SET - 1 (2083) | Lesson Note

1Rewrite the correct option in your answer sheet.

A. When solving \(ax^2 + bx + c = 0\) graphically, the real roots correspond to 1

(a) The vertex of the parabola

(b) The \(y\)-intercept where the parabola crosses the \(Y\)-axis

(c) The \(x\)-intercepts where the parabola crosses the \(X\)-axis

(d) The axis of symmetry

B. If \(f(x) = 3x - 2\), which one of the following is true? 1

(a) \(f^{-1}(3) = 7\)

(b) \(f^{-1}(7) = 3\)

(c) \(f(7) = 3\)

(d) \(f^{-1}(f^{-1}(3)) = 7\)

C. Which of the following is the value of \(\cos 20° + \cos 40°\)? 1

(a) \(2\cos 30°\cos 10°\)

(b) \(2\sin 30°\cos 30°\)

(c) \(2\sin 30°\)

(d) \(1\)

D. Which of the following correctly expresses \(\sin(A + B)\)? 1

(a) \(\sin A\cos B - \cos A\sin B\)

(b) \(\sin A\cos B + \cos A\sin B\)

(c) \(\cos A\cos B - \sin A\sin B\)

(d) \(\sin A + \sin B\)

E. What happens to the angle of elevation when an observer moves closer to a tall tree? 1

(a) It decreases.

(b) It remains the same.

(c) It increases.

(d) It first increases then decreases.

F. The equation \(x^2 + y^2 = 49\) represents a circle. What is its diameter? 1

(a) 7

(b) 14

(c) 49

(d) 98

G. Two lines are perpendicular. If one line has slope \(m\), the slope of the other is 1

(a) \(m\)

(b) \(-m\)

(c) \(\dfrac{1}{m}\)

(d) \(-\dfrac{1}{m}\)

H. Two reflections are performed successively in \(y = x\) and \(x = 0\). Which point is invariant under the combined transformation? 1

(a) Every point on \(y = x\)

(b) Every point on \(x = 0\)

(c) Only the origin

(d) Such a point does not exist.

I. If the scalar product of two non-zero vectors is zero, the angle between them is 1

(a) \(-1\)

(b) \(1\)

(c) \(0\)

(d) \(k\;(k \neq 0)\)

J. A function \(f(x)\) has equal left-hand and right-hand limits at \(x = a\), but \(f(a)\) is not defined. At \(x = a\) the function is 1

(a) Continuous because limits are equal.

(b) Continuous because limits matter more.

(c) Discontinuous because it has no limit.

(d) Discontinuous because continuity requires \(\lim = f(a)\).

K. A higher Coefficient of Variation (CV) indicates: 1

(a) More consistent data

(b) Greater variability

(c) Lower mean

(d) Higher median

2The function \(f(x) = x - 1\) and a polynomial \(p(x) = x^3 - 6x^2 + 11x - 6\) are given.

(a)Write the condition for \(f(x)\) to be a factor of \(p(x)\).1
(b)Use the rational root theorem to determine the possible roots of \(p(x)\).1

3Given \(f(x) = x + 1\) and \(g(x) = x^2 - 5x + 6\).

(a)Find the composite function \(f(g(x))\).1
(b)Convert \(g(x)\) into vertex form \(y = (x - h)^2 + k\).1
(c)Describe the combined transformation to turn \(y = x^2\) into \(g(x)\).2

4Matrices \(A = \begin{pmatrix}2 & 1\\1 & 3\end{pmatrix}\) and \(B = \begin{pmatrix}a\\b\end{pmatrix}\) are given.

(a)Find \(A^T\).1
(b)Find \(A^{-1}\).1
(c)The constraints are \(2x + y \leq 8\), \(x + 3y \leq 9\), \(x \geq 0\), \(y \geq 0\). Show the feasible region on a graph.2

5Given \(\sin(A + B) = \sin A\cos B + \cos A\sin B\).

(a)How can you convert this into \(\sin 2A = 2\sin A\cos A\)?1
(b)If \(\sin 2A = 2\sin A\cos A\), show that \(\sin 2A = \dfrac{2\tan A}{1 + \tan^2 A}\).1

6From the top of a cliff of height 100 m, two points on the same straight line on the ground are observed with angles of depression 60° and 45°.

(a)Draw a labelled diagram.1
(b)Find the distance between the two points.2

7Given a conditional identity:

\cos 2A + \cos 2B + \cos 2C = -1 - 4\cos A\cos B\cos C

(a)Write \(2\cos A\cos B\) in terms of sum or difference.1
(b)Verify the identity for \(A = B = C = 60°\).2
(c)Why can the identity not be verified for \(A = 90°, B = 60°, C = 45°\)?1

8A line \(L_1\) passes through \(A(1, 2)\) and \(B(5, 6)\).

(a)Write the matrix that transforms \(A(1, 2)\) into \(A'(-2, -1)\).1
(b)\(L_1\) makes an acute angle \(\theta\) with \(L_2 : 2x - y + 3 = 0\). Find \(\tan\theta\).2

9\(x^2 + y^2 - 6x - 4y - 12 = 0\) is a circle.

(a)Write the equation of a circle with centre \((h, k)\) and radius \(r\).1
(b)Find the centre and radius of the given circle.2
(c)Use the two-point formula to find the equation of the radius through point \(P(8, 2)\) on the circle.2

10Points \(A(2, 1)\) and \(B(6, 3)\).

(a)Apply the combined transformation \(T \circ R\) where \(R\) is reflection in the \(Y\)-axis and \(T\) is translation by \(\begin{pmatrix}3\\2\end{pmatrix}\). Find the images of \(A\) and \(B\).2
(b)Plot the object and final image on the same graph paper.1

11Triangle \(ABC\) has vertices \(A(2, 3)\), \(B(4, 1)\), \(C(6, 5)\). \(D\) and \(E\) are midpoints of \(AB\) and \(AC\).

(a)Write the position vector of \(D\) in terms of \(A\) and \(B\).1
(b)Find the position vector of \(D\) in \(\vec{i},\,\vec{j}\) form.1
(c)Prove by vector method: \(\overrightarrow{DE} = \dfrac{1}{2}\overrightarrow{BC}\).1

12The box plots below show battery life of Battery \(X\) and Battery \(Y\).

(a)Which battery has higher median battery life?1
(b)The coefficient of quartile deviation of Battery \(Y\) is 0.098. Calculate that of Battery \(X\).1
(c)Which battery would you suggest for a remote signal tower and why?1

13A function \(p(x)\) is defined as:

p(x) = \begin{cases} 2x + 1, & 1 \leq x \leq 3 \\ x + 4, & x > 3 \end{cases}

(a)Define continuity of a function.1
(b)Find \(\displaystyle\lim_{x \to 3} p(x)\).1
(c)Find \(p(3)\).1
(d)Is \(p(x)\) continuous at \(x = 3\)? Justify.1

14Triangle \(OAB\) with \(O(0,0)\), \(A(1,0)\), \(B(1,1)\) and transformation matrix \(M = \begin{pmatrix}0 & 1\\1 & 2\end{pmatrix}\).

(a)Find the image \(O'A'B'\) under \(M\).2
(b)Find the slope of the altitude from \(O\) to \(AB\).1
(c)Is \(M\) singular? If not, how can you make it singular?1
(d)Dolma claims that \(M\) and \(M^T\) give the same transformation. Do you agree? Justify with an example.2

15Shreya cuts a wooden cone to form a conic section.

(a)Name the shape when the cut is inclined more than the semi-vertical angle but less than 90°.1
(b)For a semicircle \(ABC\) with diameter \(AC\), prove using vectors that \(\angle ABC = 90°\) for any \(B\) on the arc.2
(c)Would the claim hold if \(B\) lies inside the semicircle? Explain.1
(d)If \((AB)^2 = \sqrt{x+9}\), \((BC)^2 = \sqrt{x}\) and \(AC = 3\) units, find \(x\).2

16A hiker climbs a hill with height \(h = f(x) = mx + 50\) (metres). The slope \(m\) is given by

m = \frac{\cos\theta - \cos 3\theta}{\sin 3\theta - \sin\theta}

(a)Prove that \(m = \tan 2\theta\).2
(b)If \(\tan\theta = \dfrac{1}{3}\), find \(m\).2
(c)Using \(f^{-1}(x)\), find the horizontal distance when the hiker is at height 150 m.2
(d)Bina claims: "If the angle with the \(X\)-axis doubles, the slope doubles." Verify with a specific \(\theta\).1

17A bus stop is to be placed at position \(k\) metres along a road. The table gives population in each section:

Length (in meter)	Mid value (\(m\))	Population (\(f\))
0 – 200	100	20
200 – 400	300	30
400 – 600	500	40
600 – 800	700	10

The total squared walking distance is \(s(k) = 10k^2 - 76k + 178\).

(a)Calculate the coefficient of variation of the residence distribution.3
(b)Is \(s(k)\) continuous for all \(k\)?1
(c)Given \(s(3.8) = 30\), is \(s(k)\) continuous at 3.8?1

SET - 1 (2083)

CDC Model Question – 2083