List of practice Questions

A bag contains 4 red balls, 3 blue balls, and 2 green balls. Two balls are picked at random. What is the probability that both balls are of the same color?
Let \( D = \{a, b, c\} \). How many distinct ways can \( D \) be partitioned into non-empty subsets, representing equivalence relations?
Find the value of the integral \( \int_0^{\frac{\pi}{2}} \sin^2(x) \, dx \).
For the function \( f(x) = \ln(x^2 + 1) \), what is the second derivative of \( f(x) \)?
If the equation of a circle is \( 4x^2 + 4y^2 - 12x + 8y = 0 \), what is the radius of the circle?
What is the sum of the infinite series \( S = \sum_{n=0}^{\infty} \frac{1}{3^n} \)?
In a sound wave, the displacement of the air particles follows the equation \( y = A \cos(kx - \omega t) \). What is the wave velocity?
What is the time period of a simple pendulum of length \( L \) and acceleration due to gravity \( g \)?
In a resistive circuit, the power dissipated by a resistor \( R \) when a current \( I \) flows through it is given by \( P = I^2 R \). What happens to the power when the current is doubled?
A block of mass \( m \) is moving with velocity \( v \) on a smooth horizontal surface. If a constant force \( F \) is applied to the block, what is the work done by this force after 5 seconds?
In the context of the photoelectric effect, what is the threshold frequency \( f_0 \) for a metal if the work function is \( \phi = 2 \, \text{eV} \)?
Find the dimensions of \( \frac{B}{\mu_0} \)
For \([ \text{NiCl}_4 ]^{2-}\), what is the charge on the metal and shape of the complex respectively?
Compare boiling point of given solutions:
(i) \( 10^{-4} \) NaCl
(ii) \( 10^{-3} \) NaCl
(iii) \( 10^{-2} \) NaCl
(iv) \( 10^{-4} \) urea
Which of the following is the correct order of acidic strength for the following compounds: HNO\(_3\), HClO\(_4\), and HClO\(_3\)?
What is the hybridization of the central atom in the molecule \( \text{SO}_3 \)?
In an oxidation-reduction reaction, what happens to the oxidation state of the central atom in \( \text{MnO}_4^- \) when it is reduced to \( \text{Mn}^{2+} \)?
In the reaction \( 2\text{H}_2\text{O}_2 \rightarrow 2\text{H}_2\text{O} + \text{O}_2 \), what is the role of potassium iodide (KI) in this reaction?
What is the shape of the \( \text{ClO}_2^- \) ion according to VSEPR theory?
Let {an}n=0∞ be a sequence such that a0=a1=0 and an+2=3an+1−2an+1,∀ n≥0. Then a25a23−2a25a22−2a23a24+4a22a24  is equal to
The sum of the cubes of all the roots of the equation x4 – 3x3 –2x2 + 3x +1 = 0 is
N>40000, where N is divisible by 5. How many such 5 digit numbers can be formed using 0,1,3,5,7,9 without repetition.
Let $A$ be the area of the region
$\left\{(x, y): y \geq x^2, y \geq(1-x)^2, y \leq 2 x(1-x)\right\}$. 
Then $540 A$ is equal to ______.
There are .............. stages of active listening.
‘An affluent individual who provides capital for a business start-up and early stage companies having a high-risk, high-return matrix usually in exchange for convertible debt or ownership equity.’
(i) Identify the source to raise finance available to an entrepreneur discussed in the above lines.
(ii) State any four features of the source of raising finance discussed in (i) above.