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Mathematics: CUET Mock Test - 6


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Mathematics: CUET Mock Test - 6 - Question 1

Which of these is not a type of relation?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 1

Surjective is not a type of relation. It is a type of function. Reflexive, Symmetric and Transitive are type of relations.

Mathematics: CUET Mock Test - 6 - Question 2

What will be the value of x + y + z if cos-1 x + cos-1 y + cos-1 z = 3π?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 2

The equation is cos-1 x + cos-1 y + cos-1 z = 3π
This means cos-1 x = π, cos-1 y = π and cos-1 z = π
This will be only possible when it is in maxima.
As, cos-1 x = π so, x = cos-1 π = -1 similarly, y = z = -1
Therefore, x + y + z = -1 -1 -1
So, x + y + z = -3.

Mathematics: CUET Mock Test - 6 - Question 3

If the order of the matrix is m×n, then how many elements will there be in the matrix?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 3

The number of elements for a matrix with the order m×n is equal to mn, where m is the number of rows and n is the number of columns in the matrix.

Mathematics: CUET Mock Test - 6 - Question 4

The matrix which follows the conditions m=n is called?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 4

A square matrix is a matrix in which the number of rows(m) is equal to the number of columns(n). Therefore, the matrix which follows the condition m = n is a square matrix.

Mathematics: CUET Mock Test - 6 - Question 5

Evaluate .

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 5

Expanding along R1, we get
∆=2(-1)-5(-1)=-2+5
= 3.

Mathematics: CUET Mock Test - 6 - Question 6

Which of the following matrices will not have a determinant?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 6

Determinant of the matrix A= is not possible as it is a rectangular matrix and not a square matrix. Determinants can be calculated only if the matrix is a square matrix.

Mathematics: CUET Mock Test - 6 - Question 7

Which value is similar to sin-1sin(6 π/7)?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 7

sin-1sin(6 π/7)
Now, sin(6 π/7) = sin(π – 6 π/7)
= sin(2π + 6 π/7) = sin(π/7)
= sin(3π – 6 π/7) = sin(20π/7)
= sin(-π – 6 π/7) = sin(-15π/7)
= sin(-2π + 6 π/7) = sin(-8π/7)
= sin(-3π – 6 π/7) = sin(-27π/7)
Therefore, sin-1sin(6 π/7) = sin-1(π/7).

Mathematics: CUET Mock Test - 6 - Question 8

Which of the following is a matrix of the order 2×2 where the equation of the elements is given by aij =i+j.

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 8

a11=1+1=2, a12=1+2=3, a21=2+1=3, a22=2+2=4
∴ 

Mathematics: CUET Mock Test - 6 - Question 9

Consider the matrix A=  What is the type of matrix?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 9

The matrix in which number of rows is smaller than the number of columns is called is called a horizontal matrix. In the given matrix A=  m = 3 and n = 2 i.e.
3<2. Hence, it is a horizontal matrix.

Mathematics: CUET Mock Test - 6 - Question 10

Evaluate 

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 10

Evaluating along R1, we get
∆ = 5(√3)-(-4)1 = 5√3+4.

Mathematics: CUET Mock Test - 6 - Question 11

Which of the following relations is symmetric but neither reflexive nor transitive for a set A = {1, 2, 3}.

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 11

A relation in a set A is said to be symmetric if (a1, a2)∈R implies that (a1, a2)∈R,for every a1, a2∈R.
Hence, for the given set A={1, 2, 3}, R={(1, 2), (2, 1)} is symmetric. It is not reflexive since every element is not related to itself and neither transitive as it does not satisfy the condition that for a given relation R in a set A if (a1, a2)∈R and (a2, a3)∈R implies that (a1, a3)∈ R for every a1, a2, a3∈R.

Mathematics: CUET Mock Test - 6 - Question 12

What is the value of sin-1(-x) for all x belongs to [-1, 1]?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 12

Let, θ = sin-1(-x)
So, -π/2 ≤ θ ≤ π/2
⇒ -x = sinθ
⇒ x = -sinθ
⇒ x = sin(-θ)
Also, -π/2 ≤ -θ ≤ π/2
⇒ -θ = sin-1(x)
⇒ θ = -sin-1(x)
So, sin-1(-x) = -sin-1(x)

Mathematics: CUET Mock Test - 6 - Question 13

What is the order of the matrix A = 

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 13

The number of rows (m) and the number of columns (n) in the given matrix A=   is 2. Therefore, the order of the matrix is 2×2(m×n).

Mathematics: CUET Mock Test - 6 - Question 14

The matrix A=  is ____

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 14

The given matrix A =  is of the order 3×1. The matrix has only one column (n=1). Hence, it is a column matrix.

Mathematics: CUET Mock Test - 6 - Question 15

Evaluate 

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 15

Expanding along R1, we get
∆=-sinθ(sinθ)-(-1)1=-sin2⁡θ+1=cos2⁡θ.

Mathematics: CUET Mock Test - 6 - Question 16

Which of the following relations is transitive but not reflexive for the set S={3, 4, 6}?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 16

For the above given set S = {3, 4, 6}, R = {(3, 4), (4, 6), (3, 6)} is transitive as (3, 4)∈R and (4, 6) ∈R and (3,6) also belongs to R . It is not a reflexive relation as it does not satisfy the condition (a, a) ∈ R, for every a ∈ A for a relation R in the set A.

Mathematics: CUET Mock Test - 6 - Question 17

What is the value of sin-1(sin 6)?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 17

We know that sin(x) = sin(2A * π + x) where A can be positive or negative integer.
If A is -1, then sin(6) = sin(-2π + 6);
If A is 1, then sin(6) = sin(2π + 6);

Mathematics: CUET Mock Test - 6 - Question 18

What is the value of r = 1Σn f(x) if f(r) =  where n € N?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 18

The given determinant is f(r) =  
Now, r = 1Σn (2r) = 2[(n(n + 1))/2] ……….(1)
= n2 + n
r = 1Σn(6r2 – 1) = 6[((n + 1)(2n + 1))/6] – n ……….(2)
= n(2n2 + 2n + n + 1) – n
= 2n3 + 2n2 + n2 + n – n
= 2n3 + 3n2
= r = 1Σn(4r3 – 2nr) = n3 (n + 1) ……….(3)
From (1), (2) and (3) we get
r = 1Σn f(x) = 0

Mathematics: CUET Mock Test - 6 - Question 19

The matrix which follows the condition m>n is called as _______

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 19

The matrix in which the number of columns is greater than the number of rows is called a vertical matrix. There the matrix which follows the condition m>n is a vertical matrix.

Mathematics: CUET Mock Test - 6 - Question 20

Evaluate 

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 20

Expanding along R1, we get
∆=-i(i)-(-1)(-1)=-i2-1=-(-1)-1=0.

Mathematics: CUET Mock Test - 6 - Question 21

Let R be a relation in the set N given by R={(a,b): a+b=5, b>1}. Which of the following will satisfy the given relation?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 21

(2, 3) ∈ R as 2 + 3 = 5, 3 >1, thus satisfying the given condition.
(4, 2) doesn’t belong to R as 4 + 2 ≠ 5.
(2,1) doesn’t belong to R as 2+ 1 ≠ 5.
(5, 0) doesn’tbelong to R as 0 ⊁ 1
 

Mathematics: CUET Mock Test - 6 - Question 22

What is the value of cos-1(-x) for all x belongs to [-1, 1]?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 22

Let, θ = cos-1(-x)
So, 0 ≤ θ ≤ π
⇒ -x = cosθ
⇒ x = -cosθ
⇒ x = cos(-θ)
Also, -π ≤ -θ ≤ 0
So, 0 ≤ π -θ ≤ π
⇒ -θ = cos-1(x)
⇒ θ = -cos-1(x)
So, cos-1(x) = π – θ
θ = π – cos-1(x)
⇒ cos-1(-x) = π – cos-1(x)

Mathematics: CUET Mock Test - 6 - Question 23

Which one is correct, the following system of linear equations 2x – 3y + 4z = 7, 3x – 4y + 5z = 8, 4x – 5y + 6z = 9 has?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 23

Solving the given system of equation by Cramer’s rule, we get,
x = D1/D, y = D2/D, z = D3/D where,


Now, performing, C3 = C3 – C2 and C2 = C2 – C1 we get,

As two columns have identical values, so,
D = 0
Similarly,

Now, performing, C1 = C1 – C3

Now, performing, C3 = C3 – C2

As two columns have identical values, so,
D1 = 0

Now, performing,

Now, performing, C2 = C2 – C3 and C3 = C3 – C1

As two columns have identical values, so,
D2 = 0


Now, performing, C2 = C2 – C2 and C3 = C3 – C2

As two columns have identical values, so,
D3 = 0
Since, D = D1 = D2 = D3 = 0, thus, it has infinitely many solutions.

Mathematics: CUET Mock Test - 6 - Question 24

Find the value of a,b,c,d if 

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 24

The two matrices  and  are equal matrices. Comparing the two matrices, we get a-b=3, c=2, a-b=1, 2c+d=6
Solving the above equations, we get a=2, b=1, c=2, d=2.

Mathematics: CUET Mock Test - 6 - Question 25

Evaluate 

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 25

∆= 
Expanding along the first row, we get

=1(4-5(2))-1(3-5(-1))-2(6-4(-1))
=(4-10)-(3+5)-2(6+4)
=-6-8-20=-34.

Mathematics: CUET Mock Test - 6 - Question 26

Which of the following relations is reflexive but not transitive for the set T = {7, 8, 9}?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 26

The relation R= {(7, 7), (8, 8), (9, 9)} is reflexive as every element is related to itself i.e. (a, a) ∈ R, for every a∈A. and it is not transitive as it does not satisfy the condition that for a relation R in a set A if (a1, a2)∈R and (a2, a3)∈R implies that (a1, a3) ∈ R for every a1, a2, a3 ∈ R.

Mathematics: CUET Mock Test - 6 - Question 27

The given graph is for which equation?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 27

The following graph represents 2 equations.

The pink curve is the graph of y = sinx
The blue curve is the graph for y = sin-1x
This curve passes through the origin and approaches to infinity in both positive and negative axes.

Mathematics: CUET Mock Test - 6 - Question 28

Which of the following is not a possible ordered pair for a matrix with 6 elements.

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 28

The possible orders in which the matrix with 6 elements can be formed are 2×3, 3×2, 1×6, 6×1. Therefore, the possible orders pairs are (2,3), (3,2), (1,6), (6,1). Thus, (3,1) is not possible.

Mathematics: CUET Mock Test - 6 - Question 29

Which of the following is a diagonal matrix.

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 29

The matrix is said to be a diagonal matrix if the elements along the diagonal of the matrix are non – zero.
i.e. aij=0 for i≠j and aij≠0 for i=j.
Therefore, the matrix A=  is a diagonal matrix.

Mathematics: CUET Mock Test - 6 - Question 30

Evaluate .

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 30

Expanding along the first row, we get

=5(4-6)-4(3-5)+3(18-20)
=5(-2)-4(-2)+3(-2)=-10+8-6=-8.

Mathematics: CUET Mock Test - 6 - Question 31

Let I be a set of all lines in a XY plane and R be a relation in I defined as R = {(I1, I2):I1 is parallel to I2}. What is the type of given relation?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 31

This is an equivalence relation. A relation R is said to be an equivalence relation when it is reflexive, transitive and symmetric.
Reflexive: We know that a line is always parallel to itself. This implies that I1 is parallel to I1 i.e. (I1, I2)∈R. Hence, it is a reflexive relation.
Symmetric: Now if a line I1 || I2 then the line I2 || I1. Therefore, (I1, I2)∈R implies that (I2, I1)∈R. Hence, it is a symmetric relation.
Transitive: If two lines (I1, I3) are parallel to a third line (I2) then they will be parallel to each other i.e. if (I1, I2) ∈R and (I2, I3) ∈R implies that (I1, I3) ∈R.

Mathematics: CUET Mock Test - 6 - Question 32

The given graph is for which equation?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 32

There are 2 curves.

The green curve is the graph of y = cosx
The red curve is the graph for y = cos-1x
This curve origin from some point before π/3 and approaches to infinity in both positive y axis by intersecting at a point near 1.5 in y axis.

Mathematics: CUET Mock Test - 6 - Question 33

Which of the following matrix is of the order 3×4.

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 33

The matrix A=  is a 3×4 matrix as it as 3 rows and 4 columns.

Mathematics: CUET Mock Test - 6 - Question 34

Find the value of k if the area is 7/2 sq. units and the vertices are (1,2), (3,5), (k,0).

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 34

Given that the vertices are (1,2), (3,5), (k,0)
Therefore, the area of the triangle with vertices (1,2), (3,5), (k,0) is given by

Expanding along R3, we get
(1/2){k(2-5)-0+1(5-6)}=(1/2){-3k-1}=(7/2)
⇒ -(1/2)(3k+1)=7/2
3k=-8
k = -(8/3)

Mathematics: CUET Mock Test - 6 - Question 35

Evaluate 

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 35

Expanding along the first row, we get
∆=8x+1(3x+5)-(2x-2)(x2-1)
=(24x2+43x+5)-(2x3-2x2-2x+2)
=-2x3+26x2+45x+3.

Mathematics: CUET Mock Test - 6 - Question 36

Which of the following relations is symmetric and transitive but not reflexive for the set I = {4, 5}?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 36

R= {(4, 5), (5, 4), (4, 4)} is symmetric since (4, 5) and (5, 4) are converse of each other thus satisfying the condition for a symmetric relation and it is transitive as (4, 5)∈R and (5, 4)∈R implies that (4, 4) ∈R. It is not reflexive as every element in the set I is not related to itself.

Mathematics: CUET Mock Test - 6 - Question 37

The given graph is for which equation?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 37

There are 2 curves.

The blue curve is the graph of y = tanx
The red curve is the graph for y = tan-1x
This curve passes through the origin and approaches to infinity in the direction of x axis only.
This graph lies below –x axis and above +x axis.

Mathematics: CUET Mock Test - 6 - Question 38

Consider the matrix A= . Find the element a32.

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 38

a32 is the element represented in the form aij i is the row number and j is the column number. Therefore, the element a32 is the element in the third row (i=3)and second column (j=2) which is 8.

Mathematics: CUET Mock Test - 6 - Question 39

Which of the following is a scalar matrix?

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 39

A matrix is called a scalar matrix if the elements along the diagonal of the matrix are equal and are non-zero i.e. aij=k for i=j and aij=0 for i≠j.
Therefore, the matrix A=  is a scalar matrix.

Mathematics: CUET Mock Test - 6 - Question 40

If A=  find |A|.

Detailed Solution for Mathematics: CUET Mock Test - 6 - Question 40

Given that, A= 

Evaluating along the first row, we get

∆=2(2-24)-5(12-12)+9(48-4)
∆=2(-22)-0+9(44)
∆=-44+9(44)=44(-1+9)=352

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