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Math Assignment Class XII Ch-04 | Determinants

 Math Assignment  Class XII  Ch - 04
DETERMINANTS

Extra questions of chapter 04 Determinants class XII  with answers and  hints to the difficult questions, strictly according to the CBSE syllabus. Important and useful math. assignment for the students of class XII

Strictly according to the CBSE Board

Question 1

Find the value of k for which matrix A is a singular matrix



Ans: 土 4
Question 2

If A is a square matrix of order 3 and |A| = - 4, then find the value of |adj A|

Ans: 16

Question 3

,

and |A3| = 27, then find the value of α

Ans: 土√7
Question 4

equation , then find the value of x

Ans : x = 9
Question 5


equation equation
then verify that |AB| = |A||B|

Solution Hint: |AB| = 77 and |A||B| = 7 x 11 = 77

Question 6
Evaluate:  equation
Ans: a2 + b2 + c2 + d2
Question 7

equation

Ans: 0
Question 8

For what value of x, the matrix A is singular


Solution Hint:
A matrix is said to be singular if |A| = 0
Question 9
Find the area of triangle with vertices A(5, 4), B(-2, 4), C(2, -6)
Ans: 35 sq unit

Question 10

Using determinants show that the points (2, 3), (-1, -2) and (5, 8) are collinear

Solution Hint:
Find area of triangle by taking above given points as vertices.
If area of triangle = 0 then points are collinear.

Question 11

Using determinants find the value of k so that the points (k, 2 - 2k), (- k + 1, 2k), and (- 4 - k, 6 - 2k) may be collinear

Ans: k = -1, 1/2

Question 12
Using determinants, find the equation of line joining the points (3,1), and (9,3)

Ans: x - 3y = 0

Question 13

Find the value of k, if area of triangle is 4 square units whose vertices are (-2,0), (0,4), and (0, k)

Ans: K = 0, 8

Question 14

Find the value of |AB| if matrices A and B are given below

and

Ans: 0

Question 15

Find the value of x if matrix A is a singular matrix 


Ans: x = - 4

Question 16

Find the product :

Hence solve the following system of equations
x - y = 3, 2x + 3y + 4z = 17, y + 2z = 7

Ans: x = 2, y = -1, z = 4

Solution Hint
Let given matrices are A and C

Now find the product AC we get


Now given system of equations can be written as



AX = B ⇒ X = 
A-1B
⇒ x = 2, y = -1, z = 4 is the required solution

Question 17


Find A-1  and hence solve following system of equations

3x - 4y + 2z = -1, 2x + 3y + 5z = 7, x + z = 2

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

Solution Hint:

Find |A| we find |A| = -9 ⇒ A is invertible

Find cofactors of A and then find Adj. A we get

Find A-1  we get



Given system of equations can be written as AX = B
⇒ X= A-1B



⇒ x = 3, y = 2, z = -1

Question 18



Find A-1. Use A-1 to solve the system of equations.

2x – 3y + 5z = 11, 3x + 2y – 4z = - 5, x + y - 2z = - 3

Ans: x = 1, y = 2, z = 3

Solution Hint

Find the |A| we get |A| = -1

Find adjoint of A

equation

Solve: X = A-1B we get x = 1, y = 2, z = 3

Question 19

Show that the matrix A satisfies the equation A - 4A - 5I = O and hence find A-1


Ans:
equation

Question 20
Find the matrix X for which



Answer


Question 21

  

Find  (AB)-1

Ans:  Find B-1 

equation 

equation 

equation
Question 22
Solve the following system of equations
x + y + z = 3, 2x - y + z = -1, 2x + y - 3z = -9
Ans:  x = -8/7,  y=10/7,  z = 19/7
Question 23
Solve the following system of equations

 
 
  where x, y, z ≠ 0
Ans: x = 2, y = 3, z = 5

Question 24

If A =equation,  B = equation  and X be a matrix such that  A = BX, then find X 

Answer:  equation 

Question 25

equation ,

Use it to solve the following system of equations

x - 2y = 10, 2x - y - z = 8,  - 2y + z = 7

Solution Hint

equation 

x = 0,   y = -5,   z = -3

Question 26

A scholarship is a sum of money provided to a student to help him or her pay for education. Some students are granted scholarships based on their academic achievements, while others are rewarded based on their financial needs. Every year a school offers scholarships to girl children and meritorious achievers based on certain criteria. In the session 2023 – 24, the school offered monthly scholarship of ₹ 3,000 each to some girl students and ₹ 4,000 each to meritorious achievers in academics as well as sports. In all, 50 students were given the scholarships and monthly expenditure incurred by the school on scholarships was ₹ 1,80,000.

Based on the above information, answer the following questions :
i) Express the given information algebraically using matrices.
ii) Check whether the system of matrix equations so obtained is consistent or not.
iii) (a) Find the number of scholarships of each kind given by the school, using matrices.
OR 
iii) (b) Had the amount of scholarship given to each girl child and meritorious student been interchanged, what would be the monthly expenditure incurred by the school ?

Answer (i)
Let No. of girl child scholarships = x
No. of meritorious achievers = y
x + y = 50
3000x + 4000y = 180000 ⇒ 3x + 4y = 180

equation 

Answer (ii)

equation 

Therefore the system of equations are consistent

Answer (iii) a

X  = A-1

equation 

⇒ x = 20, y = 30

Answer (iii) b

Required expenditure = ₹ [30(3000) + 20(4000)] = ₹ 1,70,000




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