Board Paper of Class 12 2018 Mathematics - Solutions

(a) Select and write the appropriate answer from the given alternatives in each of the following sub-questions·:
(i) If $\mathrm{A}=\left[\begin{array}{cc}2& -3\\ 4& 1\end{array}\right]$,  then adjoint of matrix A is

(a) $\left[\begin{array}{cc} 1& 3\\ -4& 2\end{array}\right]$

(b) $\left[\begin{array}{cc} 1& -3\\ -4& 2\end{array}\right]$

(c) $\left[\begin{array}{cc}1& 3\\ 4& -2\end{array}\right]$

(d) $\left[\begin{array}{cc}-1& -3\\ -4& 2\end{array}\right]$
 
(ii) The principal solutions or $secx =\frac{2}{\sqrt{3}}$ are _______.

(a) $\frac{\mathrm{\pi }}{3}, \frac{11\mathrm{\pi }}{6}$

(b) $\frac{\mathrm{\pi }}{6}, \frac{11\mathrm{\pi }}{6}$

(c) $\frac{\mathrm{\pi }}{4}, \frac{11\mathrm{\pi }}{4}$

(d) $\frac{\mathrm{\pi }}{6}, \frac{11\mathrm{\pi }}{4}$

(iii) The measure of the acute angle between the lines whose direction ratios are 3, 2, 6 and -2, 1, 2 is ______

(a) ${\cos }^{-1}\left(\frac{1}{7}\right)$

(b) ${\cos }^{-1}\left(\frac{8}{15}\right)$

(c) ${\cos }^{-1}\left(\frac{1}{3}\right)$

(d) ${\cos }^{-1}\left(\frac{8}{21}\right)$

(b) Attempt any Three of the following:
(i) Write the negations of the following statements:
(ii) If a line makes angles α, β, γ, with the co-ordinate axes, prove that cos2 α + cos2 β + cos2 γ + 1 = 0.

(iii) Find the distance of the point (1, 2, –1) from the plane x – 2y + 4z  – 10 = 0.

(iv) Find the vector equation of the line which passes through the point with position vector $4\hat{i}-\hat{j}+2\hat{k}$ and is in the direction of $2\hat{i}+\hat{j}+\hat{k}$.

( v) If $a=3\hat{i}-2\hat{j}+7\hat{k}, b=5\hat{i}+\hat{j}-2\hat{k} \mathrm{and} c=\hat{i}+\hat{j}-\hat{k}. \mathrm{then} \mathrm{find} \overline{a}.\left(\overline{b}\times \overline{c}\right)$. VIEW SOLUTION

(a) Attempt any TWO of the following:
(i) Using vector method prove that the medians of a triangle are concurrent.

(ii) Using the truth table, prove that following logical equivalence:
(ii) If the origin is the centroid of the triangle whose vertices are A(2, p, –3 ), B(q, –2, 5) and R(–5, 1, r), then find the value' of p, q, r.

(b) Attempt any Two of the following:
(i) Show that a homogeneous equation of degree two in x and y, i.e. $a{x}^2+2hxy+b{y}^2=0$ represents a pair of  lines passing through the origin if h² – ab ≥ 0.

(ii) In ABC, prove that $\tan \left(\frac{\mathrm{C}-\mathrm{A}}{2}\right)=\left(\frac{c-a}{c+a}\right)\cot \frac{\mathrm{B}}{2}$

(iii) Find the inverse of the matrix, $A=\left[\begin{array}{ccc}1& 2& -2\\ -1& 3& 0\\ 0& -2& 1\end{array}\right]$ using elementary row transformations. VIEW SOLUTION

(a) Attempt any TWO of the following:
(i) Find the joint equation of the pair of lines passing through the origin, which are perpendicular to the lines represented by 5x² + 2xy – 3y² = 0.
(ii) Find the angle between the lines $\frac{x-1}{4}=\frac{y-3}{1}=\frac{z}{8}$ and $\frac{x-2}{2}=\frac{y+1}{2}=z-4$
(iii) Write converse, inverse and contrapositive of the following conditional statement. If an angle is a right angle then its measure is 90°.

(b) Attempt any TWO of the following:
(i) Prove that: ${\sin }^{-1 }\left(\frac{3}{5}\right)+{\cos }^{-1 }\left(\frac{12}{13}\right)={\sin }^{-1 }\left(\frac{56}{65}\right)$

(ii) Find the vector equation of the plane passing through the points A(l, 0, 1), B(1, –1, 1) and C (4, –3, 2).

(iii) Minimize Z = 7x + y subject to 5x + y ≥ 5, x + y ≥ 3, x ≥ 0, y ≥ 0 VIEW SOLUTION