Questions

$X^2-5X+6=0$

Let the position vectors of points \(P, Q, R\) and \(S\) be \[ \vec a = \hat i + 2\hat j - 5\hat k,\quad \vec b = 3\hat i + 6\hat j + 3\hat k,\quad \vec c = \frac{17}{5}\hat i + \frac{16}{5}\hat j + 7\hat k,\quad \vec d = 2\hat i + \hat j + \hat k. \] Then which of the following statements is true? (A) The points \(P, Q, R, S\) are not coplanar.
(B) \(\displaystyle \frac{\vec b + 2\vec d}{3}\) divides \(PR\) internally in the ratio \(5:4\).
(C) \(\displaystyle \frac{\vec b + 2\vec d}{3}\) divides \(PR\) externally in the ratio \(5:4\).
(D) The square of magnitude of \(\vec b \times \vec d\) is \(95\).

Coplanarity.
[ vec{PQ}=vec b-vec a=langle 2,4,8 angle,quad vec{PR}=vec c-vec a=leftlangle frac{12}{5}, frac{6}{5},12 ight angle,quad vec{PS}=vec d-vec a=langle 1,-1,6 angle. ] [ [vec{PQ},vec{PR},vec{PS}] = 0 ;Rightarrow; ext{coplanar} ;Rightarrow; ext{(A) false}. ] Division.
[ frac{vec b+2vec d}{3}=frac{(3,6,3)+2(2,1,1)}{3} =Big( frac{7}{3}, frac{8}{3}, frac{5}{3}Big). ] [ ext{Point dividing } PR ext{ internally in } 5{:}4 ext{ from } P:; vec X=frac{5vec r+4vec p}{9}=frac{5vec c+4vec a}{9} =Big( frac{7}{3}, frac{8}{3}, frac{5}{3}Big). ] Hence (B) true and (C) false. Cross product.
[ vec b imesvec d= egin{vmatrix} hat i&hat j&hat k\ 3&6&3\ 2&1&1 end{vmatrix}=(3,3,-9),qquad lVert vec b imesvec d Vert^2=3^2+3^2+(-9)^2=99 eq95. ] Therefore only (B) is correct.

Evaluate $$sin 30^circ$$.

Standard value.

For real $$a\ne 0$$, compute $$a^0 + a^1 + a^2$$.

By definition of powers.

Find $$lceil pi ceil$$.

Since (3.14