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Question 1:

In the expansion of \( (1+x)^{p}(1+x)^{q} \), if the coefficient of \( x^{3} \) is 35, then what is the value of \( (p+q) \)?

5

6

7

8

Question 2:

What is the value of the sum \( \sum_{n=1}^{20}\bigl(i^{n-1}+i^{n}+i^{n+1}\bigr) \) where \(i=\sqrt{-1}\)?

\(-2i\)

0

1

\(2i\)

Question 3:

If \( p \) times the \( p \)th term of an AP is equal to \( q \) times the \( q \)th term \((p \neq q)\), then what is the \((p+q)\)th term equal to?

0

\( p+q \)

\( p q \)

\( p q(p+q) \)

Question 4:

Let \(p = \ln(x)\), \(q = \ln(x^3)\) and \(r = \ln(x^5)\), where \(x>1\). Which of the following statements is/are correct?
I. \(p, q\) and \(r\) are in AP.
II. \(p, q\) and \(r\) can never be in GP.
Select the answer using the code given below.

I only

II only

Both I and II

Neither I nor II

Question 5:

How many 4-digit numbers are there having all digits as odd?

625

400

196

120

Question 6:

Let \( A \) and \( B \) be two square matrices of same order. If \( A B \) is a null matrix, then which one of the following is correct?

Both \( A \) and \( B \) are null matrices

Either \( A \) or \( B \) is a null matrix

\( B \) is a null matrix if \( A \) is a nonsingular matrix

Both \( A \) and \( B \) are singular matrices

Question 7:

If \( \omega \neq 1 \) is a cube root of unity, then what is \( (1+\omega-\omega^{2})^{100} + (1-\omega+\omega^{2})^{100} \) equal to?

\( 2^{100} \omega^{2} \)

\( 2^{100} \omega \)

\( 2^{100} \)

\( -2^{100} \)

Question 8:

Consider the following statements :
I. The set of all irrational numbers between \( \sqrt{12} \) and \( \sqrt{15} \) is an infinite set.
II. The set of all odd integers less than 1000 is a finite set.

Which of the statements given above is/are correct?

I only

II only

Both I and II

Neither I nor II

Question 9:

In an arithmetic progression, if the product of the 3rd term and the 4th term is equal to the product of the 2nd term and the 5th term, what is the 7th term?

0

7

14

21

Question 10:

Consider an arithmetic progression where the sum of the 2nd and 5th terms is equal to the sum of the 3rd and 4th terms. What is the 6th term?

0

5

10

15

Question 11:

In an arithmetic progression, if the sum of the 1st and 6th terms equals the sum of the 2nd and 5th terms, what is the 4th term?

0

4

8

12

Question 12:

In an arithmetic progression, if the sum of the 3rd and 8th terms equals the sum of the 4th and 7th terms, what is the 5th term?

0

5

10

15

Question 13:

Consider an arithmetic progression where the sum of the 2nd and 7th terms equals the sum of the 3rd and 6th terms. What is the 5th term?

0

2

4

6

Question 14:

In an arithmetic progression, if the sum of the 4th and 9th terms equals the sum of the 5th and 8th terms, what is the 6th term?

0

3

6

9

tEST O1

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