Board Paper of Class 12 2017 Mathematics - Solutions

(a) Select and write the most appropriate answer from the given alternatives in each of the following sub-questions.
(i) If the points A (2, 1, 1), B (0, -1, 4) and C (k, 3, -2) are collinear, then k =______.
(A) 0

(B) 1

(C) 4

(D) – 4

(ii) The inverse of the matrix $\mathrm{A}=\left[\begin{array}{cc}-1& 5\\ -3& 2\end{array}\right]$ is

(A) $\frac{1}{13}\left[\begin{array}{cc}2& -5\\ 3& -1\end{array}\right]$

(B) $\frac{1}{13}\left[\begin{array}{cc}-1& 5\\ -3& 2\end{array}\right]$

(C) $\frac{1}{13}\left[\begin{array}{cc}-1& -3\\ 5& 2\end{array}\right]$

(D) $\frac{1}{13}\left[\begin{array}{cc}1& 5\\ 3& -2\end{array}\right]$

(iii) In △ABC, if a = 13, b = 14 and c = 15, then sin [A / 2] is

(A) $\frac{1}{5}$

(B) $\sqrt{\frac{1}{5}}$

(C) $\frac{4}{5}$

(D) $\frac{2}{5}$

(b) Attempt any THREE of the following:
(i) Find the volume of the parallelepiped whose coterminous edges are are given by vectors 2i + 3j – 4k, 5i + 7j + 5k and 4i + 5j – 2k.

(ii) In △ABC, prove that, a(b cos C – c cos B) = b² – c².

(iii) If from a point Q(a, b, c) perpendiculars QA and QB are drawn to the YZ and ZX planes respectively, then find the vector equation of the plane QAB.

(iv) Find the cartesian equation of the line passing through the points A(3, 4, –7) and B(6, –1, 1).

(v) Write the following statement in symbolic form and find its truth value: ∀ n ε N, n² + n is an even number and n² – n is an odd number. VIEW SOLUTION

(a) Attempt any TWO of the following:

(i) Using truth tables, examine whether the statement pattern (p ∧ q) ∨ (p ∧ r) is a tautology, contradiction or contingency.

(ii) Find the shortest distance between the lines:

(iii) Find the general solution of the equation sin 2x + sin 4x + sin 6x = 0.

(b) Attempt any TWO of the following:
(i) Solve the following equations by method of reduction: x – y + z = 4, 2x + y – 3z = 0, x + y + z = 2.

(ii) If θ is the measure of the acute angle between the lines represented by the equation ax² + 2hxy + by² = 0, then prove that$\tan \theta =\left|\frac{2\sqrt{h^2-ab}}{a+b}\right|$ where a + b ≠ 0 and b ≠ 0. Find the condition for coincident lines.

(iii) Using the vector method, find incentre of the triangle whose vertices are P(0, 4, 0), Q(0, 0, 3) and R(0, 4, 3). VIEW SOLUTION

(a) Attempt any TWO of the following:

(i) Construct the switching circuit for the statement (p∧q) ∨ (~p) ∨ (p∧~q).

(ii) Find the joint equation of the pair of lines passing through the origin which are perpendicular respectively to the lines represented by 5x² + 2xy – 3y² = 0.

(iii) Show that ${\cos }^{-1}\left(\frac{4}{5}\right)+{\cos }^{-1}\left(\frac{12}{13}\right)={\cos }^{-1}\left(\frac{33}{65}\right)$.

(b) Attempt any TWO of the following:
(i) If l, m, n are the direction cosines of a line, then prove that l² + m² + n² = 1.
(ii) Find the vector and cartesian equations of the plane passing through the points A( 1, 1, −2), B(1, 2, 1) and C(2, -1, 1).

(iii) Solve the following L.P.P. by the graphical method: VIEW SOLUTION