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Question 1:

The sum of the first \( k \) terms of a series \( S \) is \( 3 k^{2}+5 k \). Which one of the following is correct?

The terms of \( S \) form an arithmetic progression with common difference 14.

The terms of \( S \) form an arithmetic progression with common difference 6.

The terms of \( S \) form a geometric progression with common ratio 10/7.

The terms of \( S \) form a geometric progression with common ratio \( 11/4 \).

Question 2:

If \( x=(1111)_{2}, y=(1001)_{2} \) and \( z=(110)_{2} \), then what is \( x^{3}-y^{3}-z^{3}-3xyz \) equal to?

(1111001)_{2}

(1001111)_{2}

(1)_{2}

(O)_{2}

Question 3:

If \( p, 1, q \) are in AP and \( p, 2, q \) are in GP, then which of the following statements is/are correct?
I. \( p, 4, q \) are in HP.
II. \( (1 / p), 1 / 4, (1 / q) \) are in AP.

I only

II only

Both I and II

Neither I nor II

Question 4:

If 5th, 7th and 13th terms of an AP are in GP, then what is the ratio of its first term to its common difference?

-3

-2

2

3

Question 5:

The sum of the first 8 terms of a GP is five times the sum of its first 4 terms. If \( r \neq 1 \) is the common ratio, then what is the number of possible real values of \( r \)?

One

Two

Three

More than three

Question 6:

If one root of the equation \( x^{2}-k x+k=0 \) exceeds the other by \( 2 \sqrt{3} \), then which one of the following is a value of \( k \)?

3

6

9

12

Question 7:

If \( x+\frac{5}{y}=4 \) and \( y+\frac{5}{x}=-4 \), then what is \( (x+y) \) equal to?

0

1

4

5

Question 8:

If \(\left|\begin{array}{ccc}2 & 3+i & -1\\3-i & 0 & i-1\\-1 & -1-i & 1\end{array}\right| = A + iB\), where \(i=\sqrt{-1}\), then what is \(A+B\) equal to?

-10

-6

0

6

Question 9:

If \( x^{2}-x+1=0 \), then what is \( \left(x-\frac{1}{x}\right)^{2} + \left(x-\frac{1}{x}\right)^{4} + \left(x-\frac{1}{x}\right)^{8} \) equal to?

81

85

87

90

Question 10:

If \( \omega \) is a non-real cube root of unity, then what is a root of the following equation?
\[
\left|\begin{array}{ccc}
x+1 & \omega & \omega^{2} \\\
\omega & x+\omega^{2} & 1 \\\
\omega^{2} & 1 & x+\omega
\end{array}\right|=0
\]

\( x=0 \)

\( x=1 \)

\( x=\omega \)

\( x=\omega^{2} \)

Question 11:

If the sum of binomial coefficients in the expansion of \( (x+y)^{n} \) is 256, then the greatest binomial coefficient occurs in which one of the following terms?

Third

Fourth

Fifth

Ninth

Question 12:

If
\[
A=\left[\begin{array}{lll}
y & z & x \\
z & x & y \\
x & y & z
\end{array}\right]
\]
where \(x, y, z\) are integers, is an orthogonal matrix, then what is the value of \(x^{2}+y^{2}+z^{2}\)?

0

1

4

14

Question 13:

If
\[
\left[\begin{array}{lll}
x & 1 & 1
\end{array}\right]\left[\begin{array}{lll}
1 & 2 & 3 \\\
4 & 5 & 6 \\\
7 & 8 & 9
\end{array}\right]\left[\begin{array}{l}
1 \\\
1 \\\
x
\end{array}\right]=[45]
\]
then which one of the following is a value of \( x \) ?

-2

-1

0

1

Question 14:

How many 7-letter words (with or without meaning) can be constructed using all the letters of the word CAPITAL so that all consonants come together in each word?

360

300

288

240

Question 15:

In how many ways can the letters of the word DELHI be arranged keeping the positions of vowels and consonants unchanged?

6

9

12

24

Question 16:

If \( k< (\sqrt{2}+1)^{3} < k+2 \), where \( k \) is a natural number, then what is the value of \( k \)?

11

13

15

17

Question 17:

What is the number of rational terms in the expansion of \(\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)^{12}\)?

2

3

4

6

Question 18:

How many sides are there in a polygon which has 20 diagonals?

6

7

8

10

Question 19:

What is the number of positive integer solutions of \( x+y+z=5 \)?

3

5

6

9

Question 20:

If \( z \neq 0 \) is a complex number, then what is \( \operatorname{amp}(z)+\operatorname{amp}(\bar{z}) \) equal to?

0

\( \pi / 2 \)

\( \pi \)

\( 2 \pi \)

Bihar SI Test 01

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