List of practice Questions

Let the positive integers be written in the form :

If the $k^\text{th}$ row contains exactly $k$ numbers for every natural number $k$, then the row in which the number $5310$ will be, is _____

The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7, when the repetition of digits is not allowed, and which are not divisible by 3, is equal to ______.

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables $X$ and $Y$ respectively denote the number of blue and yellow balls. If $\bar{X}$ and $\bar{Y}$ are the means of $X$ and $Y$ respectively, then $7\bar{X} + 4\bar{Y}$ is equal to _____.

For the function $f(x) = (\cos x) - x + 1, x \in \mathbb{R}$, find the correct relationship between the following two statements
(S1) $f(x) = 0$ for only one value of x is $[0, \pi]$.
(S2) $f(x)$ is decreasing in $\left[0, \frac{\pi}{2}\right]$ and increasing in $\left[\frac{\pi}{2}, \pi\right]$.

If the set $R = {(a, b) : a + 5b = 42, a, b \in \mathbb{N}}$ has $m$ elements and $\sum_{n=1}^m (1 + i^n) = x + iy$, where $i = \sqrt{-1}$, then the value of $m + x + y$ is:

Let $[t]$ be the greatest integer less than or equal to t. Let A be the set of all prime factors of 2310 and $f: A \to \mathbb{Z}$ be the function $f(x) = \left[ \log_2 \left( x^2 + \left[ \frac{x^3}{5} \right] \right) \right]$. The number of one-to-one functions from A to the range of f is:

The equations of two sides AB and AC of a triangle ABC are $4x + y = 14$ and $3x - 2y = 5$, respectively. The point $\left(2, -\frac{4}{3}\right)$ divides the third side BC internally in the ratio 2 : 1. The equation of the side BC is:

Let $f(x) = 4\cos^3 x + 3\sqrt{3} \cos^2 x - 10$. The number of points of local maxima of $f$ in interval $(0, 2\pi)$ is:

The number of critical points of the function $f(x) = (x - 2)^{2/3}(2x + 1)$ is:

Let the circles $C_1 : (x - \alpha)^2 + (y - \beta)^2 = r_1^2$ and $C_2 : (x - 8)^2 + \left( y - \frac{15}{2} \right)^2 = r_2^2$ touch each other externally at the point $(6, 6)$. If the point $(6, 6)$ divides the line segment joining the centres of the circles $C_1$ and $C_2$ internally in the ratio $2 : 1$, then $(\alpha + \beta) + 4\left(r_1^2 + r_2^2\right)$ equals _____.

The sum of all the solutions of the equation \[(8)^{2x} - 16 \cdot (8)^x + 48 = 0\]is:

The value of $k \in \mathbb{N}$ for which the integral \[ I_n = \int_0^1 (1 - x^k)^n \, dx, \, n \in \mathbb{N}, \] satisfies $147 \, I_{20} = 148 \, I_{21}$ is:

Wavenumber for a radiation having $5800 \, \mathring{A}$ wavelength is $x \times 10 \, \text{cm}^{-1}$. The value of $x$ is _________

Total number of unpaired electrons in the complex ion [Co(NH$_3$)$_6$]$^{3+}$ and [NiCl$_4$]$^{2-}$ is

Total number of aromatic compounds among the following compounds is _______.

Two moles of benzaldehyde and one mole of acetone under alkaline conditions using aqueous NaOH after heating gives $x$ as the major product. The number of $\pi$ bonds in the product $x$ is ................

Total number of optically active compounds from the following is _______.

Number of molecules having bond order 2 from the following molecule is ...................
$C_2, O_2, Be_2, Li_2, Ne_2, N_2, He_2$

Match List-I with List-II.

List-I (Complex ion)	List-II (Spin only magnetic moment in B.M.)
(A) [Cr(NH$_3$)$_6$]$^{3+}$	(I) 4.90
(B) [NiCl$_4$]$^{2-}$	(II) 3.87
(C) [CoF$_6$]$^{3-}$	(III) 0.0
(D) [Ni(CN)$_4$]$^{2-}$	(IV) 2.83

Choose the correct answer from the options given below:

Given below are two statements:
Statement (I) : Fusion of MnO$_2$ with KOH and an oxidising agent gives dark green K$_2$MnO$_4$.
Statement (II) : Manganate ion on electrolytic oxidation in alkaline medium gives permanganate ion.
In the light of the above statements, choose the correct answer from the options given below.

The correct sequence of acidic strength of the following aliphatic acids in their decreasing order is: $CH_3CH_2COOH$, $CH_3COOH$, $CH_3CH_2CH_2COOH$, $HCOOH$

IUPAC name of following hydrocarbon (X) is :

Identify the incorrect statements about group 15 elements:
(A) Dinitrogen is a diatomic gas which acts like an inert gas at room temperature.
(B) The common oxidation states of these elements are −3, +3 and +5.
(C) Nitrogen has unique ability to form p$\pi$-p$\pi$ multiple bonds.
(D) The stability of +5 oxidation states increases down the group.
(E) Nitrogen shows a maximum covalency of 6.
Choose the correct answer from the options given below.

Match List-I with List-II.

List-I (Test)	List-II (Identification)
(A) Bayer's test	(I) Phenol
(B) Ceric ammonium nitrate test	(II) Aldehyde
(C) Phthalein dye test	(III) Alcoholic-OH group
(D) Schiff's test	(IV) Unsaturation

Choose the correct answer from the options given below:

Match List-I with List-II.

Choose the correct answer from the options given below :