List of top Questions asked in MET- 2023

If the vectors \( \vec{p} = (a+1)\hat{i} + a\hat{j} + a\hat{k},\ \vec{q} = a\hat{i} + (a+1)\hat{j} + a\hat{k} \) and \( \vec{r} = a\hat{i} + a\hat{j} + (a+1)\hat{k} \) are coplanar, then the value of \(a\) is _ _ _ _.
If \( A = \begin{bmatrix} 2 & -1 \\ -7 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} 4 & 1 \\ 7 & 2 \end{bmatrix} \), then \( B^T A^T \) is:
An unbiased coin is tossed \(n\) times. If the probability of getting 5 heads is equal to the probability of getting 6 heads, then the probability of getting 3 heads is:
If the foot of the perpendicular drawn from the point \((2,0,1)\) on a line passing through \((\alpha,5,1)\) is \(\left(\frac{7}{3}, \frac{5}{3}, \frac{11}{4}\right)\), then \(\alpha\) is equal to _ _ _ _ _ _
If the lines \( \frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{4} \) and \( \frac{x-3}{1} = \frac{y-k}{2} = \frac{z}{1} \) intersect, then \(k\) is equal to:
The area bounded by the curves \( y = \tan x,\ -\frac{\pi}{3} \le x \le \frac{\pi}{3} \), \( y = \cot x,\ \frac{\pi}{6} \le x \le \frac{\pi}{2} \) and the X-axis is:
The area of the region bounded by the curves \( y = |5 - x| \), \( x = 1 \), \( x = 6 \) and the X-axis is:
If \( \int f(x)\,dx = g(x) \), then \( \int \cos x \, f(\sin x)\,dx \) is equal to:
Set of values of \(x\) lying in \([0,2\pi]\) satisfying the inequality \( |\sin x|>2\sin^2 x \) contains:
Let \( f(x) = \frac{1}{\sqrt{1 + x^2}} \), then:
Let \(A = \{1,2,3,4,5,6,7\}\) and \(B = \{3,6,7,9\}\). Then, the number of elements in the set \(\{ C \subseteq A : C \cap B \neq \phi \}\) is:
If \( \sum_{i=1}^{9} (x_i - 5) = 9 \) and \( \sum_{i=1}^{9} (x_i - 5)^2 = 45 \), then the standard deviation of the 9 items \(x_1, x_2, \ldots, x_9\) is:
The value of \( \lim_{x \to 0} \left(\frac{e^x + 2^x + 4^x}{3}\right)^{\frac{2}{x}} \) is:
If \( f : \mathbb{R} \to \mathbb{R} \) is a differentiable function and \( f(3) = 6 \), then \[ \lim_{x \to 3} \frac{\displaystyle \int_{6}^{f(x)} \frac{2t \, dt}{t - 2}}{x - 3} \text{ is equal to} \]
The differential equation of all circles which pass through the origin and whose centres lie on Y-axis is:
Last two digits in \(9^{50}\) are _ _ _ _ _.
If the coefficients of \(x^3\) and \(x^4\) in the expansion of \((1 + ax + bx^2)(1 - 2x)^{18}\) are both zero, then \((a,b)\) is equal to
If \(a^2, b^2, c^2\) are in A.P., then \(b+c, c+a, a+b\) are in:
The number of words (with or without meaning) that can be formed from all the letters of the word "LETTER" in which vowels never come together is _ _ _ _ _ .
If complex number \(z\) lies in the interior or on the boundary of circle of radius 3 units, then maximum and minimum values of \(|z + 1|\) are:
In which form styrene exists at room temperature?
Which of the following reaction does not form benzaldehyde as product?
Find the product Z in the series of reactions. \[ \text{Phenol} \xrightarrow{\text{Zn dust}} X \xrightarrow{\text{CH}_3\text{Cl / anhyd. AlCl}_3} Y \xrightarrow{\text{alkaline KMnO}_4} Z \]
Match the complexes in Column-I with their hybridisation in Column-II.

Column-I     Column-II
A. \([ \mathrm{Ni(CO)_4} ]\)      1. \(dsp^2\)
B. \([ \mathrm{Ni(CN)_4} ]^{2-}\)      2. \(sp^3\)
C. \([ \mathrm{Ni(NH_3)_6} ]^{2+}\)      3. \(d^2sp^3\)
D. \([ \mathrm{Fe(CN)_6} ]^{3-}\)      4. \(sp^3d^2\)
Among \( \mathrm{Cu^{2+}}, \mathrm{Fe^{3+}}, \mathrm{Ti^{3+}} \) and \( \mathrm{Zn^{2+}} \) ions, the number of ions that impart colour is _ _ _ _ _ _ _