List of practice Questions

The negation of the statement
(p v q) ∧ (q v(∼r)) is

Let the probability of getting head for a biased coin be $\frac{1}{4}$ . It is tossed repeatedly until a head appears. Let N be the number of tosses required. If the probability that the equation $64x² + 5Nx + 1 = 0$ has no real root is $\frac{p}{q}$ , where p and q are co-prime, then q – p is equal to _______.

Let (a+bx+cx²)¹⁰ = $ \sum_{i=0}^{20} $ p_ixⁱ, a,b,c∈N. If p₁=20 and P₂ = 210, then 2(a+b+c) is equal to

Negation of $ p \land (q \land \neg (p \land q)) $ is:}

A person forgets his 4-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is 7 and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct code is

Three dice are rolled. If the probability of getting different numbers on the three dice is $\frac{p}{q}$, where $p$ and $q$ are co-prime, then $q - p$ is equal to:

If the equation of the normal to the curve $ y = \frac{x - a}{(x + b)(x - 2)} $ at the point $ (1, -3) $ is $ x - 4y = 13 $, then the value of $ a + b $ is:

Let $ f: [2, 4] \to \mathbb{R} $ be a differentiable function such that $ (x \log x) f'(x) + (\log x) f(x) \geq 1 $, $ x \in [2, 4] $ with $ f(2) = \frac{1}{2} $ and $ f(4) = \frac{1}{4} $. Consider the following two statements:
$ (A) \quad f(x) \geq 1 \quad \text{for all} \quad x \in [2, 4] $
$ (B) \quad f(x) \leq \frac{1}{8} \quad \text{for all} \quad x \in [2, 4] $ Then,

Let $y = y(x)$ be a solution curve of the differential equation $$(1 - x^2 y)\,dx = y\,dx + x\,dy.$$ If the line $x = 1$ intersects the curve $y = y(x)$ at $y = 2$ and the line $x = 2$ intersects the curve $y = y(x)$ at $y = \alpha$, then a value of $\alpha$ is:

Area of the region $(x, y) : x^2 + (y - 2)^2 \leq 4, \, x^2 \geq 2y$ is:

Let $a = 30!$, $b = 50!$, and $c = 100!$. Consider the following numbers: $\log_a c$, $\log_c a$, $\log_b a$, $\log_a b$
Which one of the following inequalities is CORRECT?

A duck named Donald Duck says "All ducks always lie." Based only on the information above, which one of the following statements can be logically inferred with certainty?

A line of symmetry is defined as a line that divides a figure into two parts in a way such that each part is a mirror image of the other part about that line. The figure below consists of 20 unit squares arranged as shown. In addition to the given black squares, up to 5 more may be coloured black. Which one among the following options depicts the minimum number of boxes that must be coloured black to achieve two lines of symmetry?

Based only on the truth of the statement 'Some humans are intelligent', which one of the following options can be logically inferred with certainty?

Which one of the options can be inferred about the mean, median, and mode for the given probability distribution (i.e. probability mass function), $P(x)$, of a variable $x$?

The James Webb telescope, recently launched in space, is giving humankind unprecedented access to the depths of time by imaging very old stars formed almost 13 billion years ago. Astrophysicists and cosmologists believe that this odyssey in space may even shed light on the existence of dark matter. Dark matter is supposed to interact only via the gravitational interaction and not through the electromagnetic-, the weak- or the strong-interaction. This may justify the epithet "dark" in dark matter. Based on the above paragraph, which one of the following statements is FALSE?

A square of side length 4 cm is given. The boundary of the shaded region is defined by one semi-circle on the top and two circular arcs at the bottom, each of radius 2 cm, as shown. The area of the shaded region is cm$^2$.

In the given figure, PQRSTV is a regular hexagon with each side of length 5 cm. A circle is drawn with its centre at V such that it passes through P. What is the area (in cm$^2$) of the shaded region? (The diagram is representative)

Eject : Insert :: Advance : $\underline{\hspace{1cm}}$ (By word meaning)

“I have not yet decided what I will do this evening; I $\underline{\hspace{2cm}}$ visit a friend.”

Reaction of thionyl chloride with white phosphorus forms a compound [A], which on hydrolysis gives [B], a dibasic acid $[ A ]$ and $[ B ]$ are respectively

$2 I O^ − _3 + x I ^− + 12 H ^+ ⟶ 6 I_ 2 + 6 H _2 O$ What is the value of x?

The reaction $\frac{ 1}{2}$H₂(g) + AgCl(s) → H⁺(aq) + Cl^-(aq) + Ag(s) occurs in which of the given galvanic cell.

Which of the following complex is octahedral, diamagnetic and the most stable?

The correct order of spin only magnetic moments for the following complex ions is