The maximum value of the determinant of the matrix \(\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+\sin x & 1 \\ 1+\cos x & 1 & 1\end{array}\right]\)
(where \(x\) is real) is :

For the differential equation \(x \frac{d y}{d x}+3 y=x^2 \log _e x\) which of the following statements are TRUE?
(A) Product of order and degree is 1
(B) Integrating factor is \(x^3\)
(C) Integrating factor is \(3 x\)
(D) General solution is \(y=\frac{x^3}{36}\left(6 \log _e|x|-1\right)+C x^{-3}\), C is an arbitrary constant
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The set of value of \(x\) for which the angle between the \(\vec{a}=2 x^2 \hat{i}+4 x \hat{j}+\hat{k}\) and \(\vec{b}=7 \hat{i}-2 \hat{j}+x \hat{k}\) is obtuse is :

A man rows 24 km upstream in 6 hours and 36 km downstream in 6 hours each time, then the speed of the boat in the still water is

Match List-I with List-II
An urn contains 4 white and 3 red balls. In a random draw of three balls, the probability of

List-I	List-II
(A) No red ball is	(I) \(\frac{12}{35}\)
(B) Only 1 red ball is	(II) \(\frac{1}{35}\)
(C) Exactly 2 red balls is	(III) \(\frac{4}{35}\)
(D) No white ball is	(IV) \(\frac{18}{35}\)

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If \(A\) and \(B\) are independent events and \(P(A)=\frac{1}{2}, P(B)=\frac{1}{3}\) then
Match List-I with List-II

List-I	List-II
(A) \(P(A \cap B)\)	(I) \(\frac{1}{2}\)
(B) \(P(\bar{A}) P(B)+P(A) P(\bar{B})\)	(II) \(\frac{1}{3}\)
(C) \(P(A \mid B)+P(B \mid A)\)	(III) \(\frac{1}{6}\)
(D) \(P(\overline{A \cap B})\)	(IV) \(\frac{5}{6}\)

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Which of the following statements are NOT correct about Standard Normal Distribution?
(A) The probability curve of the Standard Normal Distribution is a bell-shaped curve.
(B) The Standard Normal variate ( Z ) score describes the position of each data point in terms of its distance from the mean, when measured in standard deviation units.
(C) The Z -score is negative if the data point lies above the mean, and positive if it lies below the mean.
(D) There is a \(95.45 \%\) probability of randomly selecting a score between \(\mu-\sigma\) and \(\mu+\sigma\), when \(\sigma\) is standard deviation and \(\mu\) is mean.
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A dietician wishes to mix two types of food in such a way that the vitamin contents of the mixture contains atleast 8 units of Vitamin A and 10 units of Vitamin C. Food 'I' contains 2 units/kg of Vitamin A and 1 unit/kg of Vitamin C while food 'II' contains 1 unit/kg of Vitamin A and 2 units/kg of Vitamin C. It costs ₹5 per kg to purchase food I and ₹7 per kg to purchase food II. If 'x' kg of food I is mixed with 'y' kg of food II, then the constraint pertaining to vitamin C contents will be:

Match List-I with List-II
Let \(f: A \rightarrow B\) be a function given by \(f(x)=x^2\)

List - I (Domain and Co-domain)	List - II (Kind)
(A) \(A= R\) and \(B= R\)	(I) \(f\) is both one-one and onto
(B) \(A= R\) and \(B=[0, \infty]\)	(II) \(f\) is one-one but not onto
(C) \(A=B=[0, \infty]\)	(III) \(f\) is not one-one but onto
(D) \(A=[0, \infty], B= R\)	(IV) \(f\) is neither one-one nor ont

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The following experiment suggests that light propagates as waves

Observe the figure and answer the following:

Match List I with List II

LIST I	LIST II
A. Disrupting selection	I. A
B. Stabilising selection	II. D
C. Favouring medium sized individuals	III. C
D. Directional selection	IV. B

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The magnetic moment in transition metals is calculated using the formula:

The distance between the line \(\vec{r}=2 \hat{i}+2 \hat{j}-3 \hat{k}+\lambda(\hat{i}+\hat{j}+4 \hat{k})\) and the plane \(\vec{r} \cdot(\hat{i}-5 \hat{j}+\hat{k})-4=0\) is: