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Mathematics: CUET Mock Test - 8


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40 Questions MCQ Test CUET Science Subjects Mock Tests | Mathematics: CUET Mock Test - 8

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Mathematics: CUET Mock Test - 8 - Question 1

If f(x) =  then what is the value of 0π/2 f(x) dx = (π/4 + 8/15)?

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 1

(dy/dx) = (dx/dy)-1
So, d2y/dx2 = -(dx/dy)-2 d/dx(dx/dy)
= -(dy/dx)2(d2x/dy2)(dy/dx)
= d2y/dx2 + (dy/dx)3 d2y/dx2 = 0

Mathematics: CUET Mock Test - 8 - Question 2

What will be the value of x if  = 0?

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 2

Here, we have, 
Now, replacing C3 = C3 – 3C1, we get,

Or, (x – 1)
Now, replacing R1 = R1 + 3R3, we get,
Or, (x – 1) 
Or, (x – 1)[-1 {(11 – x)(5 – x) – 28}] = 0
Or, -(x – 1)(55 – 11x – 5x + x2 – 28)
Or, (x – 1)(x2 – 16x + 27) = 0
Thus, either x – 1 = 0 i.e. x = 1 or x2 – 16x + 27 = 0
Therefore, solving x2 – 16x + 27 = 0 further, we get,
x = 8 ± √37

Mathematics: CUET Mock Test - 8 - Question 3

Find the area of the triangle with the vertices (2,3), (4,1), (5,0).

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 3

The area of the triangle with vertices (2,3), (4,1), (5,0) is given by

Applying R2→R2-R3

Expanding along R2, we get
Δ=(1/2){-(-1)(3-0)+1(2-5)}
Δ=(1/2) (0-0)=0.

Mathematics: CUET Mock Test - 8 - Question 4

For which of the elements in the determinant Δ=  the cofactor is -37.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 4

Consider the element -3 in Δ= 
The cofactor of the element -3 is given by
A22=(-1)2+2 M22
M22 =1(5)-(-6)(-7)=5-42=-37
A22=(-1)2+2 (-37)=-37.

Mathematics: CUET Mock Test - 8 - Question 5

Find the determinant of the matrix A= .

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 5

Given that, A= 
|A| = 
|A|=-cos⁡θ (cos⁡θ )-cotθ(-tan⁡θ)
|A|=-cos2⁡θ+1=sin2⁡θ.

Mathematics: CUET Mock Test - 8 - Question 6

What will be the value of 

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

The above matrix is a skew symmetric matrix and its order is odd
And we know that for any skew symmetric matrix with odd order has determinant = 0
Therefore, the value of the given determinant = 0

Mathematics: CUET Mock Test - 8 - Question 7

What is the relation between the two determinants f(x) =  and g(x) = 

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 7

Let, D = 
Expanding D by the 1st row we get,
D = – c 
= – c(0 – ab) + b(ac – 0)
= 2abc
Now, we have adjoint of D = D’

Or, D’ = 
Or, D’ = D2
Or, D’ = D2 = (2abc)2

Mathematics: CUET Mock Test - 8 - Question 8

Find the equation of the line joining A(5,1), B(4,0) using determinants.

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

Let C(x,y) be a point on the line AB. Thus, the points A(5,1), B(4,0), C(x,y) are collinear. Hence, the area of the triangle formed by these points will be 0.
⇒ Δ = (1/2) 
Applying R1→R1-R2

Expanding along R1, we get
=(1/2) {1(0-y)-1(4-x)}=0
=(1/2){-y-4+x}=0
⇒ x-y = 4.

Mathematics: CUET Mock Test - 8 - Question 9

For which of the following elements in the determinant Δ=  the minor of the element is 2?

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 9

Consider the element 7 in the determinant Δ= 
The minor of the element 7 can be obtained by deleting R2 and C2
∴ M22 = 2
Hence, the minor of the element 7 is 2.

Mathematics: CUET Mock Test - 8 - Question 10

Evaluate 

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 10


Expanding along R1, we get

Δ=5(24-24)-0+5(8-0)
Δ=0-0+40=40.

Mathematics: CUET Mock Test - 8 - Question 11

If f(x) =  = 0,then what will be the value of p?

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 11

Here, C1 and C3 becomes equal when we put p = xn
And R1 and R3 becomes equal when we put p = n + 1
And R1 and R3 becomes equal when we put p = n + 1

Mathematics: CUET Mock Test - 8 - Question 12

Find the value of k for which the points (3, 2), (1, 2), (5, k) are collinear.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 12

Given that the vertices are (3,2), (1,2), (5,k)
Therefore, the area of the triangle with vertices (3,2), (1,2), (5,k) is given by
Δ=(1/2) 
Applying R1→R1-R2, we get
1/2 
Expanding along R1, we get
(1/2) {2(2-k)-0+0} = 0
2-k = 0
k = 2 .

Mathematics: CUET Mock Test - 8 - Question 13

For which of the following element in the determinant Δ=  the minor and the cofactor both are zero.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 13

Consider the element 2 in the determinant Δ= 
The minor of the element 2 is given by
∴ M22 = = 40-40 = 0
⇒ A22 = (-1)2+2 (0) = 0.

Mathematics: CUET Mock Test - 8 - Question 14

Which of the following conditions holds true for a system of equations to be consistent?

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 14

If a given system of equations has one or more solutions then the system is said to be consistent.

Mathematics: CUET Mock Test - 8 - Question 15

Differentiate (log⁡2x)sin⁡3x with respect to x.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 15

Consider y=(log2x)sin3x
Applying log on both sides, we get
log⁡y=log(log2x)sin3x
log⁡y=sin⁡3x log⁡(log⁡2x)
Differentiating with respect to x, we get

By using chain rule, we get

dy/dx =y(3 cos⁡3x log⁡(log⁡2x)+ 
∴ dy/dx=log⁡2xsin⁡3x(3cos3xlog(log2x)+(sin3x/xlog2x))

Mathematics: CUET Mock Test - 8 - Question 16

Find the second order derivative of y=9 log⁡ t3.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 16

Given that, y=9 log⁡t3

Mathematics: CUET Mock Test - 8 - Question 17

Differentiate 8e-x+2ex w.r.t x.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 17

To solve:y=(8e-x+2ex)
Differentiating w.r.t x we get,
(dy/dx)= 8(-e-x+2ex)
∴ (dy/dx)= 2ex - 8e-x.

Mathematics: CUET Mock Test - 8 - Question 18

If the rate of change of radius of a circle is 6 cm/s then find the rate of change of area of the circle when r=2 cm.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 18

The rate of change of radius of the circle is dr/dt = 6 cm/s
The area of a circle is A=πr2
Differentiating w.r.t t we get,
(dA/dt = d/dt) (πr2) = 2πr (dr/dt) =2πr(6)=12πr.
dA/dt |r=2=24π= 24×3.14=75.36 cm2/s

Mathematics: CUET Mock Test - 8 - Question 19

A given systems of equations is said to be inconsistent if _____

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 19

If a given system of equations has no solutions, then the system is said to be inconsistent.

Mathematics: CUET Mock Test - 8 - Question 20

Differentiate  with respect to x.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 20

Consider
Applying log on both sides, we get
log⁡y=log
log⁡y=log⁡4+  (∵log⁡ab  =log⁡a+log⁡b)
Differentiating both sides with respect to x, we get

Mathematics: CUET Mock Test - 8 - Question 21

Find d2y/dx2, if y=tan2⁡x+3 tan⁡x.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 21

Given that, y=tan2⁡⁡x+3 tan⁡x
dy/dx =2 tan⁡x sec2⁡⁡x+3 sec2⁡x=sec2⁡⁡x (2 tan⁡x+3)
By using the u.v rule, we get
 (sec2⁡⁡x).(2 tan⁡x+3)+ (d/dx) (2 tan⁡x+3).sec2⁡⁡x
 =2 sec2⁡⁡x tan⁡x (2 tan⁡x+3)+sec2⁡⁡x (2 sec⁡x tanx)
=2 sec2⁡x tan⁡x (2 tan⁡x+sec⁡x+3).

Mathematics: CUET Mock Test - 8 - Question 22

Differentiate 8ecos2x w.r.t x.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 22

Consider y=8ecos2x
Differentiating w.r.t x by using chain rule, we get

Mathematics: CUET Mock Test - 8 - Question 23

The edge of a cube is increasing at a rate of 7 cm/s. Find the rate of change of area of the cube when x=6 cm.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 23

Let the edge of the cube be x. The rate of change of edge of the cube is given by dx/dt =7cm/s.
The area of the cube is A=6x2

Mathematics: CUET Mock Test - 8 - Question 24

Find the value of x and y for the given system of equations.
3x+2y=6
5x+y=2

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 24

By using the matrix method, the given equations can be expressed in the form of the equation AX=B, where

To find the value of x and y, we need to solve the matrix X
X = A-1 B

Mathematics: CUET Mock Test - 8 - Question 25

Differentiate 9tan⁡3x with respect to x.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 25

Consider y=9tan⁡3x
Applying log on both sides, we get
log⁡y=log⁡9tan⁡3x
Differentiating both sides with respect to x, we get

 (∵ Using u.v = u′v + uv′)
(dy/dx) = y(3 sec2⁡⁡x.log⁡9+0)
(dy/dx) = 9tan⁡3x (3 log⁡9 sec2⁡x)

Mathematics: CUET Mock Test - 8 - Question 26

Differentiate 3 sin-1⁡(e2x) w.r.t x.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 26

Consider y=3 sin-1⁡(e2x)
dy/dx = d/dx (3 sin-1⁡(e2x))

Mathematics: CUET Mock Test - 8 - Question 27

The rate of change of area of a square is 40 cm2/s. What will be the rate of change of side if the side is 10 cm.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 27

Let the side of the square be x.
A = x2, where A is the area of the square
Given that, 

Mathematics: CUET Mock Test - 8 - Question 28

Differentiate (cos⁡3x)3x with respect to x.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 28

Consider y=(cos⁡3x)3x
Applying log on both sides, we get
log⁡y=log⁡(cos⁡3x)3x
log⁡y=3x log⁡(cos⁡3x)
Differentiating both sides with respect to x, we get

By using u.v=u’ v+uv’, we get
(3x)log(cos3x)+(d/dx)(log(cos3x)).3x
(dy/dx)=y(3 log⁡(cos⁡3x) +(cos3x).3x)
(dy/dx)=y(3 log⁡(cos⁡3x) +
(dy/dx)=y(3 log⁡(cos⁡3x) +
(dy/dx) = y(3 log⁡(cos⁡3x) – 9x tan⁡3x)
(dy/dx) = (cos⁡3x)3x (3 log⁡(cos⁡3x) – 9x tan⁡3x)

Mathematics: CUET Mock Test - 8 - Question 29

Find the second order derivative of y=2e2x-3 log⁡(2x-3).

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 29

Given that, y=2e2x-3 log⁡(2x-3)

Mathematics: CUET Mock Test - 8 - Question 30

Differentiate log⁡(log⁡x5) w.r.t x.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 30

Consider y=(log⁡(log⁡(x5)))

Mathematics: CUET Mock Test - 8 - Question 31

The total cost P(x) in rupees associated with a product is given by P(x)=0.4x2+2x-10. Find the marginal cost if the no. of units produced is 5.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 31

The Marginal cost is the rate of change of revenue w.r.t the no. of units produced, we get
 = 0.8x+2
cost(MC)=  =0.8x+2=0.8(5)+2=4+2=6.

Mathematics: CUET Mock Test - 8 - Question 32

Find the value of x, y, z for the given system of equations.
2x+3y+2z=50
x+4y+3z=40 
3x+3y+5z=60

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 32

The given system of equations can be expressed in the form of AX=B, where

X = A-1 B
∴ A-1 = (1/|A|) adj A

X = A-1 B

Mathematics: CUET Mock Test - 8 - Question 33

Differentiate  with respect to x.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 33

Consider y= 
log⁡y=log 
log⁡y=log⁡7+ 
log⁡y=log⁡7+2e2x log⁡x
Differentiating with respect to x on both sides, we get

 (log⁡x)2e2x (using u.v=u’ v+uv’)

Mathematics: CUET Mock Test - 8 - Question 34

Find d2y/dx2, if y=2 sin-1⁡(cos⁡x).

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 34

Given that, y=2 sin-1⁡(cos⁡x)

Mathematics: CUET Mock Test - 8 - Question 35

Differentiate  w.r.t x.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 35

Consider 

Mathematics: CUET Mock Test - 8 - Question 36

At what rate will the lateral surface area of the cylinder increase if the radius is increasing at the rate of 2 cm/s when the radius is 5 cm and height is 10 cm?

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 36

Let r be the radius and h be the height of the cylinder. Then, dr/dt =2 cm/s
The area of the cylinder is given by A=2πrh

Mathematics: CUET Mock Test - 8 - Question 37

Differentiate  with respect to x.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 37

 Consider 
Applying log on both sides, we get

Differentiating with respect to x, we get


Mathematics: CUET Mock Test - 8 - Question 38

If y=log⁡(2x3), find

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 38

Given that, y=log⁡(2x3)

Mathematics: CUET Mock Test - 8 - Question 39

Differentiate  tan⁡x w.r.t x.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 39

Consider  tan⁡x
Differentiating w.r.t x by using chain rule, we get

Mathematics: CUET Mock Test - 8 - Question 40

If the circumference of the circle is changing at the rate of 5 cm/s then what will be rate of change of area of the circle if the radius is 6cm.

Detailed Solution for Mathematics: CUET Mock Test - 8 - Question 40

The circumference of the circle is given by C=2πr, where r is the radius of the circle.

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