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# B.Math 2009 Objective Paper| Problems & Solutions

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. ## Some Useful Links: 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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### One comment on “B.Math 2009 Objective Paper| Problems & Solutions”

1. K J M Meghana says:

Solutions for Test of Mathematics at the 10 +2 Level