PW Test

 2 mark

| -0.25 mark |

 10 minutes

00:00:00

Question 1:

If \(p \tan(\theta - 30^{\circ}) = q \tan(\theta + 120^{\circ})\), then what is \((p+q)/(p-q)\) equal to?

Question 2:

What is the general solution of \( \cos^{100} x - \sin^{100} x = 1 \)?

Question 3:

Let \(x>1, y>1, z>1\) be in GP. Then \(\frac{1}{1+\ln x}, \;\frac{1}{1+\ln y}, \;\frac{1}{1+\ln z}\) are

Question 4:

The letters of the word EQUATION are arranged in such a way that all vowels as well as consonants are together. How many such arrangements are there?

Question 5:

If \( \frac{x}{\cos \theta} = \frac{y}{\cos\bigl(\tfrac{2\pi}{3}-\theta\bigr)} = \frac{z}{\cos\bigl(\tfrac{2\pi}{3}+\theta\bigr)} \), then what is \( x+y+z \) equal to?

Question 6:

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 7:

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

Question 8:

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?

Question 9:

Let \( P \) and \( Q \) be two non-void relations on a set \( A \). Which of the following statements are correct? I. \( P \) and \( Q \) are reflexive \( \Rightarrow P \cap Q \) is reflexive. II. \( P \) and \( Q \) are symmetric \( \Rightarrow P \cup Q \) is symmetric. III. \( P \) and \( Q \) are transitive \( \Rightarrow P \cap Q \) is transitive.

Question 10:

In a class of 240 students, 180 passed in English, 130 passed in Hindi and 150 passed in Sanskrit. Further, 60 passed in only one subject, 110 passed in only two subjects and 10 passed in none of the subjects. How many passed in all three subjects?

Question 11:

In an AP, the ratio of the sum of the first p terms to the sum of the first q terms is p^{2}: q^{2}. Which one of the following is correct?

Question 12:

In an arithmetic progression (AP), if the 5th term is 10 and the 10th term is 25, what is the common difference?

Question 13:

Consider an arithmetic progression where the sum of the first 4 terms is equal to the sum of the next 4 terms. What can be said about the common difference?

Question 14:

If the 7th term of an arithmetic progression is twice the 4th term, what is the ratio of the common difference to the first term?

Question 15:

समीकरण \(7x^{2}-6x+1=0\) के मूल \(\tan\alpha\) और \(\tan\beta\) हैं, जहाँ \(2\alpha\) तथा \(2\beta\) एक त्रिभुज के कोण हैं। निम्नलिखित में से कौन-सा एक सही है?

Question 16:

\(0

Question 17:

एक त्रिभुज \(A B C\) में \(\frac{a}{\cos A}=\frac{b}{\cos B}=\frac{c}{\cos C}\) है। त्रिभुज का क्षेत्रफल क्या है यदि \(a=6\) से० मी० हो?

Question 18:

एक त्रिभुज \( A B C \) में, \( \angle A=75^{\circ} \) और \( \angle B=45^{\circ} \) है। \( 2a - b \) किसके बराबर है?

Question 19:

\( \cot^{2}\bigl(\sec^{-1}2\bigr) + \tan^{2}\bigl(\operatorname{cosec}^{-1}3\bigr) \) किसके बराबर है?

Question 20:

If \( P \) is a skew-symmetric matrix of order 3, then what is \( \det(P) \) equal to?

Question 21:

If
\[
D_{n}=\left|\begin{array}{ccc}
n & 20 & 30 \\
n^{2} & 40 & 50 \\
n^{3} & 60 & 70
\end{array}\right|
\]
then what is the value of \( \sum_{n=1}^{4} D_{n} \)?

Question 22:

Consider the following in respect of the matrices
\[
P=\left[\begin{array}{rrr}
0 & c & -b \\
-c & 0 & a \\
b & -a & 0
\end{array}\right] \text{ and } Q=\left[\begin{array}{lll}
a^{2} & a b & a c \\
a b & b^{2} & b c \\
a c & b c & c^{2}
\end{array}\right]
\]
I. \(P Q\) is a null matrix.
II. \(Q P\) is an identity matrix of order 3.
III. \(P Q=Q P\)

Which of the above is/are correct?

Question 23:

In how many ways can a student choose (n-2) courses out of n courses if 2 courses are compulsory (n>4)?

Question 24:

If \( n \) is a root of the equation \( x^{2}+p x+m=0 \) and \( m \) is a root of the equation \( x^{2}+p x+n=0 \), where \( m \neq n \), then what is the value of \( p+m+n \)?

Question 25:

What is the number of solutions of the equation \( \cot 2 x \cdot \cot 3 x=1 \) for \( 0

Question 26:

The roots of the equation \(7x^{2}-6x+1=0\) are \(\tan \alpha\) and \(\tan \beta\), where \(2\alpha\) and \(2\beta\) are the angles of a triangle. Which one of the following is correct?

Question 27:

In a triangle \(A B C\)
\[\frac{a}{\cos A}=\frac{b}{\cos B}=\frac{c}{\cos C}\]
What is the area of the triangle if \(a=6\ \mathrm{cm}\)?

Question 28:

In a triangle \(A B C, \angle A=75^{\circ}\) and \(\angle B=45^{\circ}\). What is \(2a - b\) equal to?

Question 29:

If \(4\sin^{-1}x + \cos^{-1}x = \pi\), then what is \(\sin^{-1}x + 4\cos^{-1}x\) equal to?

Question 30:

What is \( \cot^{2}(\sec^{-1} 2) + \tan^{2}(\cosec^{-1} 3) \) equal to?

Question 31:

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

Question 32:

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 33:

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 34:

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