List of top Questions asked in CBSE CLASS XII

Let \(\vec a\) anb \(\vec b\) be two unit vectors and θ is the angle between them.Then,\(\vec a+\vec b\) is a unit vector if

Show that each of the given three vectors is a unit vector:
\(\frac{1}{7}(2\hat{i}+3\hat{j}+6\hat{k}),\frac{1}{7}(3\hat{i}-6\hat{j}+2\hat{k}),\frac{1}{7}(6\hat{i}+2\hat{j}-3\hat{k})\)
Also show that they are mutually perpendicular to each other.

Find the values of x and y so that the vectors \(2 \hat {i} +3\hat{j}\) and \(x \hat {i} +y\hat{j}\) are equal.

If θ is the angle between two vectors \(\vec a\) and \(\vec b\), then \(\vec a.\vec b\)≥0 only when

Find the cartesian equations of the following planes:

\(\overrightarrow r.(\hat i+\hat j-\hat k)=2\) (b) \(\overrightarrow r.(2\hat i+3\hat j-4\hat k)=1\)

(c) \(\overrightarrow r.[(s-2t)\hat i+(3-t)\hat j+(2s+t)\hat k)=15\)

Prove that (\(\vec a+\vec b\)).(\(\vec a+\vec b\))=|\(\vec a\)|²+|\(\vec b\)|², if and only if \(\vec a\),\(\vec b\) are perpendicular, given a≠0,b≠0.

Find the projection of the vector \(\hat{i}-\hat{j}\) on the vector \(\hat{i}+\hat{j}\).

Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector \(3\hat i+5\hat j-6\hat k.\)

Find the angle between the vectors \(\hat{i}-2\hat{j}+3\hat{k}\) and \(3\hat{i}-2\hat{j}+\hat{k}\).

Find the general solution of the differential equation:
\(\frac {dy}{dx}+\sqrt {\frac {1-y^2}{1-x^2}}=0\)

In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
(a) z=2 (b) x+y+z=1
(c)2x+3y-z=5 (d) 5y+8=0

Form the differential equation representing the family of curves given by:
\((x-α)^2+2y^2=α^2\) where a is an arbitrary constant.

If \(α→s\) a nonzero vector of magnitude \('α'\) and \(λ\) a nonzero scalar,then \(λ\vec{α}\) is unit vector if

Show that the lines\(\frac{x-5}{7}=\frac{y+2}{-5}=\frac{z}{1}\) and \(\frac{x}{1}=\frac{y}{2}=\frac{z}{3}\) are perpendicular to each other.

Find the values of P so the line\(\frac{1-x}{3}=\frac{7y-14}{2p}=\frac{z-3}{2}\) and \(\frac{7-7x}{3p}=\frac{y-5}{1}=\frac{6-z}{5}\) are at right angles.

Prove tan^-1\((\frac{√1+x-√1-x}{√1+x+√1-x}=\frac{π}{4}-\frac{1}{2}cos^-1x,-\frac{1}{√2}≤x≤1.\)[Hint: putx = cos 2θ]

Show that the vectors \(2\hat{i}-\hat{j}+\hat{k},\hat{i}-3\hat{j}-5\hat{k}\) and \(3\hat{i}-4\hat{j}-4\hat{k}\) from the vertices of a right-angled triangle.

Find the angle between two vectors \(\vec{a}\) and \(\vec{b}\) with magnitudes \(\sqrt3\) and \(2\), respectively having \(\vec{a}.\vec{b}=\sqrt{6}.\)

The scalar product of the vector \(\hat i+\hat j+\hat k \) with a unit vector along the sum of vectors \(2\hat i+4\hat j-5 \hat k\) and \(\lambda \hat i+2\hat j+3\hat k\) is equal to one. Find the value of λ.

Find the vector and the cartesian equations of the line that passes through the points(3,-2,-5),(3,-2,6).

In triangle ABC which of the following is not true.

Show that the points \(A, B,\) and \(C\) with position vectors, \(\vec{a}=3\hat{i}-4\hat{j}-4\hat{k}, \vec{b}=2\hat{i}-\hat{j}+\hat{k}\) and \(\vec{c}=\hat{i}-3\hat{j}-5\hat{k}\), respectively from the vertices of a right angled triangle.

Find the coordinates of the point where the line through (3,-4,-5) and (2,-3,1)crosses the plane 2x+y+z=7.

Find the coordinates of the point where the line through (5,1,6) and (3,4,1) crosses the ZX-plane.

Find the coordinates of the point where the line through (5,1,6)and (3,4,1) crosses the YZ plane.