# MATHEMATICS  ( New Syllabus )-2022

## PART-A  [Marks : 70]

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

$~~(i)~~f(x)=\frac{3x+4}{5x-7}~~\left(x \neq \frac 75\right)~~$ and $~~g(x)=\frac{7x+4}{5x-3}~~\left( x \neq \frac 35\right),~~$ show that $~~f(g(x))=g(f(x)).$

$~~(ii)~~$ 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 $~~f=g.$

$~(iii)~~$ Find the value of $~~\sec^2\left(\cot^{-1}\frac 13\right)+\csc^2\left(\tan^{-1}\frac 12\right).$

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

$~(i)~$ If $~A=\begin{pmatrix} 1 & 5 \\ 6 & 7\\ \end{pmatrix},~$ then show that $~~A-A^{T}~~$ is a skew symmetric matrix.

$~(ii)~~$ 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 $~y.$

$~(iii)~~$ If $~\begin{vmatrix} 5 & 4 \\ 3 & 2 \\ \end{vmatrix}=\begin{vmatrix} 2x & 7 \\ x & 3 \\ \end{vmatrix},~~$ find the value of $~x.$

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

$~(i)~$ If $~f(x)=x~~\text{for}~~x \geq 0\\~~f(x)=2~~\text{for}~~x < 0$

show that $~f(x)~$ is discontinuous at $~~x=0.$

$~(ii)~$ If $~ye^y=x,~~$ then show that $~~\frac{dy}{dx}=\frac{y}{x(1+y)}.$

$~(iii)~$ Evaluate $~~\displaystyle\int_{-1}^{1}~x|x|~dx.$

$~(iv)~~$ Solve : $~~\frac{dy}{dx}=e^{x-y}+x^2~e^{-y}.$

$~(v)~~x>0,~y>0~$ and $~~xy=1~,$ find the least value of $~~x+y.$

$~(vi)~~$ Find the equation of tangent of $~~x^2+y^2=32~~$ at $~(4,4).$

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

$~(i)~$ Show that the line joining $~~(1,-1,2),~(3,4,-2)~$ is perpendicular to the line through $~(0,3,2)~$ and $~(3,5,6).$

$~(ii)~$ Find the equation of the plane passing through $~(1,0,0),~(0,2,0)~~$ and $~~(0,0,4).$

$~(iii)~$ 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. De -Straight Line(Part -2) (Eng. Version)

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

$~(i)~~$ If $~P(A)=\frac{6}{13},~P(B)=\frac{5}{13}~$ and $~P(A \cup B)=\frac{7}{13},~$ find $~~P(A/B).$

$~(ii)~~$ 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 : $~~4 \times 1=4$

$~(i)~$ Find the range of the function $~~f(x)=\frac{1}{1-x^2},~x~$ is real and $~~x \neq \pm 1.$

$~(ii)~~f : R \to R~$ 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.

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

#$~(ii)~~$ 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.

$~(iii)~~$ Show that $~~\begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \\ \end{vmatrix}\\=abc\left(1+\frac 1a+\frac 1b+\frac 1c\right).$

OR

Show that $~~\begin{vmatrix} 1 & x & x^2 \\ x^2 & 1 & x \\ x & x^2 & 1 \\ \end{vmatrix}=(1-x^3)^2.$

$~(c)~$ Answer the following questions :

$~(i)~f(x)=\frac{|x|}{x}~~\text{for}~~x \neq 0\\~~~~~~f(x)=0~~\text{for}~~x = 0$

Find the point of discontinuity of $~f(x).$

OR

If $~~y=\sin(2\sin^{-1}x),~~$ then show that $~~(1-x^2)~\frac{d^2y}{dx^2}=x~\frac{dy}{dx}-4y.$

$~(ii)~~$ Evaluate : $~~\displaystyle \int \frac{dx}{x(x^2+1)}$

OR

Evaluate : $~~\displaystyle \int \frac{dx}{1+\tan x}$

$~(iii)~~$ Solve : $~~x\cos \left(\frac yx\right)~\frac{dy}{dx}=y\cos\left(\frac yx\right)+x$

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 $\frac{2x}{y^2}.$

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

$~(i)~~$  Find the equation of the line that passes through the origin and $~( 5, 2, 4).~$

$~(ii)~~$ 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 $~x-y+z=0~$.

$~(iii)~$ If sum of two unit vectors be a unit vector, then show that difference of those two vectors is $~\sqrt{3}.$

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

$~(i)~$ Find the area of the circle $~~x^2+y^2=16~$ using integral calculus.

$~(ii)~$ From the definition of definite integral find the value of $~~\displaystyle\int_{0}^{1} (2x+1)~dx$

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

$~(i)~$ 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?

$~(ii)~$ 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 : $~~5 \times 1=5$

$~(i)~$ 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.

$~(ii)~$ 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)

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

$~(i)~$ Using calculus , show that the straight line $~~lx+my+n=0~$ touches the circle $~~x^2+y^2=a^2~~$ if $~~a^2(l^2+m^2)=n^2.$

$~(ii)~~x~$ is real. Using differential calculus find the maximum and minimum values of $~~\frac{x^2-x+1}{x^2+x+1}.$

$~(iii)~$ Find differential equation by eliminating $~a~$ and $~b~$ from $~~y=e^x(a\cos x+b\sin x).$

$~(iv)~$ Evaluate $~~\lim_{n \to \infty}~\left[\frac{1}{n+1}+\frac{1}{n+1}+\cdots+\frac{1}{3n}\right].$

$~(c)~$ Answer any two questions : $~~ 5 \times 1=5$

$~(i)~$ Find the cartesian equation of the line which passes through

$~(1,2,3)~$ and parallel to the line $~~\frac{x+3}{3}=\frac{y-4}{5}=\frac{z+8}{6}.$

$~(ii)~$ Find the coordinates of the foot of the perpendicular from origin

to the plane $~x+y+z=3.$

$~(iii)~$ 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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