List of practice Questions

Let 'a' be a positive real number. If a real valued function $ f(x) = \begin{cases}\frac{6^x - 3^x - 2^x + 1}{1 - \cos\left(\frac{x}{a}\right)} & \text{if } x \neq 0 \\ \log 3 \log 4 & \text{if } x = 0 \end{cases} $ is continuous at $ x=0 $, then $ a = $

If [x] is the greatest integer function then $ \lim_{x \to 3^-} \frac{(3-|x| + \sin|3-x|) \cos(9-3x)}{|3-x|[3x-9]} = $

One of the foci of an ellipse is $ (2,-3) $ and its corresponding directrix is $ 2x+y=5 $. If the eccentricity of the ellipse is $ \frac{\sqrt{5}}{3} $ then the coordinates of the other focus are

A line L perpendicular to the line $ 5x-12y+6=0 $ makes positive intercept on the Y-axis. If the distance from the origin to the line L is 2 units and the angle made by the perpendicular drawn from the origin to the line L with positive X-axis is $ \theta $, then $ \tan \theta + \cot \theta = $

If $ X \sim B(7, p) $ is a binomial variate and $ P(X=3) = P(X=5) $ then $ p = $

Let $ z=x+iy $ and $ P(x,y) $ be a point on the Argand plane. If $ z $ satisfies the condition $ \text{Arg}\left(\frac{z-3i}{z+2i}\right) = \frac{\pi}{4} $, then the locus of P is

Water falls from a height of 200 m into a pool. Calculate the rise in temperature of the water assuming no heat dissipation from the water in the pool. (Take $ g = 10 \, \text{m/s}^2 $, specific heat of water = 4200 J/(kg K))

An ideal gas has undergone through the cyclic process as shown in the figure. Work done by the gas in the entire cycle is _____ $ \times 10^{-1} $ J. (Take $ \pi = 3.14 $)

During the melting of a slab of ice at 273 K at atmospheric pressure:

If $ f(x)=\tan \left(\frac{\pi}{\sqrt{x+1}+4}\right) $ is a real valued function then the range of $ f $ is

The focus of the parabola $ y^2 = 4x + 16 $ is the center of the circle $ C $ with radius 5. If the values of $ \lambda $, for which $ C $ passes through the point of intersection of the lines $ 3x - y = 0 $ and $ x + \lambda y = 4 $, are $ \lambda_1 $ and $ \lambda_2 $, $ \lambda_1 <\lambda_2 $, then $ 12\lambda_1 + 29\lambda_2 $ is equal to:

The radius of the smallest circle which touches the parabolas $ y = x^2 + 2 $ and $ x = y^2 + 2 $ is

The most stable carbocation from the following is:

If $ A = \begin{bmatrix} 1 & -3 & -5 \\ -2 & 4 & -6 \\ 7 & -11 & 13 \end{bmatrix} $, then $ \sqrt{|\text{Adj } A|} = $

Let $ A = \{1, 2, 3, 4\} $ and $ B = \{1, 4, 9, 16\} $. Then the number of many-one functions $ f: A \to B $ such that $ 1 \in f(A) $ is equal to:

If $ \begin{vmatrix} 9 & 25 & 16 \\ 16 & 36 & 25 \\ 25 & 49 & 36 \end{vmatrix} = K $, then $ K, K+1 $ are the roots of the equation

If the augmented matrix corresponding to the system of equations $ x+y-z=1 $, $ 2x+4y-z=0 $ and $ 3x+4y+5z=18 $ is transformed to $ \begin{bmatrix} 1 & a & 0 & -1 \\ 0 & 2 & 1 & b \\ 0 & 0 & c & 32 \end{bmatrix} $, then

If the equation of the plane passing through the points (2,1,2), (1,2,1) and perpendicular to the plane $ 2x-y+2z=1 $ is $ ax+by+cz+d=0 $ then $ \frac{a+b}{c+d} = $

The mean deviation from the median for the following data is

$ x_i $	2	9	8	3	5	7
$ f_i $	5	3	1	6	6	1

The shortest distance between the lines $ \bar{r} = (3\bar{i} - 5\bar{j} + 2\bar{k}) + t(4\bar{i} + 3\bar{j} - \bar{k}) $ and $ \bar{r} = (\bar{i} + 2\bar{j} - 4\bar{k}) + s(6\bar{i} + 3\bar{j} - 2\bar{k}) $ is

If $ \bar{a} = \bar{i} - 2\bar{j} - 2\bar{k} $ and $ \bar{b} = 2\bar{i} + \bar{j} + 2\bar{k} $ are two vectors then $ (\bar{a} + 2\bar{b}) \times (3\bar{a} - \bar{b}) = $

The major products P and Q from the following reactions are
Reaction 1: $ \text{P} \leftarrow[\text{(ii) H}_2\text{O}]{\text{(i) LiAlH}_4} \text{C}_6\text{H}_5\text{CONH}_2 $ Reaction 2: $ \text{C}_6\text{H}_5\text{CONH}_2 \xrightarrow{\text{Br}_2 / \text{NaOH}} \text{Q} $

Consider the following reaction sequence. $ \text{CH}_3\text{CHO} \xrightarrow[\text{(ii) } \text{H}_2\text{O} / \text{H}^+]{\text{(i) } \text{CH}_3\text{MgBr}} \text{(A)} \xrightarrow{\text{H}_2\text{SO}_4, \Delta} \text{(B)} \xrightarrow[\text{(ii) } \text{H}_2\text{O}_2 / \text{OH}^-]{\text{(i) } \text{B}_2\text{H}_6} \text{(C)} $ (A) and (C) are

The structure of which artificial sweetener contains aspartic acid and phenylalanine parts?

Which of the following hormones is an example of polypeptide?