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Test: Comprehension Based Questions: Matrices and Determinants


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13 Questions MCQ Test Maths 35 Years JEE Main & Advanced Past year Papers | Test: Comprehension Based Questions: Matrices and Determinants

Test: Comprehension Based Questions: Matrices and Determinants for JEE 2022 is part of Maths 35 Years JEE Main & Advanced Past year Papers preparation. The Test: Comprehension Based Questions: Matrices and Determinants questions and answers have been prepared according to the JEE exam syllabus.The Test: Comprehension Based Questions: Matrices and Determinants MCQs are made for JEE 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Comprehension Based Questions: Matrices and Determinants below.
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Test: Comprehension Based Questions: Matrices and Determinants - Question 1

A square matrix in which all the elements except at least the one element in the diagonal are zeros is said to be a

Detailed Solution for Test: Comprehension Based Questions: Matrices and Determinants - Question 1

A diagonal matrix is a matrix in which all of the elements not on the diagonal of a square matrix are 0.

Test: Comprehension Based Questions: Matrices and Determinants - Question 2

PASSAGE - 1

and U1, U2 and U3 are columns of a 3 × 3 matrix U. If column matrices U1, U2 and U3 satisfying  evaluate as directed in the following questions

Q. The sum of the elements of the matrix U–1 is

Detailed Solution for Test: Comprehension Based Questions: Matrices and Determinants - Question 2


⇒ Sum of elements of  

Test: Comprehension Based Questions: Matrices and Determinants - Question 3

PASSAGE - 1

and U1, U2 and U3 are columns of a 3 × 3 matrix U. If column matrices U1, U2 and U3 satisfying  evaluate as directed in the following questions

Q. The value of

Detailed Solution for Test: Comprehension Based Questions: Matrices and Determinants - Question 3

Test: Comprehension Based Questions: Matrices and Determinants - Question 4

PASSAGE - 2

Let  be the set of all 3× 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.

Q. The number of matrices in is

Detailed Solution for Test: Comprehension Based Questions: Matrices and Determinants - Question 4

Each element of set A is 3 × 3 symmetric matrix with five of its entries as 1 and four of its entries as 0, we can keep in diagonal either 2 zero and one 1 or no zero and three 1 so that the left over zeros and one’s are even in number.
Hence taking 2 zeros and one 1 in diagonal the possible cases are 
and taking 3 ones in diagonal the possible cases are 
∴ Total elements A can have = 9 + 3 = 12

Test: Comprehension Based Questions: Matrices and Determinants - Question 5

PASSAGE - 2

Let  be the set of all 3× 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.

Q. The number of matrices A in for which the system of linear equations

Detailed Solution for Test: Comprehension Based Questions: Matrices and Determinants - Question 5

The given system will have unique solution if | A |≠ 0 which is so for the matrices.


which are 6 in number.

Test: Comprehension Based Questions: Matrices and Determinants - Question 6

PASSAGE - 2

Let  be the set of all 3× 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.

Q. The number of matrices A in for which the system of linear equations

Detailed Solution for Test: Comprehension Based Questions: Matrices and Determinants - Question 6

For the given system to be inconsistent |A| = 0. The matrices for which | A | = 0 are




We find for A = (i)
By Cramer’s rule D1 = 0 = D2=D3
∴ infinite many solution
For A  = (ii)
By Cramer ’s rule D1 ≠ 0
⇒ no solution i.e. inconsistent.
Similarly we find the system as inconsistent in cases (iii), (v) and (vi).
Hence for four cases system is inconsistent.

Test: Comprehension Based Questions: Matrices and Determinants - Question 7

PASSAGE - 3

Let p be an odd prime number and Tp be the following set of 2 × 2 matrices :

Q. The number of A in Tp such that A is either symmetric or skew-symmetric or both, and det(A) divisible by p is

Detailed Solution for Test: Comprehension Based Questions: Matrices and Determinants - Question 7

If A is symmetrie then b = c
⇒ A = a2 – b2 = (a+b)(a –b)
Which is divisible by p if (a + b) is divisible by p or (a – b) is divisible by p.
Now (a + b) is divisible by p if (a, b) can take values (1, p – 1), (2, p – 2),(3, p – 3),...(p–1, 1)
∴ (p – 1)ways.
Also (a – b) is divisible by p only when a – b = 0 i.e. a = b, then (a, b) can take values (0,0) (1,1), (2,2)..... (pÝ, p–2)
∴ p ways.
If A is skew symmetric, then a = 0 and b = – c or b + c = 0 which gives |A| = 0 when b2
∴ b = 0, c = 0 But this possibility is already included when A is symmetric and (a, b) = (0, 0).
Again if A is both symmetric and skew symmetric, then clearly A is null matrix which case is already included.
Hence total number of ways = p + (p – 1) = 2p – 1

Test: Comprehension Based Questions: Matrices and Determinants - Question 8

PASSAGE - 3

Let p be an odd prime number and Tp be the following set of 2 × 2 matrices :

Q. The number of A in Tp such that the trace of A is not divisible by p but det (A) is divisible by p is

[Note: The trace of a matrix is the sum of its diagonal entries.]

Detailed Solution for Test: Comprehension Based Questions: Matrices and Determinants - Question 8

Trace A = a + a = 2a is not divisible by p ⇒ a is not divisible by p
⇒ a ≠ 0
But |A| is divisible by p
⇒ a2 – bc is divisible by p
It will be so if on dividing a2 by p suppose we get   then on dividing bc by p we should get  for some
integeral values of m, n and l. i.e. the remainder should be same in each case, so that 

For this to happen a can take any value from 1 to p–1, also if b takes any value from 1 to p–1 then c should take only that value corresponding to which the remainder is same.
∴ No. of ways = (p – 1) × (p – 1) × 1 = (p – 1)2.

Test: Comprehension Based Questions: Matrices and Determinants - Question 9

PASSAGE - 3

Let p be an odd prime number and Tp be the following set of 2 × 2 matrices :

Q. The number of A in Tp such that det (A) is not divisible by p is

Detailed Solution for Test: Comprehension Based Questions: Matrices and Determinants - Question 9

Total number of matrices = total number of ways a, b, c can be selected = p × p × p = p3.
Number of ways when det (A) is divisible by p and trace (A) ≠ 0 are equal to number of ways det (A) is divisible by p and trace (A) is not divisible by p = (p – 1)2

Also number of ways when det (A) is divisible by p and trace A = 0 are the ways when bc is multiple of p
⇒ b = 0 or c = 0 for b = 0, c can take values 0, 1, 2, ......., p – 1
For c = 0, b can take values 0, 1, 2, ............, p – 1
Here (b, c) = (0, 0) is coming twice.
∴ Total ways of selecting b and c = p + p – 1 = 2p – 1
∴ Total number of ways when det (A) is divisible by p = (p – 1)2 + 2p – 1 = p2 Hence the number of ways when det (A) is not divisible by p = p3 – p2.

Test: Comprehension Based Questions: Matrices and Determinants - Question 10

PASSAGE - 4

Let a, b and c be three real numbers satisfying

     ...(E)

Q. If the point P(a, b, c), with reference to (E), lies on the plan e 2x + y + z = 1, then the value of 7a + b + c is 

Detailed Solution for Test: Comprehension Based Questions: Matrices and Determinants - Question 10

From equation (E), we get
a + 8b + 7c = 0
9a + 2b + 3c = 0
a + b + c  = 0

Therefore system has infinite many solutions.
Solving these, we get b = 6a and c = – 7a
Now (a, b, c) lies on 2x + y + z = 1
⇒ b = 6, c = – 7
∴  2a + 6a – 7a = 1 ⇒ a = 1
∴ 7a + b + c = 7 + 6 – 7 = 6 ⇒ b = 6, c = – 7

Test: Comprehension Based Questions: Matrices and Determinants - Question 11

PASSAGE - 4

Let a, b and c be three real numbers satisfying

     ...(E)

Q. Let ω be a solution of x3 – 1 = 0 with Im (ω) > 0 , if a = 2 with b and c satisfying (E), then the value of   equal to

Detailed Solution for Test: Comprehension Based Questions: Matrices and Determinants - Question 11

If a = 2 then b = 12, c = –14

Test: Comprehension Based Questions: Matrices and Determinants - Question 12

PASSAGE - 4

Let a, b and c be three real numbers satisfying

     ...(E)

Q. Let b = 6, with a and c satisfying (E). If a and b are the roots of the quadratic equation ax2 + bx + c = 0, then 

Detailed Solution for Test: Comprehension Based Questions: Matrices and Determinants - Question 12

If b = 6 then a = 1, c = – 7
∴ Equation becomes x2 + 6x – 7 = 0 or (x + 7) (x – 1) = 0 whose roots are 1 and –7.
Let α = 1 and β = – 7

Test: Comprehension Based Questions: Matrices and Determinants - Question 13

Consider the system of equations

x – 2y + 3z = –1
–x + y – 2z = k
x – 3y + 4z = 1

STATEMENT - 1 : The system of equations has no solution for k ¹ 3   and

STATEMENT-2 : The determinant   k ≠ 3 .

Detailed Solution for Test: Comprehension Based Questions: Matrices and Determinants - Question 13

The given equations are
x – 2y + 3z = – 1
– x + y – 2z = k
x – 3y + 4z =1

∴ If k ≠ 3 , the system has no solutions.
Hence statement-1 is true and statement-2 is a correct explanation for statement - 1.

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