This document contains 12 mathematics questions involving matrix operations such as finding the inverse of a matrix, solving systems of linear equations using matrices, and matrix multiplication. Each question provides the necessary matrix equations and asks the student to find specific values or matrices based on the given information. The student provides step-by-step working to show the calculations and reasoning to arrive at the answers.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Derivation and Application of Six-Point Linear Multistep Numerical Method for...IOSR Journals
A six-step Continuous Block method of order (5, 5, 5, 5, 5, 5) T is proposed for direct solution of the second (2nd) order initial value problems. The main method and additional ones are obtained from the same continuous interpolant derived through interpolation and collocation procedures. The methods are derived by interpolating the continuous interpolant at 𝑥 = 𝑥𝑛+𝑗 , 𝑗 = 6 and collocating the first and second derivative of the
continuous interpolant at 𝑥𝑛+𝑗 , 𝑗 = 0 and 𝑗 = 2, 3, … 5 respectively. The stability properties of the methods are discussed and the stability region shown. The methods are then applied in block form as simultaneous numerical integrators. Two numerical experiments are given to illustrate the efficiency of the new methods.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Derivation and Application of Six-Point Linear Multistep Numerical Method for...IOSR Journals
A six-step Continuous Block method of order (5, 5, 5, 5, 5, 5) T is proposed for direct solution of the second (2nd) order initial value problems. The main method and additional ones are obtained from the same continuous interpolant derived through interpolation and collocation procedures. The methods are derived by interpolating the continuous interpolant at 𝑥 = 𝑥𝑛+𝑗 , 𝑗 = 6 and collocating the first and second derivative of the
continuous interpolant at 𝑥𝑛+𝑗 , 𝑗 = 0 and 𝑗 = 2, 3, … 5 respectively. The stability properties of the methods are discussed and the stability region shown. The methods are then applied in block form as simultaneous numerical integrators. Two numerical experiments are given to illustrate the efficiency of the new methods.
Research Inventy : International Journal of Engineering and Scienceinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Research Inventy : International Journal of Engineering and Scienceinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
APEX INSTITUTE was conceptualized in May 2008, keeping in view the dreams of young students by the vision & toil of Er. Shahid Iqbal. We had a very humble beginning as an institute for IIT-JEE / Medical, with a vision to provide an ideal launch pad for serious JEE students . We actually started to make a difference in the way students think and approach problems.
CBSE Board Class 10 Previous Year Maths Paper 2007 SolutionMATHS BLOGS
CBSE Board Class 10 Previous Year Maths Paper 2007 Solution. It is the solution of the previous year 2007 Maths Paper solution that helps students of the class 10.
1. [ SM. ST. PETER TELIPOK ] [ LEE CHIONG TEE ] [ MATHEMATICS 1449 / 2 ]
109
Q10 MATRICES
1 (a) Cari nilai k, jika
62
1 k
tidak mempunyai songsangan.
(b) (i) Find the inverse matrix of
62
21
.
(ii) Hence, using matrices, find the value of x and of y that satisfy the following matrix equation :
62
21
y
x
=
4
1
. (Ans : x = 1, y = 1)
[6 marks]
Answer :
(a)
(b) (i)
(ii)
2. [ SM. ST. PETER TELIPOK ] [ LEE CHIONG TEE ] [ MATHEMATICS 1449 / 2 ]
110
2 Given that matrix P =
31
4k
and matrix Q =
11
53
.
(a) Find the value of k, if P does not have an inverse. (Ans : 3
4 )
(b) Find the inverse matrix of Q. (Ans :
2
3
2
1
2
5
2
1
)
(c) If Q
y
x
=
2
14
, find the values of x and y using matrices. (Ans : x = 2, y = 4)
[6 marks]
Answer :
(a)
(b)
(c)
3. [ SM. ST. PETER TELIPOK ] [ LEE CHIONG TEE ] [ MATHEMATICS 1449 / 2 ]
111
3 (a) It is given that
m3
25
is the inverse matrix of
n3
21
.
Find the value of m and of n. (Ans : m = 1, n = 5)
(b) Write the following simultaneous linear equations as matrix equation :
5x + 2y = 4
3x y = 3
Hence, calculate the value of x and of y. (Ans : x = 2, y = 3)
[6 marks]
Answer :
(a)
(b)
4. [ SM. ST. PETER TELIPOK ] [ LEE CHIONG TEE ] [ MATHEMATICS 1449 / 2 ]
112
4 (a) The inverse matrix of
nm
1
2
1
is
13
24
.
Find the values of m and n. (Ans : m = 2
3
, n = 2)
(b) Using matrices, calculate the value of x and of y that satisfy the following simultaneous linear
equations :
4x 2y = 10
3x y = 6 (Ans : x = 1, y = 3)
[6 marks]
Answer :
(a)
(b)
5. [ SM. ST. PETER TELIPOK ] [ LEE CHIONG TEE ] [ MATHEMATICS 1449 / 2 ]
113
5 (a) The inverse matrix of
65
43
is m
35
6 p
.
Find the value of m and of p. (Ans : m = 2
1
, p = 4)
(b) Using matrices, calculate the value of x and of y that satisfy the following simultaneous linear
equations :
3x 4y = 1
5x 6y = 2 (Ans : x = 7, y = 5.5)
[6 marks]
Answer : [2004, No.8]
(a)
(b)
6. [ SM. ST. PETER TELIPOK ] [ LEE CHIONG TEE ] [ MATHEMATICS 1449 / 2 ]
114
6 Given that matrix
1
2
n
m
M .
(a) If M1 =
2
31
8
1
n
, find the values of m and n. (Ans : m = 3, n = 2)
(b) Hence, find the value of h and of k that satisfy the matrix equation
8
16
k
h
M using matrices.
(Ans : h = 5, k = 2)
[6 marks]
Answer :
(a)
(b)
7. [ SM. ST. PETER TELIPOK ] [ LEE CHIONG TEE ] [ MATHEMATICS 1449 / 2 ]
115
7 Given that matrix M =
25
36
, and the inverse matrix of M is
65
3
15
1 a
ab
.
(a) Find the value of a and of b. (Ans : a = 2, b = 6)
(b) Hence, using matrices, find the value of e and of f that satisfy the matrix equation
2
0
f
e
M .
(Ans : e = 2, f = 4)
[6 marks]
Answer :
(a)
(b)
8. [ SM. ST. PETER TELIPOK ] [ LEE CHIONG TEE ] [ MATHEMATICS 1449 / 2 ]
116
8 Given that the simultaneous linear equations, 2p + 7q = 2 dan 3p + 8q = 3 is write as F
3
2
q
p
, where
F is a matrix.
(a) Find matrix F. (Ans :
83
72
)
(b) Given that
3
2
3
7
5
1
n
m
q
p
.
(i) Find the value of m and of n. (Ans : m = 8, n = 2)
(ii) Hence, using matrices, find the values of p and q. (Ans : p = 5
37
, q = 5
12 )
[6 marks]
Answer :
(a)
(b) (i)
(ii)
9. [ SM. ST. PETER TELIPOK ] [ LEE CHIONG TEE ] [ MATHEMATICS 1449 / 2 ]
117
9 (a) Given that matrix R is
23
47
.
Find matrix Q such that RQ =
10
01
. (Ans :
2
7
2
3
21
)
(b) Using matrices, find the value of p and of q that satisfy the following simultaneous linear equations :
7p + 4q = 13
3p + 2q = 7 (Ans : ; p = 1, q = 5)
[6 marks]
Answer :
(a)
(b)
10. [ SM. ST. PETER TELIPOK ] [ LEE CHIONG TEE ] [ MATHEMATICS 1449 / 2 ]
118
10 Given that matrix P =
32
53
.
(a) If PQ = QP = I, find matrix Q. (Ans :
32
53
(b) Hence, using matrices, find the value of x and of y that satisfy the following simultaneous linear
equations :
3s + 5t = 8
2s + 3t = 6 (Ans : s = 6, t = 2)
[6 marks]
Answer :
(a)
(b)
11. [ SM. ST. PETER TELIPOK ] [ LEE CHIONG TEE ] [ MATHEMATICS 1449 / 2 ]
119
11 Given that matrix P =
12
31
, matrix R =
1
311
km
, and matrix PR =
10
01
.
(a) Find the value of k and of m. (Ans : m = 5, k = 2)
(b) Hence, find the value of x and of y that satisfy the following matrix equation :
5
5
12
31
y
x
(Ans : x = 2, y = 1)
[6 marks]
Answer :
(a)
(b)
12. [ SM. ST. PETER TELIPOK ] [ LEE CHIONG TEE ] [ MATHEMATICS 1449 / 2 ]
120
12 (a) Given that
10
01
62
7
42
761 s
r
.
Find the value of r and of s. (Ans : r = 10, s = 4)
(b) Hence, using matrices, find the value of x and of y that satisfy the following matrix equation :
6
3
42
76
y
x
(Ans : x = 3, y = 3)
[6 marks]
Answer :
(a)
(b)