# Board Paper of Class 12 2017 Mathematics - Solutions

**Section A**- Answer

**Question 1**(

*compulsory*) and five other questions.

**Section B and Section C**-

*Answer*

**two**

*questions from*

**either**Section B**or**Section C.All working, including rough work, should be done on the same sheet as, and adjacent to, the rest of the answer.

*The intended marks for questions or parts of questions are given in brackets [ ].*

**Mathematical tables and graph papers are provided.**

Slide rule may be used.Slide rule may be used.

- Question 1
**(a)**(i) If the matrix $\left(\begin{array}{cc}6& -{x}^{2}\\ 2x-15& 10\end{array}\right)$ is symmetric, find the value of*x*.**(b)**(ii) If*y*– 2*x*–*k*= 0 touches the conic 3*x*^{2}– 5*y*^{2}= 15, find the value of*k*.**(c)**(iii) Prove that $\frac{1}{2}{\mathrm{cos}}^{-1}\left(\frac{1-x}{1+x}\right)={\mathrm{tan}}^{-1}\sqrt{x}$**(d)**(iv) Using L’Hospital’s Rule, evaluate:$\underset{x\to \mathrm{\pi}/2}{lt}\left(x\mathrm{tan}x-\frac{\pi}{4}\xb7\mathrm{sec}x\right)$

**(e)**(v) Evaluate: $\int \frac{1}{{x}^{2}}{\mathrm{sin}}^{2}\left(\frac{1}{x}\right)dx$**(f)**(vi) Evaluate: ${\int}_{0}^{\pi /4}\mathrm{log}\left(1+\mathrm{tan}\theta \right)d\theta $**(g)**(vii) By using the data $\overline{)x}=25,\overline{)y}=30$,*b*= 1·6 and_{yx}*b*= 0·4, find:_{xy }

(a) The regression equation*y*on*x*.(b) What is the most likely value of*y*when*x*= 60?

(c) What is the coefficient of correlation between*x*and*y*?**(h)**(viii) A problem is given to three students whose chances of solving it are $\frac{1}{4},\frac{1}{5}\mathrm{and}\frac{1}{3}$ respectively. Find the probability that the problem is solved.**(i)**(ix) If $a+ib=\frac{x+iy}{x-iy}\mathrm{prove}\mathrm{that}{a}^{2}+{b}^{2}=1\mathrm{and}\frac{b}{a}=\frac{2xy}{{x}^{2}-{y}^{2}}$**(j)**(x) Solve: $\frac{dy}{dx}=1-xy+y-x$ VIEW SOLUTION

- Question 2
**(a)**Using properties of determinants, prove that: $\left|\begin{array}{ccc}a& b& b+c\\ c& a& c+a\\ b& c& a+b\end{array}\right|=\left(a+b+c\right){\left(a-c\right)}^{2}$**(b)**Given that: $\mathrm{A}=\left(\begin{array}{ccc}1& -1& 0\\ 2& 3& 4\\ 0& 1& 2\end{array}\right)\mathrm{and}B=\left(\begin{array}{ccc}2& 2& -4\\ -4& 2& -4\\ 2& -1& 5\end{array}\right)$, find AB.

Using this result, solve the following system of equation:

*x – y*= 3, 2*x*+ 3*y*+ 4*z*= 17 and*y*+ 2*z*= 7 VIEW SOLUTION

- Question 3
**(a)**Solve the equation for*x*: sin^{–1 }*x*+ sin^{–1}(1 –*x*) = cos^{–1}*x,**x*≠ 0**(b)**If A, B and C are the elements of Boolean algebra, simplify the expression (Aʹ + Bʹ) (A + Cʹ) + Bʹ(B + C). Draw the simplified circuit. VIEW SOLUTION

- Question 4
**(a)**Verify Langrange’s mean value theorem for the function:*f*(*x*) =*x*(1 – log*x*) and find the value of ‘*c*’ in the interval [1, 2]**(b)**Find the coordinates of the centre, foci and equation of directrix of the hyperbola*x*^{2}– 3*y*^{2}– 4*x*= 8. VIEW SOLUTION

- Question 5
**(a)**If*y*= cos (sin*x*), show that: $\frac{{d}^{2}y}{d{x}^{2}}+\mathrm{tan}x\frac{dy}{dx}+y{\mathrm{cos}}^{2}x=0$**(b)**Show that the surface area of a closed cuboid with square base and given volume is minimum when it is a cube. VIEW SOLUTION

- Question 6
**(a)**Evaluate: $\int \frac{\mathrm{sin}2x}{\left(1+\mathrm{sin}x\right)\left(2+\mathrm{sin}x\right)}dx$**(b)**Draw a rough sketch of the curve*y*^{2}= 4*x*and find the area of the region enclosed by the curve and the line*y = x*. VIEW SOLUTION

- Question 7
**(a)**Calculate the Spearman’s rank correlation coefficient for the following data and interpret the result:**X**35 54 80 95 73 73 35 91 83 81 **Y**40 60 75 90 70 75 38 95 75 70 **(b)**Find the line of best fit for the following data, treating*x*as dependent variable (Regression equation*x*on*y*):**X**14 12 13 14 16 10 13 12 **Y**14 23 17 24 18 25 23 24

Hence, estimate the value of*x*when*y*= 16. VIEW SOLUTION

- Question 8
**(a)**In a class of 60 students, 30 opted for Mathematics, 32 opted for Biology and 24 opted for both Mathematics and Biology. If one of these students is selected at random, find the probability that:

(i) The student opted for Mathematics or Biology.

(ii) The student has opted neither Mathematics nor Biology.

(iii) The student has opted Mathematics but not Biology.**(b)**Bag A contains 1 white, 2 blue and 3 red balls. Bag B contains 3 white, 3 blue and 2 red balls. Bag C contains 2 white, 3 blue and 4 red balls. One bag is selected at random and then two balls are drawn from the selected bag. Find the probability that the balls drawn are white and red. VIEW SOLUTION

- Question 9
**(a)**Prove that locus of*z*is circle and find its centre and radius if $\frac{z-i}{z-1}$ is purely imaginary.**(b)**Solve: (𝑥^{2}− 𝑦𝑥^{2})𝑑𝑦 + (𝑦^{2}+ 𝑥𝑦^{2}) 𝑑𝑥 = 0 VIEW SOLUTION