List of top Mathematics Questions

Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls.Find the probability that:
(i)both balls are red.
(ii)first ball is black and second is red.
(iii)one of them is black and other is red.

Find the area of the parallelogram whose adjacent sides are determined by the vector \(\vec{a}=\hat{i}-\hat{j}+3\hat{k}\) and \(\vec{b}=2\hat{i}-7\hat{j}+\hat{k}.\)

A die is tossed thrice. Find the probability of getting an odd number at least once.
Maximise Z=5x+3y
Subject to 3x+5y≤15,5x+2y≤10,x≥0,y≥0.

Find the vector equation of the line passing through the point (1, 2, -4)and perpendicular to the two lines: \(\frac {x-8}{3}=\frac {y+19}{-16}=\frac {z-10}{7}\) and \(\frac {x-15}{3}=\frac {y-29}{8} =\frac {z-5}{-5}\)

Maximise Z=3x+2y
Subject to x+2y≤10,3x+y≤15,xy≥0.
Given two independent events \(A\) and \(B\) such that \(P(A)=0.3,P(B)=0.6.\) Find:
(A)\(P(A\, and\, B)\)
(B)\(P(A\, and\, not\, B)\)
(C)\(P(A or B)\)
(D)\(P(neither\, A\, nor\, B)\)

If l1,m1,n1 and l2,m2,n2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m1n2-m2n1,n1l2-n2l1,l1m2-l2m1.

Minimise Z=3x+5y
such that x+3y≥3,x+y≥2,x,y≥0

Prove that if a plane has the intercepts a, b, c and is at a distance of P units from the origin, then \(\frac {1}{a^2}+\frac {1}{b^2}+\frac {1}{c^2} =\frac {1}{p^2}\).

Events \(A\) and \(B\) are such that \(P(A)=\frac{1}{2},P(B)=\frac{7}{12}\) and (not \(A\) or not \(B\))\(=\frac{1}{4}\).State whether \(A\) and \(B\) are independent.
Minimise Z=-3x+4y
Subject to x+2y≤8,3x+2y≤12,x≥0,y≥0.

Maximise Z=3x+4y
Subject to the constrains:x+y≤4,x≥0,y≥0.

Show that the line joining the origin to the point(2,1,1)is perpendicular to the line determined by the points(3,5,-1),(4,3,-1).

Distance between the two planes: 2x+3y+4z = 4 and 4x+6y+8z = 12 is

\(The\  planes: 2x-y+4z=5 \ and \ 5x-2.5y+10z=6\  are\)

Find the angle between the planes whose vector equations are 

\(\overrightarrow r.(2\hat i+2\hat j-3\hat k)=5\) and \(\overrightarrow r.(3\hat i-3\hat j+5\hat k)=3\)

If either \(\vec{a}=0\) or \(\vec{b}=0\),then \(\vec{a}\times\vec{b}=0\). Is the converse true? Justify your answer with an example.

Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2.
Let A and B independent events, \(P(A)=0.3\) and \(P(B)=0.4\).Find:
(A)\(P(A\cap B)\)
(B)\(P(A\cup B)\)
(C)\(P(A|B)\)
(D)\(P(B|A)\)

Find the equation of the plane through the line of intersection of the planes
x+y+z=1 and 2x+3y+4z=5 which is perpendicular to the plane x-y+z= 0

Given that \(\vec{a}.\vec{b}=0\) and \(\vec{a}\times\vec{b}=0\).What can you conclude about the vectors \(\vec{a}\) and \(\vec{b}\) ?

Given that the events \(A\) and \(B\) are such that \(P(A)=\frac{1}{2},P(A∪B)=\frac{3}{5}\) and \(P(B)=ρ.\)
Find \(ρ\) if they are(i) mutually exclusive,(ii) independent.
Let E and F be events with \(P(E)=\frac{3}{5},P(F)=\frac{3}{10}\) and \(P(E∩F)=\frac{1}{5}\) .Are E and F independent?

Show that
\((\vec{a}-\vec{b})\times(\vec{a}+\vec{b})=2(\vec{a}\times \vec{b})\)