List of top Questions asked in MET- 2024

Let \( f:\mathbb{N} \to \mathbb{N} \) be defined as \[ f(n)= \begin{cases} \frac{n+1}{2}, & \text{if } n \text{ is odd} \\ \frac{n}{2}, & \text{if } n \text{ is even} \end{cases} \] Then \( f \) is:
Given \(\frac{dy}{dx} + 2y\tan x = \sin x\), \(y=0\) at \(x=\frac{\pi}{3}\). If maximum value of \(y\) is \(1/k\), find \(k\).
Given vectors \(\vec{a}, \vec{b}, \vec{c}\) are non-collinear and \((\vec{a}+\vec{b})\) is collinear with \((\vec{b}+\vec{c})\) which is collinear with \(\vec{a}\), and \(|\vec{a}|=|\vec{b}|=|\vec{c}|=\sqrt{2}\), find \(|\vec{a}+\vec{b}+\vec{c}|\).
Let \( D = \begin{vmatrix} n & n^2 & n^3 \\ n^2 & n^3 & n^5 \\ 1 & 2 & 3 \end{vmatrix} \). Then \( \lim_{n \to \infty} \frac{M_{11} + C_{33}}{(M_{13})^2} \) is:
Let \( f(x) \) be a polynomial such that \( f(x) + f(1/x) = f(x)f(1/x) \), \( x > 0 \). If \( \int f(x)\,dx = g(x) + c \) and \( g(1) = \frac{4}{3} \), \( f(3) = 10 \), then \( g(3) \) is:
If \(x = \sin(2\tan^{-1}2)\), \(y = \sin\left(\frac{1}{2}\tan^{-1}\frac{4}{3}\right)\), then:
A real differentiable function \(f\) satisfies \(f(x)+f(y)+2xy=f(x+y)\). Given \(f''(0)=0\), then \[ \int_0^{\pi/2} f(\sin x)\,dx = \]
In \(\triangle ABC\), \(\sin A, \sin B, \sin C\) are in A.P. and \(C>90^\circ\). Then \(\cos A\) is:
If in a \(\triangle ABC\), \(\sin^2 A + \sin^2 B + \sin^2 C = 2\), then the triangle is always:
Let \( f(x) = \left[\frac{\sin x}{x}\right] + \left[\frac{2\sin x}{x}\right] + \cdots + \left[\frac{10\sin x}{x}\right] \) (where \([\,]\) is the greatest integer function). Find \( \lim_{x \to 0} f(x)\).
If two tangents from point \((h,k)\) to parabola \(y^2 = 64x\) have slopes such that one is 8 times the other, then value of \( \frac{k^2}{2h} \) is:
If the angle between the pair of straight lines formed by joining the points of intersection of \(x^2 + y^2 = 4\) and \(y = 3x + c\) to the origin is a right angle, then \(c^2\) is:
The equation of mirror image of the circle \(x^2 + y^2 - 6x - 10y + 33 = 0\) about the line \(y = x\) is:
If the number of terms in the expansion of \((x\sqrt{180} + \sqrt[3]{432})^{200}\) having integral coefficients is \(n\), then the value of \([n/6]\) is:
If the expression \(x + \frac{1}{x^2},\; x>0\) attains minimum value at \(x=\alpha\), then \(\alpha^3\) is:
If the coefficient of \(x^m\) in the expansion of \(\left(\sqrt{2x} + \sqrt[3]{\frac{3}{x^2}}\right)^9\) is equal to \(k\), then \(k\) is:
If \(N\) denotes number of 8-digit numbers that contain exactly four nines, then unit digit of \(N\) is:
Total number of even divisors of \(2079000\) which are divisible by \(15\) are:
If \( \frac{a}{b} = \frac{1}{3} \) and \( \frac{b}{c} = \frac{3}{4} \), then the value of \( \frac{a+2b}{b+2c} \) is:
The solution of the equation \( \log\left(\log_4(\sqrt{x+4} + \sqrt{x})\right) = 0 \) is:
In the following sequence of reactions, \[ CH_3Br \xrightarrow{KCN} X \xrightarrow{H_3O^+} Y \xrightarrow{LiAlH_4 / \text{ether}} Z \] The final product \(Z\) is:
The major product \(Y\) in the following reaction is:
Consider the following amino acids:
•[(i)] Lysine
•[(ii)] Glutamine
•[(iii)] Arginine
•[(iv)] Leucine
•[(v)] Serine
•[(vi)] Proline
•[(vii)] Valine Which of the given amino acids are basic in nature?
The oxidation state of Cr in \([Cr(H_2O)_6]Cl_3\), \([Cr(C_6H_6)_2]\), \(K_2[Cr(CN)_2(O)_2(O_2)(NH_3)]\) respectively are:
For a first order reaction, time required for 99% completion is \(x\) times the time required for 90% completion. Find \(x\).