IPMAT Mock Test - 1 (New Pattern) - Commerce MCQ

Given,

²ⁿC₃ : ⁿC³ = 17 : 2

⇒ n = 26

Hence, the correct option is (B).

General term: General term in the expansion of (x + y)ⁿ is given by

In the binomial expansion of (x + y)ⁿ, the rth term from end is (n − r + 2)^th term.

Calculation:

We have to find 9th term from the end in (x − 1 / x)¹²

We know that r^th term from end means (n − r + 2)^th term from start.

So. 9^th term from the end = [12 − 9 + 2]^th term from start = 5^th term from start

General term:

Hence, the correct option is (A).

Multiply and divide by √7 + 2

Hence, the correct option is (A).

When two matrices are equal, all the corresponding elements must be equal.

Given,

Comparing two elements of first column,

2x + y = 7

⇒ y = 7 − 2x ...(1)

5x − 7 = y ...(2)

Comparing value of y from equation (1) and (2) we get,

7 − 2x = 5x - 7

⇒ 7 + 7 = 5x + 2x

⇒ 14 = 7x

∴ x = 2

Putting value of x in equation (1), we get

y = 7 − 2 × 2

⇒ y = 7 − 4

⇒ y = 3

Now,

x + y = 2 + 3 = 5

So, the value of x + y is 5.

Hence, the correct option is (C).

A shopkeeper sells 200 shirts

Profit = S.P of 25 shirts

Formula:

C.P = S.P – Profit

Profit % = Profit / C.P × 100

Calculation:

Let S.P of 200 shirts be Rs. 200

S.P of 25 shirts = Rs. 25

Profit = S.P of 25 shirts = Rs. 25

C.P = Rs. 200 – Rs. 25 = Rs. 175

Profit % = 25 / 175 × 100

= 100 / 7%

= 14.28%

∴ The profit percentage is 14.28%

Hence, the correct option is (C).

m : n = 3 : 2

= 11 : 1

Hence, the correct option is (C).

= 120 + 75 + 80 + 135 + 125 + 75 = 610

Average number of males

= 610 / 6 = 101.67

Total number of Females visiting the restaurant

= 90 + 125 + 110 + 95 + 105 + 110 = 635

Average number of females

= 635 / 6 = 105.833

∴ Required difference = 105.83 − 101.67 = 4.16

Hence, the correct option is (D).

= 75 +135 + 125 = 335

Number of Females visiting the restaurant on Tuesday,

Thursday and Friday = 125 + 95 + 105 = 325

∴ Required ratio = 335 / 325 = 67 / 65 = 67 : 65

∴ The ratio between the number of Males visiting the restaurant on Tuesday, Thursday and Friday together and the number of females visiting the restaurant on the same days is 67 : 65.

Hence, the correct option is (C).

= 210 + 190 + 230 = 630

Customers who have pre-booking on Monday, Wednesday and Friday

= 40 / 100 × 630 = 252

Total customers on Tuesday, Thursday and Saturday = 200 + 230 + 185 = 615

Customers who have pre-booking on Monday, Wednesday and Friday

= 60 / 100 × 615 = 369

Required sum = 252 + 369 = 621

∴ The sum of the customers who have pre-booking in the restaurant is 621.

Hence, the correct option is (A).

= 130 / 100 × (80 + 110)

= 13 × 19

= 247

Ratio of Male to females = 8 : 5

∴ Number of males on Sunday

= 8 / 13 × 247 = 152

Number of males on Monday = 120

∴ Required percentage

=152 / 120 × 100 = 126.67%

Hence, the correct option is (B).

x −y^x−y

Here x = 2,y = −2

By substituting in x − y^x−y we get,

x − y^x−y = 2 − (−2)² − (−2)

= 2 − (−2)² + 2

= 2 − (−2)⁴

= 2 − (16)

= −14

The value of x − y^x−y is −14.

Hence, the correct option is (D).

According to the question,

a² + b² = 289

ab = 120

(a + b)² = a2 + b² + 2ab

(a + b)² = (289)² + (2 × 120)

(a + b)² = 529

(a + b) = 23

Hence, the correct option is (A).

x² < 4

⇒ x² − 4 < 0

⇒ (x − 2) × (x + 2) < 0

⇒ −2 < x < 2

⇒ x ∈ (−2, 2)

Hence, the correct option is (B).

Let the number of natural numbers be x

⇒ 15×x + 30 − 5 = 17.5×x

⇒ 15x + 25 = 17.5x

⇒ 2.5x = 25

⇒ x = 10

Hence, the correct option is (D).

= 2

Hence, the correct option is (B).

5/9 × (10ⁿ − 1)

We put n = 1,

It means Option C is satisfying the sequence so the n^th term would be

= 5/9 × (10ⁿ − 1)

Hence, the correct option is (C).

and y m/sec respectively.

Then, length of the first train = 27x metres,

and length of the second train = 17y metres.

∴ 27x + 17y / x + y = 23

⇒ 27x + 17y = 23x + 23y

⇒ 4x = 6y

⇒ x/y = 32

Hence, the correct option is (B).

{x : x is a common point to any two parallel lines}

So, it is an empty set as parallel lines do not have a common point.

Set of all even prime number = {2}

Thus, its not empty set.

{x : x² − 2 = 0 and x is a real number }

x = +2 and −2

So, its not empty set.

{x:x is a natural number, x>8 and simultaneously x < 12}

Thus, it is also non empty set.

Hence, the correct option is (D).

First term (a) =22

Last term (I) =−11

Sum =66

Sum of an AP is given by:

= Number of terms × First term + Last term / 2

n = 12

No. of terms = 12

Hence, the correct option is (B).

1/2 log10²⁵ –2log10³ + log10¹⁸

Hence, the correct option is (C).

The Sum of money becomes 5 times of itself in 16 years

Formula used:

(i) SI = PRT / 100

Where P = principal

R = rate

T = time

(ii) A = SI + P

Where, A = Amount

SI = simple interest

Let the principal be P.

For 16 years,

SI = (P × R × 16) / 100

= 16PR / 100

A = SI + P

⇒ 5P = 16PR / 100 + P

⇒ 4P = 16PR / 100

⇒ R = 400 / 16

⇒ R = 25%

∴ The rate of interest is 25%.

Hence, the correct option is (B).

x2 − y²a − b = xy/h

For x² − 2pxy − y² = 0, equation of angle bisector will be

⇒ x² − y² / 1 − (−1) = xy / − p

⇒ x² − y² + 2xy / p = 0 .....(i)

Given equation of angle bisector is:

x² − 2qxy + y² = 0 .....(ii)

Comparing both the equations:

⇒ 2 / p = −2q

⇒ pq = −1

⇒ pq + 1 = 0

Hence, the correct option is (C).

Here, we have to find (A × B) ∩ (A × C)

As we know that, A × B = {(a, b) ∣ a ∈ A and b ∈ B}

⇒ A × B = {(1, 2), (1, 3), (4, 2), (4, 3)} ......(i)

⇒ A × C = {(1, 3), (1,5 ), (4, 3), (4, 5)} ......(ii)

From (i) and (ii) we can say that,

⇒ (A × B) ∩ (A × C) = {(1, 3), (4, 3)}

Hence, the correct option is (A).

Given:

5^1/4 × (125)^0.25

= 5^0.25 × (5³)^0.25

= 5^0.25 × 5^(3×0.25)

= 5^0.25 × 5^0.75

= 5^{(0.25 + 0.75)}

= 5¹

= 5

Hence, the correct option is (B)

or,

63 is divisible by 7 for any power, so required remainder will depend on the power of 4

Required remainder

Note :

4+/7 remainder = 4

If we check for more power we will find that the remainder start repeating themselves as 4, 2, 1, 4, 2, 1 and so on. So when we get A number having greater power and to be divided by the other number B, we will break power in (4n + x) and the final remainder will depend on x i.e.

A^x / B.

Hence, the correct option is (C).

Option (D): {x ∈ N : x is even}

As we can see that, elements of {x∈N:x is even} are: 2, 4, 6, 8,…….

So, there are infinitely many elements in the set {x∈N:x is even}

Thus, {x ∈ N : x is even } is not a finite set.

Option (A) {x : x ∈ N and x² < 36}

As we can see that, elements of {x:x∈N and x² < 36} are: 1,2,3,4,5

So, there are 5 elements in the set {x:x∈N and x² < 36}

Thus, the set {x : x ∈ N and x² < 36} is a finite set.

Option (B): {x ∈ z : 0 < x < 10}

As we can see that, elements of {x ∈ z : 0 < x < 10} are: 1, 2, 3,………,9

So, there are 9 elements in the set {x ∈ z : 0 < x < 10}

Thus, the set {x ∈ z : 0 < x < 10} is a finite set.

Option (C): {x : x ∈ N and x² = x}

∵ x² = x

⇒ x² − x = 0

⇒ x(x − 1) = 0

⇒ x = 0 or 1

But since x∈N.

So, x = 0 ∉ {x : x ∈ N and x² = x}

Therefore, only x = 1 ∈ {x : x ∈ N and x² = x}

Thus, {x : x ∈ N and x² = x} is a finite set.

Hence, the correct option is (D).

R is a relation on Z and is defined as:

R = {(a, b) : a − b is divisible by 7 where a, b ∈ Z}

We know that:

A relation R on a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.

Therefore,

R is reflexive as:

a − a = 0 is divisible by 7 for all a ∈ Z.

Suppose, if (a, b) ∈ R

⇒ 7 divides a − b i.e.,

⇒ a − b = 7m, where m ∈ Z

⇒ b−a = 7n, where n = −m

⇒ 7 divides b−a too, which implies that:

(b, a) ∈ R.

Therefore, R is symmetric.

Suppose, if (a, b) ∈ R and (b, c) ∈ R, then:

a − b and b − c are divisible by 7.

⇒ a − b = 7m and b − c = 7n, where m, n ∈ Z

⇒ a − c = 7q, where q = m + n

⇒ (a, c) ∈ R

Therefore, R is transitive.

Hence, the correct option is (D).

Distance between A and B = 600 km

Time taken to meet each other (when they start together) = 12 hours

Distance = Speed × Time

Relative speed of two bodies when they travel towards each other = Sum of their speed

When they start moving towards each other, they meet in 12 hours,

Let the speed of A be ‘A’ km/hr and the speed of B be ‘B’ km/hr

⇒ 600 / (A + B) = 12

⇒ A + B = 50......(1)

If A started moving 5 hours after B, then they meet in 10 hours,

Distance covered by A and B in 10 hours = 50 × 10 = 500

Remaining distance (600 - 500 = 100 km) was already covered by B in 5 hours

Speed of B = 100 / 5 = 20 km/hr

∴ The speed of B is 20 km/hr

Hence, the correct option is (A).

First matrix =

Number of rows (m₁) = 2

Number of columns (n₁) = 3

So, order of first matrix = 2 × 3

And second matrix =

Number of rows (m₂) = 3

Number of columns (n₂) = 1

So, order of first matrix = 3 × 1

As we know,

Multiplication of matrices is possible if

Number of column (n₁) in the first matrix = Number of rows (m₂) in the second matrix

i.e., n₁ = m₂

∴n₁ = m₂ = 3

So, is possible.

As we know,

Multiplication of matrices is possible if the order of the new matrix is a row of the first matrix and column of the second matrix.

i.e., m₁ × n₂

Now,

Order of new matrix formed by multiplication = m₁ × n₂ = 2×1

Hence, the correct option is (B).

Distance = 4 km

Time = 6 minutes

Distance = Speed × Time

Speed = Distance / Time

Time = 6 minutes

⇒ Time = ( 6/60) hr = (1/10) hr

⇒ Speed = DistanceTime = 4(1/10)

⇒ Speed = 40 km/hr

According to the question,

If its speed is decreased by 2 km/hr

⇒ New speed = (40 - 2) km/hr = 38 km/hr

Time taken by the car to cover same distance,

Time = Distance / Speed

⇒ Time = (4/38) hr = (4/38) × 60 = 120/19 minutes

∴ The time taken by the car to cover same distance is 120/19 minutes.

Hence, the correct option is (A).