List of practice Questions

The number of real roots of the equation $e^{4x} - e^{3x} - 4e^{2x} - e^x + 1 = 0$ is equal to ________.
For real numbers $\alpha$ and $\beta \neq 0$, if the point of intersection of the straight lines
$\frac{x-\alpha}{1} = \frac{y-1}{2} = \frac{z-1}{3}$ and $\frac{x-4}{\beta} = \frac{y-6}{3} = \frac{z-7}{3}$
lies on the plane $x+2y-z=8$, then $\alpha-\beta$ is equal to :
If $\tan(\frac{\pi}{9})$, x, $\tan(\frac{7\pi}{18})$ are in arithmetic progression and $\tan(\frac{\pi}{9})$, y, $\tan(\frac{5\pi}{18})$ are also in arithmetic progression, then $|x-2y|$ is equal to :
If the real part of the complex number $z = \frac{3+2i\cos\theta}{1-3i\cos\theta}$, $\theta \in (0, \frac{\pi}{2})$ is zero, then the value of $\sin^2 3\theta + \cos^2 \theta$ is equal to ________.
Let C be the set of all complex numbers. Let
$S_1 = \{z \in C : |z-2| \le 1\}$ and
$S_2 = \{z \in C : z(1+i) + \bar{z}(1-i) \ge 4\}$.
Then, the maximum value of $|z-\frac{5}{2}|^2$ for $z \in S_1 \cap S_2$ is equal to :
Let $f: (a,b) \to \mathbb{R}$ be twice differentiable function such that $f(x) = \int_a^x g(t)dt$ for a differentiable function $g(x)$. If $f(x)=0$ has exactly five distinct roots in $(a,b)$, then $g(x)g'(x)=0$ has at least :
Let A and B be two $3 \times 3$ real matrices such that $(A^2 - B^2)$ is invertible matrix. If $A^5=B^5$ and $A^3B^2=A^2B^3$, then the value of the determinant of the matrix $A^3+B^3$ is equal to :
Let $\mathbb{N}$ be the set of natural numbers and a relation R on $\mathbb{N}$ be defined by R = $\{(x, y) \in \mathbb{N} \times \mathbb{N} : x^3 - 3x^2y - xy^2 + 3y^3 = 0\}$. Then the relation R is :
Let $\alpha = \max_{x \in \mathbb{R}}\{8^{2\sin 3x} \cdot 4^{4\cos 3x}\}$ and $\beta = \min_{x \in \mathbb{R}}\{8^{2\sin 3x} \cdot 4^{4\cos 3x}\}$.
If $8x^2+bx+c=0$ is a quadratic equation whose roots are $\alpha^{1/5}$ and $\beta^{1/5}$, then the value of c-b is equal to :
The value of $\lim_{x \to 0} \left(\frac{x}{\sqrt[8]{1-\sin x} - \sqrt[8]{1+\sin x}}\right)$ is equal to :
Two sides of a parallelogram are along the lines $4x+5y=0$ and $7x+2y=0$. If the equation of one of the diagonals of the parallelogram is $11x+7y=9$, then other diagonal passes through the point :
Which of the following is the negation of the statement "for all M>0, there exists x$\in$S such that x $\ge$ M" ?
The area of the region bounded by $y-x=2$ and $x^2=y$ is equal to :
The dihedral angle in staggered form of 1,1,1-Trichloro ethane is ________ degree. (Round off to the Nearest Integer).
Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x+y) + f(x-y) = 2f(x)f(y)$, $f(\frac{1}{2}) = -1$. Then, the value of $\sum_{k=1}^{20} \frac{1}{\sin(k) \sin(k+f(k))}$ is equal to :
Consider a circle C which touches the y-axis at (0, 6) and cuts off an intercept $6\sqrt{5}$ on the x-axis. Then the radius of the circle C is equal to :
2SO$_2$(g) + O$_2$(g) $\rightarrow$ 2SO$_3$(g)
The above reaction is carried out in a vessel starting with partial pressures P$_{SO_2}$ = 250 m bar, P$_{O_2}$ = 750 m bar and P$_{SO_3}$ = 0 bar. When the reaction is complete, the total pressure in the reaction vessel is ________ m bar. (Round off to the Nearest Integer).
The equilibrium constant for the reaction
A(s) $\rightleftharpoons$ M(s) + $\frac{1}{2}$O$_2$(g)
is K$_p$=4. At equilibrium, the partial pressure of O$_2$ is ________ atm. (Round off to the Nearest Integer).
The number of neutrons and electrons, respectively, present in the radioactive isotope of hydrogen is :
Number of Cl=O bonds in chlorous acid, chloric acid and perchloric acid respectively are :
To an aqueous solution containing ions such as Al$^{3+}$, Zn$^{2+}$, Ca$^{2+}$, Fe$^{3+}$, Ni$^{2+}$, Ba$^{2+}$ and Cu$^{2+}$ was added conc. HCl, followed by H$_2$S.
The total number of cations precipitated during this reaction is/are :
Select the correct statements.
(A) Crystalline solids have long range order.
(B) Crystalline solids are isotropic.
(C) Amorphous solids are sometimes called pseudo solids.
(D) Amorphous solids soften over a range of temperatures.
(E) Amorphous solids have a definite heat of fusion.
Choose the most appropriate answer from the options given below :
If the Thompson model of the atom was correct, then the result of Rutherford's gold foil experiment would have been :
Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : SO$_2$(g) is adsorbed to a larger extent than H$_2$(g) on activated charcoal.
Reason R : SO$_2$(g) has a higher critical temperature than H$_2$(g).
In the light of the above statements, choose the most appropriate answer from the options given below.
The CORRECT order of first ionisation enthalpy is :