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2023
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18 maths questions

Q3 Straight Lines & Coordinate Geometry Line Equation and Parametric Representation View
3. The perpendicular bisector of the line segment joining the points $( 2 , - 6 )$ and $( 5,4 )$ cuts the $x$-axis at the point with $x$-coordinate
A $\frac { 1 } { 20 }$
B $\frac { 1 } { 6 }$
C $\frac { 1 } { 3 }$
D $\frac { 19 } { 5 }$
E $\frac { 41 } { 6 }$
Q4 Inequalities Solve Polynomial/Rational Inequality for Solution Set View
4. The complete set of values of $x$ for which $\left( x ^ { 2 } - 1 \right) ( x - 2 ) > 0$ is
A $x < - 1,1 < x < 2$
B $x < - 1 , x > 2$
C $- 1 < x < 2$
D $x < 1 , x > 2$
E $\quad - 1 < x < 1 , x > 2$
Q5 Laws of Logarithms Solve a Logarithmic Equation View
5. Given that $y = - \log _ { 10 } ( 1 - x )$ for $x < 1$, find $x$ in terms of $y$.
A $\quad x = - \frac { 1 } { \log _ { 10 } ( 1 - y ) }$
B $x = 1 + \log _ { 10 } y$
C $x = 1 - \log _ { 10 } y$
D $\quad x = 1 - 10 ^ { - y }$
E $\quad x = 10 ^ { - y } - 1$
F $\quad x = 10 ^ { 1 - y }$
Q6 Factor & Remainder Theorem Divisibility and Factor Determination View
6. It is given that $x + 2$ is a factor of $x ^ { 3 } + 4 c x ^ { 2 } + x ( c + 1 ) ^ { 2 } - 6$.
The sum of the possible values of $c$ is
A - 10
B - 6
C 0
D 6
E 10
Q7 Probability Definitions Finite Equally-Likely Probability Computation View
7. A bag contains $n$ red balls, $n$ yellow balls, and $n$ blue balls.
One ball is selected at random and not replaced.
A second ball is then selected at random and not replaced. Each ball is equally likely to be chosen. The probability that the two balls are not the same colour is
A $\frac { n - 1 } { 3 n - 1 }$
B $\frac { 2 n - 2 } { 3 n - 1 }$
C $\frac { 2 n } { 3 n - 1 }$
D $\quad \frac { ( n - 1 ) ^ { 3 } } { 27 ( 3 n - 1 ) ^ { 3 } }$
E $\quad \frac { 3 ( n - 1 ) } { 3 n - 1 }$ F $\quad \frac { n ^ { 3 } } { 27 ( 3 n - 1 ) ^ { 3 } }$
Q8 Laws of Logarithms Solve a Logarithmic Equation View
8. Given that $a ^ { x } b ^ { 2 x } c ^ { 3 x } = 2$, where $a , b$, and $c$ are positive real numbers, then $x =$
A $\quad \log _ { 10 } \left( \frac { 2 } { a + 2 b + 3 c } \right)$
B $\frac { \log _ { 10 } 2 } { \log _ { 10 } ( a + 2 b + 3 c ) }$
C $\quad \frac { 2 } { \log _ { 10 } ( a + 2 b + 3 c ) }$
D $\frac { 2 } { a + 2 b + 3 c }$
E $\quad \log _ { 10 } \left( \frac { 2 } { a b ^ { 2 } c ^ { 3 } } \right)$ F $\quad \frac { \log _ { 10 } 2 } { \log _ { 10 } \left( a b ^ { 2 } c ^ { 3 } \right) }$ G $\quad \frac { 2 } { \log _ { 10 } \left( a b ^ { 2 } c ^ { 3 } \right) }$ H $\frac { 2 } { a b ^ { 2 } c ^ { 3 } }$
Q9 Discriminant and conditions for roots Root relationships and Vieta's formulas View
9. The roots of the equation $2 x ^ { 2 } - 11 x + c = 0$ differ by 2 . The value of $c$ is
A $\frac { 105 } { 8 }$
B $\frac { 113 } { 8 }$
C $\frac { 117 } { 8 }$
D $\frac { 119 } { 8 }$
Q10 Function Transformations View
10. The curve $y = \cos x$ is reflected in the line $y = 1$ and the resulting curve is then translated by $\frac { \pi } { 4 }$ units in the positive $x$-direction. The equation of this new curve is
A $y = 2 + \cos \left( x + \frac { \pi } { 4 } \right)$
B $y = 2 + \cos \left( x - \frac { \pi } { 4 } \right)$
C $y = 2 - \cos \left( x + \frac { \pi } { 4 } \right)$
D $\quad y = 2 - \cos \left( x - \frac { \pi } { 4 } \right)$
Q11 Exponential Functions Exponential Equation Solving View
11. The sum of the roots of the equation $2 ^ { 2 x } - 8 \times 2 ^ { x } + 15 = 0$ is
A 3
B 8
C $\quad 2 \log _ { 10 } 2$
D $\quad \log _ { 10 } \left( \frac { 15 } { 4 } \right)$
E $\quad \frac { \log _ { 10 } 15 } { \log _ { 10 } 2 }$
Q12 Applied differentiation Applied modeling with differentiation View
12. The cross-section of a triangular prism is an equilateral triangle with side $2 x \mathrm {~cm}$. The length of the prism is $d \mathrm {~cm}$.
Let the total surface area of the prism be $T \mathrm {~cm} ^ { 2 }$. Given that the volume of the prism is $T \mathrm { cm } ^ { 3 }$, which one of the following is an expression for $d$ in terms of $x$ ?
A $\frac { x } { 2 x - 3 }$
B $\frac { 3 x } { 3 x - 2 \sqrt { 3 } }$
C $\frac { 2 x } { x - 4 \sqrt { 3 } }$
D $\frac { 2 x } { x - 2 \sqrt { 3 } }$
E $\frac { 2 x } { x - \sqrt { 3 } }$
Q13 Curve Sketching Number of Solutions / Roots via Curve Analysis View
13. How many real roots does the equation $x ^ { 4 } - 4 x ^ { 3 } + 4 x ^ { 2 } - 10 = 0$ have?
A 0
B 1
C 2
D 3
E 4
Q14 Exponential Equations & Modelling Properties of Logarithmic Functions and Statement Verification View
14. $a , b , x$, and $y$ are real and positive. $a$ and $b$ are constants. $x$ and $y$ are related.
A graph of $\log y$ against $\log x$ is drawn.
For which one of the following relationships will this graph be a straight line?
A $y ^ { b } = a ^ { x }$
B $y = a b ^ { x }$
C $y ^ { 2 } = a + x ^ { b }$
D $y = a x ^ { b }$
E $y ^ { x } = a ^ { b }$
Q15 Indefinite & Definite Integrals Maximizing or Optimizing a Definite Integral View
15. The smallest possible value of $\int _ { 0 } ^ { 1 } ( x - a ) ^ { 2 } d x$ as $a$ varies is
A $\frac { 1 } { 12 }$
B $\frac { 1 } { 3 }$
C $\frac { 1 } { 2 }$
D $\frac { 7 } { 12 }$
E 2
Q16 Indices and Surds Number-Theoretic Reasoning with Indices View
16. Given that $c$ and $d$ are non-zero integers, the expression $\frac { 10 ^ { c - 2 d } \times 20 ^ { 2 c + d } } { 8 ^ { c } \times 125 ^ { c + d } }$ is an integer if
A $\quad c < 0$
B $\quad d < 0$
C $\quad c < 0$ and $d < 0$
D $\quad c < 0$ and $d > 0$
E $\quad c > 0$ and $d < 0$
F $\quad c > 0$ and $d > 0$ G $\quad d > 0$
H $\quad c > 0$
Q17 Discriminant and conditions for roots Parameter range for no real roots (positive definite) View
17. For what values of the non-zero real number $a$ does the quadratic equation $a x ^ { 2 } + ( a - 2 ) x = 2$ have real distinct roots?
A All values of $a$
B $\quad a = - 2$
C $\quad a > - 2$
D $\quad a \neq - 2$
E No values of $a$
Q18 Standard trigonometric equations Solve trigonometric inequality View
18. The angle $x$ is measured in radians and is such that $0 \leq x \leq \pi$.
The total length of any intervals for which $- 1 \leq \tan x \leq 1$ and $\sin 2 x \geq 0.5$ is
A $\frac { \pi } { 12 }$
B $\frac { \pi } { 6 }$
C $\frac { \pi } { 4 }$
D $\frac { \pi } { 3 }$
E $\frac { 5 \pi } { 12 }$ F $\frac { \pi } { 2 }$ G $\quad \frac { 5 \pi } { 6 }$
Q19 Geometric Sequences and Series Arithmetic-Geometric Sequence Interplay View
19. A geometric series has first term 4 and common ratio $r$, where $0 < r < 1$.
The first, second, and fourth terms of this geometric series form three successive terms of an arithmetic series.
The sum to infinity of the geometric series is
A $\frac { 1 } { 2 } ( \sqrt { 5 } - 1 )$
B $2 ( 3 - \sqrt { 5 } )$
C $2 ( 1 + \sqrt { 5 } )$
D $2 ( 3 + \sqrt { 5 } )$
Q20 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View
20. The coefficient of $x ^ { 2 }$ in the expansion of $\left( 4 - x ^ { 2 } \right) \left[ \left( 1 + 2 x + 3 x ^ { 2 } \right) ^ { 6 } - \left( 1 + 4 x ^ { 3 } \right) ^ { 5 } \right]$ is
A 28
B 72
C 78
D 192
E 240
F 310
G 312
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