Here are the problems and their corresponding solutions from B.Math Hons Objective Admission Test 2011.
Problem 1:
The domain of definition of \( f(x)=-\log \left(x^{2}-2 x-3\right) \) is
(a) \( (0, \infty) \)
(b) \( (-\infty,-1) \)
(c) \( (-\infty,-1) \cup(3, \infty) \)
(d) \( (-\infty,-3) \cup(1, \infty) \)
Problem 2:
\( A B C \) is a right-angled triangle with the right angle at B. If \( A B=7 \) and \( B C=24 \), then the length of the perpendicular from \( B \) to \( A C \) is
(a) \( 12.2 \)
(b) \( 6.72 \)
(c) \( 7.2 \)
(d) \( 3.36 \)
Problem 3:
If the points \( \mathbf{z}{1} \) and \( \mathbf{z}{2} \) are on the circles \( |\mathbf{z}|=2 \) and \( |\mathbf{z}|=3 \) respectively and the angle included between these vectors is \( 60^{\circ} \), then \(\left(\mathbf{z}{1}+\mathbf{z}{2}\right) /\left(\mathbf{z}{1}-\mathbf{z}{2}\right) \) equals
(a) \( \sqrt{(19 / 7)} \)
(b) \( \sqrt{19} \)
(c) \( \sqrt{7} \)
(d) \( \sqrt{133} \)
Problem 4:
Let \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) and \( \mathbf{d} \) be positive integers such that \( \log \mathrm{a}(\mathbf{b})=\mathbf{3 / 2} \) and
\( \log (\mathrm{d})=5 / 4 . \) If \( \mathrm{a}-\mathrm{c}=9 \), then \( b-d \) equals
(a) 55
(b) 23
(c) 89
(d) 93
Problem 5:
Let \( f(x)=x \sin (1 / x) \) for \( x>0 . \) Then
(A) \( f \) is unbounded;
(B) \( f \) is bounded, but \( \lim _{x \rightarrow \infty} f(x) \) does not exist;
(C) \( \lim _{x \rightarrow \infty} f(x)=1 ; \)
(D) \( \lim _{x \rightarrow \infty} f(x)=0 \).
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 mini-meeting 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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Solutions for Test of Mathematics at the 10 +2 Level