Board Paper of Class 12 2016 Mathematics - Solutions

(a) Select and write the most appropriate answer from the given alternatives in each of the following sub-questions:
(i) The negation of $p\wedge (q\to r)$ is
(A) p ∨ (~ q ∨ r)
(B) ~ p
∧ (q → r)
(C) ~ p ∧
(~ q → ~ r)
(D) ~ p ∨ (q
∧ ~ r)

(ii) If ${\sin }^{-1}\left(1-x\right)-2 {\sin }^{-1}x=\frac{\mathrm{\pi }}{2}$ then x is
(A) $-\frac{1}{2}$
(B) 1
(C) 0
(D) $\frac{1}{2}$

(iii) The joint equation of the pair of lines passing through (2, 3) and parallel to the coordinate axes is
(A) xy − 3x − 2y + 6 = 0
(B) xy + 3x + 2y + 6 = 0
(C) xy = 0
(D) xy − 3x − 2y − 6 = 0

(b) Attempt any THREE of the following:
(i) Find (AB)⁻¹ if $\mathrm{A}=\left[\begin{array}{ccc}1& 2& 3\\ 1& -2& -3\end{array}\right] \mathrm{B}=\left[\begin{array}{cc}1& -1\\ 1& 2\\ 1& -2\end{array}\right]$

(ii) Find the vector equation of the plane passing through a point having position vector $3\hat{i} - 2\hat{j}+\hat{k}$ and perpendicular to the vector $4\hat{i}+3\hat{j}+2\hat{k}$.

(iii) If $\overline{p}=\hat{i}-2\hat{j}+\hat{k} \mathrm{and} \overline{q}=\hat{i}+4\hat{j}-2\hat{k}$ are position vector (P.V.) of points P and Q, find the position vector of the point R which divides segment PQ internally in the ratio 2 : 1.

(iv) Find k, if one of the lines given by 6x² + kxy +y² = 0 is 2x + y = 0.

(v) If the lines $\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2} \mathrm{and} \frac{x-1}{3k}=\frac{y-5}{1}=\frac{z-6}{-5}$ are at the right angle then find the value of k. VIEW SOLUTION

(a) Attempt any TWO of the following:
(i) Examine whether the following logical statement pattern is a tautology, contradiction or contingency. [(p → q) ∧ q] → p

(ii) By vector method prove that the medians of a triangle are concurrent.

(iii) Find the shortest distance between the lines and $\overline{r}=\left(4\hat{i}-\hat{j}\right)+\lambda \left(\hat{i}+2\hat{j}-3\hat{k}\right) \mathrm{and} \overline{r}=\left(\hat{i}-\hat{j}+2\hat{k}\right)+\mu \left(\hat{i}+4\hat{j}-5\hat{k}\right)$ where λ and μ are parameters.

(b) Attempt any TWO of the following: (8)
(i) In ΔABC with the usual notations prove that ${\left(a-b\right)}^2{\cos }^2\left(\frac{C}{2}\right)+{\left(a+b\right)}^2 {\sin }^2\left(\frac{C}{2}\right)=c^2$.

(ii) Minimize z = 4x + 5y subject to 2x + y ≥ 7, 2x + 3y ≤ 15, x ≤ 3, x ≥ 0, y ≥ 0. Solve using graphical method.

(iii) The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is ₹ 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is ₹ 90 whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is ₹ 70. Find the cost of each item per dozen by using matrices. VIEW SOLUTION

(a) Attempt any TWO of the following:
(i) Find the volume of tetrahedron whose coterminous edges are $7\hat{i}+\hat{k}, 2\hat{i}+5\hat{j}-3\hat{k} \mathrm{and} 4\hat{i}+3\hat{j}+\hat{k}$.

(ii) Without using truth table show that $~\left(p\vee q\right)\vee \left(~p\vee q\right) = ~p$.

(iii) Show that every homogeneous equation of degree two in x and y, i.e., ax² + 2hxy + by² = 0 represents a pair of lines passing through origin if h² − ab ≥ 0.

(b) Attempt any TWO of the following:

(i) If a line drawn from the point A(1, 2, 1) is perpendicular to the line joining P(1, 4, 6) and Q(5, 4, 4) then find the co-ordinates of the foot of the perpendicular.

(ii) Find the vector equation of the plane passing through the points  $\hat{i}+\hat{j}-2\hat{k} , \hat{i}+2\hat{j}+\hat{k}, 2\hat{i}-\hat{j}+\hat{k}$. Hence find the cartesian equation of the plane.

(iii) Find the general solution of sin x + sin 3x + sin 5x = 0. VIEW SOLUTION