# CAT Model Paper 4 Questions and Answers with Explanation Part 4

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Q: 18. , where i is a complex number, find the value of

(A)

(B)

(C)

(D)

Ans: A

Solution:

Substituting for the values of ‘x’ in the equation

and reducing it is a tedious process. Hence, let us construct a quadratic equation whose roots are and .

A quadratic equation is of the form (Sum of roots)

Product of roots

Sum of roots

Product of roots

Hence Option A

Q: 19. ABCD is a parallelogram and P is any point within it. If the area of the parallelogram ABCD is 28 units then what is the sum of the areas of the APD and BPC?

(A)

(B)

(C)

(D)

Ans: C

Solution:

As P can be any point, we can take it as point of intersection of diagonals.

Now sum of areas of given 2 triangles

Hence, the sum of the areas of the APD and BPC is 14.

Q: 20. Ravi and Akash run a 2 km long race. Ravi beats Akash by 2 minutes. If Akash increases his speed by 2 km/hr and Ravi decreases his speed by 2 km/hr, Akash will beat Ravi in the same race by 2 minutes. What are the speeds of Ravi and Akash?

(A)

(B)

(C)

(D)

Ans: D

Solution:

By checking all the options, we can observe that a difference of 2 minutes can(1st case) be observed in 1^{st} and 4^{th} options. If we consider the 2^{nd} case, only option 4 satisfies the condition.

Hence option D

Q: 21. From 1 to , how many numbers will have exactly factors?

(A)

(B)

(C)

(D)

Ans: D

Solution:

.

So the numbers can be numbers.

Hence option D

Q: 22. If the factors of are multiplied and written in the form of ?

(A)

(B)

(C)

(D)

Ans: B

Solution:

Factors of . If all these are multiplied, we get

Hence option B

Q: 23. What should be added to such that they become proportional to each other?

(A)

(B)

(C)

(D)

Ans: A

Solution:

Checking the options, we get .

So 1 should be added.

Hence option A

Q: 24. In is the midpoint of and and . Find the value of

?

(A)

(B)

(C)

(D)

Ans: B

Solution:

In .

Angles opposite to equal sides in a triangle are equal. Hence,

Therefore is an equilateral triangle. So

But A (as is the median)

So . Hence,

But, (Sum of interior angles in a triangle is equal to the

exterior opposite angle). So .

Hence option B