Two events A and B are dependent if :

The maximum value of \(z=5 x+7 y\) subjected to constraints \(x+y \leq 5, x \geq 0, y \geq 0\) is :

Two persons A and B throw a die alternately till one of them gets a 'three' and wins the game. The probability of A's winning if A starts first is :

Let E and F are events associated with an experiment. If \(P(E)=0.4, P(F)=0.8\) and \(P(F \mid E)=0.6\) then \(P(E \mid F)\) is:

Choose the correct answer from the options given below:
A. Equation of the line passing through the point (1, 2, 3) and parallel to the vector \(3 \hat{i}+2 \hat{j}-2 \hat{k}\) is \(\frac{x-1}{3}=\frac{y-2}{2}=\frac{y-3}{-2}\)
B. Equation of line passing through (1, 2, 3) and parallel to the line given by \(\frac{x+3}{3}=\frac{4-y}{5}=\frac{z+8}{6}\) is \(\frac{x-1}{3}=\frac{y-2}{5}=\frac{z+3}{6}\).
C. Equation of line passing through the origin and (5, -2,3) is \(\frac{x}{5}=\frac{y}{-2}=\frac{z}{3}\).
D. Equation of plane passing through the point (1, 2, 3) and perpendicular to the line with direction ratio's 2, 3, -1 is 2(x - 1) + 3(y - 2) - (z - 3) = 0.
E. Equation of plane with intercepts 2, 3 and 4 on x, y and z-axis respectively is 2x + 3y + 4z = 1

Let \(\vec{a}=\hat{i}+\hat{j}, \vec{b}=\hat{i}-\hat{j}\) and \(\vec{c}=\hat{i}+\hat{j}+\hat{k}\). If \(\hat{m}\) is a unit vector perpendicular to both \(\vec{a}\) and \(\vec{b}\), then \(|\vec{c} \cdot \hat{m}|\) is equal to :

Match List I with List II and choose the correct answer from the options given below:

List - I	List - II
(A) \(\sin ^{-1} x+\cos ^{-1} x, x \in[-1,1]\)	(I) \(-\frac{\pi}{2}\)
(B) \(\tan ^{-1} \sqrt{3}-\cot ^{-1}(-\sqrt{3})\)	(II) \(-\frac{\pi}{6}\)
(C) \(\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)\)	(III) \(\frac{\pi}{2}\)
(D) \(\sin ^{-1}\left(-\frac{1}{2}\right)\)	(IV) \(\frac{\pi}{6}\)

If the difference of two unit vectors is again a unit vector, then the angle between them is ______________

Consider the curves, \(x^2+y^2=1\) and \((x-1)^2+y^2=1\).
Points of intersection of two curves are :

Read the passage carefully and answer the Questions
Aldehydes, carboxylic acids and ketones are widespread in the animal and plant kingdoms.
They play a pivotal role in various biochemical processes of life. Their presence in nature adds fragrance and flavor. These compounds are widely used in food products, pharmaceuticals, paints, resins and other important product industries. These compounds are prepared by various laboratory methods, which mainly include oxidation, formylation, acylation and reduction. Due to the polar nature of the carbonyl group in aldehydes and ketones, they can exhibit different reactions like nucleophilic addition. They do exhibit redox and various condensation reactions which lead to the formation of various important compounds. On the other hand, the carboxylic acids are mainly prepared by oxidation and hydrolysis of different compounds. The carboxylic acid consists of a carbonyl group and the hydroxy group (attached to the carbonyl carbon atom). This makes it possible for the carboxylic acid to participate in various chemical reactions which involve cleavage of the C-OH bond and the O_H bond along with the reactions involving the complete -COOH group.
The products formed when 2,2-Dimethylpropan-1-al is heated with concentrated KOH are :

\(\text{Ni}^{2+}\) in \([\text{NiCl}_2(\text{H}_2\text{O})_4]\) and \(\text{Fe}^{3+}\) in \([\text{Fe}(\text{CN})_6]^{3-}\) are called:

Which one of the given hormones is secreted from hypothalamus?

The refractive index of a glass prism is minimum for