List of practice Questions

A battery of emf \( 8V \) and internal resistance \( 0.5 \Omega \) is being charged by a \( 120V \) DC supply using a series resistor of \( 15.5 \Omega \). The terminal voltage of the \( 8V \) battery during charging is:

If \( x_1, x_2, x_3, \dots, x_n \) are \( n \) observations such that \( \sum (x_i + 2)^2 = 28n \) and \( \sum (x_i - 2)^2 = 12n \), then the variance is:

Evaluate the sum: \[ \frac{1}{3 \cdot 7} + \frac{1}{7 \cdot 11} + \frac{1}{11 \cdot 15} + \dots \text{ up to 50 terms=} \]

If \( A(1,0,2) \), \( B(2,1,0) \), \( C(2,-5,3) \), and \( D(0,3,2) \) are four points and the point of intersection of the lines \( AB \) and \( CD \) is \( P(a,b,c) \), then \( a + b + c = ? \)

If \(\int \frac{\log(1+x^4)}{x^3} dx = f(x) \log(\frac{1}{g(x)}) + \tan^{-1}(h(x)) + c\), then \(h(x) [f(x) + f(\frac{1}{x})] =\)

The acceleration of a particle which moves along the positive \( x \)-axis varies with its position as shown in the figure. If the velocity of the particle is \( 0.8 \, {m/s} \) at \( x = 0 \), then its velocity at \( x = 1.4 \, {m} \) is: \includegraphics[]{83.png}

The triangle \( PQR \) is inscribed in the circle \[ x^2 + y^2 = 25. \] If \( Q = (3,4) \) and \( R = (-4,3) \), then \( \angle QPR \) is:

If \( A(1,2,0), B(2,0,1), C(-3,0,2) \) are the vertices of \( \triangle ABC \), then the length of the internal bisector of \( \angle BAC \) is:

If \( z_1 = 10 + 6i \), \( z_2 = 4 + 6i \) and \( z \) is any complex number such that the argument of \( \frac{z-z_1}{z-z_2} \) is \( \frac{\pi}{4} \), then the value of \( |z - 7 - 9i| \) is:

There are 6 different novels and 3 different poetry books on a table. If 4 novels and 1 poetry book are to be selected and arranged in a row on a shelf such that the poetry book is always in the middle, then the number of such possible arrangements is:

The sum of the series \( 1 - \frac{2}{3} + \frac{2.4}{3.6} - \frac{2.4.6}{3.6.9} + \cdots \infty \) is:

The circumference of a circle passing through the point \( (4, 6) \) with two normals represented by \( 2x - 3y + 4 = 0 \) and \( x + y - 3 = 0 \) is:

If \( z = x+iy \), \( x^2+y^2 = 1 \) and \( z_1 = e^{i\theta} \), then the expression \( \frac{z_1^{2n-1} - 1}{z_1^{2^n-1} + 1} \) simplifies to:

If \(2.4^{2n+1} + 3^{3n+1}\) is divisible by \(k\) for all \(n \in \mathbb{N}\), then \(k\) is:

\( x \) g of an organic compound was analysed by Kjeldahl’s method for the estimation of nitrogen. The ammonia gas evolved from the compound was passed into 100 mL of 0.2 M H\(_2\)SO\(_4\). The excess of acid required 40 mL of 0.5 M NaOH solution for complete neutralisation. What is \( x \) when the percentage of nitrogen in it is 46%?

If \((\alpha + \beta)\) is not a multiple of \(\frac{\pi}{2}\) and \(3 \sin(\alpha - \beta) = 5 \cos(\alpha + \beta)\), then \[ \tan\left(\frac{\pi}{4} + \alpha\right) + 4\tan\left(\frac{\pi}{4} + \beta\right) = \]

What is Y in the following reaction sequence?

Arrange the following in the increasing order of number of unpaired electrons present in the central metal ion:
I. \([MnCl_6]^{4-}\)
II. \([FeF_6]^{3-}\)
III. \([Mn(CN)_6]^{3-}\)
IV. \([Fe(CN)_6]^{3-}\)

If \([x]\) is the greatest integer function, then evaluate the integral \( \int_{0}^{5} [x] \, dx \):

If \( L_1 \) and \( L_2 \) are two lines which pass through origin and have direction ratios \( (3, 1, -5) \) and \( (2, 3, -1) \) respectively, then the equation of the plane containing \( L_1 \) and \( L_2 \) is:

The largest among the distances from the point \(P(15,9)\) to the points on the circle \(x^2 + y^2 - 6x - 8y - 11 = 0\) is:

What happens in the dark reaction of photosynthesis?

If \( \vec{f} = i + j + k \) and \( \vec{g} = 2i - j + 3k \), then the projection vector of \( \vec{f} \) on \( \vec{g} \) is:

Let \( \mathbf{a} = 3\hat{i} + 4\hat{j} - 5\hat{k} \), \( \mathbf{b} = 2\hat{i} + \hat{j} - 2\hat{k} \). The projection of the sum of the vectors \( \mathbf{a}, \mathbf{b} \) on the vector perpendicular to the plane of \( \mathbf{a}, \mathbf{b} \) is:

The length of the latus rectum of \( 16x^2 + 25y^2 = 400 \) is: