List of practice Questions

If \( \tan \theta = \frac{3}{4} \) and \( \theta \) is acute, find \( \sin 2\theta \):
The equation of the line passing through the point \((1, 2)\) and perpendicular to the line \(3x + 4y - 12 = 0\) is:
If \( x \) satisfies the equation \( 3x^2 - 7x + 2 = 0 \), then what is the value of \( \frac{1}{x_1} + \frac{1}{x_2} \), where \( x_1 \) and \( x_2 \) are the roots?
A gas occupies 2 liters at a pressure of 3 atm. What will be its volume when the pressure is reduced to 1.5 atm at constant temperature?
A concave mirror forms an image at twice the focal length from the mirror. What is the nature and size of the image compared to the object?
What is the molarity of a solution prepared by dissolving 10 grams of NaOH (molar mass = 40 g/mol) in 500 mL of solution?
A compound contains 40% carbon, 6.7% hydrogen, and 53.3% oxygen by mass. What is its empirical formula?
The mean of 10 numbers is 18. If one number is excluded, the mean becomes 17. What is the excluded number?
If \( \vec{a} = 3\hat{i} - 2\hat{j} \) and \( \vec{b} = \hat{i} + 4\hat{j} \), find the magnitude of \( 2\vec{a} - 3\vec{b} \).
How many 4-digit numbers (between 1000 and 9999) can be formed using the digits \(2, 3, 5, 7, 8\) such that:
  • Each number contains both \(3\) and \(7\) exactly once,
  • No digit is repeated.
Calculate the enthalpy change ($\Delta H$) for the combustion of 1 mole of methane ($\mathrm{CH}_4$) given the reaction: $$ \mathrm{CH}_4(g) + 2 \mathrm{O}_2(g) \rightarrow \mathrm{CO}_2(g) + 2 \mathrm{H}_2 \mathrm{O}(l) $$ Given bond energies: C–H = 413 kJ/mol, O=O = 498 kJ/mol, C=O = 803 kJ/mol, O–H = 467 kJ/mol.
If the roots of the quadratic equation \( 2x^2 - 4x + k = 0 \) are real and equal, find the value of \( k \).
If \( \tan \theta = \frac{4}{3} \), find the value of \( \sin \theta \) in the first quadrant.
If the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are in the ratio 3:4, then which of the following relationships holds between the coefficients?
Find the derivative of the function \( f(x) = \sin^{-1} \big( 2x \sqrt{1 - x^2} \big) \) with respect to \( x \), for \( x \in \left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) \).
If \( \sin \theta + \cos \theta = \sqrt{2} \cos \theta \), then the value of \( \tan \theta \) is
The equation of the circle passing through the points (1, 1), (2, 2), and (3, 1) is
A particle is projected at an angle of \( 60^\circ \) to the horizontal with an initial speed of \( 20 \, \text{m/s} \). If \( g = 10 \, \text{m/s}^2 \), what is the maximum height reached by the particle?
Two capacitors of capacitances \( 4 \, \mu\text{F} \) and \( 6 \, \mu\text{F} \) are connected in series across a potential difference of \( 300 \, \text{V} \). What is the potential difference across the \( 4 \, \mu\text{F} \) capacitor?
A light ray passes from air (\( \mu = 1 \)) into a glass slab (\( \mu = 1.5 \)) at an angle of incidence of \( 30^\circ \). What is the angle of refraction inside the glass slab?
What is the IUPAC name of the compound \( \text{CH}_3\text{CH}_2\text{COCH}_2\text{CH}_3 \)?
Which of the following elements has the highest first ionization energy?
The enthalpy change (ΔH) for the reaction N2(g) + 3H2(g) → 2NH3(g) is -92 kJ/mol.

What is the enthalpy change for the decomposition of 1 mol of NH3(g) into its elements?
Find the midpoint of the line segment joining the points (3, 4) and (7, -2).
A bag contains 5 red and 3 blue balls. If one ball is drawn at random, what is the probability that it is red?