List of practice Questions

(i) For any integer a, what is (–1) × a equal to?
(ii) Determine the integer whose product with (–1) is
(a) –22
(b) 37
(c) 0

If \[ f(x) = 2 \left( \cos x + i \sin x \right) \left( \cos 3x + i \sin 3x \right) \cdots \left( \cos (2n-1)x + i \sin (2n-1)x \right) \] where \( n \in \mathbb{N} \), then what is the value of \( f''(x) \) ?

A ladder has rungs 25 cm apart. (see Fig. 5.7). The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and the bottom rungs are \(2\frac 12\) m apart, what is the length of the wood required for the rungs?

The van’t Hoff factor for a solution of \( \text{K}_2\text{SO}_4 \) in water is:

Find the area bounded between the parabola \( y^2 = 4x \) and the line \( y = 2x - 3 \).

The EAN of cobalt in the complex \([Co(NH_3)_6]^{3+\) is:}

Which of the following statements is \emph{correct regarding the coordination compound \([Fe(CN)_6]^{4-}\) and fructose structure?}

A population \( P(t) \) of 1000 bacteria introduced to a nutrient medium grows according to the relation \[ P(t) = \frac{1000t + 1000t}{100 + t^2} \] The maximum size of this bacterial population is:

If \[ [2\vec{p} - 3\vec{q} \ \vec{q} \ \vec{s}] + [3\vec{p} + 2\vec{q} \ \vec{r} \ \vec{s}] = m[\vec{p} \ \vec{r} \ \vec{s}] + n[\vec{q} \ \vec{r} \ \vec{s}] + l[\vec{p} \ \vec{q} \ \vec{s}], \] then the values of \( m, n, l \) respectively are:

Let \[ f(x) = (\cos x + \sin x) \cdot \cos(3x + i \sin x) \cdot \left[ (2n - 1)x + i \sin((2n - 1)x) \right], \] where \( n \in \mathbb{N} \), and \( i = \sqrt{-1} \). Then: \[ f''(x) = \, ? \]

The value of \[ \int_1^4 \log(\lfloor x \rfloor) \, dx \] where \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \), is equal to:

A plane passes through the point \( (1, -2, 1) \) and is perpendicular to both the planes \[ 2x - 2y - 2z = 5 \quad \text{and} \quad x - y + 2z = 24 \] Then, the distance of the point \( (1, 2, 2) \) from this plane is:

In a triangle \( ABC \), with usual notation, if \[ \frac{b + c}{11} = \frac{c + a}{12} = \frac{a + b}{13} \] then the ratio \( \cos A : \cos B : \cos C \) is:

Smallest angle of a triangle whose sides are \( 6 + \sqrt{12}, \sqrt{48}, \sqrt{54} \) is: