1.02g - OCR Spec

CAIE P1 2023 June Q2

4 marks Standard +0.3

2 The function f is defined for $x \in \mathbb { R }$ by $\mathrm { f } ( x ) = x ^ { 2 } - 6 x + c$, where $c$ is a constant. It is given that $\mathrm { f } ( x ) > 2$ for all values of $x$. Find the set of possible values of $c$.

CAIE P1 2020 March Q11

9 marks Standard +0.3

11

Solve the equation $3 \tan ^ { 2 } x - 5 \tan x - 2 = 0$ for $0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$.
Find the set of values of $k$ for which the equation $3 \tan ^ { 2 } x - 5 \tan x + k = 0$ has no solutions.
For the equation $3 \tan ^ { 2 } x - 5 \tan x + k = 0$, state the value of $k$ for which there are three solutions in the interval $0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$, and find these solutions.

CAIE P1 2020 November Q1

3 marks Moderate -0.3

1 Find the set of values of $m$ for which the line with equation $y = m x - 3$ and the curve with equation $y = 2 x ^ { 2 } + 5$ do not meet.

CAIE P1 2003 June Q11

13 marks Moderate -0.3

11 The equation of a curve is $y = 8 x - x ^ { 2 }$.

Express $8 x - x ^ { 2 }$ in the form $a - ( x + b ) ^ { 2 }$, stating the numerical values of $a$ and $b$.
Hence, or otherwise, find the coordinates of the stationary point of the curve.
Find the set of values of $x$ for which $y \geqslant - 20$. The function g is defined by $\mathrm { g } : x \mapsto 8 x - x ^ { 2 }$, for $x \geqslant 4$.
State the domain and range of $\mathrm { g } ^ { - 1 }$.
Find an expression, in terms of $x$, for $\mathrm { g } ^ { - 1 } ( x )$.

CAIE P1 2004 June Q10

12 marks Moderate -0.8

10 The functions $f$ and $g$ are defined as follows: $$\begin{array} { l l } \mathrm { f } : x \mapsto x ^ { 2 } - 2 x , & x \in \mathbb { R } , \\ \mathrm {~g} : x \mapsto 2 x + 3 , & x \in \mathbb { R } . \end{array}$$

Find the set of values of $x$ for which $\mathrm { f } ( x ) > 15$.
Find the range of f and state, with a reason, whether f has an inverse.
Show that the equation $\operatorname { gf } ( x ) = 0$ has no real solutions.
Sketch, in a single diagram, the graphs of $y = \mathrm { g } ( x )$ and $y = \mathrm { g } ^ { - 1 } ( x )$, making clear the relationship between the graphs.

CAIE P1 2010 June Q9

11 marks Moderate -0.8

9 The function f is defined by $\mathrm { f } : x \mapsto 2 x ^ { 2 } - 12 x + 7$ for $x \in \mathbb { R }$.

Express $\mathrm { f } ( x )$ in the form $a ( x - b ) ^ { 2 } - c$.
State the range of f .
Find the set of values of $x$ for which $\mathrm { f } ( x ) < 21$. The function g is defined by $\mathrm { g } : x \mapsto 2 x + k$ for $x \in \mathbb { R }$.
Find the value of the constant $k$ for which the equation $\operatorname { gf } ( x ) = 0$ has two equal roots.

CAIE P1 2011 June Q7

7 marks Moderate -0.8

7 A curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { ( 1 + 2 x ) ^ { 2 } }$ and the point $\left( 1 , \frac { 1 } { 2 } \right)$ lies on the curve.

Find the equation of the curve.
Find the set of values of $x$ for which the gradient of the curve is less than $\frac { 1 } { 3 }$.

CAIE P1 2011 June Q2

5 marks Standard +0.3

2 Find the set of values of $m$ for which the line $y = m x + 4$ intersects the curve $y = 3 x ^ { 2 } - 4 x + 7$ at two distinct points.

CAIE P1 2012 June Q10

9 marks Moderate -0.3

10 It is given that a curve has equation $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + x$.

Find the set of values of $x$ for which the gradient of the curve is less than 5 .
Find the values of $\mathrm { f } ( x )$ at the two stationary points on the curve and determine the nature of each stationary point.

CAIE P1 2014 June Q2

4 marks Moderate -0.8

2

Express $4 x ^ { 2 } - 12 x$ in the form $( 2 x + a ) ^ { 2 } + b$.
Hence, or otherwise, find the set of values of $x$ satisfying $4 x ^ { 2 } - 12 x > 7$.

CAIE P1 2014 June Q10

15 marks Moderate -0.3

10 Functions $f$ and $g$ are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x - 3 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \mapsto x ^ { 2 } + 4 x , \quad x \in \mathbb { R } . \end{aligned}$$

Solve the equation $\mathrm { ff } ( x ) = 11$.
Find the range of g .
Find the set of values of $x$ for which $\mathrm { g } ( x ) > 12$.
Find the value of the constant $p$ for which the equation $\mathrm { gf } ( x ) = p$ has two equal roots. Function h is defined by $\mathrm { h } : x \mapsto x ^ { 2 } + 4 x$ for $x \geqslant k$, and it is given that h has an inverse.
State the smallest possible value of $k$.
Find an expression for $\mathrm { h } ^ { - 1 } ( x )$.

CAIE P1 2014 June Q8

8 marks Moderate -0.8

8

Express $2 x ^ { 2 } - 10 x + 8$ in the form $a ( x + b ) ^ { 2 } + c$, where $a , b$ and $c$ are constants, and use your answer to state the minimum value of $2 x ^ { 2 } - 10 x + 8$.
Find the set of values of $k$ for which the equation $2 x ^ { 2 } - 10 x + 8 = k x$ has no real roots.

CAIE P1 2016 June Q11

11 marks Moderate -0.3

11 The function f is defined by $\mathrm { f } : x \mapsto 6 x - x ^ { 2 } - 5$ for $x \in \mathbb { R }$.

Find the set of values of $x$ for which $\mathrm { f } ( x ) \leqslant 3$.
Given that the line $y = m x + c$ is a tangent to the curve $y = \mathrm { f } ( x )$, show that $4 c = m ^ { 2 } - 12 m + 16$. The function g is defined by $\mathrm { g } : x \mapsto 6 x - x ^ { 2 } - 5$ for $x \geqslant k$, where $k$ is a constant.
Express $6 x - x ^ { 2 } - 5$ in the form $a - ( x - b ) ^ { 2 }$, where $a$ and $b$ are constants.
State the smallest value of $k$ for which g has an inverse.
For this value of $k$, find an expression for $\mathrm { g } ^ { - 1 } ( x )$. {www.cie.org.uk} after the live examination series. }

CAIE P1 2017 June Q7

8 marks Moderate -0.8

7 A curve for which $\frac { \mathrm { d } y } { \mathrm {~d} x } = 7 - x ^ { 2 } - 6 x$ passes through the point $( 3 , - 10 )$.

Find the equation of the curve.
Express $7 - x ^ { 2 } - 6 x$ in the form $a - ( x + b ) ^ { 2 }$, where $a$ and $b$ are constants.
Find the set of values of $x$ for which the gradient of the curve is positive.

CAIE P1 2018 June Q9

11 marks Moderate -0.3

9 Functions f and g are defined for $x \in \mathbb { R }$ by $$\begin{aligned} & \mathrm { f } : x \mapsto \frac { 1 } { 2 } x - 2 \\ & \mathrm {~g} : x \mapsto 4 + x - \frac { 1 } { 2 } x ^ { 2 } \end{aligned}$$

Find the points of intersection of the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { g } ( x )$.
Find the set of values of $x$ for which $\mathrm { f } ( x ) > \mathrm { g } ( x )$.
Find an expression for $\mathrm { fg } ( x )$ and deduce the range of fg .
The function h is defined by $\mathrm { h } : x \mapsto 4 + x - \frac { 1 } { 2 } x ^ { 2 }$ for $x \geqslant k$.
Find the smallest value of $k$ for which h has an inverse.

CAIE P1 2018 June Q10

8 marks Moderate -0.3

10

Solve the equation $2 \cos x + 3 \sin x = 0$, for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
Sketch, on the same diagram, the graphs of $y = 2 \cos x$ and $y = - 3 \sin x$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
Use your answers to parts (i) and (ii) to find the set of values of $x$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$ for which $2 \cos x + 3 \sin x > 0$.

CAIE P1 2003 November Q4

6 marks Easy -1.2

4 A curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 4 x + 1$. The curve passes through the point $( 1,5 )$.

Find the equation of the curve.
Find the set of values of $x$ for which the gradient of the curve is positive.

CAIE P1 2005 November Q9

11 marks Standard +0.3

9 The equation of a curve is $x y = 12$ and the equation of a line $l$ is $2 x + y = k$, where $k$ is a constant.

In the case where $k = 11$, find the coordinates of the points of intersection of $l$ and the curve.
Find the set of values of $k$ for which $l$ does not intersect the curve.
In the case where $k = 10$, one of the points of intersection is $P ( 2,6 )$. Find the angle, in degrees correct to 1 decimal place, between $l$ and the tangent to the curve at $P$.

CAIE P1 2006 November Q10

10 marks Moderate -0.8

10 The function f is defined by $\mathrm { f } : x \mapsto x ^ { 2 } - 3 x$ for $x \in \mathbb { R }$.

Find the set of values of $x$ for which $\mathrm { f } ( x ) > 4$.
Express $\mathrm { f } ( x )$ in the form $( x - a ) ^ { 2 } - b$, stating the values of $a$ and $b$.
Write down the range of f .
State, with a reason, whether f has an inverse. The function g is defined by $\mathrm { g } : x \mapsto x - 3 \sqrt { } x$ for $x \geqslant 0$.
Solve the equation $\mathrm { g } ( x ) = 10$.

CAIE P1 2013 November Q10

10 marks Moderate -0.3

10 A curve has equation $y = 2 x ^ { 2 } - 3 x$.

Find the set of values of $x$ for which $y > 9$.
Express $2 x ^ { 2 } - 3 x$ in the form $a ( x + b ) ^ { 2 } + c$, where $a , b$ and $c$ are constants, and state the coordinates of the vertex of the curve. The functions f and g are defined for all real values of $x$ by $$\mathrm { f } ( x ) = 2 x ^ { 2 } - 3 x \quad \text { and } \quad \mathrm { g } ( x ) = 3 x + k$$ where $k$ is a constant.
Find the value of $k$ for which the equation $\mathrm { gf } ( x ) = 0$ has equal roots.

CAIE P1 2013 November Q1

3 marks Easy -1.2

1 Solve the inequality $x ^ { 2 } - x - 2 > 0$.

CAIE P1 2015 November Q8