$( 2k + 2m + 1 = 101^2; 2k - 2m + 1 = 1)$ Subtracting this pair of equations we get $(4m = 101^2 - 1)$ or $(4m = 100 \times 102$) or m = 50 × 51
This gives N = 2601 (ignoring extraneous solutions)
Case 2:
$(2k + 2m + 1 = 101 ; 2k - 2m + 1 = 101)$ which gives m = 0 or N = 101. This solution we ignore as it makes $N(N- 101) = 0$ (a non positive square).
Hence the only solution is $N = 2601$ and there are no other values of N which makes $N(N-101)$ a perfect square.
ISI Entrance Paper 2013 - B.Stat, B.Math Subjective
Here, you will find all the questions of ISI Entrance Paper 2013 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.
Let $i=\sqrt{-1}$ and $S=\{i+i^{2}+\cdots+i^{n}: n \geq 1\}$. The number of distinct real numbers in the set $S$ is (A) $1$ (B) $2$ (C) $3$ (D) infinite.
Problem 2:
From a square of unit length, pieces from the corners are removed to form a regular octagon. Then, the value of the area removed is (A) $1 / 2$ (B) $1 / \sqrt{2}$ (C) $\sqrt{2}-1$ (D) $(\sqrt{2}-1)^{2}$.
Problem 3:
We define the dual of a line $y=m x+c$ to be the point $(m,-c)$. Consider a set of $n$ non-vertical lines, $n>3$, passing through the point $(1,1)$ . Then the duals of these lines will always (A) be the same (B) lie on a circle (C) lie on a line (D) form the vertices of a polygon with positive area.
Problem 4:
Suppose $\alpha, \beta$ and $\gamma$ are three real numbers satisfying
Suppose $f$ is a differentiable and increasing function on $[0,1]$ such that
$f(0)<$ $0<f(1) .$ Let $F(t)=\int_{0}^{t} f(x) d x .$ Then
(A) $F$ is an increasing function on $[0,1]$ (B) $F$ is a decreasing function on $[0,1]$ (C) $F$ has a unique maximum in the open interval $(0,1)$ (D) $F$ has a unique minimum in the open interval $(0,1)$.
Problem 8:
In an isosceles triangle $\Delta A B C,$ the angle $\angle A B C=120^{\circ} .$ Then the ratio of two sides $A C: A B$ is (A) $2: 1$ (B) $3: 1$ (C) $\sqrt{2}: 1$ (D) $\sqrt{3}: 1$
Problem 9:
Let $x, y, z$ be positive real numbers. If the equation $x^{2}+y^{2}+z^{2}=(x y+y z+z x) \sin \theta$ has a solution for $\theta,$ then $x, y$ and $z$ must satisfy
(A) $x=y=z$ (B) $x^{2}+y^{2}+z^{2} \leq 1$ (C) $x y+y z+z x=1$ (D) $0<x, y, z \leq 1$
Problem 10:
Suppose $\sin \theta=\frac{4}{5}$ and $\sec \alpha=\frac{7}{4}$ where $0 \leq \theta \leq \frac{\pi}{2}$ and $-\frac{\pi}{2} \leq \alpha \leq 0 .$ Then $\sin (\theta+\alpha)$ is
Let $i=\sqrt{-1}$ and $z_{1}, z_{2}, \ldots$ be a sequence of complex numbers defined by $z_{1}=i$ and $z_{n+1}=z_{n}^{2}+i$ for $n \geq 1$. Then $\left|z_{2013}-z_{1}\right|$ is
(A) $0$ (B) $1$ (C) $2$ (D) $\sqrt{5}$.
Problem 12:
The last digit of the number $2^{100}+5^{100}+8^{100}$ is
(A) $1$ (B) $3$ (C) $5$ (D) $7$ .
Problem 13:
The maximum value of $|x-1|$ subject to the condition $|x^{2}-4| \leq 5$ is
(A) $2$ (B) $3$ (C) $4$ (D) $5$ .
Problem 14:
Which of the following is correct?
(A) $e x \leq e^{x}$ for all $x$.
(B) $e x<e^{x}$ for $x<1$ and $e x \geq e^{x}$ for $x \geq 1$
(C) $e x \geq e^{x}$ for all $x$
(D) $e x<e^{x}$ for $x>1$ and $e x \geq e^{x}$ for $x \leq 1$.
Problem 15:
The area of a regular polygon of $12$ sides that can be inscribed in the circle $x^{2}+y^{2}-6 x+5=0$ is
(A) $6$ units (B) $9$ units (C) $12$ units (D) $15$ units.
Problem 16:
Let $f(x)=\sqrt{\log _{2} x-1}+\frac{1}{2} \log _{\frac{1}{2}} x^{3}+2$. The set of all real values of $x$ for which the function $f(x)$ is defined and $f(x)<0$ is
(A) $x>2$ (B) $x>3$ (C) $x>e$ (D) $x>4$
Problem 17:
Let $a$ be the largest integer strictly smaller than $\frac{7}{8} b$ where $b$ is also an integer. Consider the following inequalities: (1) $\frac{7}{8} b-a \leq 1$ (2) $\frac{7}{8} b-a \geq \frac{1}{8}$
and find which of the following is correct. (A) Only (1) is correct. (B) Only (2) is correct. (C) Both (1) and (2) are correct. (D) None of them is correct.
Problem 18:
The value of $\lim _{x \rightarrow-\infty} \sum_{k=1}^{1000} \frac{x^{k}}{k !}$ is
(A) $-\infty$ (B) $\infty$ (C) $0$ (D) $e^{-1}$.
Problem 19:
For integers $m$ and $n$, let $f_{m . n}$ denote the function from the set of integers to itself, defined by $f_{m, n}(x)=m x+n$ Let $\mathcal{F}$ be the set of all such functions, $\mathcal{F}=\{f_{m, n}: m, n \text { integers }\}$
Call an element $f \in \mathcal{F}$ invertible if there exists an element $g \in \mathcal{F}$ such that $g(f(x))=f(g(x))=x$ for all integers $x$. Then which of the following is true?
(A) Every element of $\mathcal{F}$ is invertible. (B) $\mathcal{F}$ has infinitely many invertible and infinitely many non-invertible elements. (C) $\mathcal{F}$ has finitely many invertible elements. (D) No element of $\mathcal{F}$ is invertible.
Problem 20:
Consider six players $P_{1}, P_{2}, P_{3}, P_{4}, P_{5}$ and $P_{6}$. A team consists of two players. (Thus, there are 15 distinct teams.) Two teams play a match exactly once if there is no common player. For example, team $\{P_{1}, P_{2}\}$ can not play with $\{P_{2}, P_{3}\}$ but will play with $\{P_{4}, P_{5}\}$. Then the total number of possible matches is
(A) $36$ (B) $40$ (C) $45$ (D) $54$
Problem 21:
The minimum value of $f(\theta)=9 \cos ^{2} \theta+16 \sec ^{2} \theta$ is
(A) $25$ (B) $24$ (C) $20$ (D) $16$ .
Problem 22:
The number of $0^{\prime}$ s at the end of the integer $100 !-101 !+\cdots-109 !+110 !$ is (A) $24$ (B) $25$ (C) $26$ (D) $27$
Problem 23:
We denote the largest integer less than or equal to $z$ by $[z]$. Consider the identity $(1+x)(10+x)\left(10^{2}+x\right) \cdots\left(10^{10}+x\right)=10^{a}+10^{b} x+a_{2} x^{2}+\cdots+a_{11} x^{11}$ Then (A) $[a]>[b]$ (B) $[a]=[b]$ and $a>b$ (C) $[a]<[b]$ (D) $[a]=[b]$ and $a<b$.
Problem 24:
The number of four tuples $(a, b, c, d)$ of positive integers satisfying all three equations
$a^{3} =b^{2} \\ c^{3} =d^{2} \\ c-a =64$ is
(A) $0$ (B) $1$ (C) $2$ (D) $4$ .
Problem 25:
The number of real roots of $e^{x}=x^{2}$ is
(A) $0$ (B) $1$ (C) $2$ (D) $3$.
Problem 26:
Suppose $\alpha_{1}, \alpha_{2}, \alpha_{3}$ and $\alpha_{4}$ are the roots of the equation $x^{4}+x^{2}+1=0$. Then the value of $\alpha_{1}^{4}+\alpha_{2}^{4}+\alpha_{3}^{4}+\alpha_{4}^{4}$ is
(A) $-2$ (B) $0$ (C) $2$ (D) $4$.
Problem 27:
Among the four time instances given in the options below, when is the angle between the minute hand and the hour hand the smallest?
Suppose all roots of the polynomial $P(x)=a_{10} x^{10}+a_{9} x^{9}+\cdots+a_{1} x+a_{0}$ are real and smaller than $1 .$ Then, for any such polynomial, the function $f(x)=a_{10} \frac{e^{10 x}}{10}+a_{9} \frac{e^{9 x}}{9}+\cdots+a_{1} e^{x}+a_{0} x, x>0$
(A) is increasing (B) is either increasing or decreasing (C) is decreasing (D) is neither increasing nor decreasing.
Problem 29:
Consider a quadrilateral $A B C D$ in the XY-plane with all of its angles less than $180^{\circ} .$ Let $P$ be an arbitrary point in the plane and consider the six triangles each of which is formed by the point $P$ and two of the points $A, B, C, D .$ Then the total area of these six triangles is minimum when the point $P$ is
(A) outside the quadrilateral (B) one of the vertices of the quadrilateral (C) intersection of the diagonals of the quadrilateral (D) none of the points given in (A), (B) or (C).
Problem 30:
The graph of the equation $x^{3}+3 x^{2} y+3 x y^{2}+y^{3}-x^{2}+y^{2}=0$ comprises (A) one point (B) union of a line and a parabola (C) one line (D) union of a line and a hyperbola.
Let $a, b, c$ be real numbers greater than $1$. Let $S$ denote the sum
$S =\log_{a}bc + \log_{b}ca + \log_{c}ab$. Find the smallest possible value of $S$.
Problem 2:
For $x>0$ define $f(x)=\frac{1}{x+2 \cos (x)}$. Determine the set ${y \in \mathbb{R}: y=f(x), x \geq 0}$
Problem 3:
Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ be a function satisfying $|f(x+y)-f(x-y)-y| \leq y^{2}$ for all $x, y \in \mathbb{R}$. Show that $f(x)=\frac{x}{2}+c$, where $c$ is a constant.
Problem 4:
In a badminton singles tournament, each player played against all the others exactly once and each game had a winner. After all the games, each player listed the names of all the players she defeated as well as the names of all the players defeated by the players defeated by her. For instance, if $A$ defeats $B$ and $B$ defeats $C$. then in the list of $A$ both $B$ and $C$ are included. Prove that at least one player listed the names of all other players.
Problem 5:
Let $A D$ be a diameter of a circle of radius $r$. Let $B, C$ be points on the semicircle (with $C$ distinct from $A$ ) so that $A B=B C=\frac{r}{2}$. Determine the ratio of the length of the chord $C D$ to the radius.
Problem 6:
Let $p(x), q(x)$ be distinct polynomials with real coefficients such that the sum of the coefficients of each of the polynomials equals s. If $(p(x))^{3}-(q(x))^{3}=$ $p\left(x^{3}\right)-q\left(x^{3}\right),$ then prove the following:
(1) $p(x)-q(x)=(x-1)^{a} r(x)$ for some integer $a \geq 1$ and a polynomial $r(x)$ with $r(1) \neq 0$.
(2) $s^{2}=3^{a-1}$ where $a$ is as given in $(a)$.
Problem 7:
Let $N$ be a positive integer such that $N(N-101)$ is the square of a positive integer. Then determine all possible values of $N$. (Note that $101$ is a prime number).
Problem 8:
Let $A B C D$ be a square with the side $A B$ lying on the line $y=x+8$. Suppose $C, D$ lie on the parabola $x^{2}=y$. Find the possible values of the length of the side of the square.
ISI B.Stat & B.Math Paper 2012 Objective| Problems & Solutions
Here, you will find all the questions of ISI Entrance Paper 2012 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.
Consider the equation $x^2 + y^2 = 2007$. How many solutions $(x,y)$ exist such that $x$ and $y$ are positive integer?
(A) None (B) Exactly two (C) More than two but finitely many (D) Infinitely many.
Discussion:
Problem 3:
Consider the functions $f_{1}(x)=x, f_{2}(x)=2+\log _{e} x, x>0$ (where $e$ is the base of natural logarithm ). The graphs of the functions intersect
(A) once in (0,1) and never in $(1, \infty)$
(B) once in (0,1) and once in $\left(e^{2}, \infty\right)$
(C) once in (0,1) and once in $\left(e, e^{2}\right)$
(D) more than twice in $(0, \infty)$.
Problem 4:
Consider the sequence
$$ u_{n}=\sum_{r=1}^{n} \frac{r}{2^{r}}, n \geq 1 $$
Then the limit of $u_{n}$ as $n \rightarrow \infty$ is (A) $1$ (B) $2$ (C) $e$ (D) $1 / 2$.
Problem 5:
Suppose that $z$ is any complex number which is not equal to any of $\{3,3 \omega, 3 \omega^{2}\}$ where $\omega$ is a complex cube root of unity. Then
Consider all functions $f:\{1,2,3,4\} \rightarrow \{1,2,3,4\}$ which are one-one, onto and satisfy the following property: if $f(k)$ is odd then $f(k+1)$ is even, $k=1,2,3$. The number of such functions is (A) $4$ (B) $8$ (C) $12$ (D) $16$ .
Problem 7:
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined by
$f(x)=\left\{\begin{array}{cl}e^{-\frac{1}{x}}, & x \geq 0 \\ 0 & x \leq 0\end{array}\right.$ Then (A) $f$ is not continuous (B) $f$ is differentiable but $f^{\prime}$ is not continuous (C) $f$ is continuous but $f^{\prime}(0)$ does not exist (D) $f$ is differentiable and $f^{\prime}$ is continuous.
Problem 8:
The last digit of $9 !+3^{9966}$ is (A) $3$ (B) $9$ (C) $7$ (D) $1$ .
Problem 9:
Consider the function $f(x)=\frac{2 x^{2}+3 x+1}{2 x-1}, 2 \leq x \leq 3$. Then
(A) maximum of $f$ is attained inside the interval $(2,3)$ (B) minimum of $f$ is $\frac{28}{5}$ (C) maximum of $f$ is $\frac{28}{5}$ (D) $f$ is a decreasing function in $(2,3)$ .
Problem 10:
A particle $P$ moves in the plane in such a way that the angle between the two tangents drawn from $P$ to the curve $y^{2}=4 a x$ is always $90^{\circ} .$ The locus of $P$ is (A) a parabola (B) a circle (C) an ellipse (D) a straight line.
Problem 11:
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x)=\left|x^{2}-1\right|, x \in \mathbb{R}$. Then
(A) $f$ has a local minima at $x=\pm 1$ but no local maximum (B) $f$ has a local maximum at $x=0$ but no local minima (C) $f$ has a local minima at $x=\pm 1$ and a local maximum at $x=0$ (D) none of the above is true.
Problem 12:
The number of triples $(a, b, c)$ of positive integers satisfying $2^{a}-5^{b} 7^{c}=1$ is
(A) infinite (B) $2$ (C) $1$ (D) $0$ .
Problem 13:
Let $a$ be a fixed real number greater than $-1 .$ The locus of $z \in \mathbb{C}$ satisfying $|z-i a|=Im(z)+1$ is (A) parabola (B) ellipse (C) hyperbola (D) not a conic.
Problem 14:
Which of the following is closest to the graph of $\tan (\sin x), x>0 ?$
Problem 15:
Consider the function $f: \mathbb{R} \backslash{1} \rightarrow \mathbb{R} \backslash{2}$ given by $f(x)=\frac{2 x}{x-1}$. Then
(A) $f$ is one-one but not onto (B) $f$ is onto but not one-one (C) $f$ is neither one-one nor onto (D) $f$ is both one-one and onto.
Problem 16:
Consider a real valued continuous function $f$ satisfying $f(x+1)=f(x)$ for all $x \in \mathbb{R} .$ Let $g(t)=\int_{0}^{t} f(x) d x, \quad t \in \mathbb{R}$. Define $h(t)=\lim _{n \rightarrow \infty} \frac{q(t+n)}{n}$, provided the limit exists. Then
(A) $h(t)$ is defined only for $t=0$ (B) $h(t)$ is defined only when $t$ is an integer (C) $h(t)$ is defined for all $t \in \mathbb{R}$ and is independent of $t$ (D) none of the above is true.
Problem 17:
Consider the sequence $a_{1}=24^{1 / 3}, a_{n+1}=\left(a_{n}+24\right)^{1 / 3}, n \geq 1$. Then the integer part of $a_{100}$ equals (A) $2$ (B) $10$ (C) $100$ (D) $24$ .
Problem 18:
Let $x, y \in(-2,2)$ and $x y=-1 .$ Then the minimum value of $\frac{4}{4-x^{2}}+\frac{9}{9-y^{2}}$ is
What is the limit of $\left(1+\frac{1}{n^{2}+n}\right)^{n^{2}+\sqrt{n}}$ as $n \rightarrow \infty$ ? (A) $e$ (B) $1$ (C) $0$ (D) $\infty$.
Problem 20:
Consider the function $f(x)=x^{4}+x^{2}+x-1, x \in(-\infty, \infty) .$ The function (A) is zero at $x=-1,$ but is increasing near $x=-1$ (B) has a zero in $(-\infty,-1)$ (C) has two zeros in (-1,0) (D) has exactly one local minimum in (-1,0) .
Problem 21:
Consider a sequence of 10 A's and 8 B's placed in a row. By a run we mean one or more letters of the same type placed side by side. Here is an arrangement of $10 A$ 's and $8 B$ 's which contains 4 runs of $A$ and 4 runs of $B:$
AAABBABBBAABAAAABB
In how many ways can $10 A$ 's and $8 B$ 's be arranged in a row so that there are 4 runs of $A$ and 4 runs of $B ?$ (A)$2 {{9} \choose {3}}$ ${{7} \choose {3}}$ (B) ${{9} \choose {3}}$ ${{7} \choose {3}}$ (C) ${{10} \choose {4}}$ ${{8} \choose {4}}$ (D) ${{10} \choose {5}}$ ${{8} \choose {5}}$.
Problem 22:
Suppose $n \geq 2$ is a fixed positive integer and $f(x)=x^{n}|x|, x \in \mathbb{R}$. Then
(A) $f$ is differentiable everywhere only when $n$ is even (B) $f$ is differentiable everywhere except at 0 if $n$ is odd (C) $f$ is differentiable everywhere (D) none of the above is true.
Problem 23:
The line $2 x+3 y-k=0$ with $k>0$ cuts the $x$ axis and $y$ axis at points $A$ and $B$ respectively. Then the equation of the circle having $A B$ as diameter is (A) $x^{2}+y^{2}-\frac{k}{2} x-\frac{k}{3} y=k^{2}$ (B) $x^{2}+y^{2}-\frac{k}{3} x-\frac{k}{2} y=k^{2}$ (C) $x^{2}+y^{2}-\frac{k}{2} x-\frac{k}{3} y=0$ (D) $x^{2}+y^{2}-\frac{k}{3} x-\frac{k}{2} y=0$.
Problem 24:
Let $\alpha>0$ and consider the sequence $x_{n}=\frac{(\alpha+1)^{n}+(\alpha-1)^{n}}{(2 \alpha)^{n}}, n=1,2, \ldots$
Then $lim_{x \to \infty} x_n$ is
(A) $0$ for any $\alpha>0$ (B) $1$ for any $\alpha>0$ (C) $0$ or $1$ depending on what $\alpha>0$ is (D) $0,1$ or $\infty$ depending on what $\alpha>0$ is.
Consider a cardboard box in the shape of a prism as shown below. The length of the prism is 5 . The two triangular faces $A B C$ and $A^{\prime} B^{\prime} C^{\prime}$ are congruent and isosceles with side lengths 2,2,3 . The shortest distance between $B$ and $A^{\prime}$ along the surface of the prism is (A) $\sqrt{29}$ (B) $\sqrt{28}$ (C) $\sqrt{29-\sqrt{5}}$ (D) $\sqrt{29-\sqrt{3}}$
Problem 27:
Assume the following inequalities for positive integer $k : \frac{1}{2 \sqrt{k+1}}<\sqrt{k+1}-\sqrt{k}<\frac{1}{2 \sqrt{k}}$. The integer part of $\sum_{k=2}^{9099} \frac{1}{\sqrt{k}}$
equals (A) $198$ (B) $197$ (C) $196$ (D) $195$.
Problem 28:
Consider the sets defined by the inequalities $A=\{(x, y) \in \mathbb{R}^2: x^4+y^2 \leq 1\}, B=\{(x, y) \in \mathbb{R}^2: x^6+y^4\leq 1\}$ Then (A) $B \subseteq A$ (B) $A \subseteq B$ (C) each of the sets $A-B, B-A$ and $A \cap B$ is non-empty (D) none of the above is true.
Problem 29:
The number $\left(\frac{2^{10}}{11}\right)^{11}$ is
If the roots of the equation $x^{4}+a x^{3}+b x^{2}+c x+d=0$ are in geometric progression then (A) $b^{2}=a c$ (B) $a^{2}=b$ (C) $a^{2} b^{2}=c^{2}$ (D) $c^{2}=a^{2} d$.
Let $X, Y, Z$ be the angles of a triangle. (1) Prove that $\tan \frac{X}{2} \tan \frac{Y}{2}+\tan \frac{X}{2} \tan \frac{Z}{2}+\tan \frac{Z}{2} \tan \frac{Y}{2}=1$
(2) Using (1) or otherwise prove that $\tan \frac{X}{2} \tan \frac{Y}{2} \tan \frac{Z}{2} \leq \frac{1}{3 \sqrt{3}}$
Problem 2:
Let $\alpha$ be s real number. Consider the function $g(x)=(\alpha+|x|)^{2} e^{(5-|x|)^{2}},-\infty<x<\infty \cdot \infty .$
(i) Determine the values of $\alpha$ for which $g$ is continuous at all $x$. (ii) Determine the values of $\alpha$ for which $g$ is differentiable at all $x$.
Problem 3:
Write the set of all positive integers in triangular array as
Find the row number and column number where $20096$ occurs. For example $8$ appears in the third row and second column.
Problem 4:
Show that the polynomial $x^{8}-x^{7}+x^{2}-x+15$ has no real root.
Problem 5:
Let $m$ be a natural number with digits consisting entirely of 6 's and 0 's. Prove that $m$ is not the square of a natural number.
Problem 6:
Let $0<a<b$. (i) Show that amongst the triangles with base $a$ and perimeter $a+b$ the maximum area is obtained when the other two sides have equal length $\frac{b}{2}$. (ii) Using the result (i) or otherwise show that amongst the quadrilateral of given perimeter the square has maximum area.
Problem 7:
Let $0<a<b$. Consider two circles with radii $a$ and $b$ and centers $(a, 0)$ and $(0, b)$ respectively with $0<a<b$. Let $c$ be the center of any circle in the crescent shaped region $M$ between the two circles and tangent to both (See figure below). Determine the locus of $c$ as its circle traverses through region $M$ maintaining tangency.
Problem 8:
Let $n \geq 1$, and $S={1,2, \ldots, n}$.For a function $f: S \rightarrow S$, a subset $D \subset S$ is said $t$ be invariant under $f,$ if $f(x) \in D$ for all $x \in D$. Note that the empty set and $S$ are invariant for all $f $. Let $\deg(f)$ be the number of subsets of $S$ invariant under $f$. (i) Show that there is a function $f: S \rightarrow S$ such that $\deg(f)=2$. (ii) Further show that for any $k$ such that $1 \leq k \leq n$ there is a function $f: S \rightarrow S$ such that $\deg(f)=2^{k}$
