Which of the following functions f(x) are differentiable at \(x = 0\)?
(A) \(|x|\)
(B) \(| x - 1|\)
(C) [x],where [t] denotes the greatest integer ≤ t
(D) \(| x + 1|\)
(E) \(x^2\)
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The maximum value of \(z=4 x+3 y\) subject to constraint \(x+y \leq 6, x, y \geq 0\) is:

Which of the following inequalities holds true?
(A) \(\sqrt{5}+\sqrt{3}>\sqrt{6}+\sqrt{2}\).
(B) If \(a>b\) and \(c<0\), then \(\frac{a}{c}<\frac{b}{c}\).
(C) \(\frac{1}{x^2}>\frac{1}{x}>1\), if \(0<x<1\).
(D) If \(a\) and \(b\) are positive integers and \(\frac{a-b}{6.25}=\frac{4}{2.5}\), then \(b>a\).
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The probability distribution of the random variable \(X\) is given by

X	0	1	2	3
P(X)	0.2	k	2k	2k

The variance of the random variable \(X\) is given by:

The area (in sq. units) of the region enclosed by the curve \(9 x^2+4 y^2=36\) is

The equation of trend line for five years (2004-2008) for the profit of a company is \(y_t=150+8 x\)The predicted profit for the year 2011 is:

Linear inequalities corresponding to the shaded feasible region \(O A B C O\) in the given figure are

The equation of normal at the point (1, 1) on the curve \(2y + x^3 = 3\) is:

A charge q moves with a velocity of \(2 m s^{-1}\) along x-axis in a uniform magnetic field \(\vec{B} = \hat{i} + 2\hat{j} + 3\hat{k}\), then charge will experience a force

In a feedback amplifier, if the feedback voltage is in opposite phase, the gain is less than 1. Then it will:

Match List I with List II.

List-I	List-II
A. Vitamin C	I. Rickets
B. Vitamin A	II. Beri Beri
C. Vitamin D	III. Scurvy
D. Vitamin \(B _{ 1 }\)	IV. Night blindness

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One face of a prism of refractive index 1.5 and angle \(75^{\circ}\) is covered with a liquid of refractive index \(\frac{3 \sqrt{2}}{4}\).
What should be the angle of incidence of light on the clear face of prism for which light is just totally reflected at the liquid covered face?

The following data shows the percentage of rural and urban Indian households who have a high speed internet connection.

Year(x)	Rural Household \((y \%)\)	Urban Household\((y \%)\)
2020	2	6
2021	3	7
2022	3	9
2023	4	15
2024	5	20

Based on the above data, which of the following statements are correct?
(A) A straight line trend by the method of least squares for rural Indians \(y_t=3.4+0.7(x-2022)\)
(B) A straight line trend by the method of least squares for urban Indians is \(y_t=11.4+4.3(x-2022)\)
(C) The forecast for the year 2025 for urban group using the trend equation is \(22.2 \%\)
(D) The forecast for the year 2025 for rural group using the trend equation is \(5.5 \%\)
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