List of practice Questions

The distance of the point \((-1, 2, -2)\) from the line of intersection of the planes \(2x + 3y + 2z = 0\) and \(x - 2y + z = 0\) is :
The mean and variance of 7 observations are 8 and 16 respectively. If two observations are 6 and 8, then the variance of the remaining 5 observations is :
The number of solutions of the equation \(32^{\tan^2 x} + 32^{\sec^2 x} = 81\), \(0 \le x \le \frac{\pi}{4}\) is :
The domain of the function \(f(x) = \sin^{-1} \left( \frac{3x^2 + x - 1}{(x - 1)^2} \right) + \cos^{-1} \left( \frac{x - 1}{x + 1} \right)\) is :
If \( \alpha = \lim_{x \to \pi/4} \frac{\tan^3 x - \tan x}{\cos(x + \frac{\pi}{4})} \) and \( \beta = \lim_{x \to 0} (\cos x)^{\cot x} \) are the roots of the equation, \( ax^2 + bx - 4 = 0 \), then the ordered pair \( (a, b) \) is :
Let \( f \) be any continuous function on \( [0, 2] \) and twice differentiable on \( (0, 2) \). If \( f(0) = 0, f(1) = 1 \) and \( f(2) = 2 \), then :
If \( [x] \) is the greatest integer \( \le x \), then \( \pi^2 \int_{0}^{2} \left( \sin \frac{\pi x}{2} \right) (x - [x])^{[x]} dx \) is equal to :
If \( \frac{dy}{dx} = \frac{2^x y + 2^y \cdot 2^x}{2^x + 2^{x+y} \log_e 2}, y(0) = 0 \), then for \( y = 1 \), the value of \( x \) lies in the interval :
If \( y \frac{dy}{dx} = x \left[ \frac{\phi(y^2/x^2)}{\phi'(y^2/x^2)} + \frac{y^2}{x^2} \right], x>0, \phi>0, \) and \( y(1) = -1 \), then \( \phi\left(\frac{y^2}{4}\right) \) is equal to :
Let \( A \) be the set of all points \( (\alpha, \beta) \) such that the area of triangle formed by the points \( (5, 6), (3, 2) \) and \( (\alpha, \beta) \) is 12 square units. Then the least possible length of a line segment joining the origin to a point in \( A \), is :
The locus of mid-points of the line segments joining \( (-3, -5) \) and the points on the ellipse \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \) is :
Let \( f : \mathbb{N} \to \mathbb{N} \) be a function such that \( f(m + n) = f(m) + f(n) \) for every \( m, n \in \mathbb{N} \). If \( f(6) = 18 \), then \( f(2) \cdot f(3) \) is equal to :
If \( z \) is a complex number such that \( \frac{z - i}{z - 1} \) is purely imaginary, then the minimum value of \( |z - (3 + 3i)| \) is :
The sum of the roots of the equation, \( x + 1 - 2\log_2(3 + 2^x) + 2\log_4(10 - 2^{-x}) = 0 \), is :
If \( \alpha + \beta + \gamma = 2\pi \), then the system of equations
\( x + (\cos \gamma)y + (\cos \beta)z = 0 \)
\( (\cos \gamma)x + y + (\cos \alpha)z = 0 \)
\( (\cos \beta)x + (\cos \alpha)y + z = 0 \)
has :
Let \( a_1, a_2, a_3, \dots \) be an A.P. If \( \frac{a_1 + a_2 + \dots + a_{10}}{a_1 + a_2 + \dots + a_p} = \frac{100}{p^2}, p \neq 10 \), then \( \frac{a_{11}}{a_{10}} \) is equal to :
An angle of intersection of the curves, \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) and \( x^2 + y^2 = ab, a>b \), is :
The empirical formula for a compound with a cubic close packed arrangement of anions and with cations occupying all the octahedral sites is $A_x B$. The value of $x$ is ________. (Integer answer)
The value of magnetic quantum number of the outermost electron of $\text{Zn}^+$ ion is ________. (Integer answer)
1.22 g of an organic acid is separately dissolved in 100 g of benzene ($K_b = 2.6 \text{ K kg mol}^{-1}$) and 100 g of acetone ($K_b = 1.7 \text{ K kg mol}^{-1}$). The acid is known to dimerize in benzene but remain as a monomer in acetone. The boiling point of the solution in acetone increases by $0.17^\circ\text{C}$. The increase in boiling point of solution in benzene in $^\circ\text{C}$ is $x \times 10^{-2}$. The value of $x$ is ________. (Nearest integer) [Atomic mass : C = 12.0, H = 1.0, O = 16.0]
The pH of a solution obtained by mixing 50 mL of 1 M HCl and 30 mL of 1 M NaOH is $x \times 10^{-1}$. The value of $x$ is ________. (Nearest integer) [log 2.5 = 0.3979]
$\text{CH}_4$ is adsorbed on 1 g charcoal at $0^\circ\text{C}$ following the Freundlich adsorption isotherm. 10.0 mL of $\text{CH}_4$ is adsorbed at 100 mm of Hg, whereas 15.0 mL is adsorbed at 200 mm of Hg. The volume of $\text{CH}_4$ adsorbed at 300 mm of Hg is $10^x$ mL. The value of $x$ is ________ $\times 10^{-2}$. (Nearest integer) [Use $\log_{10} 2 = 0.3010, \log_{10} 3 = 0.4771$]
In the electrolytic refining of blister copper, the total number of main impurities, from the following, removed as anode mud is ________.
Pb, Sb, Se, Te, Ru, Ag, Au and Pt.
Which among the following is not a polyester ?
The deposition of X and Y on ground surfaces is referred as wet and dry depositions, respectively. X and Y are :