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QB1 Number Theory Irrationality and Transcendence Proofs View

Let $x$ be an irrational number. If $a , b , c$ and $d$ are rational numbers such that $\frac { a x + b } { c x + d }$ is a rational number, which of the following must be true?
(A) $a d = b c$
(B) $a c = b d$.
(C) $a b = c d$.
(D) $a = d = 0$

QB2 Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View

Let $z = x + i y$ be a complex number, which satisfies the equation $( z + \bar { z } ) z = 2 + 4 i$. Then
(A) $y = \pm 2$.
(B) $x = \pm 2$.
(C) $x = \pm 3$.
(D) $y = \pm 1$.

QB3 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Define $S _ { n } = \frac { 1 } { 2 } \cdot \frac { 3 } { 4 } \cdots \cdot \frac { 2 n - 1 } { 2 n }$ where $n$ is a positive integer. Then
(A) $S _ { n } < \frac { 1 } { \sqrt { 4 n + 2 } }$ for some $n > 2$.
(B) $S _ { n } < \frac { 1 } { \sqrt { 2 n + 1 } }$ for all $n \geq 2$.
(C) $S _ { n } < \frac { 1 } { \sqrt { 2 n + 5 } }$ for all $n \geq 2$.
(D) $S _ { n } > \frac { 1 } { \sqrt { 4 n + 2 } }$ for all $n \geq 2$.

QB4 Number Theory Quadratic Diophantine Equations and Perfect Squares View

If $n ^ { 2 } + 19 n + 92$ is a perfect square, then the possible values of $n$ may be
(A) - 19
(B) - 8
(C) - 4
(D) - 11

QB5 Solving quadratics and applications Counting solutions or configurations satisfying a quadratic system View

Let $a , b$ and $c$ be three real numbers. Then the equation $\frac { 1 } { x - a } + \frac { 1 } { x - b } + \frac { 1 } { x - c } = 0$
(A) always have real roots.
(B) can have real or complex roots depending on the values of $a , b$ and $c$.
(C) always have real and equal roots.
(D) always have real roots, which are not necessarily equal.

QB6 Combinations & Selection Subset Counting with Set-Theoretic Conditions View

Let $X$ be the set $\{ 1,2,3 , \ldots , 10 \}$ and $P$ the subset $\{ 1,2,3,4,5 \}$. The number of subsets $Q$ of $X$ such that $P \cap Q = \{ 3 \}$ is
(A) 1
(B) $2 ^ { 4 }$
(C) $2 ^ { 5 }$
(D) $2 ^ { 9 }$

QB7 Roots of polynomials Determine coefficients or parameters from root conditions View

Suppose that the equations $x ^ { 2 } + b x + c a = 0$ and $x ^ { 2 } + c x + a b = 0$ have exactly one common non-zero root. Then
(A) $a + b + c = 0$.
(B) the two roots which are not common must necessarily be real.
(C) the two roots which are not common may not be real.
(D) the two roots which are not common are either both real or both not real.

QB8 Addition & Double Angle Formulae Trigonometric Equation Solving via Identities View

Let $\frac { \tan ( \alpha - \beta + \gamma ) } { \tan ( \alpha + \beta - \gamma ) } = \frac { \tan \beta } { \tan \gamma }$. Then
(A) $\sin ( \beta - \gamma ) = \sin ( \alpha - \beta )$.
(B) $\sin ( \alpha - \gamma ) = \sin ( \beta - \gamma )$.
(C) $\sin ( \beta - \gamma ) = 0$.
(D) $\sin 2 \alpha + \sin 2 \beta + \sin 2 \gamma = 0$

QB9 Straight Lines & Coordinate Geometry Point-to-Line Distance Computation View

Let $K$ be the set of all points $(x , y)$ such that $| x | + | y | \leq 1$. Given a point $A$ in the plane, let $F _ { A }$ be the point in $K$ which is closest to $A$. Then the points $A$ for which $F _ { A } = ( 1,0 )$ are
(A) all points $A = ( x , y )$ with $x \geq 1$.
(B) all points $A = ( x , y )$ with $x \geq y + 1$ and $x \geq 1 - y$.
(C) all points $A = ( x , y )$ with $x \geq 1$ and $y = 0$.
(D) all points $A = ( x , y )$ with $x \geq 0$ and $y = 0$.

QB10 Addition & Double Angle Formulae Qualitative Reasoning about Double Angle Signs or Inequalities View

Let $\frac { \tan 3 \theta } { \tan \theta } = k$. Then
(A) $k \in ( 1 / 3,3 )$
(B) $k \notin ( 1 / 3,3 )$
(C) $\frac { \sin 3 \theta } { \sin \theta } = \frac { 2 k } { k - 1 }$.
(D) $\frac { \sin 3 \theta } { \sin \theta } > \frac { 2 k } { k - 1 }$

Q1 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View

The value of the integral $\int _ { \pi / 2 } ^ { 5 \pi / 2 } \frac { e ^ { \tan ^ { - 1 } ( \sin x ) } } { e ^ { \tan ^ { - 1 } ( \sin x ) } + e ^ { \tan ^ { - 1 } ( \cos x ) } } d x$ equals
(a) 1 .
(B) $\pi$.
(C) $e$.
(D) none of these.

Q2 Quadratic trigonometric equations View

The set of all solutions of the equation $\cos 2 \theta = \sin \theta + \cos \theta$ is given by
(a) $\theta = 0$.
(b) $\theta = n \pi + \frac { \pi } { 2 }$, where $n$ is any integer.
(c) $\theta = 2 n \pi$ or $\theta = 2 n \pi - \frac { \pi } { 2 }$ or $\theta = n \pi - \frac { \pi } { 4 }$, where $n$ is any integer.
(d) $\theta = 2 n \pi$ or $\theta = n \pi + \frac { \pi } { 4 }$, where $n$ is any integer.

Q3 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View

For $k \geq 1$, the value of $\binom { n } { 0 } + \binom { n + 1 } { 1 } + \binom { n + 2 } { 2 } + \cdots + \binom { n + k } { k }$ equals
(a) $\binom { n + k + 1 } { n + k }$.
(B) $( n + k + 1 ) \binom { n + k } { n + 1 }$.
(C) $\binom { n + k + 1 } { n + 1 }$.
(D) $\binom { n + k + 1 } { n }$.

Q4 Reciprocal Trig & Identities View

The value of $\sin ^ { - 1 } \cot \left[ \sin ^ { - 1 } \left\{ \frac { 1 } { 2 } \left( 1 - \sqrt { \frac { 5 } { 6 } } \right) \right\} + \cos ^ { - 1 } \sqrt { \frac { 2 } { 3 } } + \sec ^ { - 1 } \sqrt { \frac { 8 } { 3 } } \right]$ is
(a) 0 .
(B) $\pi / 6$.
(C) $\pi / 4$.
(D) $\pi / 2$.

Q5 Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View

If $a _ { n } = \left( 1 + \frac { 1 } { n ^ { 2 } } \right) \left( 1 + \frac { 2 ^ { 2 } } { n ^ { 2 } } \right) ^ { 2 } \left( 1 + \frac { 3 ^ { 2 } } { n ^ { 2 } } \right) ^ { 3 } \cdots \left( 1 + \frac { n ^ { 2 } } { n ^ { 2 } } \right) ^ { n }$, then $$\lim _ { n \rightarrow \infty } a _ { n } ^ { - 1 / n ^ { 2 } }$$ is
(a) 0 .
(B) 1 .
(C) $e$.
(D) $\sqrt { e } / 2$.

Q6 Differentiating Transcendental Functions Higher-order or nth derivative computation View

If $f ( x ) = e ^ { x } \sin x$, then $\left. \frac { d ^ { 10 } } { d x ^ { 10 } } f ( x ) \right| _ { x = 0 }$ equals
(a) 1 .
(B) - 1 .
(C) 10 .
(D) 32 .

Q7 Circles Chord Length and Chord Properties View

Consider a circle with centre $O$. Two chords $A B$ and $C D$ extended intersect at a point $P$ outside the circle. If $\angle A O C = 43 ^ { \circ }$ and $\angle B P D = 18 ^ { \circ }$, then the value of $\angle B O D$ is
(a) $36 ^ { \circ }$.
(B) $29 ^ { \circ }$.
(C) $7 ^ { \circ }$.
(D) $25 ^ { \circ }$.

Q8 Combinations & Selection Selection with Group/Category Constraints View

A box contains 10 red cards numbered $1 , \ldots , 10$ and 10 black cards numbered $1 , \ldots , 10$. In how many ways can we choose 10 out of the 20 cards so that there are exactly 3 matches, where a match means a red card and a black card with the same number?
(a) $\binom { 10 } { 3 } \binom { 7 } { 4 } 2 ^ { 4 }$.
(B) $\binom { 10 } { 3 } \binom { 7 } { 4 }$.
(C) $\binom { 10 } { 3 } 2 ^ { 7 }$.
(D) $\binom { 10 } { 3 } \binom { 14 } { 4 }$.

Q9 Conic sections Tangent and Normal Line Problems View

Let $P$ be a point on the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$ which does not lie on the axes. If the normal at the point $P$ intersects the major and minor axes at $C$ and $D$ respectively, then the ratio $P C : P D$ equals
(a) 2 .
(B) $1 / 2$.
(C) 4 .
(D) $1 / 4$.

Q10 Complex Numbers Argand & Loci Locus Identification from Modulus/Argument Equation View

The set of complex numbers $z$ satisfying the equation $$( 3 + 7 i ) z + ( 10 - 2 i ) \bar { z } + 100 = 0$$ represents, in the Argand plane,
(a) a straight line. (B) a pair of intersecting straight lines. (C) a pair of distinct parallel straight lines. (D) a point.

Q11 Sine and Cosine Rules Ambiguous case and triangle existence/uniqueness View

The number of triplets $( a , b , c )$ of integers such that $a < b < c$ and $a , b , c$ are sides of a triangle with perimeter 21 is
(a) 7 .
(B) 8.
(C) 11 .
(D) 12 .

Q12 Geometric Sequences and Series Arithmetic-Geometric Sequence Interplay View

Suppose $a , b$ and $c$ are three numbers in G.P. If the equations $a x ^ { 2 } + 2 b x + c = 0$ and $d x ^ { 2 } + 2 e x + f = 0$ have a common root, then $\frac { d } { a } , \frac { e } { b }$ and $\frac { f } { c }$ are in
(a) A.P.
(B) G.P.
(C) H.P.
(D) none of the above.

Q13 Reciprocal Trig & Identities View

The number of solutions of the equation $\sin ^ { - 1 } x = 2 \tan ^ { - 1 } x$ is
(a) 1 .
(B) 2 .
(C) 3 .
(D) 5 .

Q14 Sine and Cosine Rules Multi-step composite figure problem View

Suppose $A B C D$ is a quadrilateral such that $\angle B A C = 50 ^ { \circ } , \angle C A D = 60 ^ { \circ } , \angle C B D = 30 ^ { \circ }$ and $\angle B D C = 25 ^ { \circ }$. If $E$ is the point of intersection of $A C$ and $B D$, then the value of $\angle A E B$ is
(a) $75 ^ { \circ }$.
(B) $85 ^ { \circ }$.
(C) $95 ^ { \circ }$.
(D) $110 ^ { \circ }$.

Q15 Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View

Let $\mathbb { R }$ be the set of all real numbers. The function $f : \mathbb { R } \rightarrow \mathbb { R }$ defined by $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 6 x - 5$ is
(a) one-to-one, but not onto.
(B) one-to-one and onto.
(C) onto, but not one-to-one.
(D) neither one-to-one nor onto.

Q16 Trig Proofs Ordering or Comparing Trigonometric Expressions View

Suppose $x , y \in ( 0 , \pi / 2 )$ and $x \neq y$. Which of the following statements is true?
(a) $2 \sin ( x + y ) < \sin 2 x + \sin 2 y$ for all $x , y$.
(b) $2 \sin ( x + y ) > \sin 2 x + \sin 2 y$ for all $x , y$.
(c) There exist $x , y$ such that $2 \sin ( x + y ) = \sin 2 x + \sin 2 y$.
(d) None of the above.

Q17 Circles Circle-Related Locus Problems View

A triangle $A B C$ has a fixed base $B C$. If $A B : A C = 1 : 2$, then the locus of the vertex $A$ is
(a) a circle whose centre is the midpoint of $B C$.
(b) a circle whose centre is on the line $B C$ but not the midpoint of $B C$.
(c) a straight line.
(d) none of the above.

Q18 Number Theory Modular Arithmetic Computation View

Let $N$ be a 50 digit number. All the digits except the 26th one from the right are 1. If $N$ is divisible by 13, then the unknown digit is
(a) 1 .
(B) 3 .
(C) 7 .
(D) 9 .

Q19 Indefinite & Definite Integrals Maximizing or Optimizing a Definite Integral View

Suppose $a < b$. The maximum value of the integral $$\int _ { a } ^ { b } \left( \frac { 3 } { 4 } - x - x ^ { 2 } \right) d x$$ over all possible values of $a$ and $b$ is
(a) $3 / 4$.
(B) $4 / 3$.
(C) $3 / 2$.
(D) $2 / 3$.

Q20 Sequences and Series Estimation or Bounding of a Sum View

For any $n \geq 5$, the value of $1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \cdots + \frac { 1 } { 2 ^ { n } - 1 }$ lies between
(a) 0 and $n / 2$.
(B) $n / 2$ and $n$.
(C) $n$ and $2 n$.
(D) none of the above.

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