Questions for 1st Sem

Topic: Beta and Gamma Function

 Q1. Evaluate $\int_0^1 x^4 (1-\sqrt{x})dx$

Q2. Evaluate $\int_0^1 (1-x^3)^{-\frac{1}{2}}dx$

Q3. Show that $\int_0^{\frac{\pi}{2}} \sin^p \theta \cos^q  \theta d\theta=\frac{\Gamma(\frac{p+1}{2})\Gamma(\frac{q+1}{2})}{2 \Gamma(\frac{p+q+2}{2})}$

Q4. Show that $\int_0^{\frac{\pi}{2}} \sqrt{\cot \theta} d\theta =\frac{1}{2}\Gamma(\frac{1}{4})\Gamma(\frac{3}{4})$.

Q5. Show that $\Gamma(n)\Gamma(1-n)=\frac{\pi}{\sin n\pi}, 0<n<1$

Q6. Show that $\int_0^{\frac{\pi}{2}} \tan^p \theta d\theta =\frac{\pi}{2} \sec \frac{p\pi}{2}$

Q7. Express the follwing integrals in terms of Gamma Function

    a) $\int_0^1 \frac{dx}{\sqrt{1-x^4}}$                b) $\int_0^{\frac{\pi}{2}} \sqrt(\tan \theta)d\theta$

    c) $\int_0^{\infty}\frac{x^c}{c^x}dx$                d) $\int_0^{\infty} a^{-bx^2}dx$

    e) $\int_0^1 x^5 [log(1/x)]^3 dx$

Q8. Prove that $\Gamma(m)\Gamma(m+\frac{1}{2})=\frac{\sqrt{\pi}}{2^{2m-1}}\Gamma(2m)$

Questions from Differential Equations

 Q1. For the diffrential equation $\frac{dy}{dt}+5y=0$ with $y(0)=1,$ the general solution is

    a) $e^{5t}$         b) $e^{-5t}$        c) $5e^{-5t}$            d) $e^{\sqrt{-5t}}$

Q2. $y=e^{-2x}$ is a solution of the differential equation $y''+y'-2y=0$

a) True          b) False

Q3.  The differetial equation $\frac{dy}{dx}+Py=Q$, is a linear equation of first order only if

a) P is a constant But Q is a function of y

b) P and Q are functions of y (or) constants.

c) P is a function of y but Q is a constant.

d) P and Q are function of x (or) constant.

Q4. If c is a constant, then the solution of $\frac{dy}{dx}=1+y^2$ is

a)$y=sin(x+c)$          b) $y=cos (x+c)$         c) $y=tan(x+c)$          d) $y=e^x+c$

Q5. The solution of the differential equation $\frac{dy}{dx}+y^2=0$ is

a) $y=\frac{1}{x+c}$         b) $y=-\frac{x^3}{3}+c$ c) $y=ce^x$ d) Unsolvable as equation is non-liear

Q6. Biotransformation of an organic compound having concentration (x) can be modeled using an ordinary differential equation $\frac{dx}{dt}+kx^2=0$, where $k$ is the reaction rate constant. If $x=-a$ at $t=0$ then solution of the equation is

a)$x=ae^{-kt}$         b)$\frac{1}{x}=\frac{1}{a}+kt$         c) $x=a(1-e^{-kt})$         d)$x=a+kt$

 Q7. The differential equation $\left[1+\left(\frac{dy}{dx}\right)^2\right]^3=C^2\left[\frac{d^2y}{dx^2}\right]^2$

a) 2nd order and 3rd degree  b) 3rd order 2nd degree 

c) 2nd order 2nd degree  d) 3rd order 3rd degree.

Q8. The solution of the first order differential equation $\frac{dx}{dt}=-3x,~x(0)=x_0$ is

a) $x(t)=x_0e^{-3t}$      b) $x(t)=x_0e^{3t}$      c)$x(t)=x_0e^{-t/3}$       d) $x(t)=x_0e^{-t}$

Q9. Transformation to linear form by substitutiong $v=y^{1-n}$ of the equation $\frac{dy}{dt}+p(t)y=q(t)y^n, n>0$ will be

a) $\frac{dv}{dt}+(1-n)pv=(1-n)q$

b) $\frac{dv}{dt}+(1+n)pv=(1+n)q$

c) $\frac{dv}{dt}+(1+n)pv=(1-n)q$

d) $\frac{dv}{dt}+(1+n)pv=(1-n)q$

Q10. If $x^2\left(\frac{dy}{dx}\right)+2xy= \frac{2ln x}{x}$ and $y(1)=0$ then what is $y(e)$?

a) $e$          b) 1          c) $\frac{1}{e}$          d) $\frac{1}{e^2}$

Q11. The solution of the differential equation $x^2\frac{dy}{dx}+2xy-x+1=0$ given that at $x=1, y=0$ is

a) $\frac{1}{2}-\frac{1}{x}+\frac{1}{2x^2}$ 

        b) $\frac{1}{2}-\frac{1}{x}-\frac{1}{2x^2}$

         c) $\frac{1}{2}+\frac{1}{x}+\frac{1}{2x^2}$ 

        d) -$\frac{1}{2}+\frac{1}{x}+\frac{1}{2x^2}$

Q12. The solution of the differential equation $\frac{dy}{dx}+2xy=e^{-x^2}$ with $y(0)=1$ is

a) $(1+x)e^{x^2}$     b)$(1+x)e^{-x^2}$        c) $(1-x)e^{x^2}$        d)$(1+x)e^{-x^2}$

Q13. The solution for the differential equation $\frac{dy}{dx}=x^2y$ with the condition that $y=1$ at $x=0$ is

a) $y=e^{\frac{1}{2x}}$         b) $ln(y)=\frac{x^3}{3}+4$          c) $ln(y)=\frac{x^2}{2}$         d) $y=e^{\frac{x^3}{3}}$

Q14. The solution for the differential equation $\frac{dy}{dx}=y^2$ with initial value $y(0)=1$ is bounded in the internal is

a)$-\infty \leq x\leq \infty$    b) $-\infty \leq x\leq 1$     c) $x<1, x>1$         d) $-2 \leq x\leq 2$

Q15. Consider the differential equation $\frac{dy}{dx}=1+y^2$. Which one of the following can be particular solution of this differential equation?

a)$y=tan(x+3)$            b)$y=tanx +3$        c)$x=tan(y+3)$         d)$x=tany+3$

Q16. Which of the following is a solution to the differential equation $\frac{d}{dt}x(t)+3x(t)=0, x(0)=2?$

a)$x(t)=3e^{-t}$     b)$x(t)=2e^{-3t}$       c)$x(t)=\frac{-3}{2}t^2$       d) $x(t)=3t^2$

Q16. Solution of the differential equation $3y\frac{dy}{dx}+2x=0$ represents a family of \\

a) Ellipse         b) Circles         c)Parabolas     d) Hyperbolas.

Q17. The order of the differential equation $\frac{d^2y}{dt^2}+\left(\frac{dy}{dt}\right)^3+y^4=e^{-t}$ is

a) 1         b) 2             c)3            d)4

Q18.The solution of $x\frac{dy}{dx}+y=x^4$ with condition $y(1)=\frac{6}{5}$.

Q19. The solution of the differential equation $\frac{dy}{dx}-y^2=1$ satisfying the condition $y(0)=1$ is.

Q20.Which one of the following differential equations has a solution given by the function $y=5 sin\left(3x+\frac{\pi}{3}\right)$.

Q21. The order and degree of a differential equation $\frac{d^3y}{dx^3}+4\sqrt{\left(\frac{dy}{dx}\right)^3+y^2}=0$ are respectively.\\

a) 3 and 2         b) 2 and 3              c)3 and 3              d) 3 and 1

Q22.  Consider the differential equation $\frac{dy}{dx}+y=e^x$ with $y(0)=1$. Then find the value of $y(1)$ is

Q23.  With K as constant, the possible solution for the first order differential equation $\frac{dy}{dx}=3e^{-3x}$ is

Q24. The solution of the differential equation $\frac{dy}{dx}=ky, ~y(0)=c$ is

Q25. Consider the differential equation $\frac{dy}{dx}=(1+y^2)x$. The general solution with constant C is

Q26. The solution of the differential equation $\frac{dy}{dx}+\frac{y}{x}=x$ with the condition that $y=1$ at $x=1$ is

Q27. The integrating factor for the differential equation $\frac{dP}{dt}+k_2P=k_1L_0e^{-k_1t}$ is

Q28. Which one of the following is a linear non-homogeneous differential equation, where $x$ and $y$ are independent and dependent variable respectively?

a) $\frac{dy}{dx}+xy=e^{-x}$         b)$\frac{dy}{dx}+xy=0$         c)$\frac{dy}{dx}+xy=e^{-y}$         d) $\frac{dy}{dx}+e^{-y}=0$

Q29.  The solution of the initial value problem $\frac{dy}{dx}=-2xy; y(0)=2$ is

Q30. The general solution of the differential equation $\frac{dy}{dx}=cos(x+y),$ with c as a constant, is

Q31. The general solution of the differential equation $\frac{dy}{dx}=\frac{1+cos 2y}{1-cos 2x}$

Q32. Consider the differential equation $\frac{dx}{dt}=10-0.2x$ with initial condition $x(0)=1$. The response $x(t)$ for $t>0$ is

Q33. Consider the following differential equation $\frac{dy}{dt}=-5y$ initial condition: $y=2$ at $t=3$.

Q34. Consider the following differential equation $x(ydx+xdy)cos \frac{y}{x}=y(xdy-ydx)sin\frac{y}{x}$ which of the following is the solution of the above equation.

Q35. The solution of the equation $\frac{dQ}{dt}+Q=1$ with $Q=0~at~t=0$ is

Q36. A curve passes through the point $(x=1, y=0)$ and satisfies the differential equation $\frac{dy}{dx}=\frac{x^2+y^2}{2y}+\frac{y}{x}.$ The equation that describes the curve is 

a)$ln\left(1+\frac{y^2}{x^2}\right)=x-1$

b)$\frac{1}{2}ln\left(1+\frac{y^2}{x^2}\right)=x-1$

c)$ln\left(1+\frac{y}{x}\right)=x-1$

d)$\frac{1}{2}ln\left(1+\frac{y^2}{x^2}\right)=x-1$

Q37. The solution of the equation $x\frac{dy}{dx}+y=0$ passing through the point $(1, 1)$.

Q38. If $y$ is the solution of the differential equation $y^3\frac{dy}{dx}+x^3=0, y(0)=1$ The value of $y(-1)$ is

a) -2         b) -1         c)0        d)1

Q39. For the equation $\frac{dy}{dx}+7x^2y=0$, if $y(0)=3/7$ then the value of $y(1)$ 

Q40  The differential equation $\frac{dy}{dx}+4y=5$ is valid in the domain $0\leq x\leq 1$ with $y(0)=2.25$ The solution of the differential equation is.

Q41. The family of curves represented by the solution of the equation $\frac{dy}{dx}=-\left(\frac{x}{y}\right)^n$ For $n=-1$ and $n=+1$ respectively, are

a) Hyperbola and Parabolas b) Circles and Hyperbolas

c)Parabolas and Circles  d) Hyperbolas and circles.

Q42. What is the solution of the differential equation $\frac{dy}{dx}=\frac{x}{y}$, with the initial condition, at $x=0, y=1$?

Q43.  Find the solution of the given differential equation $\left(\frac{dy}{dx}\right)xln x=y$

Q44. The solution of the ordinary differential equation $\frac{dy}{dx}+3y=1$ Subject to the initial condition $y=1$ at $x=0$ is

Q45. One of the points which lies on the solution curves of the following differential equation $2xydx+(x^2+y^2)dy=0$ with the initial condition $y(1)=1$ is

a) (1,1)     b)(0,0)         c)(0,1)     d)(2,1).

Q46. Obtain the differential equation of all circles each of which touches the axis of $x$ at the origin.

Q47. Solve the differential equations given below:

A. $(y^2e^{xy^2}+4x^3)dx+(2xye^{xy^2}-3y^2)dy=0$.

B. $(x^3-3x^2y+2xy^2)dx-(x^3-2x^2y+y^3)dy=0$

C. $(xy^2-e^{\frac{1}{x^3}})dx-x^2ydy=0$.

D. $2ydx-xdy=xy^3dy$

E. $\frac{dy}{dx}+2xy=x^2+y^2$

F. $x^2dy+y(x+y)dx=0$

Important Links:

1) AKU previous year questions and solution from Matrices

AKU Previous year questions from Matrices:

 [AKU- Year 2020]

Q1. If $A=\begin{bmatrix}2 &0&0\\0&2&0\\0&0&2\end{bmatrix}~ and~ B=\begin{bmatrix}1&2&3\\0&1&3\\0&0&2\end{bmatrix}$                                        

Then the determinant of $AB$ has the value

a) 4            b) 8        c) 16        d) 32

Q2. The sum and product of the eigenvalues of 

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

a) 7 and 5            b) 5 and 7        c) 12 and 3        d) 3 and 12

[AKU- Year 2018]

Q3. If the nullity of the matrix $\begin{bmatrix}k&1&2\\1&-1&-2\\1&1&4 \end{bmatrix}$ is 1. Then the value of $k$ is                                                                                                                                                  

a) -1                    b) 0            c)  1        d) 2                                                       

Q4.  Let $A=\begin{bmatrix}3&0&0\\0&6&2\\0&2&6\end{bmatrix}$ and let $\lambda_1 \geq \lambda_2 \geq \lambda_3$ be the eigenvalues of $A$. Then the triple $(\lambda_1, \lambda_2, \lambda_3)$equals

a) (9, 4, 2)        b) (8,4,3)    c) (9,3,3)    d)(7,5,3)                         

Q5. If $A=\begin{bmatrix}1&0&0\\1&0&1\\0&1&0\end{bmatrix}$ then $A^{50}$ is                                                                             

a) $\begin{bmatrix}1&0&0\\50&0&0\\50&0&1\end{bmatrix}$ b) $\begin{bmatrix}1&0&0\\48&0&0\\48&0&1\end{bmatrix}$ c) $\begin{bmatrix}1&0&0\\25&1&0\\25&0&1\end{bmatrix}$ d)$\begin{bmatrix}1&0&0\\24&1&0\\24&0&1\end{bmatrix}$ 

[AKU- Year 2019]

Q6. If the eigen values of the given matrix                                

$\begin{bmatrix}-2&2&-3\\2&1&-6\\-1&-2&0\end{bmatrix}$

is 3. then the eigen value of $adj (A)$ will be

a) $-\frac{1}{3}$      b) $-\frac{1}{5}$        c)$-\frac{1}{15}$         d) -3

Q7. Write down the quadratic forms corresponding to the given matrix

$\begin{bmatrix}2&4&5\\4&3&1\\5&1&1\end{bmatrix}$     

Subjective Questions:    

AKU- YEAR 2020   

Q1. Determine the value of $p$ such that the rank of 

$\begin{bmatrix}1&1&-1&0\\4&4&-3&1\\p&2&2&2\\9&9&p&3\end{bmatrix}$ is 3

Solution: Click here for solution. 

Q2. Use Gauss-Jordan method to find the inverse of the matrix

$\begin{bmatrix}2&3&4\\4&3&1\\1&2&4\end{bmatrix}$ 

Solution: Click here for solution. 

Q3. Find the non-singular matrices $P$ and $Q$ such that

$\begin{bmatrix}1&2&3&4\\2&1&4&3\\3&0&5&-10\end{bmatrix}$

is reduced to normal form. Also find its rank.

Solution: Click here for solution. 

Q4. Solve the given equations by Cramer's rule

$x+y+z=4$

$x-y+z=0$

$2x+y+z=5$

Solution: Click here for solution.

 Q5. Verify Cayley-Hamilton theorem for the matrix

$\begin{bmatrix}2&-1&1\\-1&2&-1\\1&-1&2\end{bmatrix}$ and find the inverse. 

Solution: Click here for solution.

Q6. Find the eigen vectors of the matrix

$\begin{bmatrix}6&-2&2\\-2&3&-1\\2&-1&3\end{bmatrix}$

Hence deduce 

$6x^2+3y^2+3z^2-2yz+4zx-4xy$

to a 'sum of squares'. Also write nature of the matrix.

Solution: Click here for solution. 

Q7. Find the eigen values and eigen vectors of the matrix

$\begin{bmatrix}-2&2&-3\\2&1&-6\\-1&-2&0\end{bmatrix}$

Solution: Click here for solution.

Q8. For what values of $k$, the equations

$x+y+z=1,~2x+y+4z=k,~4x+y+10z=k^2$

Have solution? Solve them completely in each case.

Solution: Click here for solution.

Q9. Reduce the matrix

$\begin{bmatrix}-1&2&-2\\1&2&1\\-1&-1&0\end{bmatrix}$ to the diagonal form.

 Solution: Click here for solution.

Q10. Find the rank of the matrix

$\begin{bmatrix}2&3&-1&-1\\1&-1&-2&-4\\3&1&3&-2\\6&3&0&-7\end{bmatrix}$

Solution: Click here for solution.

Q11. Find the characteristic equation of the matrix

$A=\begin{bmatrix}2&1&1\\0&1&0\\1&1&2\end{bmatrix}$

and hence find the matrix represented by

$A^8-5A^7+7A^6-3A^5+A^4-5A^3+8A^2-2A+I$

Solution: Click here for solution.

Q12. State and prove Cayley-Hamilton theorem.

Solution: Click here for solution.

Q13. Reduce the quadratic form $3x^2+5y^2+3z^2-2xy-2yz+2zx$ to canonical forms.

Solution: Click here for solution.

AKU- YEAR 2018  

 Q14. Let $T:R^3\rightarrow R^4$ be a linear transformation defined by

$\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}x+y\\z+y\\x+z\\x+y+z\end{bmatrix}$

Find the matrix representation of $T$ with respect to the ordered basis

$x=\lbrace \begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\1\\0\end{bmatrix},\begin{bmatrix}0\\1\\1\end{bmatrix}\rbrace$ in $R^3$ and $y=\lbrace \begin{bmatrix}0\\1\\1\\1\end{bmatrix},\begin{bmatrix}1\\0\\1\\1\end{bmatrix},\begin{bmatrix}1\\1\\0\\1\end{bmatrix},\begin{bmatrix}1\\1\\1\\0\end{bmatrix}\rbrace $ in $R^4$

Solution: Click here for solution.

 

Central Tendency

Central Tendency:

Data can be classified in various forms. One way to distinguish between data is in terms of grouped and ungrouped data.

What is ungrouped data?

When the data has not been placed in any categories and no aggregation/summarization has taken placed on the data then it is known as ungrouped data. Ungrouped data is also known as raw data.

Height of students:

 (171,161,155,155,183,191,185,170,172,177,183,190,139,149,150,

150,152,158,159,174,178,179,190,170,143,165,167,187,169,182,

163,149,174,174,177,181,170,182,170,145,143):

This is raw/ungrouped data.

When raw data have been grouped in different classes then it is said to be grouped data.

Height of students:

(139, 143, 143, 145, 149, 149,150,150,152, 155,155, 158,159, 161, 163, 165,167, 169, 170, 170, 170 170, 171, 172, 174, 174,174, 177, 177, 178,179, 181, 182, 182, 183, 183, 185, 187, 190, 190, 191)

Before we study more about grouped and ungrouped data it is important to understand what do we mean by “Central Tendencies”?

 

Measures of central tendency

These are statistical constants which give us an idea about the concentration of the values in the central part of the distribution. The various measures of central tendency are:

1. Arithmetic Mean
2. Median
3. Mode
4. Gerometric Mean
5. Harmonic Mean

• Arithmetic Mean

    Arithmetic mean of a set of observations is their sum divided by

 the number of observations, E.g., the Arithmetic mean $\bar{x}$ of $n$ observations $x_1, x_2, x_3,........, x_n$ is given by

                  $\bar{x}=\frac{x_1+x_2+x_3+....+x_n}{n}=\frac{1}{n}\sum_{i=1}^n x_i$ (Ungrouped data) 

        In case of frequency distribution $\ x_i, f_i, i=1,2,3,......, n$ where $\ f_i$ is the frequency of the variable $\ x_i$

$\bar{x}$=$\frac{f_1x_1+f_2x_2+f_3x_3+.......+f_nx_n}{f_1+f_2+f_3+.......+f_nx_n}$=$\frac{\sum_{i=1}^n f_ix_i}{\sum_{i=1}^n f_i}$ (Grouped Data with frequency Distribution)  

       In case of continuous frequency distribution, $x$ is taken as the mid value of the corresponding class.

 

Example: Find the arithmetic mean of the following distribution

        (a) x: 150,200,300,650,250,180,400,500,550,220

Solution (a): Here the data is group data. So the arithmetic mean is

           $\bar{x}$ = $\frac{150+200+300+650+250+180+400+500+550+220}{10}$=340

Solution (b) :  Here the given data are grouped data with discrete frequency distribution. So the arithmetic mean is

                  $\bar{x}$=$\frac{1\times5+2\times9+3\times12+4\times17+5\times14+6\times10+7\times6}{5+9+12+17+14+10+6}$=4.09

Example: Find the arithmetic mean of the following distribution

                            If the values of $x$ or $f$ are large, the calculation of mean is quite time consuming and tedious. The arithmetic is required to a great extent by taken deviations of the given values from any arbitrary point "A".

                      

                      $d_i$ = $x_i - A$ ⇒$\sum_{i=1}^n d_i$ = $\sum_{i=1}^n x_i - A_n$

 

            Now , 

                         $\frac{\sum_{i=1}^n d_i}{n}$ = $\frac{\sum_{i=1}^n x_i}{n}$ - $A$

                     

                         $\bar{x}$ = $A$ + $\frac{\sum_{i=1}^n d_i}{n}$         (for ungrouped data)  

 

         

                            $\bar{x}$ = $A$+$\frac{1}{N}{\sum f_id_i}$     where $N$=$\sum f_i$

 

 

                            $d_i$ = $\frac{x_i-A}{h}$

                        $\bar{x}$= $A$+$\frac{h}{N}{\sum f_id_i}$      where, $N$= $\sum f_i$

 

 Example : Calculate the mean for the following frequency distribution

 

Solution :

  

                            

Example : The average salary of male employees in a firm was Rs. 5,200 and that of female was Rs. 4,200. The mean salary of all the employees was Rs. 5000. Find the percentage of male and female employees.