Sunday, July 28, 2019

Matrices and Determinants II

BCA

Practice Questions

Matrices and Determinants

Q1 Find A^-1

^{Q2 IF}

^{a) A+B, Check if A+B=B+A}
^{b) A-B}
c) B-A, Check if A-B=B-A

d) AXB, Check if AXB=BXA

e) (A)'

f ) A²
g) 3A + A²

Q 3 IF
$A=\begin{bmatrix} \frac{3}{2} & 2 &8 &1\\ 2& \frac{-1}{2} &5 &2\\ 1&3 & 11&3 \\ 1&1 & 9 & 1 \end{bmatrix} B=\begin{bmatrix} \frac{3}{5} & 2 &8& 2\\ 2& \frac{-3}{2} &5&1 \\ 1&3 & 9&5 \\ 1&3 & 9 &3 \end{bmatrix}$
a) A-B

b) A X B

Q4 Find the determinant of the following

$a) \begin{vmatrix} 1 &8 \\ 5&3 \end{vmatrix}$

$b) \begin{vmatrix} 1 &8&2 \\ 5&3&1 \\ 4&3&2 \end{vmatrix}$

$c) \begin{vmatrix} 1 &8&2 &1\\ 5&3&1&2 \\ 4&3&2&1\\ 4&3&2&1 \end{vmatrix}$

Solutions

Ans 1

$A^{-1}=\begin{bmatrix} \frac{-39}{2} &6 & 14\\ -13 &\frac{11}{2} &\frac{17}{2} \\ \frac{13}{2}& \frac{-5}{2} & \frac{-19}{2} \end{bmatrix}$

Ans 2

a)
$A+B=\begin{bmatrix} \frac{21}{10} & 4 & 16 \\ 4& -2 & 10\\ 2& 6 & 18 \end{bmatrix}$

$B+A=\begin{bmatrix} \frac{21}{10} & 4 & 16 \\ 4& -2 & 10\\ 2& 6 & 18 \end{bmatrix}$

hence A+B=B+A

b)
$A-B=\begin{bmatrix} \frac{9}{10} & 0 & 0 \\ 0& 1 & 0\\ 0& 0 & 0 \end{bmatrix}$

c)
$A-B=\begin{bmatrix} \frac{9}{10} & 0 & 0 \\ 0& 1 & 0\\ 0& 0 & 0 \end{bmatrix}$

$B-A=\begin{bmatrix} \frac{-9}{10} & 0 & 0 \\ 0& -1 & 0\\ 0& 0 & 0 \end{bmatrix}$

hence A-B is not equal to B-A

d)
$A X B=\begin{bmatrix} \frac{129}{10} & 24 & 94\\ \frac{26}{5}& \frac{79}{4} &\frac{117}{2} \\ \frac{78}{5}& \frac{49}{2} & 104 \end{bmatrix}$

$B X A= \begin{bmatrix} \frac{129}{10} & \frac{121}{5} & \frac{434}{5}\\ 5 & \frac{79}{4} & \frac{107}{2} \\ \frac{33}{2} & \frac{35}{2}& 104 \end{bmatrix}$

Hence AXB is not equal to BXA
e)
$A'=\begin{bmatrix} \frac{3}{2} &2 &1 \\ 2& \frac{-1}{2} &3 \\ 8& 5 &9 \end{bmatrix}$

f)
$A^2=\begin{bmatrix} \frac{57}{4} &26 &94 \\ 7& \frac{77}{4} &\frac{117}{2} \\ \frac{33}{2}& \frac{55}{2} &104 \end{bmatrix}$

g)

$3A+A^2=\begin{bmatrix} \frac{75}{4}& 32 & 118 \\ 13& \frac{71}{4} & \frac{147}{2}\\ \frac{39}{2}&\frac{73}{2} & 131 \end{bmatrix}$

Ans 3

a)
$A-B=\begin{bmatrix} \frac{9}{10} & 0 & 0 & -1\\ 0& 1 &0 &1 \\ 0 &0 &2 &-2 \\ 0& -2 & 0 & -2 \end{bmatrix}$

b)
$A X B=\begin{bmatrix} \frac{139}{10} & 27 & 103 & 48\\ \frac{36}{5}& \frac{103}{4} &\frac{153}{2} &\frac{69}{2} \\ \frac{103}{5}&\frac{79}{2} &149 &69 \\ \frac{63}{5}&\frac{61}{2} & 103 & 51 \end{bmatrix}$

Ans 4

a) -37

b)
$\frac{-13}{4}$

c) 0

Saturday, July 27, 2019

Result

Test Dated 27/07/2019

S#	Candidate Name	Topic	Max Marks 50
1	Ravkirat Gulati	Linear equations and quadrilateral	36
2	Rhea Khanna	Integers,data handling and lines & angles	31
3	Kunal	Rational no, square roots and cube roots	44

Thursday, July 25, 2019

Matrices and Determinants

Q1 If

$A=\begin{bmatrix} 1 &3 & 5\\ -6& 1 & 8\\ -1& 3 & 4 \end{bmatrix} B=\begin{bmatrix} 4 &-1 &3 \\ 2 & 5 &4 \\ -9& 2 & 6 \end{bmatrix}$

Find

a) A+B

b) A-B

c) AxB

d) Adj(A)

e) A^-1

^f) (A')

^{Q2 Find the product of the following matrices}

$a)\begin{bmatrix} 3 \\ -1 \\ 9 \end{bmatrix} \begin{bmatrix} 1 & 8 & 6 \end{bmatrix}$

$b)\begin{bmatrix} 1 & 8 & 6 \end{bmatrix} \begin{bmatrix} 3 \\ -1 \\ 9 \end{bmatrix}$

$c) \begin{bmatrix} 1 & 3 &2 \\ -1& 7 &-1 \\ 2& 3& 6 \end{bmatrix} \begin{bmatrix} 2 & 5& 6\\ 1& 3& -2 \end{bmatrix}$

$d) \begin{bmatrix} 1 & 6 &2 \\ 2 & -4 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 &6 \\ 4& 3 & 1\\ 8&6 & 9 \end{bmatrix}$

$e) \begin{bmatrix} 1 & 6 &2 \\ \end{bmatrix} \begin{bmatrix} 1 & 2 &6 \\ 4& 3 & 1\\ 1&6 & 9 \end{bmatrix}$

Q3 Solve the following equation using cramer's rule

2x +   y + z = 3
  x –   y – z = 0
  x + 2y + z = 0

Q4 Find the determinant of the following

$a) \begin{vmatrix} -1 & 4\\ 2& 3 \end{vmatrix}$

$b) \begin{vmatrix} 1 & 3 & -2\\ 6 & 1 & 2\\ 4 & 3 & 7 \end{vmatrix}$

Solutions

Ans 1

$a) \begin{bmatrix} 5 & 2 & 8\\ -4 & 6 & 12\\ -10 & 5 & 10 \end{bmatrix}$

$b) \begin{bmatrix} -3 & 4 & 2\\ -8 & -4 & 4\\ 8 & 1 & -2 \end{bmatrix}$

$c) \begin{bmatrix} -35 & 24 & 45\\ -94 & 27 & 34\\ -34 & 24 & 33 \end{bmatrix}$

$d) \begin{bmatrix} -20 & 3 & 19\\ 16 & 9 & -38\\ -17 & -6 & 19 \end{bmatrix}$

$e) \frac{1}{57}\begin{bmatrix} -20 & 3 & 19\\ 16 & 9 & -38\\ -17 & -6 & 19 \end{bmatrix}$

$f) \begin{bmatrix} 1 & -6 & -1\\ 3 & 1 & 3\\ 5 & 8 & 4 \end{bmatrix}$

Ans 2

$a) \begin{bmatrix} 3 & 24 & 18\\ -3 & -8 & -6\\ 9 & 72 & 54 \end{bmatrix}$

b) [49]

c) Not defined

$d) \begin{bmatrix} 41 & 32 & 30\\ -6 & -2 & 17\\ \end{bmatrix}$

$e) \begin{bmatrix} 27 & 32 & 30\\ \end{bmatrix}$

Ans 3

x=1, y=-2, z=3

Ans 4

a) -11 b) -129

Tuesday, July 23, 2019

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Wednesday, July 17, 2019

Polygons

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Grade 8

Polygons, Quadrilaterals and Parallelograms

Practice Questions

Q1 Find the measure of exterior angle of a regular polygon of

a)8 sides b) 5 sides c) 12 sides

Q2 Is it possible to have a regular polygon each of whose exterior angles is 25 degree?

Q3 What is the minimum interior angle possible for a regular polygon?

Q4 Find the measure of each interior angle of a regular polygon.
a)Triangle b) Polygon with 13 sides c) Polygon with 11 sides

Q5 What is the sum of all interior angles of a polygon of following sides?
a) 5 sides b) 9 sides c) k+1 sides

Q6 What is the number of diagonals in the following..?

a)Octagon b) Decagon c) Polygon with 14 sides

Q7 Find the number of sides of a regular polygon whose each exterior angle measures 60 degree.

Q8 Find the measure of exterior and interior angle of octagon.

Q9 The four angles of Quadrilateral are in ratio of 2:3:5:8. Find each angle of Quadrilateral.

Q10 The measure of two angles of Quadrilateral are 110 and 70 degree. Also the other two angles are equal. Find the measure of the equal angles.

Q 11 Prove that the sum of the angles of a quadrilateral is 360 degree.

Q12 Three angles of Quadrilateral are equal and the fourth angle is of 60 degrees. Find the measure of three remaining angles.

Q13 ABCD is a parallelogram in which $\angle A$ = 100 degree. Find the measure of each of the remaining angles( $\angle B \angle C \angle D$ )

Q14 If one adjacent angle is half the other. Find the measure of all the angles of a parallelogram.

Q15 If the two adjacent angles of Parallelogram are equal. What is the measure of each of these angles.

Q16 The sum of two opposite angles of a parallelogram is 120 degrees. Find the measure of each of its angle.

Q17 Two adjacent angles of a Parallelogram are (2x+8) and (4x+4). Find the measure of each angle .

Q18 The length of diagonals of Rhombus are 6cm and 8 cm respectively. Find the length of each side.

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Sunday, July 28, 2019

Matrices and Determinants II

Saturday, July 27, 2019

Result

Thursday, July 25, 2019

Matrices and Determinants

Tuesday, July 23, 2019

Courses@INFATE Academy

INFATE Academy

Wednesday, July 17, 2019

Polygons

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