List of practice Questions

Let \[ f(x) = \begin{cases} 1 + \dfrac{2x}{\pi}, & -\pi \le x \le 0
1 - \dfrac{2x}{\pi}, & 0<x \le \pi \end{cases} \] The constant term of the Fourier series of \(f(x)\) is:
The particular integral of \[ (D^2 - 6D + 13)\, y = 8e^{3x} \sin 2x \] is:
The characteristic polynomial of a \(3 \times 3\) matrix \( A \) is \[ |A - \lambda I| = \lambda^3 - 9\lambda^2 + 23\lambda - 15. \] Let \( X = \text{trace}(A) \) and \( Y = \det(A) \), then:
In a homologous series of any general anesthetic, increasing the chain length increases the lipid solubility and produces a corresponding increase in anesthetic potency, is proposed by:
If the points with position vectors \[ (\mathbf{a}i + 10j + 13k), \quad (6i + 11j + 11k), \quad \left(\frac{9}{2}i + \beta j - 8k\right) \] are collinear, then evaluate \( (19a - 6\beta)^2 \).
Evaluate the sum: \[ \sin^2 18^\circ + \sin^2 24^\circ + \sin^2 36^\circ + \sin^2 42^\circ + \sin^2 78^\circ + \sin^2 90^\circ + \sin^2 96^\circ + \sin^2 102^\circ + \sin^2 138^\circ + \sin^2 162^\circ. \]
If \( H \) is the orthocenter of \( \triangle ABC \) and \( AH = x \), \( BH = y \), \( CH = z \), then evaluate: \[ \frac{abc}{xyz} \]
Evaluate the expression: \[ \sin^2 76^\circ + \sin^2 16^\circ - \sin 76^\circ \sin 16^\circ \]
Evaluate the expression: \[ \tan 6^\circ \tan 42^\circ \tan 66^\circ \tan 78^\circ \]

The absolute value of the difference of the coefficients of $x^4$ and $x^6$ in the expansion of  
$\frac{2x^2}{(x^2+1)(x^2+2)}$ 
is:

Cells present in the mature pollen grains are___:
When a convex lens is immersed in a liquid of refractive index equal to 80% of the refractive index of the material of the lens, the focal length of the lens increases by 100%. The refractive index of the liquid is:
While solving a system of linear equations \( AX = B \) using Cramer’s rule, if \includegraphics[]{6.png}
Identify the order of reaction if the rate constant is k=2×10−2s−1:
In a class of 10 girls and 20 boys, Jaya’s rank is ‘4’ among the girls and ‘18’ in the class. What is Jaya’s rank among boys in the class?

In a Carnot engine, the absolute temperature of the source is 25% more than the absolute temperature of the sink. The efficiency of the engine is

The kinetic energy of a satellite in its orbit around Earth is \( E \). What should be the kinetic energy of the satellite to escape Earth's gravity?
In an \( S_{\text{N}2} \) substitution reaction of the type: \[ R - Br + Cl^- \longrightarrow R - Cl + Br^- \]
Which one of the following has the highest relative rate?
5 boys and 6 girls are arranged in all possible ways. Let \(X\) denote the number of linear arrangements in which no two boys sit together, and \(Y\) denote the number of linear arrangements in which no two girls sit together. If \(Z\) denotes the number of ways of arranging all of them around a circular table such that no two boys sit together, then \(X:Y:Z\) = ?
Let \( S = (-1, \infty) \) and \( f : S \rightarrow \mathbb{R} \) be defined as \[ f(x) = \int_{-1}^{x} (e^t - 1)^{11} (2t - 1)^5 (t - 2)^7 (t - 3)^{12} (2t - 10)^{61} \, dt \] Let \( p = \) Sum of squares of the values of \( x \), where \( f(x) \) attains local maxima on \( S \). And \( q = \) Sum of the values of \( x \), where \( f(x) \) attains local minima on \( S \). Then, the value of \( p^2 + 2q \) is ______
If the number of real roots of \( x^9 - x^5 + x^4 - 1 = 0 \) is \( n \), the number of complex roots having argument on imaginary axis is \( m \), and the number of complex roots having argument in the second quadrant is \( k \), then \( m.n.k \) is:
If \( \alpha \) and \( \beta \) are two distinct negative roots of the equation \( x^5 - 5x^3 + 5x^2 - 1 = 0 \), then the equation of least degree with integer coefficients having \( \sqrt{-\alpha} \) and \( \sqrt{-\beta} \) as its roots is:
If \( \alpha, \beta \) are the roots of the equation \( x^2 - 6x - 2 = 0 \), \( \alpha > \beta \), and \( a_n = \alpha^n - \beta^n, n \geq 1 \), then the value of \( \frac{a_{10} - 2 a_8}{2 a_9} \) is:
Iron at 20◦C is BCC with atoms of atomic radius 0.124 nm. The lattice constant a for the cube edge of the unit cell is
The smallest change in measurant that will result in a measurable change in the transducer output is called