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April 25, 2017

B.Stat 2005 Objective Paper| Problems & Solutions

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.

BStat Hons Objective Admission Test 2005 - Product 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.


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