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Mathematics: CUET Mock Test - 9


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Mathematics: CUET Mock Test - 9 - Question 1

Find the values of x and y for the given system of equations.
3x-2y=3
2x+2y=4

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 1

The given system of equations can be expressed in the form of AX=B,
⇒ X= A-1 B

We know that, A-1= (1/|A|) adj A
A-1= (1/10) 
∴ X = A-1 B= 

Mathematics: CUET Mock Test - 9 - Question 2

Differentiate 2(tan⁡x)cot⁡x with respect to x.

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 2

Consider y=2(tan⁡x)cot⁡x
Applying log in both sides,
log⁡y=log⁡2(tan⁡x)cot⁡x
log⁡y=log⁡2+log⁡(tan⁡x)cot⁡x
log⁡y=log⁡2+cot⁡x log⁡(tan⁡x)
Differentiating both sides with respect to x, we get

dy/dx = 2(tan⁡x)cot⁡x (−csc2xlog(tanx)+cot2x+1)
dy/dx = 2(tan⁡x)cot⁡x (−csc2xlog(tanx)+csc2x)
dy/dx = 2(tan⁡x)cot⁡x (csc2⁡x (1-log⁡(tan⁡x))
∴ dy/dx =2 csc2⁡x.tan⁡xcot⁡x (1-log⁡(tan⁡x))

Mathematics: CUET Mock Test - 9 - Question 3

Find 

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 3

Given that, y=4x4+2x
dy/dx = 16x3+2
d2y/dx2 =48x2
48x2−96x3−12
=12(4x2-8x-1)

Mathematics: CUET Mock Test - 9 - Question 4

Differentiate  w.r.t x.

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 4

Consider y= 
y=5x3 (∴log⁡ex=x)

∴ dy/dx = 5(3x2)=15x2

Mathematics: CUET Mock Test - 9 - Question 5

The total cost N(x) in rupees, associated with the production of x units of an item is given by N(x)=0.06x3-0.01x2+10x-43. Find the marginal cost when 5 units are produced.

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 5

The marginal cost is given by the rate of change of revenue.
Hence, (dN(x)/dt) =0.18x2-0.02x+10.
 = 0.18(5)2-0.02(5)+10
= 4.5-0.1+10
= Rs. 14.4

Mathematics: CUET Mock Test - 9 - Question 6

The cost of 8kg apple and 3kg is Rs 70. The cost of 10kg apple and 6kg orange is 90. Find the cost of each item if x is the cost of apples per kg and y is the cost of oranges per kg.

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

The given situation can be represented by a system of equations as:
8x+3y=70
10x+5y=90
Consider, A= 
It can be expressed in the form of AX=B
⇒ X=A-1 B
We know that, A-1
 
∴ x=3, y=2.
i.e. The cost of apples is Rs 3 per kg and the cost of oranges is Rs 2 per kg.

Mathematics: CUET Mock Test - 9 - Question 7

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

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 7

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

 log⁡3+log⁡(cos⁡x)-x tan⁡x
dy/dx =y(log⁡(3 cos⁡x)-x tan⁡x)
dy/dx =(3 cos⁡x)x (log⁡(3 cos⁡x)-x tan⁡x)

Mathematics: CUET Mock Test - 9 - Question 8

Find the second order derivative y=e2x+sin-1⁡ex.

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

Given that, y=e2x+sin-1⁡ex



Mathematics: CUET Mock Test - 9 - Question 9

Differentiate 7  w.r.t x.

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

Consider y=7 


Mathematics: CUET Mock Test - 9 - Question 10

The length of the rectangle is changing at a rate of 4 cm/s and the area is changing at the rate of 8 cm/s. What will be the rate of change of width if the length is 4cm and the width is 1 cm.

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 10

Let the length be l, width be b and the area be A.
The Area is given by A=lb

Given that, dl/dt =4cm/s and dA/dt =8 cm/s
Substituting in the above equation, we get

Given that, l=4 cm and b=1 cm

Mathematics: CUET Mock Test - 9 - Question 11

For a given system of equations if |A|=0 and (adj A)B≠O(zero matrix), then which of the following is correct regarding the solutions of the given equations?

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 11

If A is a singular matrix, then |A|=0
In this case, if (adj A) B≠O, then solution does not exist and the system of equations is called inconsistent.

Mathematics: CUET Mock Test - 9 - Question 12

Differentiate  with respect to x.

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 12

Consider y= 
Applying log to both sides, we get
log⁡y=log 
log⁡y= 
log⁡y= (1/2) (log⁡(x+1)-log⁡(3x-1))
Differentiating with respect to x, we get

Mathematics: CUET Mock Test - 9 - Question 13

Find the second order derivative of y=3x2 1 + log⁡(4x)

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 13

Given that, y=3x2+log⁡(4x)


Mathematics: CUET Mock Test - 9 - Question 14

Differentiate  log⁡x w.r.t x.

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 14

Consider  log⁡x

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

Mathematics: CUET Mock Test - 9 - Question 15

For which of the values of x, the rate of increase of the function y=3x2-2x+7 is 4 times the rate of increase of x?

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 15

Given that, 
y=3x2-2x+7

4=6x-2
6x=6
⇒ x=1

Mathematics: CUET Mock Test - 9 - Question 16

Find the value of x and y for the given system of equations.
3x+4y=6
5x-4y=4

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 16

The given system of equations can be expressed in the form of AX=B,
⇒X=A-1 B
We know that, A-1=1/|A| adj A

  

Mathematics: CUET Mock Test - 9 - Question 17

Differentiate  with respect to x.

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 17

Consider y= 
Applying log on both sides, we get
log⁡y=3e3x log⁡x
Differentiating both sides with respect to x, we get

Mathematics: CUET Mock Test - 9 - Question 18

Find the second order derivative if y= 

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 18

Given that, 

By using u.v rule, we get

Mathematics: CUET Mock Test - 9 - Question 19

Differentiate log(cos(sinw.r.t x.

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 19

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


Mathematics: CUET Mock Test - 9 - Question 20

The volume of a cube of edge x is increasing at a rate of 12 cm/s. Find the rate of change of edge of the cube when the edge is 6 cm.

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 20

Let the volume of cube be V.
V=x3

Mathematics: CUET Mock Test - 9 - Question 21

What is the slope of the tangent to the curve y = 2x/(x2 + 1) at (0, 0)?

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 21

We have y = 2x/(x2 + 1)
Differentiating y with respect to x, we get
dy/dx = d/dx(2x/(x2 + 1))
= 2 * [(x2 + 1)*1 – x * 2x]/(x2 + 1)2
= 2 * [1 – x2]/(x2 + 1)2
Thus, the slope of tangent to the curve at (0, 0) is,
[dy/dx](0, 0) = 2 * [1 – 0]/(0 + 1)2
Thus [dy/dx](0, 0) = 2.

Mathematics: CUET Mock Test - 9 - Question 22

Find the approximate value of 

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 22

Let y = √x. Let x=64 and Δx=0.3
Then, Δy= 

dy is approximately equal to Δy is equal to:

dy=0.3/16=0.01875
∴ The approximate value of  is 8+0.01875=8.01875

Mathematics: CUET Mock Test - 9 - Question 23

Find ∫7 cos⁡mx dx.

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 23

Using Integration by Substitution, Let xm=t
Differentiating w.r.t x, we get
mdx=dt

Replacing t with mx again we get,

Mathematics: CUET Mock Test - 9 - Question 24

Find the integral of 8x3+1.

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 24

∫8x3+1dx
Using
∫8x3+1dx=∫8x3dx+∫1dx

=2x4+x+C.

Mathematics: CUET Mock Test - 9 - Question 25

What form of rational function  represents?

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 25

The given function  can also be written as  and is further used to solve integration by partial fractions numerical.

Mathematics: CUET Mock Test - 9 - Question 26

The value of f’(x) is -1 at the point P on a continuous curve y = f(x). What is the angle which the tangent to the curve at P makes with the positive direction of x axis?

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 26

Let, Φ be the angle which the tangent to the curve y = f(x) at P makes with the positive direction of the x axis.
Then,
tanΦ = [f’(x)]p = -1
= -tan(π/4)
So, it is clear that this can be written as,
= tan(π – π/4)
= tan(3π/4)
So, Φ = 3π/4
Therefore, the required angle which the tangent at P to the curve y = f(x) makes with positive direction of x axis is 3π/4.

Mathematics: CUET Mock Test - 9 - Question 27

Find the approximate value of 

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 27

Let y= Let x=49 and Δx=0.1
Then, Δy= 

dy is approximately equal to Δy is equal to

dy=0.1/14=0.00714
∴ The approximate value of  is 7+0.00714=7.00714

Mathematics: CUET Mock Test - 9 - Question 28

Integrate 3x2(cosx3+8).

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 28

By using the method of integration by substitution,
Let x3=t
Differentiating w.r.t x, we get
3x2 dx=dt
∫3x2(cosx3+8)dx=∫(cost+8)dt
∫(cost+8)dt=sint+8t
Replacing t with x3,we get
∫3x2(cosx3+8)dx=sinx3+8x3+C

Mathematics: CUET Mock Test - 9 - Question 29

Find ∫ 7x2-x3+2x dx.

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 29

To find ∫7x2−x3+2xdx
∫7x2−x3+2xdx=∫7x2dx−∫x3dx+2∫xdx
Using 

Mathematics: CUET Mock Test - 9 - Question 30

Find 

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 30

As it is not proper rational function, we divide numerator by denominator and get

Let 
So that, 5x–5 = A(x-3) + B(x-2)
Now, equating coefficients of x and constant on both sides, we get A + B = 5 and 3A + 2B = 5. Solving these equations, we get A=-5 and B=10.
Therefore, 

= x – 5log|x-2| + 10log|x-3|+C

Mathematics: CUET Mock Test - 9 - Question 31

What will be the differential function of √(x2 + 2)?

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 31

Let, y = f(x) = √(x2 + 2)
So, f(x) = (x2 + 2)1/2
On differentiating it we get,
f’(x) = d/dx[(x2 + 2)1/2]
f’(x) = 1/2 * 1/√(x2 + 2) * 2x
So f’(x) = x/√(x2 + 2)
So the differential equation is:
dy = f’(x)dx
Hence, dy = x/√(x2 + 2) dx

Mathematics: CUET Mock Test - 9 - Question 32

Find the approximate value of f(5.03), where f(x)=4x2-7x+2.

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 32

Let x=5 and Δx=0.03
Then, f(x+Δx)=4(x+Δx)2-7(x+Δx)+2
Δy=f(x+Δx)-f(x)
∴f(x+Δx)=Δy+f(x)
Δy=f’ (x)Δx
⇒ f(x+Δx)=f(x)+f’ (x)Δx
f(5.03)=(4(5)2-7(5)+2)+(8(5)-7)(0.03) (∵ f’ (x)=8x-7)
f(5.03)=(100-35+2)+(40-7)(0.03)
f(5.03)=67+33(0.03)
f(5.03)=67+0.99=67.99

Mathematics: CUET Mock Test - 9 - Question 33

Find ∫6x(x2+6)dx .

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 33

Let x2=t
Differentiating w.r.t x, we get
2x dx=dt
∫6x(x2+6)dx=3∫(t+6)dt
3∫(t+6)dt=3 
Replacing t with x2
∫6x(x2+6)dx= +18x2+C

Mathematics: CUET Mock Test - 9 - Question 34

Find the integral of 2 sin⁡2x+3.

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 34

To find ∫ 2 sin⁡2x+3 dx
∫2sin2x+3dx=∫2sin2xdx+∫3dx
∫2sin2x+3dx=2∫sin2xdx+3∫dx
∫2sin2x+3dx=  + 3x
∴∫2 sin⁡2x+3 dx = -cos⁡2x+3x+C

Mathematics: CUET Mock Test - 9 - Question 35

Find 

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 35

It is a proper rational function. Therefore,

Where real numbers are determined, 1 = A(x+2) + B(x+1), Equating coefficients of x and the constant term, we get A+B = 0 and 2A+B = 1.
Solving it we get A=1, and B=-1.
Thus, it simplifies to, 
= log|x+1| – log|x+2| + C

Mathematics: CUET Mock Test - 9 - Question 36

What will be the differential function of log(x2 + 4)?

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 36

Let, y = f(x) = log(x2 + 4)
So f(x) = log(x2 + 4)
On differentiating it we get,
f’(x) = d/dx[log(x2 + 4)]
So f’(x) = 2x/(x2 + 4)
So the differential equation is:
dy = f’(x)dx
Hence, dy = 2x/(x2 + 4) dx

Mathematics: CUET Mock Test - 9 - Question 37

Find the approximate value of 

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 37

Let y=√x. Let x=9 and Δx=2
Then, Δy= 

dy is approximately equal to Δy is equal to

dy=2/6=0.34
∴ The approximate value of √11 is 3+0.34=3.34.

Mathematics: CUET Mock Test - 9 - Question 38

Find the integral of 

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 38


Let log⁡x=t
Differentiating w.r.t x, we get

Replacing t with log⁡x, we get

Mathematics: CUET Mock Test - 9 - Question 39

Find the integral of 

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 39

To find 


Mathematics: CUET Mock Test - 9 - Question 40

An improper integration fraction is reduced to proper fraction by _____

Detailed Solution for Mathematics: CUET Mock Test - 9 - Question 40

An improper integration factor can be reduced to proper fraction by division, i.e., if the numerator and denominator have same degree, then they must be divided in order to reduce it to proper fraction.

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