CUET PG Computer Science Mock Test - 2 - CUET PG MCQ

Given:

ⁿC₁₀ = ⁿC₁₅ -------(1)

Concept:
ⁿC_r =

Calculation:

We can write,
nC₁₀ = ------(2)
ⁿC₁₅ = ------(3)

By equation (1) comparing equations (2) and (3), we get

------(4)

Comparing both sides of equation (4), we get

10 = n - 15, and

⇒ 15 = n - 10

⇒ n = 25

Now, 27Cn for n = 25, will be

²⁷C₂₅ = =

⇒ ²⁷C_n = = = 351

∴ ²⁷C_n or ²⁷C₂₅ equals to 351.

The word "HEALING" contains 7 letters, of which there are 3 vowels (E, A, I) and 4 consonants (H, L, N, G). When the vowels AEI are always together, they can be supposed to form one letter. Then, we have to arrange the letters HLNAG(AEI).
The number of ways to arrange the HLNAG(AEI) is 5! = 120.
The vowels (AEI) can be arranged among themselves in 3! = 6 different ways.
Therefore, the total number of ways to arrange the letters of 'HEALING' so that the vowels always come together is 120 × 6 = 720.
So, the correct option is 2) 720.

Concept:

Cube Roots of unity are 1, ω and ω²

Here, ω = and ω² =

Property of cube roots of unity

1. ω³ = 1

2. 1 + ω + ω² = 0

3. ω = 1 / ω ² and ω² = 1 / ω

4. ω³ⁿ = 1

Calculation:

We know that ω = and ω² =

= 1 + 1 = 2

Concept:
Maximum value of sin x is 1
Maximum value of cos x is 1
sin 2x = 2 sin x cos x
Calculation:
To Find: maximum value of sin x ⋅ cos x
Let f(x) = sin x ⋅ cos x

As we know, the maximum value of sin x is 1
Therefore the maximum value of sin 2x is 1
Hence maximum value of f(x) is

The logic followed here is:
Logic: Study of specific natural objects.
Here,
Botany : Plants → Botany is the study of Plants.
Similarly,
Conchology : Shells → Conchology is the study of Shells.
Hence, the correct answer is "Option 2".

Concept:

If then determinant of A is given by:

• |A| = a₁₁ × {(a₂₂ × a₃₃) – (a₂₃ × a₃₂)} - a₁₂ × {(a₂₁ × a₃₃) – (a₂₃ × a₃₁)} + a₁₃ × {(a₂₁ × a₃₂) – (a₂₂ × a₃₁)}
• Elementary row or column transformations do not change the value of the determinant of a matrix.

Calculation:

Given:

Apply R₂ → R₂ – R₁ and R₃ → R₃ – R₁, we get

Expanding along R₁, we get

Δ = 1 (x²y²z² – 0) – 0 + 0

∴ Δ = x²y²z²

Given:

20% of the pens are red,4% pens are red and defective.

Formula used:

Bayes Theorem:- Let E1, E2,... En be n mutually exclusive and exhaustive events associated with a random experiment and let S be the sample space. Let A be any event which occurs together with any one of E1 or E2 or... or En such that P(A) ≠ 0. Then
Bayes Formula:

P(Ei | A) = , i = 1, 2, ... n
Calculation:

Let A be the event that the pens are red

B be the event that the pens are defective

P(A) = =

Probability that the pen is red and defective is:

P(A ∩ B) = =

Probability that the pen is defective, given that it is red,

P() = =

⇒

⇒ 0.2

The Correct Answer is 0.2

Concept:

• Matrix product is commutative if both are diagonal matrices of same order.
  If A and B are diagonal matrices of same order then;
• The product of any square matrix and the appropriate identity matrix is always the original matrix.
  AI = IA = A
• I² = I, Here I is identity matrix.

Calculation:
1. Statement 1 is wrong because A and B are not diagonal matrices of same order.
2. (A – I) (I + A) = O
⇒ AI – I² + A² – IA = 0
⇒ - I² + A² = 0
⇒ A² = I² =I
So, statement 2 is correct.
∴ Option 2 is correct.

The logic followed here is:​
'VOCAL' is coded as '90327'
'WAVES' is coded as '31758'
'VOTED' is coded as '42631'

Thus, E is coded as 1.
Hence, "Option 1" is the correct answer.

The position of letters according to the English alphabet series:

The logic follows here is:

Hence, the correct answer is 'Option (2)'.

 Concept:

• Equation of a line perpendicular to a given line ax + by + c = 0 is bx - ay + λ = 0, where λ is a constant.
• The intercept form of the line ⇔ , Where a is the x- intercept and b is the y- intercept.

Calculation:
Given: Area of triangle is 5 square units
Given equation of line is 5x – y = 0
Equation of a line perpendicular to a given line is
⇒ x + 5y = λ …. (1)

Area of triangle = 5

⇒ λ² = 50
∴ λ = ± 5√2
Put the value of λ in equation 1^st
So, x + 5y = ± 5√2
⇒ x + 5y ± 5√2 = 1 

Concept:

Let A and B be the subsets of X, then then Venn Diagram to represent that is:

Union of sets (A ∪ B):

Intersection of subsets (A ∩ B):

Complement of subset (A’):

Difference of subsets (A – B):

Calculation:

Let figure given below denotes A and B as subsets of X.

Now, (A ∩ B’) is A intersection not B i.e., (A ∩ B’) = A – (A ∩ B) as shown below.

Similarly, (A’ ∩ B) is B intersection not A i.e., (A’ ∩ B) = B – (A ∩ B) as shown below.

Now, C = (A ∩ B’) ∪ (A’ ∩ B) means (A ∪ B) – (A ∩ B) as shown below.

Concept:
Sin (A + B) = sin (A) cos (B) + cos (A) sin (B)

Calculation:

Let, a = r sin α and b = r cos α
∴




We have, a = r sin α and b = r cos α
Squaring and adding,


Hence, option (b) is correct.

Concept:
According to Bayes’ theorem, if multiple events A_i form an exhaustive set with another event B.

Where,
Calculation:
Let P(A) be the probability of choosing a ball from box A
P(A) = 1/2
Let P(B) be the probability of choosing a ball from box B
P(B) = 1/2
P(R) be the probability of getting a red ball.
P(R/A) be the probability of getting red given that we are drawing a ball from box A
P(R/A) = 3/5
P(R/B) be the probability of getting red given that we are drawing a ball from box B
P(R/B) = 5/9
From the total probability
P(R) = P(R/A) P(A) + P(R/B) P(B)

Let P(B/R) be the probability of chosen red ball given that it was from box B,

Concept:

Parabola: The locus of a point which moves such that its distance from a fixed point is equal to its distance from a fixed straight line. (Eccentricity = e =1)

Calculation:

Given: x2 = 16y

⇒ x2 = 4 × 4 × y

Compare with standard equation of parabola x2 = 4ay

So, a = 4

Therefore, Focus = (0, a) = (0, 4)

Concept:

Number of ways to select 3 points out of the n collinear points =

Calculation:

Number of triangles that can be formed is equal to the number of ways to select 3 non-collinear points.

⇒ Number of ways to select 3 points from 15 points = ¹⁵c₃

Let n points be collinear.

⇒ Number of ways to select 3 points out of the n collinear points = ⁿc₃

So, Number of ways to select 3 non-collinear points = (Number of ways to select 3 points using all the points - Number of ways to select 3 points using the collinear points)

⇒ Number of ways to select 3 non-collinear points = ¹⁵c₃ - ⁿc₃

⇒ Number of triangles that can be formed = ¹⁵c₃ - ⁿc₃

⇒ 445 = ¹⁵c₃ - ⁿc₃

⇒ ⁿc₃ = ¹⁵c₃ – 445 = 455 – 445 = 10

⇒ n (n – 1) (n – 2) = 60

∴ n = 5

Concept:
Perpendicular Distance of a Point from a Line:
Let us consider a plane given by the Cartesian equation, Ax + By + C = 0
And a point whose coordinate is, (x₁, y₁)
Now, distance =
Calculation:
Given a straight line x = y + 2 touches the circle 4(x² + y²) = r²
x = y + 2 ⇒ x – y – 2 = 0
⇒ x² + y² = r²/4
Centre = (0, 0)
Radius = r/2

From above figure,
CP = Radius

⇒ r/2 = √2
∴ r = 2√2  

Case: 1
If a, b, c are in AP,
Then b – a = c – b
⇒ a – b = b – c
∴
Case: 2
If a, b, c are in GP,
Then
Subtracting 1 from both sides,



∴
Case: 3
If a, b, c are in HP

⇒ ba + bc = 2ac
⇒ ba + bc = ac + ac
⇒ ba – ac = ac – bc
⇒ a (b – c) = c (a – b)

∴

Concept:

Calculation:

Let us consider 1 + ln (x) = t.

Now by differentiating both the sides with respect to x, we get

-----(1)

Now by using the equation (1), we get

As we know that,

Now by substituting 1 + ln (x) = t in the above equation we get,

Concept:

The degree of a differential equation is represented by the power of the highest order derivative in the given differential equation.

Calculation:

Given:

For the given differential equation the highest order derivative is 4.

Now, the power of the highest order derivative is 1.

∴ The degree of the given differential equation is 1.

The least possible Venn diagram for the given statements is as shown below :

Conclusions:

I. Some males are not females → Does not follow (As some fathers are females and all fathers are males. As no direct relation is given between males and females, therefore it is false.)

II. Some mothers are females → Does not follow (As some fathers are females and some fathers are mothers. So it is possible but not definite.)

III. Some males are mothers → Follow (As all fathers are males and some fathers are mothers. As whole fathers comes in males and has some part common with mothers, therefore it is true.)

∴ Here, Only conclusion III follows.​

Hence, the correct answer is "Option 3".

The correct answer is 0
Given:
Every element in the central column of the matrix has a simple arithmetic relationship with the pairs on the left and right in the corresponding row.
The matrix is

Calculation:
We observe that the element in the central column (1, X, 3, 2) is the difference between the sums of the pairs on the left and right in the corresponding row.
For the first row:
(17 - 12) - (23 - 19) = 5 - 4 = 1
For the third row:
(24 - 17) - (36 - 32) = 7 - 4 = 3
For the fourth row:
(35 - 28) - (24 - 19) = 7 - 5 = 2
Now, for the second row:
(23 - 21) - (20 - 18) = 2 - 2 = 0
Therefore, the value of X should be 0

Statement I: Teddy started walking towards the north direction from his home till 11 meters and returned back in the same direction and stopped at the stop after walking for 4 meters.

So, this statement is enough for us to understand that Teddy 7 m away from his home.
11 m – 4 m = 7 m
Statement II: Teddy’s school is 5 meters to the south of Teddy’s home.

This statement is not enough for us to predict how far is Teddy from his home.
Hence, “Statement I alone is sufficient” is the correct answer.

Concept:

1. A ∆ B is the symmetric difference that represents the objects that belong to A or B but not to their intersection.
   • A Δ B = (A ∪ B) - (A ∩ B)
2. A ⊆ B subset A is a subset of B. set A is included in set B.
3. A ⋃ B, ⋃ is a union, that represents the objects that belong to set A or set B
4. A ⋂ B, ⋂ is an intersection, that represents the objects that belong to set A and set B
5. a ∈ A means x is an element of A, i.e. belongs to set membership

Calculation:

Given: A, B, C be subsets of X

Statement 1: A ⊂ C ⇒ (A ∩ B) ⊂ (C ∩ B), (A ∪ B) ⊂ (C ∪ B)

We know that,

If A ∪ B = A ∪ C and A ∩ B=A ∩ C, then B = C which gives

⇒ A ⊂ C

⇒ A ⊂ C ⇒ (A ∩ B) ⊂ (C ∩ B), (A ∪ B) ⊂ (C ∪ B)

Hence statement 1 is true.

Statement 2: (A ∩ B) ⊂ (C ∩ B) for all sets B ⇒ A ⊂ C

Let A = {1, 2, 3}, B = {3, 4, 5} and C = {1, 3, 6, 7, 8}

⇒ A ∩ B = {3} ⊂ (C ∩ B) but A ⊂ C is not true.

Hence, statement 2 is false.

Statement 3: (A ∪ B) ⊂ (C ∪ B) for all sets B ⇒ A ⊂ C

Concept:

• The greatest integer function is discontinuous at all the integers.
• For function to be continuous it should be first defined

Calculation:

Statement (1) is correct as the greatest integer function is discontinuous at all the integers.

2. f(x) = cot x

⇒ f(x) = cos x /sin x

At x = nπ, sin x = 0

∴ cot x is not defined at x = nπ

Hence, cot x is discontinuous at x = nπ

So this statement (2) is correct.

Hence, option (3) is correct.

Concept:

Calculation:

Given differential equation is

Integrating both sides, we get

(∵ -c = c)

Hence option 4 is correct.

Calculation:

Let’s analyze each statement

1. Let, f(x) = |x|,

Left hand derivative ≠ Right hand derivative

At x = 0, f’(x) doesn’t exist, so statement 1 is true.

2. Let, f(x) = tan^-1(x)

Now, f’(x) = 1/ (1 + x²) = finite value (At some points)

So Derivative of f(x) may exist finitely at some point.

Hence statement 2 is also true

3. We know that derivative of function is undefined at some points it means derivative of f(x) may be infinite (geometrically) at some point.

So, statement 3 is also true

Hence, option (4) is correct.

Logic:

Here, in all the options except option 4 proper quadrilateral is present inside and outside.

1.  → Here, a quadrilateral is present inside and outside.

2. → Here, a quadrilateral is present inside and outside.

3.→ Here, a quadrilateral is present inside and outside.

4. → Here, a quadrilateral is not present inside and outside.

Hence, option 4 is a different one.

Concept:
L-Hospital Rule: Let f(x) and g(x) be two functions
Suppose that we have one of the following cases,

Then we can apply L-Hospital Rule 
Calculation:

Here, we have 0/0 form, so applying L-Hospital Rule,

Hence, option (a) is correct.

Concept:

1. Integration by parts:

Integration by parts is a method to find integrals of products

The formula for integrating by parts is given by;

Where u is the function u(x) and v is the function v(x)

2. ILATE Rule:

Usually, the preference order of this rule is based on some functions such as Inverse, Logarithm, Algebraic, Trigonometric and Exponent.

Calculation:

Let I = ∫ (e^{log x} + sin x) cos x dx

(∵ e^{log x} = x)

⇒ I = I₁ + I₂ .... (1)

Now, I₁ =

Applying by parts, we get

⇒ x sin x - + c

⇒ x sin x + cos x + c

Now, I₂ =

Let sin x = t

Differentiating with respect to x, we get

⇒ cos x dx = dt

⇒ I₂ =

⇒

Put the value of I₁ and I₂ in equation (1), we get

∴ The required integral (I) is .