List of practice Questions

Identify the CORRECT set of details from the following:
A. $[\text{Co}(\text{NH}_3)_6]^{3+}$: Inner orbital complex; $d^2sp^3$ hybridized
B. $[\text{MnCl}_6]^{3-}$: Outer orbital complex; $sp^3d^2$ hybridized
C. $[\text{CoF}_6]^{3-}$: Outer orbital complex; $d^2sp^3$ hybridized
D. $[\text{FeF}_6]^{3-}$: Outer orbital complex; $sp^3d^2$ hybridized
E. $[\text{Ni}(\text{CN})_4]^{2-}$: Inner orbital complex; $sp^3$ hybridized}

Iodoform test can differentiate between

A. Methanol and Ethanol
B. CH$_3$COOH and CH$_3$CH$_2$COOH
C. Cyclohexene and cyclohexanone
D. Diethyl ether and pentan-3-one
E. Anisole and acetone

Choose the correct answer from the options given below:

Identify the INCORRECT statements from the following:

A. Notation $ {}^{24}_{12}\mathrm{Mg} $ represents 24 protons and 12 neutrons.
B. Wavelength of a radiation of frequency $ 4.5 \times 10^{15}\ \text{s}^{-1} $ is $ 6.7 \times 10^{-8}\ \text{m} $.
C. One radiation has wavelength $ \lambda_1 $ (900 nm) and energy $ E_1 $. Other radiation has wavelength $ \lambda_2 $ (300 nm) and energy $ E_2 $. $ E_1 : E_2 = 3 : 1 $.
D. Number of photons of light of wavelength 2000 pm that provides 1 J of energy is $ 1.006 \times 10^{16} $.

Choose the correct answer from the options given below:

The number of elements in the set \[ S = \left\{ x : x \in [0,100] \text{ and } \int_{0}^{x} t^2 \sin(x - t)\,dt = x^2 \right\} \] is

Let \[ A = \begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 1 \\ 3 & -1 & 0 \end{bmatrix} \] and $ B $ be a matrix such that \[ B(I - A) = I + A. \] Then the sum of the diagonal elements of $ B^{T}B $ is equal to

If the image of the point $ P(a,2,a) $ in the line \[ \frac{x}{2} = \frac{y+a}{1} = \frac{z}{1} \] is $ Q $ and the image of $ Q $ in the line \[ \frac{x-2b}{2} = \frac{y-a}{1} = \frac{z+2b}{-5} \] is $ P $, then $ a + b $ is equal to

Let \[ \sum_{k=1}^{n} a_k = \alpha n^2 + \beta n. \] If $ a_{10} = 59 $ and $ a_6 = 7a_1 $, then $ \alpha + \beta $ is equal to

Consider two sets \[ A = \{ x \in \mathbb{Z} : |(|x-3|-3)| \le 1 \} \] and \[ B = \left\{ x \in \mathbb{R} - \{1,2\} : \frac{(x-2)(x-4)}{x-1}\,\log_e(|x-2|) = 0 \right\}. \] Then the number of onto functions $ f : A \to B $ is equal to

If $ z = \dfrac{\sqrt{3}}{2} + \dfrac{i}{2} $, $ i = \sqrt{-1} $, then \[ \left(z^{201} - i\right)^8 \] is equal to

Let $ \dfrac{\pi}{2} < \theta < \pi $ and $ \cot \theta = -\dfrac{1}{2\sqrt{2}} $. Then the value of \[ \sin\!\left(\frac{15\theta}{2}\right)(\cos 8\theta + \sin 8\theta) + \cos\!\left(\frac{15\theta}{2}\right)(\cos 8\theta - \sin 8\theta) \] is equal to

A regular hexagon is formed by six wires each of resistance $ r \,\Omega $ and the corners are joined to the centre by wires of same resistance. If the current enters at one corner and leaves at the opposite corner, the equivalent resistance of the hexagon between the two opposite corners will be

Let $ PQ $ be a chord of the hyperbola \[ \frac{x^2}{4} - \frac{y^2}{b^2} = 1, \] perpendicular to the $x$-axis such that $ OPQ $ is an equilateral triangle, $ O $ being the centre of the hyperbola. If the eccentricity of the hyperbola is $ \sqrt{3} $, then find the area of the triangle $ OPQ $.

Let $ \vec{a} = \hat{i} - 2\hat{j} + 3\hat{k} $, $ \vec{b} = 2\hat{i} + \hat{j} - \hat{k} $, $ \vec{c} = \lambda \hat{i} + \hat{j} + \hat{k} $ and $ \vec{v} = \vec{a} \times \vec{b} $. If $ \vec{v} \cdot \vec{c} = 11 $ and the length of the projection of $ \vec{b} $ on $ \vec{c} $ is $ p $, then find the value of $ 9p^2 $.

Let $ \vec{a}, \vec{b}, \vec{c} $ be three vectors such that \[ \vec{a} \times \vec{b} = 2(\vec{a} \times \vec{c}). \] If $ |\vec{a}| = 1 $, $ |\vec{b}| = 4 $, $ |\vec{c}| = 2 $, and the angle between $ \vec{b} $ and $ \vec{c} $ is $60^\circ$, then $ |\vec{a} \cdot \vec{c}| $ is equal to.

If \[ f(x) = \begin{cases} \dfrac{a|x| + x^2 - 2(\sin|x|)(\cos|x|)}{x}, & x \ne 0, \\ b, & x = 0, \end{cases} \] is continuous at $ x = 0 $, then $ a + b $ is equal to.

Let $ A = \{0,1,2,\ldots,9\} $. Let $ R $ be a relation on $ A $ defined by $(x,y) \in R$ if and only if $ |x - y| $ is a multiple of $3$. Given below are two statements:
Statement I: $ n(R) = 36 $.
Statement II: $ R $ is an equivalence relation.
In the light of the above statements, choose the correct answer from the options given below.

The sum of all the real solutions of the equation \[ \log_{(x+3)}\left(6x^2 + 28x + 30\right) = 5 - 2\log_{(6x+10)}\left(x^2 + 6x + 9\right) \] is equal to .

Let \[ I(x) = \int \frac{3\,dx}{(4x+6)\sqrt{4x^2 + 8x + 3}} \] and \[ I(0) = \frac{\sqrt{3}}{4} + 20. \] If \[ I\left(\frac{1}{2}\right) = \frac{a\sqrt{2}}{b} + c, \] where $a, b, c \in \mathbb{N}$ and $\gcd(a,b)=1$, then find the value of \[ a + b + c. \]

Let \[ f(x)=\int \frac{7x^{10}+9x^8}{(1+x^2+2x^9)^2}\,dx \] and $f(1)=\frac14$. Given that

Three parallel plate capacitors each with area $A$ and separation $d$ are filled with two dielectric ($k_1$ and $k_2$) in the following fashion. ($k_1>k_2$) Which of the following is true?

Let the set of all values of $ r $, for which the circles $ (x + 1)^2 + (y + 4)^2 = r^2 $ and $ x^2 + y^2 - 4x - 2y - 4 = 0 $ intersect at two distinct points be the interval $ (\alpha, \beta) $. Then $ \alpha\beta $ is equal to

The magnetic field at the centre of a current carrying circular loop of radius $R$ is $16\,\mu\text{T}$. The magnetic field at a distance $x=\sqrt{3}R$ on its axis from the centre is ____ $\mu\text{T}$.

Let $S$ be a set of $5$ elements and $P(S)$ denote the power set of $S$. Let $E$ be the event of choosing an ordered pair $(A,B)$ from $P(S)\times P(S)$ such that $A\cap B=\varnothing$. If the probability of the event $E$ is $\dfrac{3^p}{2^q}$, where $p,q\in\mathbb{N}$, then $p+q$ is equal to

Let
\[ f(x)=\int \frac{7x^{10}+9x^8}{(1+x^2+2x^9)^2}\,dx,\quad x > 0, \] and
\[ A= \begin{bmatrix} 0 & 0 & 1 \\ \frac{1}{4} & f'(1) & 1 \\ \alpha & 4 & 1 \end{bmatrix}. \] If $ B=\operatorname{adj}(\operatorname{adj} A) $, then the value of $ \alpha $ for which $ \det(B)=1 $ is

Let mirror image of parabola $x^2 = 4y$ in the line $x-y=1$ be $(y+a)^2 = b(x-c)$. Then the value of $(a+b+c)$ is