MATHEMATICS

(Maximum Marks: 100)

(Time Allowed: Three Hours)

They must NOT start writing during this time)

The Question Paper consists of three sections A, B and C

Candidates are required to attempt all questions from Section A and all question EITHER from Section B OR Section C

Section A: Internal choice has been provided  in three questions  of four marks each and two questions of six marks each.

Section B: Internal choice has been provided in two question of four marks each.

Section C: Internal choice has been provided in two question of four marks each.

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 graphs papers are provided.

SECTION – A (80 Marks)

Question 1:        [10 × 2]

(i) The binary operation $\ast : R \times R \ \rightarrow R$ is defined as $a \ast b = 2a+ b$ Find $(2 \ast 3) \ast 4$.

(ii) If $\begin{bmatrix} 5 & a \\ b & 0 \end{bmatrix}$ and $A$ is symmetric matrix, show that $a = b$

(iii) Solve : $3 \ tan^{-1}x + cot^{-1}x = \pi$

(iv) Without expanding at any stage, find the value of:

$\left| \begin{array}{ccc} a & b & c \\ a+2x & b+2y & c+2z \\ x & y & z \end{array} \right|$

(v) Find the value of constant $k$ so that the function $f (x)$ defined as:

$f(x) = \Bigg\{$ $\begin{array}{ccc} \frac{a}{b}, x \neq -1 \\ \ \\ k, x = -1 \end{array}$

is continuous at $x=-1$.

(vi) Find the approximate change in the volume $V$ of a cube of side $x$ meters caused by decreasing the side by $1\%$.

(vii) Evaluate : $\int \limits_{}^{}$ $\frac{x^3+5x^2+4x+1}{x^2}$ $dx$

(viii) Find the differential equation of the family of concentric circles $x^2+y^2 = a^2$

(ix) If $A$ and $B$ are events such that $P(A) =$ $\frac{1}{2}$ $, P(B) =$ $\frac{1}{3}$ and $P(A \cap B) =$ $\frac{1}{4}$,  then find:

(a) $P(A/ B)$

(b) $P(B / A)$

(x) In a race, the probabilities of A and B winning the race are $\frac{1}{3}$ and $\frac{1}{6}$ respectively.  Find the probability of neither of them winning the race.

Question 2:  If the function $f (x) = \sqrt{2x-3}$ is invertible then find its inverse. Hence prove that $(fof^{-1})(x) = x$.       [4]

Question 3:  If $tan^{-1} a + tan^{-1} b + tan^{-1} c = \pi$  , prove that $a + b + c = abc$.        [4]

Question 4: Use properties of determinants to solve for $x$:

$\left| \begin{array}{ccc} x+a & b & c \\ c & x+b \\ a & b & x+c \end{array} \right|= 0$ and $x \neq 0$        [4]

Question 5:        [4]

(a) Show that the function  $f(x) = \Bigg\{$ $\begin{array}{ccc} x^2, x \leq 1 \\ \ \\ \frac{1}{2}, x > 1 \end{array}$ is continuous at $x =1$ but not differentiable.

OR

(b) Verify Rolle’s theorem for the following function: $f(x) = e^{-x} sin \ x \ on \ [0, \pi ]$

Question 6: If $x = tan \Big($ $\frac{1}{a}$ $log \ y \Big)$, prove that $(1+x^2)$ $\frac{d^2y}{dx^2}$ $+(2x-a)$ $\frac{dy}{dx}$ $= 0$        [4]

Question 7: Evaluate: $\int \limits_{}^{}$ $tan^{-1}\sqrt{x} \ dx$        [4]

Question 8:        [4]

(a) Find the points on the curve $y = 4x^3 - 3x + 5$ at which the equation of the tangent is parallel to the x-axis.

OR

(b) Water is dripping out from a conical funnel of semi-vertical angle $\frac{\pi}{4}$at the uniform rate of $2 \ cm^2/sec$ in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.

Question 9:        [4]

(a) Solve: $sin \ x$ $\frac{dy}{dx}$ $- y = sin \ x. tan$ $\frac{x}{2}$

OR

(b) The population of a town grows at the rate of $10\%$ per year. Using differential equation, find how long will it take for the population to grow $4$ times.

Question 10:       [6]

(a) Using matrices, solve the following system of equations :

$2x - 3y + 5z = 11$

$3x + 2y - 4z = -5$

$x + y - 2z = -3$

OR

(b) Using elementary transformation, find the inverse of the matrix :

$\begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{bmatrix}$

Question 11: $A$ speaks truth in $60\%$ of the cases, while $B$ is $40\%$ of the cases. In what percent of cases are they likely to contradict each other in stating the same fact ?       [4]

Question 12: A cone is inscribed in a sphere of radius $12 \ cm$. If the volume of the cone is maximum, find its height.       [6]

Question 13:       [6]

(a) Evaluate: $\int \limits_{}^{}$ $\frac{x-1}{\sqrt{x^2-x}}$ $\ dx$

OR

(b) Evaluate: $\int \limits_{0}^{\pi / 2}$ $\frac{cos^2 \ x}{1 + sin \ x \ cos \ x}$ $dx$

Question 14: From a lot of $6$ items containing $2$ defective items, a sample of $4$ items are drawn at random. Let the random variable $X$ denote the number of defective items in the sample. If the sample is drawn without replacement, find :
(a) The probability distribution of $X$
(b) Mean of $X$
(c) Variance of $X$       [6]

SECTION B (20 Marks)

Question 15:       [3 × 2]

(a) Find $\lambda$ if the scalar projection of $\overrightarrow{a} = \lambda \hat{i} + \hat{j} + 4 \hat{k}$ and $\overrightarrow{b} = 2 \hat{i} + 6\hat{j} + 3 \hat{k}$ is $4$ units.

(b) The Cartesian equation of line is : $2x - 3 = 3y +1 = 5 - 6z$ . Find the vector equation of a line passing through $(7, -5, 0)$ and parallel to the given line.

(c) Find the equation of the plane through the intersection of the planes $\overrightarrow{r} . (\hat{i}+3 \hat{j} - \hat{k})= 9$ and $\overrightarrow{r} . (2\hat{i}- \hat{j} +1\hat{k})= 3$ and passing through the origin.

Question 16:       [4]

(a) If $A, B, C$ are three non- collinear points with position vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ respectively, then show that the length of the perpendicular from $C \ on \ AB$ is  $\frac{ | (\overrightarrow{a} \times \overrightarrow{b} +\overrightarrow{b} \times \overrightarrow{c} +\overrightarrow{b} \times \overrightarrow{a} ) | }{| \overrightarrow{a} - \overrightarrow{b} | }$

OR

(b) Show that the four points A,B, C and D with position vectors $4\hat{i} + 5\hat{j} + \hat{k}, - \hat{j} - \hat{k}, 3\hat{i} + 9\hat{j} + 4\hat{k}$ and $4( -\hat{i} + \hat{j} + \hat{k})$

Question 17:       [4]

(a) Draw a rough sketch of the curve and find the area of the region bounded by curve $y^2 = 8x$ and the line $x =2$.

OR

(b) Sketch the graph of $y = |x + 4|$. Using integration, find the area of the region bounded by the curve $y = |x + 4|$ and $x = -6$ and $x = 0$.

Question 18: Find the image of a point having position vector : $3\hat{i} - 2\hat{j} + \hat{k}$ in the plane $\overrightarrow{r}.(3\hat{i} - \hat{j} + 4\hat{k})$       [6]

SECTION C (20 Marks)

Question 19:       [3 × 2]

(a) Given the total cost function of $x$ units  of a commodity as

$C(x) =$ $\frac{1}{3}$ $x^3 +3x^2 -16x + 2$

Find:  (i) Marginal cost function (ii) Average cost function

(b) Find the coefficient of correlation from the regression lines: $x-2y+3 = 0$ and $4x-5y+1 = 0$.

(c) The average cost function associated with producing  and marketing $x$ units  of an item is given by $AC = 2x - 11 +$ $\frac{50}{x}$. Find the range of values  of the output $x$, from which $AC$ is increasing.

Question 20:        [4]

(a) Find the line of regression of $y$ on $x$ from the following table.

 $x$ 1 2 3 4 5 $y$ 7 6 5 4 3

Hence estimate the value of $y$ when $x = 6$.

OR

(b) From the given data:

 Variable $x$ $y$ Mean 6 8 Standard Deviation 4 6

And the correlation coefficient: $\frac{2}{3}$. Find

(i) Regression coefficients $b_{yx}$ and $b_{xy}$

(ii) Regression line $x$ on $y$

(iii) Most likely value of $x$ when $y = 14$

Question 21:        [4]

(a) A product can be manufactured at a total cost $C(x) =$ $\frac{x^2}{100}$ $+ 100x + 40$, where $x$ is the number of units produced. The price at which each unit can be sold  is given by $P = \Big(200 -$ $\frac{x}{400}$ $\Big)$. Determine the production level $x$ at which the profit is maximum.What is the price per unit and profit at the level of production.

OR

(b) A manufacturer’s marginal cost function is $\frac{500}{\sqrt{2x + 25}}$. Find the cost involved to increase  production from $100$ units to $300$ units.

Question 22: A manufacturing company produces two type of teaching aids  $A$ and $B$ of Mathematics for Class X. Each type of $A$ requires $9$ hours  for fabricating  and $1$ hour for finishing. Each type of $B$ requires $12$ hours  for fabricating  and $3$ hours for finishing. For fabricating and finishing, the maximum labor hours available for a week are $180$ and $130$ respectively. The company makes a profit of $Rs. 80$ on each piece of type $A$ and $Rs. 120$ on each piece of type $B$. How many pieces of type $A$ and type $B$ should be manufactured  per week to get a maximum profit? Formulate this as a linear programming problem and solve it. Identify the feasible region  from the rough sketch.       [6]