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Here, you will find all the questions of ISI Entrance Paper 2009 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

Two train lines intersect each other at a junction at an acute angle $ \mathbf{\theta}$. A train is passing along one of the two lines. When the front of the train is at the junction, the train subtends an angle $\mathbf{\alpha}$ at a station on the other line. It subtends an angle $\mathbf{\beta (<\alpha)}$ at the same station, when its rear is at the junction. Show that $\mathbf{ \tan\theta=\frac{2\sin\alpha\sin\beta}{\sin(\alpha-\beta)}}$

**Problem 2: **

Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $\mathbf{[0,2\pi]}$ and $\mathbf{f''(x) \geq 0 }$ for all $x$ in $ \mathbf{[0,2\pi]}$. Show that

$\mathbf{\int_{0}^{2\pi} f(x)\cos x dx \geq 0}$

**Problem 3:**

** **Let $ABC$ be a right-angled triangle with $BC=AC=1$. Let $P$ be any point on $AB$. Draw perpendiculars $PQ$ and $PR$ on $AC$ and $BC$ respectively from $P$. Define $M$ to be the maximum of the areas of $BPR$, $APQ$ and $PQCR$. Find the minimum possible value of $M$.

**Problem 4: **

A sequence is called an arithmetic progression of the first order if the differences of the successive terms are constant. It is called an arithmetic progression of the second order if the differences of the successive terms form an arithmetic progression of the first order. In general, for $ \mathbf{k\geq 2}$, a sequence is called an arithmetic progression of the $k$-th order if the differences of the successive terms form an arithmetic progression of the $(k-1)$-th order.

The numbers $4,6,13,27,50,84$ are the first six terms of an arithmetic progression of some order. What is its least possible order? Find a formula for the $n$-th term of this progression.

**Problem 5:**

** **A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimetres and its base has radius $30$ centimetres, find the volume of the largest box that can be completely enclosed inside the container with the lid on.

**Problem 6:**

** **Let $f(x)$ be a function satisfying $xf(x)=\ln x$ for $x>0$

Show that $\mathbf{f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)}$ where $ \mathbf{f^{(n)}(x) }$ denotes the $n$-th derivative evaluated at $x$.

**Problem 7: **

Show that the vertices of a regular pentagon are concyclic. If the length of each side of the pentagon is $x$, show that the radius of the circumcircle is $ \mathbf{\frac{x}{2}cosec 36^{\circ}}$.

**Problem 8: **

Find the number of ways in which three numbers can be selected from the set $ \mathbf{{1,2,\cdots ,4n}}$, such that the sum of the three selected numbers is divisible by $4$.

**Problem 9:**

** **Consider $6$ points located at $\mathbf{P_0=(0,0), P_1=(0,4), P_2=(4,0), P_3=(-2,-2), P_4=(3,3), P_5=(5,5)}$. Let $R$ be the region consisting of all points in the plane whose distance from $P_0$ is smaller than that from any other $ \mathbf{P_i, i=1,2,3,4,5}$. Find the perimeter of the region $R$.

**Problem 10: **

Let $\mathbf{x_n}$ be the $n$-th non-square positive integer. Thus $ \mathbf{x_1=2, x_2=3, x_3=5, x_4=6}$ , etc. For a positive real number $x$, denotes the integer closest to it by $\langle x\rangle$ . If $x=m+0.5$, where $m$ is an integer, then define $\langle x\rangle=m$. For example, $\langle 1.2\rangle=1,\langle 2.8\rangle=3,\langle 3.5\rangle=3 . \text { Show that } x_{n}=n+\langle\sqrt{n}\rangle$

Here, you will find all the questions of ISI Entrance Paper 2009 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

Two train lines intersect each other at a junction at an acute angle $ \mathbf{\theta}$. A train is passing along one of the two lines. When the front of the train is at the junction, the train subtends an angle $\mathbf{\alpha}$ at a station on the other line. It subtends an angle $\mathbf{\beta (<\alpha)}$ at the same station, when its rear is at the junction. Show that $\mathbf{ \tan\theta=\frac{2\sin\alpha\sin\beta}{\sin(\alpha-\beta)}}$

**Problem 2: **

Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $\mathbf{[0,2\pi]}$ and $\mathbf{f''(x) \geq 0 }$ for all $x$ in $ \mathbf{[0,2\pi]}$. Show that

$\mathbf{\int_{0}^{2\pi} f(x)\cos x dx \geq 0}$

**Problem 3:**

** **Let $ABC$ be a right-angled triangle with $BC=AC=1$. Let $P$ be any point on $AB$. Draw perpendiculars $PQ$ and $PR$ on $AC$ and $BC$ respectively from $P$. Define $M$ to be the maximum of the areas of $BPR$, $APQ$ and $PQCR$. Find the minimum possible value of $M$.

**Problem 4: **

A sequence is called an arithmetic progression of the first order if the differences of the successive terms are constant. It is called an arithmetic progression of the second order if the differences of the successive terms form an arithmetic progression of the first order. In general, for $ \mathbf{k\geq 2}$, a sequence is called an arithmetic progression of the $k$-th order if the differences of the successive terms form an arithmetic progression of the $(k-1)$-th order.

The numbers $4,6,13,27,50,84$ are the first six terms of an arithmetic progression of some order. What is its least possible order? Find a formula for the $n$-th term of this progression.

**Problem 5:**

** **A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimetres and its base has radius $30$ centimetres, find the volume of the largest box that can be completely enclosed inside the container with the lid on.

**Problem 6:**

** **Let $f(x)$ be a function satisfying $xf(x)=\ln x$ for $x>0$

Show that $\mathbf{f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)}$ where $ \mathbf{f^{(n)}(x) }$ denotes the $n$-th derivative evaluated at $x$.

**Problem 7: **

Show that the vertices of a regular pentagon are concyclic. If the length of each side of the pentagon is $x$, show that the radius of the circumcircle is $ \mathbf{\frac{x}{2}cosec 36^{\circ}}$.

**Problem 8: **

Find the number of ways in which three numbers can be selected from the set $ \mathbf{{1,2,\cdots ,4n}}$, such that the sum of the three selected numbers is divisible by $4$.

**Problem 9:**

** **Consider $6$ points located at $\mathbf{P_0=(0,0), P_1=(0,4), P_2=(4,0), P_3=(-2,-2), P_4=(3,3), P_5=(5,5)}$. Let $R$ be the region consisting of all points in the plane whose distance from $P_0$ is smaller than that from any other $ \mathbf{P_i, i=1,2,3,4,5}$. Find the perimeter of the region $R$.

**Problem 10: **

Let $\mathbf{x_n}$ be the $n$-th non-square positive integer. Thus $ \mathbf{x_1=2, x_2=3, x_3=5, x_4=6}$ , etc. For a positive real number $x$, denotes the integer closest to it by $\langle x\rangle$ . If $x=m+0.5$, where $m$ is an integer, then define $\langle x\rangle=m$. For example, $\langle 1.2\rangle=1,\langle 2.8\rangle=3,\langle 3.5\rangle=3 . \text { Show that } x_{n}=n+\langle\sqrt{n}\rangle$

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can someone explain to me the solution to the last problem.

Please discuss problem no.1 and forward video