List of practice Questions

In \( \triangle ABC \), if \( r_1 - r = \frac{a}{3} \) and \( r_2 - r = \frac{b}{3} \), then \( r_1 + r_2 - r \) is:
In \( \triangle ABC \), if \( (a - b)(s - c) = (b - c)(s - a) \), then \( r_1, r_2, r_3 \) are in:
If the position vectors of the points \( A \) and \( B \) are \( 2i + 3j - k \) and \( i - j + 2k \) respectively, then the unit vector along \( \overrightarrow{BA} \) and in the direction of \( AB \) is:
If \( 10 \sin^4 \alpha + 15 \cos^4 \alpha = 6 \), then \( 16 \tan^6 \alpha + 27 \cot^6 \alpha \) is:
In \( \triangle ABC \), if \( \cos^2 A + \cos^2 B + \cos^2 C = 1 \), then \( \triangle ABC \) is:
If \( \alpha \) and \( \beta \) are the roots of the equation \( 2x^3 - 3(2x^2) + 32 = 0 \) with \( \beta<1 \), then \( 2\alpha + 3\beta \) is:
If \( \alpha \), \( \beta \), and \( \gamma \) are the roots of the equation \( x^3 - ax^2 + bx - c = 0 \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is:
If \( \alpha_1, \alpha_2, \alpha_3 \) are the roots of the equation \( x^3 + 3x + 2 = 0 \), then \( \alpha_1^5 + \alpha_2^5 + \alpha_3^5 \) is:
Find the value of \( (3 + \sqrt{8})^5 + (3 - \sqrt{8})^5 \):
If \( C_j \) stands for \( ^nC_j \), then \[ \frac{C_0}{2} + \frac{C_1}{2.2^2} + \frac{C_2}{3.2^3} + \dots + C_n = \frac{3^n}{2^{n+1} (n+1)} \]
If \( z_1 = 2 + 5i \), \( z_2 = -1 + 4i \) and \( z_3 = i \), then
\[ \frac{|z_1 - z_3|}{|z_3 - z_2|} = ? \]
The locus of the variable point \( z = x + iy \) whose amplitude is always equal to \( \theta \), is:
For \( x \in \mathbb{R} \), the minimum value of \( \frac{x^2 + 2x + 5}{x^2 + 4x + 10} \) is:
If \( f(0) = 0, f(1) = 1, f(2) = 2 \) and \( f(x) = f(x-2) + f(x-3) \) for \( x = 3, 4, 5, \dots \), then find \( f(10) \).
If \( f(x) = \frac{\cos^2 x + \sin^4 x}{\sin^2 x + \cos^4 x} \) for \( x \in \mathbb{R} \), then find \( f(2023) \).
The rank of the matrix \( A = \begin{bmatrix} 1 & 1 & 3 \\ 2 & 2 & 1 \\ 1 & 1 & 1 \end{bmatrix} \) is:
If A is a non singular square matrix of order 3 such that \(A^3 = 4A^2\) then value of |A| is:
Derivative of \(2x^2\) with respect to \(5x^4\) is:
Le L be the set of all lines in a plane and R be the relation in L. defined as R = {(\(L_1, L_2\)): \(L_1\) is perpendicular to \(L_2\)} then R is:
A) Reflexive
B) Symmetric
C) Neither reflexive nor transitive
D) Transitive
E) Neither reflexive nor symmetric
Choose the correct answer from the options given below:
Consider the non-empty set consisting of children in a family and a relation R is defined as aRb if a is a brother of b. Then R is:
The value of the expression sin \([cot^{-1}(cos (tan^{-1}1))]\) is:
If \(x = \sqrt{a^{sin^{-1} t}}\), \(y = \sqrt{a^{ cos^{-1}t}}\) then \(\frac{dy}{dx}\) is:
Two cards are drawn without replacement. The probability distribution of number of aces is given by:
The solution set of the inequality \(2x + 3y < 4\) is:
The corner points of the feasible region determined by the system of linear inequalities are (0,0), (4, 0), (2, 4) and (0.5). If the maximum value of Z = ax + by where a, \(b > 0\) occurs at both (2, 4) and (4.0), then