INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More

Here are the problems and their corresponding solutions from BStat Hons Objective Admission Test 2005.

**Problem 1: **

How many three-digit numbers of distinct digits can be formed by using the digits 1, 2, 3, 4, 5, 9 such that the sum of the digits is at least 12?

(A) $61$ (B) $66$ (C) $60$ (D) $11$

Discussion:

We will use the method of compliment of counting. First we count the number ways of selecting three distinct numbers from the six digits:

$$ {{6} \choose {3} }= \frac {6 \times 5 \times 4 }{3\times 2 \times 1} = 20 $$

We want to count the cases that do not work (sum becomes less than 12).

Note that if 9 is one of the three digits then the sum of the digits will be 12 or more (least case is 9+1+2 =12).

Now if we restrict to the remaining five available digits, none of the cases work except {5, 4, 3} (as 5 + 4 + 3 = 12).

Therefore we select 3 distinct digits from the five digits and delete the one selection of {5, 4, 3}. This gives us the total number of bad cases.

$$ {{5}\choose {3}} - 1 = 9 $$

Hence the good selections are: total - bad = 20 - 9 =11.

Next note that each selection of three distinct integers can be permuted in 3!= 6 ways. Hence the total number of three -digit numbers that can be formed with the desired property = $( 11 \times 6 = 66 )$

Ans: (B) 66

**Problem 2: **

If $( \sqrt{3} + 1 )$ is a root of the equation $3x^3 + ax^2 + bx + 12 = 0$ where $a$ and $b$ are rational numbers, then b is equal to

(A) -6; (B) 2; (C) 6; (D) 10;

Discussion:

Clearly if $( 1 + \sqrt {3} )$ is one of the roots, then it's conjugate $(1 - \sqrt {3} )$ is another root (as all the coefficients are rational numbers).

Suppose the third root is $( \gamma )$. (By Fundamental theorem of algebra there are three roots of a cubic).

By Vieta's Theorem, the product of the roots is $(\frac{-12}{3} = -4)$.

Hence:

$$ (1 + \sqrt{3} ) \times (1 - \sqrt {3}) \times \gamma = - 4 $$

This implies $( \gamma = 2 )$ .

Now Vieta's Theorem says, $( \frac{b}{3} )$ is sum of the product of roots taken two at a time. Thus:

$$ \frac{b}{3} = (1 + \sqrt{3} ) \times (1 - \sqrt {3}) + \gamma \times { (1 + \sqrt{3} ) + (1 - \sqrt {3}) } = 2 $$.

This implies b = 6

Ans: (C) 6;

**Problem 3: **

The sum of all integers from $1$ to $1000$ that are divisible by $2$ or $5$ but not divisible by $4$ equals

(A) $175000$; (B) $225500$; (C) $149500$; (D) $124000$;

Discussion:

This is a problem related to Inclusion and Exclusion Principle in Combinatorics.

First lets add numbers from 1 to 1000 divisible by 2 but not by 4. So starting from 2 we add every fourth number. It is an arithmetic progression with first term 2, last term 998 and number of terms 250 (there are 500 even numbers from 1 to 1000 and ( $\frac{1000}{4} = 250$ ) is divisible by 4; hence the remaining 250 are divisible by 2 and not 4).

Therefore sum of the terms is:

$$ \frac {250}{2} (2 + 998) = 500 \times 250 = 125000 $$

Next we add the odd numbers that are divisible by 5 (we have already added the even ones).

So starting from 5, 15, 25, ... , upto 995.

There are 100 such numbers and this is an arithmetic progression with common difference 10, first term 5 and last term 995.

Hence the sum is:

$$ \frac{100}{2}(5 + 995)= 500 \times 100 = 50000 $$

Thus the total required sum is 125000+50000 = 175000.

Ans: (A) 175000;

**Problem 4: **

The value of $\{ \frac {1}{2} ( -1 + \sqrt{3} i ) \}^{15} + \{ \frac {1}{2} ( -1 - \sqrt{3} i ) \}^{15}$ is

(A) $-1$; (B) $0$; (C) $( \frac{1}{2^{14}} )$; (D) $2$;

Discussion:

This is an application of De Moivre's Theorem.

$\lbrace \frac {1}{2} ( -1 + \sqrt{3} i ) \rbrace ^{15} + \lbrace \frac {1}{2} ( -1 - \sqrt{3} i ) \rbrace^{15} $

$ = \left (\cos \frac{2 \pi}{3} + i \sin \frac {2 \pi}{3} \right )^{15} + \left (\cos \frac{4\pi}{3} + i \sin \frac {4\pi}{3} \right )^{15} $

$= \cos \frac{2 \times 15 \times \pi}{3} + i \sin \frac {2 \times 15 \times \pi}{3} + \cos \frac{4\times 15\times \pi}{3} + i \sin \frac {4\times 15 \times \pi}{3} $

$= \cos 10 \pi + i \sin 10 \pi + \cos 20 \pi + i \sin 20 \pi = 2$

Ans: (D) 2;

**Problem 5: **

The equation x(x+3) = y(y-1) -2 represents

(A) a hyperbola; (B) a pair of straight lines;

(C) a point; (D) none of the foregoing curves;

Discussion:

Note that

$x(x+3) = y(y-1) - 2 \ \Rightarrow (x^2 + 3x) - (y^2 - y ) = -2 $

$\Rightarrow (x^2 + 2 \left( \frac{3}{2} \right ) x + \left ( \frac{3}{2} \right )^2 ) - ( y^2 - 2 \left (\frac{1}{2} \right ) y + \left( \frac {1}{2} \right )^2 ) = 0 $

$\Rightarrow \left (x + \frac{3}{2} \right )^2 = \left (y - \frac{1}{2} \right )^2 $

$\Rightarrow x + \frac{3}{2} = y - \frac{1}{2} \text { or } x + \frac{3}{2} = -y + \frac{1}{2} $

Hence we get a pair of straight lines.

Ans: (B) a pair of straight lines;

**Problem 6: **

Let $a$ be the $81$- digit number all digits of which are equal to $1$. Then the number $a$ is

(A) divisible by $9$ but not divisible by $27$;

(B) divisible by $27$ but not divisible by $81$;

(C) divisible by $81$ but not divisible by $243$;

(D) divisible by $243$.

**Problem 7: **

Let $P(x)$ be a polynomial of degree $11$ such that $P(x) = \frac{1}{x+1}$, for $x = 0,1,2, \cdots11$.

Then the value of $P(12)$

(A) equals 0;

(B) equals 1;

(C) equals $\frac{1}{13}$;

(D) cannot be determined from the given information.

**Problem 8: **

If $x=\log _{e}(\frac{1}{\sqrt{\tan 15^{\circ}}})$, then the value of $\frac{\sum_{n=0}^{\infty} e^{-2 n x}}{\sum_{n=0}^{\infty}(-1)^{n} e^{-2 n x}}$

equals

(A) $\sqrt{3}$

(B) $\frac{1}{\sqrt{3}}$

(C) $\frac{\sqrt{3}+1}{\sqrt{3}-1}$;

(D) $\frac{\sqrt{3}-1}{\sqrt{3}+1}$.

**Problem 9:**

Define $f(x)=\sum_{n=0}^{\infty} \frac{x^{2 n+1}}{(2 n+1) !}$ and $g(x)=\sum_{n=0}^{\infty} \frac{x^{2 n}}{(2 n) !}$, where $x$ is a real number.

Then

(A) $f(x)>g(x)$ for all $x$;

(B) $f(x)<g(x)$ for all $x$;

(C) $f(x)=g(x)$ for alt $x$;

(D) none of the above statements need necessarily hold for all $x$.

**Problem 10:**

The number of roots of the equation $ \sin \pi x=x^{2}-x+\frac{5}{4}$ is

(A) $0$;

(B) $1$;

(c) $2$;

(D) $4$.

**Problem 11:**

Let $P=(0, a), Q=(b, 0), R=(c, d),$ be three points such that $a, b, c$ and $d$ are all positive and the origin and the point $R$ are on the opposite sides of $P Q$. Then the area of the triangle $P Q R$ is equal to

(A) $\frac{a d+b c-a b}{2} ;$

(B) $\frac{a b+a c-b d}{2} ;$

(C) $\frac{a b+b d-a c}{2} ;$

(D) $\frac{a c+b d-a b}{2}$.

**Problem 12:**

Let $A_{1}, A_{2}, \cdots, A_{n}$ be the interior angles of an $n$ -sided convex polygon. Then the value of $\frac{\cos \left(A_{1}+A_{2}+\cdots+A_{k}\right)}{\cos \left(A_{k+1}+A_{k+2}+\cdots+A_{n}\right)}$ , where $\cos \left(\sum_{i=1}^{k} A_{i}\right) \neq 0$ for any $k=1,2, \ldots, n-1$

(A) is independent of both $k$ and $n$;

(B) is independent of $k$ but depends on $n$ :

(C) is independent of $n$ but depends on $k$ :

(D) depends on both $k$ and $n$.

**Problem 13:**

Let $S$ denote the set of all complex numbers of the form $\frac{z +1}{z-3}$ where $z$ varies over the set of all complex numbers with $|z| = 1$. Then

(A) the set $S$ is a straight line in the complex plane;

(B) the set $S$ is a circle of radius $\frac{1}{2}$ in the complex plane;

(C) the set $S$ is a circle of radius $\frac{1}{4}$ in the complex plane:

(D) the set $S$ is an ellipse with axes $\frac{1}{2}$ and $\frac{1}{4}$ in the complex plane.

**Problem 14:**

The value of $\int_{0}^{2 \pi}|1+2 \sin x| d x$ is

(A) $2 \pi ;$

(B) $\frac{2 \pi}{3}$;

(C) $4+\frac{\pi}{3}$ :

(D) $4 \sqrt{3}+\frac{2 \pi}{3}$.

**Problem 15:**

Let $f(x) =\begin{cases} 0 & \quad \text { if } x \leq 1 \\ \log_{2}x & \quad \text { if } x >1 \end

{cases}$

and let $f^{(2)}(x)=f(f(x)), f^{(3)}(x)=f\left(f^{(2)}(x)\right), \ldots,$ and generally, $f^{(n+1)}(x)= f\left(f^{(n)}(x)\right) . $Let $N(x)=\min \{n \geq 1: f^{(n)}(x)=0\}$.Then the value of $N(425268)$ is

(A) $4$;

(B) $5$;

(C)$6$;

(D) $7$

**Problem 16:**

Let $f$ be a positive differentiable function defined on $(0,\infty)$. Then

$\lim _{n \rightarrow \infty}\left(\frac{f\left(x+\frac{1}{n}\right)}{f(x)}\right)^{n}$

(A) equals $1$ ;

(B) equals $\frac{f^{\prime}(x)}{f(x)}$;

(C) equals $e^{\left(\frac{f^{\prime}(x)}{f(x)}\right)}$;

(D) may not exist for some $f$.

**Problem 17:**

Let $ABC$ be a right angled triangle with $BC =3$ and $AC = 4$. Let $D$ be a point on the hypotenuse $AB$ such that $\angle BCD = 30^{\circ}$. The length of $CD$ is

(A) $\frac{24}{3+4 \sqrt{3}}$;

(B) $\frac{3 \sqrt{3}}{2}$

(C) $6 \sqrt{3}-8$

(D) $\frac{25}{12}$.

**Problem 18: **

Let $a$ be a positive number. Then

$\lim _{n \rightarrow \infty}\left[\frac{1}{a+n}+\frac{1}{2 a+n}+\ldots+\frac{1}{a n+n}\right]$ equals

(A) 0

(B) $\log _{e}(1+a)$

(C) $\frac{1}{a} \log _{e}(1+a)$

(D) none of these expressions.

**Problem 19: **

The area of the region in the first quadrant bounded by the $x$-axis and the curves $y = 2-x^2$ and $x=y^{2}$ is

(A) $\frac{4 \sqrt{2}}{3}$;

(B) $\frac{4 \sqrt{2}}{3}-1$;

(C) $\frac{2}{3} \sqrt[4]{8}$;

(D) $1+\frac{2}{3} \sqrt[4]{8}$

**Problem 20:**

Let $f(x)$ be the function defined on the interval $(0,1)$ by

$f(x)=\begin{cases}x(1-x) & \text { if } x \text { is rational, } \\ \frac{1}{4}-x(1-x) & \text { if } x \text { is not rational }\end{cases}$.

Then $f$ is continuous

(A) at no point in $(0,1)$;

(B) at exactly one point in $(0,1)$;

(C) at exactly two points in (0,1);

(D) at more than two points in $(0,1)$.

**Problem 21:**

Consider a circle of radius $a$. Let $P$ be a point at a distance $b(>a)$ from the center of the circle. The tangents from the point $P$ to the circle meet the circle at $Q$ and $R$. Then the area of the triangle $PQR$ is

(A) $\frac{a\left(b^{2}-a^{2}\right)^{3 / 2}}{b^{2}}$;

(B)$\frac{a^{2} \sqrt{b^{2}-a^{2}}}{b}$;

(C) $\frac{b^{2} \sqrt{b^{2}-a^{2}}}{a}$;

(D) $\frac{b\left(b^{2}-a^{2}\right)^{3 / 2}}{a^{2}}$

**Problem 22:**

Suppose two complex numbers $z=a+i b$ and $w=c+i d$ satisfy the equation

$\frac{z+w}{z}=\frac{w}{z+w}$. Then

(A) both $a$ and $c$ are zero;

(B) both $b$ and $d$ are zero;

(C) both $b$ and $d$ must be non-zero;

(D) at least one of $b$ and $d$ is non-zero.

**Problem 23:**

$\lim _{n \rightarrow \infty}\{(1+\frac{1}{n})^{n}-(1+\frac{1}{n})\}^{-n}$ is

(A) $1$;

(B) $\frac{1}{e-1} ;$

(C) $1-e^{-1}$;

(D) $0$ .

**Problem 24:**

Let $f(x)=e^{x}$

$g(x)=\begin{cases} x^{2} & \text { if } x<1 / 2 \\ x-\frac{1}{4} & \text { if } x \geq 1 / 2

\end{cases}$

and $h(x)=f(g(x))$. The derivative of $h$ at $x=1 / 2$

(A) is $e$;

(B) is $e^{1 / 2}$;

(C) is $e^{1 / 4}$;

(D) does not exist.

**Problem 25:**

The value of

$\frac{2+6}{4^{100}}+\frac{2+2 \times 6}{4^{99}}+\frac{2+3 \times 6}{4^{94}}+\cdots+\frac{2+99 \times 6}{4^{2}}+\frac{2+100 \times 6}{4}$

is equals to

(A) $\frac{1}{3}(604-\frac{1}{4^{98}})$;

(B) $\frac{1}{3}(600-\frac{1}{4^{98}})$;

(C) $\frac{604}{3}$;

(D) $200$.

**Problem 26:**

Let $a, b$ and $c$ be the sides of a right-angled triangle, where $a$ is the hypotenuse.

Let $d$ be the diameter of the inscribed circle. Then

(A) $d+a = b+c$;

(B) $d+a < b+c$;

(C) $d+a > b+c$;

(D) none of the above relations need always be true.

**Problem 27:**

Let $P$ be a point in the first quadrant lying on the parabola $y=4-x^{2}$. Let $A B$ be the tangent to the parabola at $P$ menting the at $B$. If $O$ is the origin, then the minimeeting the $x$ -axis at $A$ and the $y$ -axis is

(A) $\frac{64}{3 \sqrt{3}}$;

(B) $\frac{32}{3 \sqrt{3}}$

(C) $64(3 \sqrt{3})$

(D) $32(3 \sqrt{3})$

**Problem 28:** The value of the expression

$$

\sum_{0 \leq i<j \leq n} \sum (-1)^{i-j+1}\left(\begin{array}{c}

n \\

i

\end{array}\right)\left(\begin{array}{c}

n \\

j

\end{array}\right)

$$ is

(A) $\left(\begin{array}{c}2 n-1 \\ n\end{array}\right)$;

(B) $\left(\begin{array}{l}2 n \\ n\end{array}\right)$;

(C) $\left(\begin{array}{c}2 n+1 \\ n\end{array}\right)$;

(D) none of these expressions

**Problem 29:**

A man standing at a point $O$ finds that a balloon at a height $h$ metres due east of him has an angle of elevation $60^{\circ}$. He walks due north while the balloon moves north-west $\left(45^{\circ}\right.$ west of north) remaining at the same height. After he has walked $100$ metres the balloon is vertically above him. Then the value of $h$ in metres is

(A) $50$ ;

(B) $50 \sqrt{3}$

(C) $100 \sqrt{3}$;

(D) $\frac{100}{\sqrt{3}}$

**Problem 30:**

About the dolls in a shop a customer said "It is not true that some dolls have neither black hair nor blue eyes". The customer means that

(A) some dolls have both black hair and blue eyes;

(R) all dolls have both black hair and blue eyes;

(c) some dolls have either black hair or blue eyes;

(n) all dolls have either black hair or blue eyes.

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

JOIN TRIAL
Solutions for Test of Mathematics at the 10 +2 Level