Differentiation (Part-38) | S N De

MATHEMATICS (New Syllabus) – 2022

PART-A [Marks : 70]

1. (a) Answer any one question: $2 \times 1 = 2$

1. $f(x) = \frac{3x+4}{5x-7} \left(x \neq \frac{7}{5}\right)$ and $g(x) = \frac{7x+4}{5x-3} \left(x \neq \frac{3}{5}\right)$, show that $f(g(x)) = g(f(x))$.
2. Let $A = \{1, 2\}$, $B = \{1, 8\}$ and $f : A \to B, g : A \to B$ are two mappings defined as $f(x) = x^3$ and $g(x) = 6x^2 - 11x + 6$, then prove that $f = g$.
3. Find the value of $\sec^2\left(\cot^{-1}\frac{1}{3}\right) + \csc^2\left(\tan^{-1}\frac{1}{2}\right)$.

1. (b) Answer any one question: $2 \times 1 = 2$

1. If $A = \begin{pmatrix} 1 & 5 \\ 6 & 7 \end{pmatrix}$, then show that $A - A^{T}$ is a skew symmetric matrix.
2. If $\begin{pmatrix} 1 & 5 \\ 6 & 7 \end{pmatrix} \times \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 \\ -1 \end{pmatrix}$, then find the values of $x$ and $y$.
3. If $\begin{vmatrix} 5 & 4 \\ 3 & 2 \end{vmatrix} = \begin{vmatrix} 2x & 7 \\ x & 3 \end{vmatrix}$, find the value of $x$.

1. (c) Answer any three questions: $2 \times 3 = 6$

1. If $f(x) = x \text{ for } x \geq 0$ and $f(x) = 2 \text{ for } x < 0$, show that $f(x)$ is discontinuous at $x = 0$.
2. If $ye^y = x$, then show that $\frac{dy}{dx} = \frac{y}{x(1+y)}$.
3. Evaluate $\displaystyle\int_{-1}^{1} x|x| \, dx$.
4. Solve : $\frac{dy}{dx} = e^{x-y} + x^2 e^{-y}$.
5. $x > 0, y > 0$ and $xy = 1$, find the least value of $x + y$.
6. Find the equation of tangent of $x^2 + y^2 = 32$ at $(4, 4)$.

1. (d) Answer any one question: $2 \times 1 = 2$

1. Show that the line joining $(1, -1, 2), (3, 4, -2)$ is perpendicular to the line through $(0, 3, 2)$ and $(3, 5, 6)$.
2. Find the equation of the plane passing through $(1, 0, 0), (0, 2, 0)$ and $(0, 0, 4)$.
3. If $\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$ and $\vec{b} = \hat{i} + 3\hat{j} - 5\hat{k}$, then show that $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ are perpendicular to each other.

S.N. Dey - Straight Line (Part - 2) (English Version)

1. (e) Answer any one question: $2 \times 1 = 2$

1. If $P(A) = \frac{6}{13}$, $P(B) = \frac{5}{13}$ and $P(A \cup B) = \frac{7}{13}$, find $P(A/B)$.
2. A die is thrown. If $E$ is the event 'the number appearing is a multiple of $3$' and $F$ be the event 'the number appearing is even'; then show that $E$ and $F$ are independent.

2. (a) Answer any one question: $4 \times 1 = 4$

1. Find the range of the function $f(x) = \frac{1}{1-x^2}, x$ is real and $x \neq \pm 1$.
2. $f : R \to R$ is a mapping where $f(x) = x^3 - 6$, for all $x \in R, R =$ set of real numbers. Prove that $f$ is a bijective mapping.

2. (b) Answer any two from the following questions: $4 \times 2 = 8$

1. If $A = \begin{pmatrix} x & -2 \\ 2 & 1 \end{pmatrix}, B = \begin{pmatrix} 3 & 4 \\ 0 & 1 \end{pmatrix}, C = \begin{pmatrix} -1 & -2 \\ y & 2 \end{pmatrix}$ and $A + B = BC$, then find the values of $x$ and $y$.
   OR
   If $A + 2B = \begin{bmatrix} 1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1 \end{bmatrix}$ and $2A - B = \begin{bmatrix} 2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2 \end{bmatrix}$, find the matrices $A$ and $B$.
2. Show that $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$ matrix satisfies the equation $A^2 - 4A - 5I_3 = O$. Hence find $A^{-1}$. $\left[I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\right]$
   OR
   If $A = \begin{pmatrix} 1 & -1 & 0 \\ -1 & 2 & 1 \\ 0 & 1 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 1 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{pmatrix}$, then show that $B^{T}AB$ is a diagonal matrix.
3. Show that $\begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix} = abc\left(1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right)$.
   OR
   Show that $\begin{vmatrix} 1 & x & x^2 \\ x^2 & 1 & x \\ x & x^2 & 1 \end{vmatrix} = (1 - x^3)^2$.

2. (c) Answer the following questions:

1. $f(x) = \frac{|x|}{x} \text{ for } x \neq 0$ and $f(x) = 0 \text{ for } x = 0$. Find the point of discontinuity of $f(x)$.
   OR
   If $y = \sin(2\sin^{-1}x)$, then show that $(1-x^2) \frac{d^2y}{dx^2} = x \frac{dy}{dx} - 4y$.
2. Evaluate : $\displaystyle \int \frac{dx}{x(x^2+1)}$
   OR
   Evaluate : $\displaystyle \int \frac{dx}{1+\tan x}$
3. Solve : $x\cos \left(\frac{y}{x}\right) \frac{dy}{dx} = y\cos\left(\frac{y}{x}\right) + x$
   OR
   Find the equation of the curve passing through the point $(-2, 3)$ given the slope of the tangent to the curve at any point $(x, y)$ is $\frac{2x}{y^2}$.

2. (d) Answer any one question: $4 \times 1 = 4$

1. Find the equation of the line that passes through the origin and $(5, 2, 4)$.
2. Find the equation of the plane through the line of intersection of the planes $x+y+z=1$ and $2x+3y+4z=5$ which is perpendicular to the plane $x-y+z=0$.
3. If sum of two unit vectors be a unit vector, then show that difference of those two vectors is $\sqrt{3}$.

2. (e) Answer any one question: $4 \times 1 = 4$

1. Find the area of the circle $x^2 + y^2 = 16$ using integral calculus.
2. From the definition of definite integral find the value of $\displaystyle\int_{0}^{1} (2x+1) \, dx$.

2. (f) Answer any one question: $4 \times 1 = 4$

1. Ten cards numbered $1$ to $10$ are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than $3$, what is the probability that it is an even number?
2. A die is thrown twice and the sum of the numbers appearing is observed to be $6$, what is the probability that the number $4$ has appeared at least once?

3. (a) Answer any one question: $5 \times 1 = 5$

1. A dietician wishes to mix two types of foods in such a way that the mixture contain at least $8$ units of Vitamin A and $10$ units of Vitamin C. Food-I contains $2$ units/kg of Vitamin A and $1$ unit/kg of Vitamin C, while Food-II contains $1$ unit/kg of Vitamin A and $2$ units/kg of Vitamin C. It costs Rs. $50$ per kg to purchase Food-I and Rs. $70$ per kg to purchase Food-II. Formulate the above as a LPP to minimise the cost of such a mixture.
2. Solve the linear programming problem graphically : Maximise $Z = 4x + y$ where $x + y \leq 50, 3x + y \leq 90, x \geq 0$ and $y \geq 0$. (Graph sheet is not required)

3. (b) Answer any two questions: $5 \times 2 = 10$

1. Using calculus, show that the straight line $lx + my + n = 0$ touches the circle $x^2 + y^2 = a^2$ if $a^2(l^2 + m^2) = n^2$.
2. $x$ is real. Using differential calculus find the maximum and minimum values of $\frac{x^2-x+1}{x^2+x+1}$.
3. Find differential equation by eliminating $a$ and $b$ from $y = e^x(a\cos x + b\sin x)$.
4. Evaluate $\displaystyle\lim_{n \to \infty} \left[\frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{3n}\right]$.

3. (c) Answer any two questions: $5 \times 2 = 10$

1. Find the cartesian equation of the line which passes through $(1, 2, 3)$ and parallel to the line $\frac{x+3}{3} = \frac{y-4}{5} = \frac{z+8}{6}$.
2. Find the coordinates of the foot of the perpendicular from origin to the plane $x + y + z = 3$.
3. A plane has the intercepts on axes are $a, b, c$ respectively and $p$ be the perpendicular distance from origin to the plane. Show that $\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} = \frac{1}{p^2}$.

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