List of practice Questions

Let the line $L$ be the projection of the line \[ \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z - 4}{2} \] in the plane $x - 2y - z = 3$. If $d$ is the distance of the point $(0, 0, 6)$ from $L$, then $d^2$ is equal to _________.

The mean and standard deviation of 20 observations were calculated as 10 and 2.5 respectively. It was found that by mistake one data value was taken as 25 instead of 35. If \(\alpha\) and \(\sqrt{\beta}\) are the mean and standard deviation respectively for correct data, then \((\alpha, \beta)\) is :

A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is $k$ (meter), then $\left( \frac{4}{\pi} + 1 \right)k$ is equal to _________.

A plane \(P\) contains the line \(x + 2y + 3z + 1 = 0 = x - y - z - 6\), and is perpendicular to the plane \(-2x + y + z + 8 = 0\). Then which of the following points lies on \(P\) ?

If the truth value of the Boolean expression \(((p \vee q) \wedge (q \to r) \wedge (\sim r)) \to (p \wedge q)\) is false, then the truth values of the statements \(p, q, r\) respectively can be :

If the sum of an infinite GP \(a, ar, ar^2, ar^3, \dots\) is 15 and the sum of the squares of its each term is 150, then the sum of \(ar^2, ar^4, ar^6, \dots\) is :

Let \(f(x) = \cos\left(2\tan^{-1} \sin \left(\cot^{-1} \sqrt{\frac{1-x}{x}}\right)\right)\), \(0<x<1\). Then :

The value of \(\lim_{n \to \infty} \frac{1}{n} \sum_{r=0}^{2n-1} \frac{n^2}{n^2 + 4r^2}\) is :

Let \(ABC\) be a triangle with \(A(-3, 1)\) and \(\angle ACB = \theta, 0<\theta \leq \frac{\pi}{2}\). If the equation of the median through \(B\) is \(2x + y - 3 = 0\) and the equation of angle bisector of \(C\) is \(7x - 4y - 1 = 0\), then \(\tan \theta\) is equal to :

Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set :

The sum of the series \(\frac{1}{x+1} + \frac{2}{x^2+1} + \frac{2^2}{x^4+1} + \dots + \frac{2^{100}}{x^{2^{100}}+1}\) when \(x = 2\) is :

The \(OH^-\) concentration in a mixture of 5.0 mL of 0.0504 M \(NH_4Cl\) and 2 mL of 0.0210 M \(NH_3\) solution is \(x \times 10^{-6}\) M. The value of \(x\) is _________. (Nearest integer)
[Given \(K_w = 1 \times 10^{-14}\) and \(K_b = 1.8 \times 10^{-5}\)]

The number of 4f electrons in the ground state electronic configuration of \(Gd^{2+}\) is _________. [Atomic number of Gd = 64] (Integer answer)

The polymer formed on heating Novolac with formaldehyde is :

Given below are two statements :
Statement I : In the titration between strong acid and weak base methyl orange is suitable as an indicator.
Statement II : For titration of acetic acid with NaOH phenolphthalein is not a suitable indicator.
In the light of the above statements, choose the most appropriate answer from the options given below :

Which one of the following when dissolved in water gives coloured solution in nitrogen atmosphere ?

The conversion of hydroxyapatite occurs due to presence of \(F^-\) ions in water. The correct formula of hydroxyapatite is :

Which one of the following is correct for the adsorption of a gas at a given temperature on a solid surface ?

Given below are two statements.
Statement I : The choice of reducing agents for metals extraction can be made by using Ellingham diagram, a plot of \(\Delta G\) vs temperature.
Statement II : The value of \(\Delta S\) increases from left to right in Ellingham diagram.
In the light of the above statements, choose the most appropriate answer from the options given below :

Which one of the following methods is most suitable for preparing deionized water ?

What are the products formed in sequence when excess of \(CO_2\) is passed in slaked lime ?

The incorrect statement is :

Two short magnetic dipoles \(m_1\) and \(m_2\) each having magnetic moment of \(1 \text{ Am}^2\) are placed at point O and P respectively. The distance between OP is 1 meter. The torque experienced by the magnetic dipole \(m_2\) due to the presence of \(m_1\) is _________ \(\times 10^{-7} \text{ Nm}\).
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An amplitude modulated wave is represented by \(C_m(t) = 10(1 + 0.2 \cos 12560 t) \sin(111 \times 10^4 t)\) volts. The modulating frequency in kHz will be _________.

Two spherical balls having equal masses with radius of 5 cm each are thrown upwards along the same vertical direction at an interval of 3 s with the same initial velocity of 35 m/s, then these balls collide at a height of _________ m.
(take \(g = 10 \text{ m/s}^2\))