Mathematics: CUET Mock Test - 1 - CUET MCQ

Probability of 3 students,
P(A) = 1/2, P(Ā) = 1/2

P(B) = 1/3, P(B̄) = 2/3

P(C) = 1/4, P(C̄) = 3/4

So, probability that no one solves the question is = 1/2 × 2/3 × 3/4 = 1/4
⇒ P(None) = 1/4

Then, the probability to solve the question is = 1 – 1/4 = 3/4

Let H₁H₂H₃H₄ B be the horses in which B is the winning horse.
Required probability

Any person can apply for any house.
Required probability

Let A be the event that minimum is 3 and let B be the event that their maximum is 6

To calculate the degree or the order of a differential equation, the powers of derivatives should be an integer.
On squaring both sides, we get a differential equation with the integral power of derivatives.

⇒ Order (the highest derivative) = 2
⇒ Degree (the power of highest degree) = 2

y = Acos(αx) + Bsin(αx)

dy/dx = -Aαsin(αx) + Bαcos(αx)

d²y/dx² = -Aα²cos(αx) - Bα²sin(αx)

= -α²(Acos(αx) + Bsin(αx))

= -α² * y

d²y/dx² + α²*y = 0

Distributive law is given by : 

On comparing the given equations with :
In the cartesian form two lines


we get ;

x₁ = -1, y₁ = -1,z₁ = -1, ; a₁ = 7, b₁ = -6, c₁ = 1 and 

x₂ = 3, y₂ = 5, z₂ = 7; a₂ = 1, b₂ = -2, c₂ = 1

Now the shortest distance between the lines is given by :

By definition , The angle θ between the planes A₁x + B₁y + C₁z + D₁ = 0 and A₂ x + B₂ y + C₂ z + D₂ = 0 is given by :

The equation of the plane through the line of intersection of the planes

Given inequality:

3(a - 6) < 4 + a

Step 1: Expand the left-hand side

3a - 18 < 4 + a

Step 2: Move all terms involving aaa to one side and constants to the other side

Subtract aaa from both sides:

3a - a - 18 < 4

This simplifies to:

2a - 18 < 4

Step 3: Add 18 to both sides

2a < 22

Step 4: Divide both sides by 2

a < 11

The correct option is D: a < 11.

Given:

Eliminate the denominators by multiplying both sides of the inequality by 35 (the least common multiple of 5 and 7):

This simplifies to:
7 × 2(x − 1) ≤ 5 × 3(2 + x)
Which further simplifies to:
14(x − 1) ≤ 15(2 + x)
Distribute the constants:
14x − 14 ≤ 30 + 15x
Move all terms involving x to one side and constants to the other side:
14x − 15x ≤ 30 + 14
This simplifies to:
−x ≤ 44
Solve for x by dividing by -1 (remember to reverse the inequality):
x ≥ −44x
The solution set is (−44, ∞)

Step 1: Simplify the inequality
The given inequality:
3x + 22x ≥ 5x
Combine like terms on the left-hand side:
(3x + 22x) = 25x
So, the inequality becomes:

Step 2: Subtract 5x from both sides
Subtract 5x from both sides to simplify further.
This simplifies to:
20x ≥ 0

Step 3: Solve for x
Divide through by 20 (a positive number, so the inequality direction remains unchanged):
x ≥ 0

Step 4: Interpret the solution
The solution to the inequality is:
x ≥ 0
This means that x can take any value in the interval [0, ∞).

Final Answer:
The solution set for x is:
[0, ∞)

x² ― 3|x| + 2 < 0
⇒|x|
2 ― 3|x| + 2 < 0
⇒(|x| ― 1)(|x| ― 2) < 0
⇒1 < |x| < 2
⇒ ― 2 < x < ―1 or 1 < x < 2
∴ X E ( ―2, ― 1) ∪ (1,2)

Hence equation of tangent to the given curve at (0 , 1) is :

(y−1) = 2(x−0),i.e..y−1 = 2x 

Answer: D
Explanation: Given that, y=9 log⁡t³

We assume that the particle moves with uniform acceleration 2f m/sec.
Let, x m be the distance of the particle from a fixed point on the straight line at time t seconds.
Let, v be the velocity of the particle at time t seconds, then,
So, dv/dt = 2f
Or ∫dv = ∫2f dt
Or v = 2ft + b  ……….(1)
Or dx/dt = 2ft + b
Or ∫dx = 2f∫tdt + ∫b dt
Or x = ft² + bt + a   ……….(2)
Where, a and b are constants of integration.
Given, x = 21, when t = 2; x = 43, when t = 4 and x = 91, when t = 7.
Putting these values in (2) we get,
4f + 2b + a = 21  ……….(3)
16f + 4b + a = 43  ……….(4)
49f + 7b + a = 91   ……….(5)
Solving (3), (4) and (5) we get,
a = 7, b = 5 and f = 1
Therefore, from (2) we get,
x = t² + 5t + 7
Therefore, the distance described by the particle in 3 seconds,
= [x]_{t = 3} = (3² + 5*3 + 7)m = 31m

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

Answer: B

Solution:

2^3x3 = 2⁹ = 512.

The number of elements in a 3 X 3 matrix is the product 3 X 3=9.

Each element can either be a 0 or a 1.

Given this, the total possible matrices that can be selected is 2⁹=512

Concept used:
The expression "a = b mod n" means that the remainder obtained when b is divided by n is assigned to the variable a. In other words, "mod n" is the modulo operation, which calculates the remainder of the division of b by n.

For example, if we have b = 17 and n = 5, then 17 divided by 5 gives a quotient of 3 with a remainder of 2. Therefore, a = 2, because 17 mod 5 is equal to 2.

Calculation:
(3)³ = 27 ≡ 0 (mod 9) So, b ≡ 0 (mod 9)
(2)⁵ = 32 ≡ 2 (mod 15) So, b ≡ 2 (mod 15)
(4)³ = 64 ≡ 4 (mod 10) So, b ≡ 4 (mod 10)
(5)³ = 125 ≡ 5 (mod 12) So, b ≡ 5 (mod 12)
Hence, option B is correct.

The product of any matrix with itself can be found only when it is a square matrix.i.e. m = n.

Concept use:

speed of the boat while going upstream is the speed of the boat in still water minus the speed of the current.

Calculation:

When the motorboat goes upstream, it moves against the current, so the effective speed of the boat is reduced. The effective speed of the boat while going upstream is the speed of the boat in still water minus the speed of the current.

In this case, the speed of the boat in still water is 15 km/h, and the speed of the current is 3 km/h. So, the effective speed of the boat while going upstream is:

15 km/h - 3 km/h = 12 km/h

The time it takes to travel a certain distance is the distance divided by the speed. So, the time it takes for the boat to go 36 km upstream is:

36 km / 12 km/h = 3 hours

So, the boat takes 3 hours to go 36 km upstream.

Concept:

A function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.

Explanation:

Statement 1: The relation f defined by

is a function.

For 0, 1, 2 for the function x³ the values are 0, 1, 8

For 2, 3, 4, 5, 6, 7, 8 for the function 4x the values are

8, 12, 16, 20, 24, 28, 32

So, for x = 2, the function has the same value i.e., 8 (for x3 and 4x).

Hence f(x) is a function.

Statement 2: The relation g defined by

is a function.

For 0, 1, 2, 3, 4 for the function x2 the values are 0, 1, 4, 9, 16.

For 4, 5, 6, 7, 8 for the function 3x the values are 12, 15, 18, 21, 24.

So, for x = 4 the function has different values i.e., 16 (for x2) and

12 (for 3x).

Hence g(x) is a not function.

∴ Correct answer is option (1)

Concept:

n(A × B) = Number of elements in A × Number of elements in B

Calculation:

Given that

A = Finite set, B = Finite set

A number of elements in the Cartesian product (n(A × B)):

n(A × B) = Number of elements in A × Number of elements in B

Therefore, n(A × B) = n (A) × n (B)

Concept:

Codomain - A codomain is the group of possible values that the dependent variable can take.

Range - The range is all the elements from set B that have the corresponding pre-image in set A.

Calculation:

Given,

For f(x) to be defined, 3x + 5 ≠ 0

⇒ x =

∴ A =

Now, y = f(x) is onto

⇒ y =

⇒ 3xy + 5y = 2x + 3

⇒ 3xy - 2x = 3 - 5y

⇒ x(3y - 2) = 3 - 5y

⇒ x =

For x to be defined for

Since, for onto functions co-domain = range

∴ B =

∴

Concept:

F of G of x is a composite function made of two functions f(x) and g(x).

It is denoted by f(g(x)) or (f ∘ g)(x) and it means that x = g(x) should be substituted in f(x).

It is an operation that combines two functions to form another new function.

For finding f(g(x)), we have to first find g(x) and then take g(x) as input of f(x) and simplify.

Formula Used:

(a + b)² = a² + b² + 2ab

Calculation:

We have,

⇒ f(x) = x²

⇒ g(x) = x + 3

⇒ f(g(x))

⇒ f(x + 3)

⇒ (x + 3)²

⇒ x² + 9 + 6x

∴ Then the value of F(g) is x² + 6x + 9.

Calculation:
log32 ⋅ log43 ⋅ log54 ⋅ log65 ⋅ log76 ⋅ log87
⇒ (log32 ⋅ log43) (log54 ⋅ log65) (log76 ⋅ log87)
[logbM × logab = logaM]
⇒ log₄2 .log₆4. log₈6
⇒ (log₄2 .log₆4) log₈6
⇒ log62 ⋅ log₈6
⇒ log₈2
⇒ 1/log₂8 
⇒ 1/log₂2³ = 1/3log₂2 =
∴ log32 ⋅ log43 ⋅ log54 ⋅ log65 ⋅ log76 ⋅ log87 = 1/3