How Cheenta works to ensure student success?
Explore the Back-Story

ISI Entrance 2006 - B.Stat Subjective Paper| Problems & Solutions

Here, you will find all the questions of ISI Entrance Paper 2006 from Indian Statistical Institute's B. Stat Entrance. You will also get the solutions soon of all the previous year problems.

Problem1 :

If the normal to the curve \displaystyle{ x^{\frac{2}{3}}+y^{\frac23}=a^{\frac23} } at some point makes an angle \displaystyle{\theta} with the X-axis, show that the equation of the normal is

\displaystyle{y\cos\theta-xsin\theta =a\cos 2\theta}

Problem 2 :

Suppose that a is an irrational number.
(a) If there is a real number b such that both (a+b) and ab are rational numbers, show that a is a quadratic surd. (a is a quadratic surd if it is of the form \displaystyle{r+\sqrt{s}} or \displaystyle{r-\sqrt{s}} for some rationals r and s, where s is not the square of a rational number).
(b) Show that there are two real numbers \displaystyle{b_1} and \displaystyle{b_2} such that
i) \displaystyle{a+b_1} is rational but \displaystyle{ab_1} is irrational.
ii) \displaystyle{a+b_2} is irrational but \displaystyle{ab_2} is rational.
(Hint: Consider the two cases, where a is a quadratic surd and a is not a quadratic surd, separately).

Problem3 :

Prove that \displaystyle{n^4 + 4^{n}} is composite for all values of n greater than 1.
Discussion

Problem4 :

In the figure below, E is the midpoint of the arc ABEC and the segment ED is perpendicular to the chord BC at D. If the length of the chord AB is \displaystyle{l_1}, and that of the segment BD is \displaystyle{l_2}, determine the length of DC in terms of \displaystyle{l_1, l_2}.
B.Stat Entrance 2006 Geometry Problem
Discussion

Problem 5 :

Let A,B and C be three points on a circle of radius 1.
(a) Show that the area of the triangle ABC equals \frac{1}{2}(\sin (2 \angle A B C)+\sin (2 \angle B C A)\\+\sin (2 \angle C A B))
(b) Suppose that the magnitude of \displaystyle{\angle ABC} is fixed. Then show that the area of the triangle ABC is maximized when \displaystyle{\angle BCA=\angle CAB}
(c) Hence or otherwise, show that the area of the triangle ABC is maximum when the triangle is equilateral.

Problem 6 :

(a) Let \displaystyle{f(x)=x-xe^{-\frac1x}, x>0 }. Show that f(x) is an increasing function on \displaystyle{(0,\infty)}, and \displaystyle{\lim_{x\to\infty} f(x)=1}.
(b) Using part (a) or otherwise, draw graphs of \displaystyle{y=x-1, y=x, y=x+1} , and y=xe^{-\frac{1}{|x|}} for \displaystyle{-\infty < x < \infty} using the same X and Y axes.

Problem 7 :

For any positive integer n greater than 1, show that \displaystyle{2^n < {{2n} \choose{n}} <\frac{2^n}{\prod\limits_{i=0}^{n-1} \left(1-\frac{i}{n}\right)}}

Problem 8:

Show that there exists a positive real number \displaystyle{x\neq 2} such that \displaystyle{\log_2x=\frac{x}{2}}. Hence obtain the set of real numbers c such that \displaystyle{\frac{\log_2x}{x}=c} has only one real solution.

Problem 9 :

Find a four digit number M such that the number \displaystyle{N=4\times M} has the following properties.
(a) N is also a four digit number
(b) N has the same digits as in M but in reverse order.

Problem 10 :

Consider a function f on nonnegative integers such that f(0)=1, f(1)=0 and f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2) for \displaystyle{n \ge 2}. Show that \displaystyle{\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}}

Some useful link :

Here, you will find all the questions of ISI Entrance Paper 2006 from Indian Statistical Institute's B. Stat Entrance. You will also get the solutions soon of all the previous year problems.

Problem1 :

If the normal to the curve \displaystyle{ x^{\frac{2}{3}}+y^{\frac23}=a^{\frac23} } at some point makes an angle \displaystyle{\theta} with the X-axis, show that the equation of the normal is

\displaystyle{y\cos\theta-xsin\theta =a\cos 2\theta}

Problem 2 :

Suppose that a is an irrational number.
(a) If there is a real number b such that both (a+b) and ab are rational numbers, show that a is a quadratic surd. (a is a quadratic surd if it is of the form \displaystyle{r+\sqrt{s}} or \displaystyle{r-\sqrt{s}} for some rationals r and s, where s is not the square of a rational number).
(b) Show that there are two real numbers \displaystyle{b_1} and \displaystyle{b_2} such that
i) \displaystyle{a+b_1} is rational but \displaystyle{ab_1} is irrational.
ii) \displaystyle{a+b_2} is irrational but \displaystyle{ab_2} is rational.
(Hint: Consider the two cases, where a is a quadratic surd and a is not a quadratic surd, separately).

Problem3 :

Prove that \displaystyle{n^4 + 4^{n}} is composite for all values of n greater than 1.
Discussion

Problem4 :

In the figure below, E is the midpoint of the arc ABEC and the segment ED is perpendicular to the chord BC at D. If the length of the chord AB is \displaystyle{l_1}, and that of the segment BD is \displaystyle{l_2}, determine the length of DC in terms of \displaystyle{l_1, l_2}.
B.Stat Entrance 2006 Geometry Problem
Discussion

Problem 5 :

Let A,B and C be three points on a circle of radius 1.
(a) Show that the area of the triangle ABC equals \frac{1}{2}(\sin (2 \angle A B C)+\sin (2 \angle B C A)\\+\sin (2 \angle C A B))
(b) Suppose that the magnitude of \displaystyle{\angle ABC} is fixed. Then show that the area of the triangle ABC is maximized when \displaystyle{\angle BCA=\angle CAB}
(c) Hence or otherwise, show that the area of the triangle ABC is maximum when the triangle is equilateral.

Problem 6 :

(a) Let \displaystyle{f(x)=x-xe^{-\frac1x}, x>0 }. Show that f(x) is an increasing function on \displaystyle{(0,\infty)}, and \displaystyle{\lim_{x\to\infty} f(x)=1}.
(b) Using part (a) or otherwise, draw graphs of \displaystyle{y=x-1, y=x, y=x+1} , and y=xe^{-\frac{1}{|x|}} for \displaystyle{-\infty < x < \infty} using the same X and Y axes.

Problem 7 :

For any positive integer n greater than 1, show that \displaystyle{2^n < {{2n} \choose{n}} <\frac{2^n}{\prod\limits_{i=0}^{n-1} \left(1-\frac{i}{n}\right)}}

Problem 8:

Show that there exists a positive real number \displaystyle{x\neq 2} such that \displaystyle{\log_2x=\frac{x}{2}}. Hence obtain the set of real numbers c such that \displaystyle{\frac{\log_2x}{x}=c} has only one real solution.

Problem 9 :

Find a four digit number M such that the number \displaystyle{N=4\times M} has the following properties.
(a) N is also a four digit number
(b) N has the same digits as in M but in reverse order.

Problem 10 :

Consider a function f on nonnegative integers such that f(0)=1, f(1)=0 and f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2) for \displaystyle{n \ge 2}. Show that \displaystyle{\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}}

Some useful link :

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
Menu
Trial
Whatsapp
magic-wandrockethighlight