IPMAT Mock Test - 10 (New Pattern) - Commerce MCQ

⇒ -(5 ≤ (x – 1) ≤ 5), (x ≤ -2 or x ≥ 2)

⇒ -(4 ≤ x ≤ 6), (x ≤ -2 or x ≥ 2)

Now, required solution is

x ∈ [-4, -2] ∪ [2, 6]

Hence, the correct option is (C).

As we know that,

As we know that, cotx is negative in 2nd and 4th quadrant.

Therefore, are the solution of the given equation.

Hence, the correct option is (C)

Involuntary Matrix: Any square matrix of order n is said to be an involuntary matrix if A² = I, where I is the identity matrix of order n.

Nilpotent Matrix: Any square matrix of order n is said to be nilpotent matrix if there exist least positive integer m such that A^m = O, where O is the null matrix of order n.

Idempotent Matrix: Any square matrix of order n is said to be an idempotent matrix if A² = A.

Hence, A is an involuntary matrix as A² = I.

Hence, the correct option is (B).

Hence, the correct option is (B).

A = {x ∈ R : x² = 2}

And, B = {y ∈ R : y² − 5y + 6 = 0}

Then,

x² − 2 = 0 (Given)

Similarly,

y² − 5y + 6 = 0 (Given)

⇒ y² − 3y − 2y + 6 = 0

⇒ (y − 3)(y − 2) = 0

⇒ y = 2, 3

⇒ B = {2, 3}

∴ n(B) = 2.

As we know that, if n(A) = p, n(B) = q, then:

n(A × B) = n(A) × n(B) = p × q.

⇒ n(A × B) = n(A) × n(B)

= 2 × 2

= 4

Hence, the correct option is (B).

Speed of train starting from station A = 80 km/hr

Train from station B started after 45 minutes at the speed of 90 km/hr

Distance between station A and station B = 400 km

Distance = Speed x Time

Relative speed is defined as the speed of a moving object with respect to another. When two objects are moving in the same direction, the relative speed is calculated as their difference. When the two objects are moving in opposite directions, the relative speed is computed by adding the two speeds.

Train from station B towards station A started after 45 minutes of the train from station A

Speed of train from station A = 80 km/hr

For 45 minutes it runs alone,

⇒ Distance covered in 45 minutes

Total distance =400 km

⇒ Distance remaining = 400 − 60 = 340 km

Now, when the two objects are moving in opposite directions, the relative speed is computed by adding the two speeds relative speed = (80 + 90)

⇒ Relative speed = 170 km/hr

⇒ Time taken to cover remaining distance = 340/170 = 2 hours

So, they will meet after 2 hr

Now, distance covered by train from station A in 2 hr = (80 × 2) = 160 km

Therefore, Total distance at which they will meet from A = (160 + 60) km

⇒ Distance from station A at which both trains meet = 220 km.

Hence, the correct option is (C).

On comparing with std equation a² = 9, b² = 25

b > a

⇒ Length of Latus rectum

⇒ Length of Latus rectum = 18/5

Hence, the correct option is (C).

Then, subsets of A,

B = Φ, {1}, {2}, {3}, {4}, {1, 2}, {2, 3}, {1, 3}, {2, 4}, {1, 4}, {3, 4}, {1, 2, 3}, {1, 2, 4} {2, 3, 4}, {1, 3, 4}, {1, 2, 3, 4}

And we know that a subset B of a set A is called a proper subset of A

if A ≠ B.

Then, proper subset of A

= {1}, {2}, {3}, {4}, {1, 2}, {2, 3}, {1, 3}, {2, 4}, {1, 4}, {3, 4}, {1, 2, 3}, {1, 2, 4} {2, 3, 4}, {1, 3, 4}, {1, 2, 3, 4}

We know that whenever a set B is a subset of set A, we say that A is a superset of B or A ⊇ B

So, super set of {3} in subset of B

= {1}, {2}, {4}, {1, 2}, {2, 4}, {1, 4}, {1, 2, 4}

Hence, the correct option is (C).

The hexagon is composed of 6 equilateral triangles, each with a side of 2.

Area of an equilateral triangle

⇒ Area of regular hexagon =6 x area of small equilateral triangles

Hence, the correct option is (C).

We know, a square matrix in which every element except the principal diagonal elements is zero is called a Diagonal Matrix.

The inverse of a diagonal matrix is obtained by replacing each element in the diagonal with its reciprocal which is again a diagonal matrix as well as a symmetric matrix.

Consider a diagonal matrix

Its inverse is obtained by replacing each element in the diagonal with its reciprocal.

which is again diagonal matrix and the symmetric matrix also.

Hence, the correct option is (C).

= 7 ⁶ − 6⁶

= (7³)² − (6³)²

= (7³ − 6³)(7³ + 6³)

= (343 − 216) × (343 + 216)

= 127 × 559

= 127 × 13 × 43

Clearly, it is divisible by 127,13 as well as 559

Hence, the correct option is (D).

⇒ x² = 20 - x

⇒ x² + x - 20 = 0

⇒ x² + 5x - 4x - 20 = 0

⇒ x(x + 5) - 4(x + 5) = 0

⇒ (x + 5)(x - 4) = 0

⇒ x = -5, 4

A square root value cannot be negative. So x = -5 is not possible.

Hence, the correct option is (A).

x² + y² – 8x – 4y – 5 = 0

Standard equation is x² + y² + 2gx + 2fy + c = 0

⇒ On comparing 2g = – 8, 2f = – 4, c = – 5

⇒ g = – 4 , f = – 2 , c = – 5

∴ Radius of circle is 5 units.

Hence, the correct option is (B).

New C.P. =125% of Rs. 100= Rs. 125

New S.P. = Rs. 420 .

Profit = Rs. (420−125)= Rs.

∴ Required percentage

Hence, the correct option is (B).

Profit earned on 1 eraser = 735/245 = Rs. 3

Total eraser sold by Shop La and Shop Pa = 65 + 95 = 160

Required profit = 160 × 3 = Rs 480

Hence, the correct option is (D).

Selling price = 12 x 125% = Rs 15

Marked price

Discount = 20 − 15 = Rs 5

Total discount

Hence, the correct option is (D).

Number of Pens sold by shop Pa and Na = 80 + 90 = 170

Ratio = 215 : 170 = 43 : 34

Hence, the correct option is (A).

x ∈ A ∩ B

⇒ x = 4n and x = 6n, n ∈ Z

⇒ x is a multiple of 4 and x is a multiple of 6

⇒ x is a multiple of 4 and 6 both

⇒ x is a multiple of 12

⇒ x = 12n, n ∈ Z

So, A ∩ B = {x : x = 12n, n ∈ Z}

Hence, the correct option is (C).

Compound interest in 2 years = 16800

Rate = 10%

Simple interest for Time = 3 years

As we know,

Compound interest

Simple interest

∴ The simple interest is Rs. 24,000.

Hence, the correct option is (A).

Then, 2n = 16 = 24 ⇒n = 4

∴ log₂ 16 = 4

Hence, the correct option is (B).

length of a rectangle is decreased by 4 cm and the width is increased by 3 cm

Formula used:

Area of rectangle = (l × b) sq. unit and Area of square = (side)² sq. unit

Perimeter of rectangle = 2(l + b) units

Let the length and breadth of the rectangle be x and y

⇒ Area of rectangle = (l × b) sq. unit

⇒ xy cm²

⇒ new length and breadth = (x - 4) and (y + 3)

⇒ (x - 4) = (y + 3)

⇒ x - y = 7 .....(1)

⇒ Area of square = (x - 4) × (y + 3)

According to question

⇒ Area of rectangle = Area of square

⇒ xy = (x - 4) × (y + 3)

⇒ 3x - 4y = 12 .....(2)

⇒ On solving these equation

⇒ x = 16 and y = 9

⇒ Perimeter of rectangle = 2(l + b) units

⇒ 2(16 + 9)

⇒ 2 × 25

⇒ 50 cm

∴ perimeter of the original rectangle is 50 cm

Hence, the correct option is (C).

A = {1, 2, 3, 4, 5, 6} and R is a relation on A such that:

R = {(x, y) : y = x + 1, where x and y ∈ R}

When x = 1, then:

y = x + 1

= 1 + 1

= 2 ∈ A

⇒ (1, 2) ∈ R

When x = 2, then:

y = x + 1

= 2 + 1

= 3 ∈ A

⇒ (2, 3) ∈ R

When x = 3, then:

y = x + 1

= 3 + 1

= 4 ∈ A

⇒ (3, 4) ∈ R

When x = 4, then:

y = x + 1

= 4 + 1

= 5 ∈ A

⇒ (4, 5) ∈ R

When x = 5, then:

y = x + 1

= 5 + 1

=6∈A

⇒ (5, 6) ∈ R

When x = 6, then:

y = x + 1

= 6 + 1

= 7 ∉ A

⇒(6, 7) ∉ R

So,

R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}

As we know that,

Domain (R) = {a : (a, b) ∈ R}

⇒ Domain (R) = {1, 2, 3, 4, 5}

Hence, the correct option is (B).

log (x + 2) + log (x − 2) = log 5

⇒ log [(x + 2) (x – 2)] = log 5 (∵ log_a (mn₎ = log_a m + log_a n)

⇒ log (x² – 4) = log 5

⇒ x² – 4 = 5

⇒ x² = 9

∴ x = ± 3

Here x ≠ -3 is not possible because (x – 2) should be greater than zero.

∴ x = 3

Hence, the correct option is (B).

Cost price of two necklace = Rs. 1,39,500.00

Selling price of one of them = Rs. 75,000.00

Selling price of other one = Rs. 80,000.00

Selling price of two necklace = Rs. 155,000.00

Gain = Selling price − Cost price

= Rs. 155,000.00− Rs. 1,39,500.00

= Rs. 15,500

So, she gain Rs. 15,500.

Hence, the correct option is (B).

Hence, the correct option is (B).

CD < />

⇒ m∈(−∞, −√3) ∪ (√3, ∞)

If the line passing through the point A(−4, 0), B(4, 0)

then

∠APB = ∠AQB = π/2 is not formed.

∴m ≠ ±2

Hence, the correct option is (A).

LCM of 12, 15, 18 and 20;

12 = 2 × 2 × 3

15 = 3 × 5

18 = 2 × 3 × 3

20 = 2 × 2 × 5;

Hence, LCM = 2 × 2 × 3 × 5 × 3

Since, the soldiers are in the form of a solid square.

Hence, LCM must be a perfect square. To make the LCM a perfect square, We have to multiply it by 5,

hence,

The required number of soldiers

= 2 x 2 x 3 x 3 x 5 x 5

= 900

Hence, the correct option is (A).

4 girls and 5 boys

Let 4 girls be one unit and now there are 6 units in all.

They can be arranged in 6! ways.

In each of these arrangements, 4 girls can be arranged in 4! ways.

Total number of arrangements in which girls are always together = 6! × 4! = 720 x 24 = 17280

[∵ 6! = 6 x 5 x 4 x 3 x 2 x 1 and 4! = 4 x 3 x 2 x 1]

Hence, the correct option is (B).

A and B are the domain and range respectively for the relation R such that R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}}

So, the relation R can be re-written as: R = {(0, 5), (1, 6), (2, 7), (3, 8), (4, 9), (5, 10)}

As we know that, Domain (R) = {a: (a, b) ∈ R}

⇒ A = {0, 1, 2, 3, 4, 5}

We also know that, Range (R) = {b: (a, b) ∈ R}

⇒ B = {5, 6, 7, 8, 9, 10}

Hence, the correct option is (A).

⇒ 1920 − x = x − 1280

⇒ 2x = 3200

⇒ x = 1600

∴ Required S.P. = 125% of Rs. 1600

Hence, the correct option is (A).