List of practice Questions

If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is:

Let
A = $\begin{bmatrix} 1 \\ 1 \\ 1 \\ \end{bmatrix}$ and B = $\begin{bmatrix} 9^2 & -10^2 & 11^2 \\ 12^2 & 13^2 & 14^2 \\ -15^2 & 16^2 & 17^2 \\ \end{bmatrix}$
then the value of A'BA is

The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 39 and x – y = 3, respectively and P(2, 3) is its circumcentre. Then which of the following is NOT true?

The integral
$\frac{24}{\pi} \int_{0}^{\sqrt{2}} \frac{2 - x^2}{(2 + x^2) \sqrt{4 + x^4}} \, dx$
is equal to _______.

In the given figure, the diameter of the circle is 3 cm. AB and MN are two diameters such that MN is perpendicular to AB. In addition, CG is perpendicular to AB such that $ AE : EB = 1 : 2 $, and DF is perpendicular to MN such that $ NL : LM = 1 : 2 $. The length of DH in cm is:

The degree measure of $\frac{\pi}{32}$ is equal to

The depression in the freezing point observed for a formic acid solution of concentration 0.5 mL L^–1 is 0.0405°C. Density of formic acid is 1.05 g mL^–1. The Van’t Hoff factor of the formic acid solution is nearly (Given for water k_f = 1.86 k kg mol^–1)

When 800 mL of 0.5 M nitric acid is heated in a beaker, its volume is reduced to half and 11.5 g of nitric acid is evaporated. The molarity of the remaining nitric acid solution is $x × 10^{–2} M$. (Nearest integer)
(Molar mass of nitric acid is 63 g mol^–1)

Let the coefficients of the middle terms in the expansion of$\begin{array}{l} \left(\dfrac{1}{\sqrt 6}+ \beta x\right)^4, \left(1 – 3\beta x\right)^2 \text{and}\left(1 – \dfrac{\beta}{2}x\right )^6, \beta > 0,\end{array}$respectively form the first three terms of an A.P. If d is the common difference of this A.P., then $\begin{array}{l} 50-\frac{2d}{\beta^2} \end{array}$is equal to _______.

The differential equation of the family of circles passing through the points (0, 2) and (0, –2) is

The value of the integral $∫^{ \frac{π}{2}}_{ \frac{-π}{2}} \frac{dx}{(1+e^x)(sin^6x+cos^6x)}$ is equal to

A steel wire of length 3.2 m (Y_s = 2.0 × 10¹¹ Nm^-2) and a copper wire of length 4.4 m (Y_c = 1.1 × 10¹¹ Nm^-2), both of radius 1.4 mm are connected end to end. When stretched by a load, the net elongation is found to be 1.4 mm. The load applied, in Newton, will be:
$(Given: π = \frac{22}{7})$

The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is _________.

A girl standing on road holds her umbrella at 45° with the vertical to keep the rain away. If she starts running without umbrella with a speed of $15\sqrt2$ $kmh^{–1}$, the rain drops hit herhead vertically. The speed of rain drops with respect to the moving girl is

Let $PQRS$ be a quadrilateral in a plane, where $QR =1, \angle PQR =\angle QRS =70^{\circ}, \angle PQS =15^{\circ}$ and $\angle PRS =40^{\circ}$. If $\angle RPS =\theta^{\circ}, PQ =\alpha$ and $PS =\beta$, then the interval(s) that contain(s) the value of $4 \alpha \beta \sin \theta^{\circ}$ is/are

If a quadratic equation $2x^2 + kx+3=0$ have two equal roots then $k=$

At 25°C and 1 atm pressure, the enthalpy of combustion of benzene (I) and acetylene (g) are –3268 kJ mol^–1 and –1300 kJ mol^–1, respectively. The change in enthalpy for the reaction 3C₂H₂(g) → C₆H₆(I), is

Let a, b be two non-zero real numbers. If p and r are the roots of the equation x² – 8ax + 2a = 0 and q and s are the roots of the equation x² + 12bx + 6b = 0, such that $\frac{1}{p}$,$\frac{1}{q}$,$\frac{1}{r}$,$\frac{1}{s}$ are in A.P., then a^–1 – b^–1 is equal to _________.

If $\lim_{n\rightarrow \infty}$ $\frac{(n+1)^{k-1}}{n^{k+1}}[(nk+1)+(nk+2)+....+(nk+n)]=33.\lim_{n\rightarrow \infty}\frac{1}{n^{k+1}}.[1^k+2^k+3^k+....+n^k]$
then the integral value of k is equal to _______.

A solid cylinder length is suspended symmetrically through two massless strings, as shown in the figure. The distance from the initial rest position, the cylinder should be unbinding the strings to achieve a speed of 4 m/s, is ______ cm. (Take g = 10 m/s²).

Let ⊙ be a binary operation on Q - {0} defined by a⊙b=$\frac{a}{b}$. Then 1⊙(2⊙(3⊙4)) is equal to

How many pairs of letters are there in the word ‘CHILDREN’ such that each pair has as many letters between them in the word ‘CHILDREN’ as they have in the English alphabet series?

Let P and Q be any points on the curves (x – 1)² + (y + 1)² = 1 and y = x², respectively. The distance between P and Q is minimum for some value of the abscissa of P in the interval

Let for some real numbers $α$ and $β$, $a = α – iβ$. If the system of equations $4ix + (1 + i) y = 0$ and $8\bigg(cosx\frac{2π}{3}+i\;sin\frac{2π}{3}\bigg)x+\bar ay=0 $has more than one solution, then $\frac{α}{β}$ is equal to

The pressure of a moist gas at 27°C is 4 atm. The volume of the container is doubled at the same temperature. The new pressure of the moist gas is ______ ×10^–1 atm. (Nearest integer)(Given: The vapor pressure of water at 27°C is 0.4 atm.)