List of practice Questions

The least value of \( n \) such that \( {}^{n-1}C_6 + {}^{n-1}C_7<{}^nC_8 \) is

If a seven digit number formed with distinct digits 4, 6, 9, 5, 3, \( x \) and \( y \) is divisible by 3, then the number of such ordered pairs \( (x, y) \) is

\( \sin^4 \frac{\pi}{8} + \sin^4 \frac{3\pi}{8} + \sin^4 \frac{5\pi}{8} + \sin^4 \frac{7\pi}{8} = \)

In \( \triangle ABC \), if \( \cos A \cos B \cos C = \frac{1}{5} \), then \( \tan A \tan B + \tan B \tan C + \tan C \tan A = \)

If \( \cos(\theta - \alpha), \cos \theta \) and \( \cos(\theta + \alpha) \) are in harmonic progression, then \( 2 \tan^2 \theta = \)

If \( \omega \) is a complex cube root of unity, then \( \cos \left( \sum_{k=1}^{2} (k - \omega)(k - \omega^2) \frac{\pi}{175} \right) = \)

Let the two values of \( z = \frac{1 - i}{\sqrt{1 + i}} \) be \( z_1 \) and \( z_2 \). If \( -\frac{\pi}{2}<\text{Arg}(z_1)<\text{Arg}(z_2)<\pi \), then \( \text{Arg}(z_1) + \text{Arg}(z_2) = \)

If \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 + x + 1 = 0 \), then the quadratic equation whose roots are \( \alpha^{2023} \) and \( \beta^{2012} \) is

If \( \alpha \) and \( \beta \) are the roots of the equation \( 2x^2 - 4x + 3 = 0 \), then \( \frac{2(\alpha^4 + \beta^4) + 3(\alpha^2 + \beta^2)}{\alpha + \beta} = \)

If \( \alpha \) and \( \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \), then the equation whose roots are \( \alpha + \beta \) and \( \frac{1}{\alpha} + \frac{1}{\beta} \) is

If the roots of the equation \( 16x^3 - 44x^2 + 36x - 9 = 0 \) are in harmonic progression, then its greatest root is

If \( 2^n \) divides \( 16! \) and \( 2^{n+1} \) does not divide \( 16! \), then \( n = \)

Let \( f(x) = |x| \) and \( g(x) = |x| + a \), \( (a>0) \). For \( 0 \le x \le b \), \( \{(x, y) / g(x) \le y \le f(x)\} \) represents all the points in the interior of

If there exists a \( k^{th} \) order non-singular sub-matrix in a matrix \( P \) of order \( m \times n \), then the rank \( (\rho) \) of \( P \)

If \( A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} \) and \( B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} \), then which one of the following is True?

In a matrix \( A \), if all the sub matrices of \( k^{th} \) order are singular and there is one non-singular sub matrix of order \( r \) \( (r<k) \), then the rank \( (\rho) \) of the matrix \( A \)

If \( \sqrt{-3 - 4i} = re^{i\theta} \), then \( r^2 \tan \theta = \)

If \( z_1 = 2 - 3i \) and the roots of the equation \( z^3 + bz^2 + cz + d = 0 \) are \( i, z_1 \) and \( \overline{z_1} \), then \( b + c + d = \)

Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a function defined by \( f(x) = \frac{2x+1}{3} \). If \( \alpha \) is an element in the domain of \( f \) whose image is \( \frac{1}{\alpha} \), then the sum of all possible values of such \( \alpha \) is

When vector \(\overrightarrow A = 2\hat i+3\hat J+2\hat k\) is subtracted from vector \(\overrightarrow B\)  , it gives a vector equal to \(2\hat j\) . Then the magnitude of vector \(\overrightarrow B\) will be :

The dissolution of Partnership may take place due to:
A. Retirement of Partner
B. Death of Partner
C. Insolvency of Partner
D. Admission of New Partner
E. Completion of Venture, if partnership formed for that purpose
Choose the correct answer from the options given below:

Identify the statement which are not true for the natural ecosystem
(A) Grassland and desert are example of terrestrial ecosystem. 
(B) Oceans and rivers are man-made ecosystems. 
(C) Wetland and estuary are examples of aquatic ecosystem. 
(D) Crop fields and aquarium are natural ecosystems.
Choose the correct answer from the option given below

The greatest biodiversity on the earth is seen in

Which of the following modern states were part of 'Tamilakam' country?
(A) Kerala 
(B) Andhra-Pradesh 
(C) Maharashtra 
(D) Madhya-Pradesh 
Choose the correct answer from the options given below: