. cosine series in 0< x <2

6. 3. Find the Fourier series for the periodic function f(x) defined as 3 [APPLY] CO3
−π, −π < x < 0
f(x) = {
x, 0 < x < π

7 Obtain the Fourier series of the function f ( x) =| x | when −  x   3 [APPLY] CO3


8 3 [APPLY] CO3
1 2  3
Test the convergence of the series  3  +  3  +  3  + .....
2  3  4 

9 Test the convergence of the series , where x is positive : 3 [APPLY] CO3