List of practice Questions

Let \( [\,\cdot\,] \) denote the greatest integer function, and let \[ f(x) = \min\{\sqrt{2}\,x, x^2\}. \] Let \[ S = \{x \in (-2,2) : \text{the function } g(x) = x[x^2] \text{ is discontinuous at } x\}. \] Then \[ \sum_{x \in S} f(x) \text{ equals} \]
If the mean deviation about the median of the numbers \[ k,\,2k,\,3k,\,\ldots,\,1000k \] is \(500\), then \(k^2\) is equal to
Let \( P(10, 2\sqrt{15}) \) be a point on the hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \] whose foci are \( S \) and \( S' \). If the length of its latus rectum is \(8\), then the square of the area of \( \triangle PSS' \) is equal to
The area of the region \[ A = \{(x,y) : 4x^2 + y^2 \le 8 \;\text{and}\; y^2 \le 4x\} \] is
Let the locus of the mid-point of the chord through the origin \(O\) of the parabola \(y^2 = 4x\) be the curve \(S\). Let \(P\) be any point on \(S\). Then the locus of the point, which internally divides \(OP\) in the ratio \(3:1\), is
If \[ X=\begin{bmatrix}x\\y\\z\end{bmatrix} \] is a solution of the system of equations $AX=B$, where \[ \text{adj }A= \begin{bmatrix} 4 & 2 & 2\\ -5 & 0 & 5\\ 1 & -2 & 3 \end{bmatrix} \quad \text{and} \quad B=\begin{bmatrix}4\\0\\2\end{bmatrix}, \] then $|x+y+z|$ is equal to
Let \( C_r \) denote the coefficient of \( x^r \) in the binomial expansion of \( (1+x)^n \), where \( n \in \mathbb{N} \) and \( 0 \le r \le n \). If \[ P_n = C_0 - C_1 + \frac{2^2}{3} C_2 - \frac{2^3}{4} C_3 + \cdots + \frac{(-2)^n}{n+1} C_n, \] then the value of \[ \sum_{n=1}^{25} \frac{1}{2n} P_n \] equals
The number of elements in the relation \[ R=\{(x,y): 4x^2+y^2<52,\; x,y\in\mathbb{Z}\} \] is
Let \[ f(x) = [x]^2 - [x+3] - 3,\quad x \in \mathbb{R}, \] where \([\,\cdot\,]\) denotes the greatest integer function. Then
Let \[ S = \{ z \in \mathbb{C} : 4z^2 + \bar{z} = 0 \}. \] Then \[ \sum_{z \in S} |z|^2 \] is equal to
Let \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \vec{b} = \lambda \hat{j} + 2\hat{k} \), where \( \lambda \in \mathbb{Z} \), be two vectors. Let \( \vec{c} = \vec{a} \times \vec{b} \) and \( \vec{d} \) be a vector of magnitude \(2\) in the \(yz\)-plane. If \( |\vec{c}| = \sqrt{53} \), then the maximum possible value of \( (\vec{c}\cdot\vec{d})^2 \) is equal to
If \[ \lim_{x\to 0} \frac{e^{(a-1)x}+2\cos bx+(c-2)e^{-x}} {x\cos x-\log_e(1+x)} =2, \] then \(a^2+b^2+c^2\) is equal to
Let \(f\) and \(g\) be functions satisfying \[ f(x+y) = f(x)f(y), \quad f(1) = 7 \] \[ g(x+y) = g(xy), \quad g(1) = 1, \] for all \(x,y \in \mathbb{N}\). If \[ \sum_{x=1}^{n} \left(\frac{f(x)}{g(x)}\right) = 19607, \] then \(n\) is equal to
Let \( L \) be the line \[ \frac{x+1}{2} = \frac{y+1}{3} = \frac{z+3}{6} \] and let \( S \) be the set of all points \( (a,b,c) \) on \( L \), whose distance from the line \[ \frac{x+1}{2} = \frac{y+1}{3} = \frac{z-9}{0} \] along the line \( L \) is \( 7 \). Then \[ \sum_{(a,b,c)\in S} (a+b+c) \] is equal to
Let \(S\) and \(S'\) be the foci of the ellipse \[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \] and \(P(\alpha,\beta)\) be a point on the ellipse in the first quadrant. If \[ (SP)^2 + (S'P)^2 - SP \cdot S'P = 37, \] then \(\alpha^2 + \beta^2\) is equal to
Suppose \(a, b, c\) are in A.P. and \(a^2, 2b^2, c^2\) are in G.P. If \(a<b<c\) and \(a+b+c = 1\), then \(9(a^2+b^2+c^2)\) is equal to ____________.
Let \(S\) be the set of the first 11 natural numbers. Then the number of elements in \[ A = \{ B \subseteq S : n(B) \ge 2 \text{ and the product of all elements of } B \text{ is even} \} \] is ____________.
Let \( \cos(\alpha + \beta) = -\dfrac{1}{10} \) and \( \sin(\alpha - \beta) = \dfrac{3}{8} \), where \( 0<\alpha<\dfrac{\pi}{3} \) and \( 0<\beta<\dfrac{\pi}{4} \).
If \[ \tan 2\alpha = \frac{3(1 - r\sqrt{5})}{\sqrt{11}(s + \sqrt{5})}, \quad r, s \in \mathbb{N}, \] then the value of \( r + s \) is ______________.
Let \([\,]\) be the greatest integer function. If \[ \alpha=\int_{0}^{64}\left(x^{1/3}-[x^{1/3}]\right)\,dx, \] then \[ \frac{1}{\pi}\int_{0}^{\alpha\pi}\frac{\sin^2\theta}{\sin^6\theta+\cos^6\theta}\,d\theta \] is equal to ____________.

Let a vector \[ \vec a=\sqrt{2}\,\hat i-\hat j+\lambda \hat k,\ \lambda>0, \] make an obtuse angle with the vector \[ \vec b=-\lambda^2\hat i+4\sqrt{2}\hat j+4\sqrt{2}\hat k \] and an angle \( \theta \), \(\frac{\pi}{6}<\theta<\frac{\pi}{2}\), with the positive \(z\)-axis. If the set of all possible values of \( \lambda \) is \((\alpha,\beta)-\{\gamma\}\), then\(\alpha+\beta+\gamma\) is equal to ____________. 

Which of the following are true for a single slit diffraction? A. Width of central maxima increases with increase in wavelength keeping slit width constant.
B. Width of central maxima increases with decrease in wavelength keeping slit width constant.
C. Width of central maxima increases with decrease in slit width at constant wavelength.
D. Width of central maxima increases with increase in slit width at constant wavelength.
E. Brightness of central maxima increases for decrease in wavelength at constant slit width.

200 cc of $x \times 10^{-3}$ M potassium dichromate is required to oxidise 750 cc of 0.6 M Mohr's salt solution in acidic medium. Here x = ______ .

Total number of unpaired electrons present in the central metal atoms/ions of \([Ni(CO)_4]\), \([NiCl_4]^{2-}\), \([PtCl_2(NH_3)_2]\), \([Ni(CN)_4]^{2-}\) and \([Pt(CN)_4]^{2-}\) is

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: