Untitled

 2 mark

| -0.25 mark |

 60 minutes

00:00:00

Question 1:

Let \( X \) be a matrix of order \( 3 \times 3, Y \) be a matrix of order \( 2 \times 3 \) and \( Z \) be a matrix of order \( 3 \times 2 \). Which of the following statements are correct?
I. \( (Z Y) X \) is defined and is a square matrix of order 3.
II. \( Y(X Z) \) is defined and is a square matrix of order 2.
III. \( X(Y Z) \) is not defined.

Select the answer using the code given below.

Question 2:

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?

Question 3:

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

Question 4:

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

Question 5:

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

Question 6:

If \(
Z=\frac{1}{3}\left|\begin{array}{ccc}
i & 2 i & 1 \
2 i & 3 i & 2 \
3 & 1 & 3\end{array}\right|=x+i y ; i=\sqrt{-1}
\) then what is modulus of \( Z \) equal to?

Question 7:

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

Question 8:

What is the value of the sum \[ \sum_{n=1}^{20}\left(i^{n-1}+i^{n}+i^{n+1}\right) \] where \( i=\sqrt{-1} \) ?

Question 9:

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

Question 10:

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?

Question 11:

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

Question 12:

If \[ Z=\frac{1}{3}\left|\begin{array}{ccc} i & 2i & 1 \\ 2i & 3i & 2 \\ 3 & 1 & 3\end{array}\right|=x+iy ; i=\sqrt{-1} \] then what is modulus of \( Z \) equal to?

Question 13:

What is the value of the sum \[
\sum_{n=1}^{20}\left(i^{n-1}+i^{n}+i^{n+1}\right)\]
where \( i=\sqrt{-1} \)?

Question 14:

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?

Question 15:

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

Question 16:

In the expansion of \((1+x)^{p}(1+x)^{q}\), if the coefficient of \(x^{2}\) is 15, what is the value of \(p+q\)?

Question 17:

What is the coefficient of \(x^{4}\) in the expansion of \((1+x)^{3}(1+x)^{4}\)?

Question 18:

In the expansion of \((1+x)^{a}(1+x)^{b}\), if the coefficient of \(x^{5}\) is 56, what is the value of \(a+b\)?

Question 19:

Let \( X \) be a matrix of order \( 3 \times 3, Y \) be a matrix of order \( 2 \times 3 \) and \( Z \) be a matrix of order \( 3 \times 2 \). Which of the following statements are correct? I. \( (Z Y) X \) is defined and is a square matrix of order 3. II. \( Y(X Z) \) is defined and is a square matrix of order 2. III. \( X(Y Z) \) is not defined. Select the answer using the code given below.

Question 20:

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?

Question 21:

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

Question 22:

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

Question 23:

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

Question 24:

मान लीजिए \( p=\ln (x), q=\ln \left(x^{3}\right) \) और \( r=\ln \left(x^{5}\right) \) है, जहाँ \( x>1 \) है। निम्नलिखित में से कौन-सा/ कौन-से कथन सही है/हैं ? I. \( p, q \) और \( r \mathrm{AP} \) में हैं। II. \( p, q \) और \( r \) कभी भी GP में नहीं हो सकते हैं। का मान क्या है, जहाँ \( i=\sqrt{-1} \) है? नीचे दिए गए कूट का प्रयोग कर उत्तर चुनिए।

Question 25:

यदि एक AP के \( p \) वें पद का \( p \) गुना, \( q \) वें पद के \( q \) गुना के बराबर है \( (p \neq q) \), तो \( (p+q) \) वाँ पद किसके बराबर है?

Question 26:

\( (1+x)^{p}(1+x)^{q} \) के प्रसार में, यदि \( x^{3} \) का गुणांक 35 है, तब \( (p+q) \) का मान क्या है ?

Question 27:

यदि \( \omega \neq 1 \) एक का एक घनमूल है, तो \[ (1+\omega-\omega^{2})^{100}+(1-\omega+\omega^{2})^{100} \] किसके बराबर है?

Question 28:

मान लीजिए \( A \) और \( B \) समान कोटि के दो वर्ग आव्यूह हैं। यदि \( A B \) एक शून्य आव्यूह है, तब निम्नलिखित में से कौन-सा एक सही है?

Question 29:

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?

Question 30:

मान लीजिए \( x>1, y>1, z>1 \mathrm{GP} \) में हैं। तब \( \frac{1}{1+\ln x}, \quad \frac{1}{1+\ln y}, \quad \frac{1}{1+\ln z} \)

Question 31:

In the expansion of \((1+x)^{p}(1+x)^{q}\), if the coefficient of \(x^{4}\) is 70, what is the value of \((p+q)\)?

Question 32:

If \((1+x)^{p}(1+x)^{q}\) is expanded, and the coefficient of \(x^{2}\) is 15, what is the value of \((p+q)\)?

Question 33:

In the expansion of \((1+x)^{p}(1+x)^{q}\), if the coefficient of \(x^{5}\) is 126, what is the value of \((p+q)\)?

Question 34:

मान लीजिए \( p=\ln (x), q=\ln \left(x^{3}\right) \) और \( r=\ln \left(x^{5}\right) \) है, जहाँ \( x>1 \) है। निम्नलिखित में से कौन-सा/ कौन-से कथन सही है/हैं ?
I. \( p, q \) और \( r \mathrm{AP} \) में हैं।
II. \( p, q \) और \( r \) कभी भी GP में नहीं हो सकते हैं।

नीचे दिए गए कूट का प्रयोग कर उत्तर चुनिए।

Question 35:

यदि
\[
Z=\frac{1}{3}\left|\begin{array}{ccc}
i & 2 i & 1 \\
2 i & 3 i & 2 \\
3 & 1 & 3\end{array}\right|=x+i y \; i=\sqrt{-1}
\]
है, तो \( Z \) का मापांक (मॉड्यूलस) किसके बराबर है ?

Question 36:

योग \[ \sum_{n=1}^{20}\left(i^{n-1}+i^{n}+i^{n+1}\right) \] का मान क्या है, जहाँ \( i=\sqrt{-1} \) है ?

Question 37:

यदि एक AP के \( p \) वें पद का \( p \) गुना, \( q \) वें पद के \( q \) गुना के बराबर है \( (p \neq q) \), तो \( (p+q) \) वाँ पद किसके बराबर है?

Question 38:

(1+x)^{p}(1+x)^{q} के प्रसार में, यदि x^{3} का गुणांक 35 है, तब (p+q) का मान क्या है?

Question 39:

यदि \[ \omega=-\frac{1}{2}+i \frac{\sqrt{3}}{2} \] है, तब \[ \left|\begin{array}{ccc} 1+\omega & 1+\omega^{2} & \omega+\omega^{2} \\ 1 & \omega & \omega^{2} \\ \frac{1}{\omega} & \frac{1}{\omega^{2}} & 1 \end{array}\right| \] किसके बराबर है?
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