List of practice Questions

Evaluate: $$ \lim_{n \to \infty} \frac{1}{n^{k+1}} \left[ 2^k + 4^k + 6^k + \dots + (2n)^k \right]. $$ 

The acceleration \( f \) (in ft/sec\(^2\)) of a particle after a time \( t \) seconds starting from rest is given by:
\[ f = 6 - \sqrt{1.2t}. \]
Then the maximum velocity \( v \) and the time \( T \) to attain this velocity are:

Let \( \Gamma \) be the curve \( y = b e^{-x/a} \) and \( L \) be the straight line:
\[ \frac{x}{a} + \frac{y}{b} = 1, \quad a, b \in \mathbb{R}. \]
Then:

If \( n \) is a positive integer, the value of:
\[ (2n + 1) \binom{n}{0} + (2n - 1) \binom{n}{1} + (2n - 3) \binom{n}{2} + \dots + 1 \cdot \binom{n}{n} \] is:

If the quadratic equation \( ax^2 + bx + c = 0 \) (\( a > 0 \)) has two roots \( \alpha \) and \( \beta \) such that \( \alpha < -2 \) and \( \beta > 2 \), then:

If $a_i, b_i, c_i \in \mathbb{R}$ ($i = 1, 2, 3$) and $x \in \mathbb{R}$, and
$\begin{vmatrix} a_1 + b_1x & a_1x + b_1 & c_1 \\ a_2 + b_2x & a_2x + b_2 & c_2 \\ a_3 + b_3x & a_3x + b_3 & c_3 \end{vmatrix} = 0$,
then:

The function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = e^x + e^{-x} \) is:

A square with each side equal to \( a \) lies above the \( x \)-axis and has one vertex at the origin. One of the sides passing through the origin makes an angle \( \alpha \) (\( 0 < \alpha < \frac{\pi}{4} \)) with the positive direction of the \( x \)-axis. The equation of the diagonals of the square is:

If \( \triangle ABC \) is an isosceles triangle and the coordinates of the base points are \( B(1, 3) \) and \( C(-2, 7) \), the coordinates of \( A \) can be:

The points of extremum of \[ \int_{0}^{x^2} \frac{t^2 - 5t + 4}{2 + e^t} \, dt \] are:

Choose the correct statement:

In the chapter 'A Baker from Goa' the narrator talks about his childhood in Goa and his fond memories. In about 120 words write a presentation draft comparing the childhood of the narrator in 'A Baker from Goa' to that of Nelson Mandela.
You may begin this way:
Childhood memories remind us of the days gone by.....
Reference: A Baker from Goa and Nelson Mandela: Long Walk

Describe any three causes of 'Civil Disobedience Movement'.

Rajesh and Vimal own 20 hectares and 30 hectares of agricultural land, respectively, which are entirely covered by wheat and mustard crops. The cultivation area of wheat and mustard in the land owned by Vimal are in the ratio of 5 : 3. If the total cultivation area of wheat and mustard are in the ratio 11 : 9, then the ratio of cultivation area of wheat and mustard in the land owned by Rajesh is

Three vectors \(\overrightarrow{OP}\), \(\overrightarrow{OQ}\), and \(\overrightarrow{OR}\) each of magnitude \(A\) are acting as shown in figure. The resultant of the three vectors is \(A \sqrt{x}\). The value of \(x\) is ______.

Explain the irony in the chapter, “A Letter to God”.

From the compounds given below, the number of compounds which give a positive Fehling’s test is____.
Compounds: Benzaldehyde, Acetaldehyde, Acetone, Acetophenone, Methanal, 4-nitrobenzaldehyde, cyclohexane carbaldehyde.

If the foot is perpendicular from (1, 2, 3) to the line \(\frac{x+1}{2} = \frac{y-2}{5} = \frac{z-1}{1}\) is \(( a, \beta, \gamma)\), then find \(a + \beta + \gamma\)

Two resistors of 1 Ω and 3 Ω are connected in parallel in a circuit. If a current of 1 ampere is flowing through the circuit, the current flowing through the 3 Ω resistor is:

Two vertices of the parallelogram \( ABCD \) are given as \( A(-1, 2, 1) \) and \( B(1, -2, 5) \). If the equation of the line passing through \( C \) and \( D \) is: \[ \frac{x - 4}{1} = \frac{y + 7}{-2} = \frac{z - 8}{2}, \] find the distance between sides \( AB \) and \( CD \). Hence, find the area of the parallelogram \( ABCD \).

In the given figure, the charge stored in \(6\mu F\) capacitor, when points A and B are joined by a connecting wire is _______\(\mu C\).

“During the colonial period the agricultural sector showed massive stagnation.” Do you agree with the given statement? Justify your answer with valid arguments.