List of top Mathematics Questions

The cartesian equation of a line is \(\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}\). Write its vector form.

A die marked 1,2,3 in red and 4,5,6 in green is tossed.Let A be the event ‘number is even’ and B be the event ‘number is red’. Are A and B independent?

Find the cartesian equation of the line that passes through the point(-2,4,-5)and parallel to the line given by \(\frac{x+3}{3}\)=\(\frac{y-4}{5}\)=\(\frac{z+8}{6}\)

A fair coin and an unbiased die are tossed.Let A be the event ‘head appears on the coin’ and B be the event ‘3 on the die’.Check whether A and B are independent events or not.

Find \( λ \) and \( μ\) if \((2\hat{i}+6\hat{j}+7\hat{k})\times(\hat{i}+λ\hat{j}+μ\hat{k})=\vec{0}.\)

Find the vector equation of the plane passing through the intersection of the planes

\(\overrightarrow r.(2\hat i+2\hat j-3\hat k)=7\),\(\overrightarrow r.(2\hat i+5\hat j+3\hat k)=9\) and through the point ( 2, 1, 3 ).

Find the equation of the line in vector and in cartesian form that passes through the point with position vector 2\(\hat i\)-\(\hat j\)+4\(\hat k\) and is in the direction \(\hat i\)+2\(\hat j\)-\(\hat k\).

A box of oranges is inspected by examining three randomly selected oranges drawn without replacement.If all the three oranges are good, the box is approved for sale otherwise it is rejected.Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.
Prove that the greatest integer function defined by f(x)=[x],0<x<3 is not differentiable at x=1 and x=2.
Show that \(|\vec{a}|\vec{b}+|\vec{b}|\vec{a}\)is perpendicular to \(|\vec{a}|\vec{b}-|\vec{b}|\vec{a}\) for any two nonzero vectors \(\vec{a}\) and \(\vec{b}\)

Find the equation of the plane through the intersection of the planes
3x-y+2z-4 =0 and x+y+z-2=0 and the point (2, 2, 1).

If \(\vec{a}=2\hat{i}+2\hat{j}+3\hat{k}\),\(b=-\hat{i}+2\hat{j}+\hat{k}\) and \(c=3\hat{i}+\hat{j}\) are such that \(\vec{a}+λ\vec{b}\) is perpendicular to \(\vec{c}\) ,then find the value of \(λ\).

Find the equation of the line that passes through the point(1,2,3)and is parallel to the vector 3\(\hat i\)+2\(\hat j\)-2\(\hat k\).

Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX plane.

Find the intercepts cut off by the plane 2x+y-z = 5

Show that the line through the points(4,7,8)(2,3,4)is parallel to the line through the points(-1,-2,1),(1,2,5).
Find \(|\vec{x}|\),if for a unit vector \(\vec{a},(\vec{x}-\vec{a}).(\vec{x}+\vec{a})=12\)

Find the equation of the planes that passes through three points.

(a) (1,1,-1),(6,4,-5),(-4,-2,-3)

(b) (1,1,0),(1,2,1),(-2,2,-1).

Find the magnitude of two vectors \(\vec{a}\) and \(\vec{b}\) ,having the same magnitude and such that the angle between them is \(60\degree\)and their scalar product is \(\frac{1}{2}\).
Evaluate the product\((3\vec{a}-5\vec{b}).(2\vec{a}+7\vec{b}).\)
Show that the lines through the points(1,-1,2)(3,4,-2)is perpendicular to the line through the points(0,3,2)and(3,5,6).

Show that the three lines with direction cosines
\(\frac{12}{13}\),\(-\frac{3}{13}\),\(-\frac{4}{13}\) ; \(\frac{4}{13}\),\(\frac{12}{13}\),\(\frac{3}{13}\);\(\frac{3}{13}\),\(-\frac{4}{13}\),\(\frac{12}{13}\) are mutually perpendicular.

Find the angle between the following pairs of lines: 

(i)\(\frac{x-2}{2}=\frac{y-1}{5}=\frac{z+3}{-3}\) and \(\frac{x+2}{-1}=\frac{y-4}{8}=\frac{z-5}{4}\) 

(ii \(\frac{x}{2}=\frac{y}{2}=\frac{z}{1}\) and \(\frac{x-5}{4}=\frac{y-2}{1}=\frac{z-3}{8}\)

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin. 

(a) 2x+3y+4z-12=0 (b)  3y+4z-6=0

(c)x+y+z=1  (d) 5y+8=0

Find the vector and cartesian equation of the planes 

(a) that passes through the point (1,0,-2)and the normal to the plane is \(\hat i+\hat j-\hat k\). 

(b) that passes through the point(1,4,6)and the normal vector to the plane is \(\hat i-2\hat j+\hat k\).