SPM Add Math · 2025 · Pahang · Percubaan Kertas 1 (Pahang)

the coordinates of $N$,

the equation of the straight line $AD$.

Given that $h(x)=(1-2x)^{3}$, evaluate $h'(1)$.

A curve has an equation $y=3x+\frac{4}{x}$. If $x$ is decreasing at a constant rate of 3 units per second, find the rate of change of $y$ when $x=2$.

Express $a$ in terms of $b$.

Solve the system of linear equations.

Sketch the graph of $y=2+\tan 2x$ for $0\leq x\leq\pi$.

Diagram 2 shows an angle of $-\frac{10}{9}\pi$ rad on a Cartesian plane. State the reference angle in term of $\pi$ rad.

图 2: 角 $-\frac{10}{9}\pi$ rad,终边在第二象限。

Diagram 3.1 shows a sector of a circle with centre $O$ and an arc of $AD$ which subtends an angle of $\theta$ rad at $O$. It is given that the radius of the sector of circle is $r$ unit and the area of sector $AOD$ is $A$ unit$^{2}$. Based on Diagram 3.1, derive $A=\frac{1}{2}r^{2}\theta$.

图 3.1: 圆心为 O、半径为 r、圆心角为 $\theta$ 的扇形 AOD。

Diagram 3.2 shows another arc $EF$ with the same centre $O$ and radius of 16 cm drawn on Diagram 3.1. Given that $OBE$ and $OCF$ are two straight lines such that $B$ and $C$ lie on arc $AD$. Arc lengths of $AB$, $BC$ and $CD$ are equal. Lines $AE$ and $DF$ are the tangents to arc $AD$ at points $A$ and $D$ respectively. [Use $\pi=3.142$] Given length of $OA=12$ cm, calculate the area, in cm$^{2}$, of the whole Diagram 3.2.

图 3.2: 在图 3.1 上叠加半径 16 cm 的弧 EF;OBE、OCF 为直线,AE、DF 为切线。

State $h(1)$.

Find the domain for $h(x)\leq 5$.

Hence, sketch the graph of the function for $h(x)$.

Diagram 4 shows a standard normal distribution curve. Given $P(0<z<k)=0.3125$. State $P(z>k)$.

图 4: 标准正态分布曲线,阴影区域由 $0$ 至 $k$。

Find the mass of the student which gives a standard score of $0.5$,

Find the percentage of students with mass greater than $48$ kg.

The amount of caffeine in a person's body after drinking coffee is given by $C=C_{0}e^{-0.15t}$ such that $C_{0}$ is the initial caffeine content, in mg, and $C$ is the caffeine content, in mg, after $t$ hours. Find the time needed for the caffeine level to reduce to half of its original value. Round off your answer correct to the nearest whole number.

Diagram 5 shows a triangular metal $ABC$ designed by an engineer to support a building structure. Determine whether triangle $ABC$ is a right-angled triangle at $C$ or not. Show your calculations.

图 5: 三角形 ABC,$AC=\sqrt{6}$ m,$BC=\frac{\sqrt{2}}{\sqrt{3}+\sqrt{2}}$ m,$AB=\sqrt{16-4\sqrt{6}}$ m。

Find the equation of the curve that has the gradient function $\frac{dy}{dx}=4x+9$ and passes through the point $A(2,-3)$.

Given $\int_{1}^{3}f(x)\,dx=4$, $\int_{3}^{6}f(x)\,dx=5$ and $\int_{1}^{6}f(x)\,dx=k$. Find the value of $k$,

Find $\int_{1}^{3}7-2f(x)\,dx$.

Given $p$ is an integer such that $-2<p<2$, state the value of $p$.

Using the answer from 11(a), find the value of $q$ when the graph touches the $x$-axis at one point.

For Plan A, if he makes a saving of RM240 in the 3rd month, calculate his savings in the 6th month.

Determine which plan allows the teacher to reach the savings target of RM10320 earlier. Justify your answer.

Given the graph of quadratic function $f(x)=rx^{2}+2x+r$ does not intersect the $x$-axis. Find the range of values of $r$ by using number line method.

Diagram 6 shows a graph of $f(x)=2x^{2}-4mx+2m^{2}+2n$ with $a=2$, $b=-4m$, and $c=2m^{2}+2n$. On Diagram 6, sketch a graph of $y=g(x)$ when the variable of $a$ becomes 1.

图 6: 抛物线 $y=f(x)$,与 $x$ 轴交于 $x=1$ 与 $x=5$,最低点在 $y=-8$。

Express $f(x)=2x^{2}-4mx+2m^{2}+2n$ in the form $a(x-h)^{2}+k$, such that $h$ and $k$ are constants. Hence, find the value of $m$ and of $n$.

Diagram 7 shows a parallelogram $DEFG$. Point $P$ lies on straight line $DE$ and point $Y$ lies on straight line $GF$. $PT$ is a straight line passing through the point $F$. Given that $\vec{DY}=\underset{\sim}{a}+6\underset{\sim}{b}$, and $\vec{DT}=\frac{3}{2}\underset{\sim}{a}+9\underset{\sim}{b}$. A straight line is drawn by connecting point $D$ to point $T$. Using the vector method, determine whether the line passes through point $Y$ or not.

图 7: 平行四边形 DEFG,Y 在 GF 上,P 在 DE 上,T 在 PF 延长线外侧。

Diagram 8 shows a regular hexagon $PQRSTU$ drawn on a Cartesian plane with centre at origin $O$. Express $\vec{PQ}+\vec{TS}$ as a single vector.

图 8: 以原点 O 为中心的正六边形 PQRSTU,P、Q 在上,U、R 在左右,T、S 在下。

Given $\vec{OR}=5\underset{\sim}{i}$, $\vec{OT}=-\frac{5}{2}\underset{\sim}{i}-\frac{5\sqrt{3}}{2}\underset{\sim}{j}$, find the unit vector in the direction of $\vec{RT}$, in terms of $\underset{\sim}{i}$ and $\underset{\sim}{j}$.

$f(x)$ in terms of $k$,

the value of $k$ such that $f(1)=4fg(\sqrt{2})$,

hence, find $g(x)$ and determine whether the inverse function of $g(x)$ exists or not by using a horizontal line test. Give your justification.