New Millennium Academy

First Terminal Examination – 2083

Subject: Maths (Additional) F.M.: 75
Class: 10            Time: 3 hrs Date: 2083-03-32
Instruction: Candidates are required to give their answers in their own words as far as practicable. All questions are compulsory.
MCQ (Multiple Choice Questions) (11 marks)
1.In the given mapping diagram, how is the function defined from $A$ to $C$ denoted? [1K]
Figure
a) $fg(x)$ b) $gf(x)$
c) $ff(x)$ d) $g(f(x))$
2.If $\begin{vmatrix} 2 & x \\ 1 & x \end{vmatrix} = 0$, then what is the value of $x$? [1A]
a) 2 b) 1
c) -2 d) 0
3.Which one is equal to $\sin(A + B)$? [1K]
  • a) $\sin A \cos B - \cos A \sin B$
  • b) $\cos A \cos B - \sin A \sin B$
  • c) $\sin A \cos B + \cos A \sin B$
  • d) $\sin A \cos A + \cos B \sin B$
4.If $A = 30^\circ$ and $B = 60^\circ$, what is the value of $\sin(A + B)$? [1A]
a) 0 b) 1
c) $\frac{1}{2}$ d) $\frac{\sqrt{3}}{2}$
5.If $\tan(A + B) = \frac{1}{7}$ and $\tan(A - B) = \frac{2}{3}$, the value of $\tan 2A$ is: [1HA]
a) 1 b) $\frac{15}{21}$
c) $\frac{11}{18}$ d) $\frac{17}{19}$
6.Which one is the equation of a straight line in two-point form? [1K]
  • a) $y = mx + c$
  • b) $\frac{x}{a} + \frac{y}{b} = 1$
  • c) $y - y_1 = m(x - x_1)$
  • d) $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1)$
7.If a point $(x, y)$ is translated by $T = \begin{pmatrix} -3 \\ 5 \end{pmatrix}$, which of the following is the resulting point? [1A]
a) $(x - 3, y - 5)$ b) $(x + 3, y + 5)$
c) $(x - 3, y + 5)$ d) $(x + 3, y - 5)$
8.The equation of the line joining points $(-1, 2)$ and $(a, 3)$ is $x - 3y + 7 = 0$, what is the value of '$a$'? [1HA]
a) 2 b) 4
c) 3 d) -3
9.If $\vec{a} = (2, 3)$, what is the value of $\vec{a}^2$? [1U]
a) 5 b) -13
c) 13 d) 25
10.The sum of $Q_1$ and $Q_3$ is 60 and the difference is 30, then the coefficient of quartile deviation is _______ [1U]
a) 0.5 b) 0.25
c) 0.75 d) none
11.Find the left-hand limit of the function at $x = 4$. [1U]
Figure
a) 2 b) 1
c) 3 d) 4
WCA (Within Content Area) (40 marks)
12.A restaurant's vending machine uses a mathematical rule to calculate the total cost of snacks. The first rule is $f(x) = 2x - 3$, and the second rule is $g(x) = \frac{2x - 7}{3}$.
  • Define composite function. [1K]
  • Find $g^{-1}(x)$. [1U]
  • For what value of '$x$', is $ff(x)$ equal to $g^{-1}(x)$? [2A]
13.A student of class 10 NMA is editing a digital photo on his laptop. The changes are saved as matrices, $$ A = \begin{bmatrix} 2m & 7 \\ 5 & 9 \end{bmatrix}, \quad B = \begin{bmatrix} 9 & n \\ -5 & 4 \end{bmatrix} $$
  • Write the condition for a matrix to have its inverse. [1K]
  • For what values of '$m$' and '$n$' are the matrices $A$ and $B$ the inverse of each other? [2HA]
  • Prove that: $(A^T)^T = A$ [1A]
14.A teacher asked two students to write two functions in ordered pair form.
Student-1 : $f = \{(a, 1), (b, 2)\}$
Student-2 : $g = \{(1, p), (2, q)\}$
  • Find $g \circ f(a)$ and $g \circ f(b)$. [1U]
  • Show '$g \circ f$' in a mapping diagram. [1A]
15.Two speakers play sound waves that combine using compound angles $A$ and $B$.
  • Define compound angles. [1K]
  • Simplify: $\cos(A + B) - \cos(A - B)$ [1U]
  • If $A + B = \frac{\pi}{4}^c$, prove that: $(1 + \tan A)(1 + \tan B) = 2$ [2A]
16.The angle between two shifting solar panels or lines on a grid is $50^\circ$ and $40^\circ$.
  • Expand: $\tan(50^\circ + 40^\circ)$ [1K]
  • Prove that: $\tan 50^\circ - 2 \tan 10^\circ = \tan 40^\circ$ [2A]
17.Two ladders lean against two different walls. The first ladder makes an angle $A$ with the ground such that $\sin A = \frac{3}{5}$. The second ladder makes an angle $B$ with the ground such that $\cos B = \frac{7}{5\sqrt{2}}$.
  • Find the value of $\sin(A + B)$. [1U]
  • Gita says, “The sum of angles $A$ and $B$ is one-fourth of $\pi^c$.” Is she correct? Justify. [1HA]
18.A vehicle on a straight highway passes through two checkpoints $A$ and $B$. A milestone $C$ is installed at the coordinate $(5, 5)$. The checkpoints are located at $A(p, 0)$ and $B(0, q)$.
  • What does it mean to say that checkpoints $A$, $B$, and milestone $C$ are on the same straight road? [1K]
  • Using the equation of the road passing through $A$ and $B$,
    prove that: $\frac{1}{p} + \frac{1}{q} = \frac{1}{5}$ [2A]
19.Three security posts of a triangular park are located at $A(2, 2)$, $B(2, 8)$, and $C(-6, 2)$. The park authority wants to construct a straight walking path from post $A$ to the mid-point of side $BC$.
  • Show that the equation of the walking path is $3x + 4y = 14$. [2A]
  • What is the length of the middle path? [1U]
20.A triangular object is placed on graph paper. The coordinates of vertices are noted as $X(-1, 3)$, $Y(1, -1)$, and $Z(5, 1)$.
  • Displace the object by $T = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and write the coordinates of its new position on the graph. [2A]
  • Show both positions of the object on the graph. [2U]
  • What do we call $T = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ in transformation? [1K]
21.A civil engineer is designing a crossroad where two roads must intersect at $90^\circ$ for smooth traffic movement. The direction of each road is represented by vectors $\vec{m} = 3\vec{i} + m\vec{j}$ and $\vec{n} = 6\vec{i} - 2\vec{j}$.
  • Write the condition for vectors $\vec{m}$ and $\vec{n}$ to be perpendicular. [1K]
  • Since the roads are perpendicular, what is the value of '$m$'? [1A]
  • The engineer claims that a service lane, the direction of which is given by vector $\vec{i}$, makes an angle of $70^\circ$ with $\vec{m}$. Is his claim correct? Give the correct angle. [1HA]
22.The health department recorded the ages (in years) of patients visiting a community health camp. To understand how the ages are distributed, the data was analyzed using quartiles. It was found that the upper quartile is 65 years and the coefficient of quartile deviation is 0.71.
  • Write the formula to calculate quartile deviation. [1K]
  • Calculate the lower quartile $Q_1$. [1U]
  • If another health camp reports a coefficient of quartile deviation of 0.30, which health camp is more consistent? Justify your answer. [1HA]
23.A smart water tank automatically controls the flow of water based on the water level measured by a sensor. The water flow function is defined as, $$ f(x) = \begin{cases} 2x + 1 & \text{if } x < 4 \\ 7 & \text{if } x = 4 \\ x + 5 & \text{if } x > 4 \end{cases} $$ where '$x$' is the water level (in meters).
  • State the condition for a function to be continuous at a point $x = a$. [1K]
  • Calculate $\lim_{x \to 4^-} f(x)$ and $\lim_{x \to 4^+} f(x)$. [1U]
  • Show the relation between $\lim_{x \to 4} f(x)$ and $f(4)$. [1A]
  • Is the water flow continuous at $x = 4$? If not, redefine $f(x)$. [1HA]
CCA (Cross Content Area) (24 marks)
24.Students of grade 8 prepare a triangular flower garden with vertices $A(2, 1)$, $B(5, 3)$, and $C(6, 2)$. To redesign the school campus, the garden is shifted 3 units to the right and 2 units upward.
  • Find the coordinates after the translation. [2U]
  • If $P$ is the mid-point of side $BC$, find the slope of the median $AP$. [1A]
  • Find the measurement of $\angle ABC$. [2A]
  • If $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$, what is the relation between $\vec{a}$ and $\vec{b}$? Establish. [1HA]
25.A robotics engineer designs a sensor that measures the angle. The engineer uses these functions:
$g(\theta) = \cos 18^\circ$
$f(x) = \frac{x - \sin 18^\circ}{x + \sin 18^\circ}$ and $m = f \circ g(\theta)$.
  • Find $f \circ g(\theta)$. [2U]
  • Prove that: $m = \tan 27^\circ$ [2A]
  • Is $(f \circ g(\theta))^{-1} = 27^\circ$? Justify how. [1HA]
  • If $f(x) = \frac{x + \sin 18^\circ}{x - \sin 18^\circ}$ and $m_1 = f \circ g(\theta)$, then is $m \cdot m_1 = 1$? How? [1HA]
26.A drone inspects a newly constructed bridge. The drone's onboard computer converts the measured altitude ($x$) (in meters) into a processed value using $f(x) = 2x + 5$. The processed value is then converted into a display value by $g(x) = 3x - 4$. While flying, the drone changes its direction by first turning $45^\circ$ and then another $30^\circ$. Finally, the drone travels from checkpoint $A(2, -1)$ to checkpoint $B(8, 5)$. During the flight, it must pass through a monitoring station located at $C(5, 2)$.
  • If $g \circ f(x) = \{(0, 11), (2, q), (p, -1), (-1, 5)\}$, find the values of $p$ and $q$. [2A]
  • Without using a calculator, find $\cos(45^\circ + 30^\circ)$. [2U]
  • Does the monitoring station lie on the drone's path from $A$ to $B$? Justify. [2HA]
27.A factory records the weights of 50 packets of rice produced in one hour.
wt (kg) 20-24 24-28 28-32 32-36 36-40
Frequency 6 10 14 12 8
The quality control system is modelled by, $$ f(x) = \begin{cases} 2x + 1 & \text{if } x < \text{Q.D.} \\ 8.9 & \text{if } x = \text{Q.D.} \\ 3x - 2.95 & \text{if } x > \text{Q.D.} \end{cases} $$ The factory has two warehouses whose equation, if joined by a line segment, is $1.975x + y = 7.9$.
  • Calculate the quartile deviation of production. [3A]
  • Find the L.H.L. and R.H.L. of $f(x)$ at $x = \text{Q.D.}$ [1A]
  • Is $f(x)$ continuous at $x = \text{Q.D.}$? Discuss. [1HA]
  • Is the point $(2, \text{Q.D.})$ on the same straight line as the two warehouses? Justify. [1HA]
Best of luck!