## Model Question: Optional Mathematics

MODEL QUESTION

Subject: Optional I: Mathematics Full Marks: 100

Time: 3 hrs

*Candidates are required to answer in their own words as far as practicable. Credit will be given to originality, not to rote-learning.

Attempt ALL the questions.

Group 'A' [8×(2+2)=32]

1. a. Define composite function. If functions f = {(1, x), (2, y), (3, z)} and g = {(a, 1), (b, 2), (c, 3)}, find the composite function fog.

b. State factor theorem. If x + 1 is a factor of 2x3 – px2 – 8x + 5, find the value of p.

2. a. Find the sum of the series –2 – 4 – 6 – ……… to 30 terms.

b. If A=$$\begin{bmatrix}1&1\\ 8&3\\ \end{bmatrix}$$ , then prove that A2 – 4A = 5I where I is an identity matrix.

3. a. If M = $$\begin{bmatrix}8p & (-2)\\ 4&6\\ \end{bmatrix}$$ and its determinant is 26, find the value of p.

b. Find the value of k if the lines given by the equations  $$\frac{3}{y}$$ - $$\frac{4}{x}$$ +$$\frac{7}{xy}$$ = 0 and 4x – ky – 5 = 0 are perpendicular to each other.

4. a. Find the obtuse angle between the lines represented by $$\sqrt{3}$$x2 + 4xy + $$\sqrt{3}$$y2 = 0 .

b. Find the centre and radius of the circle.  x2 + y2 – 4x – 10y + 20 = 0

5. a. Calculate the value of cos105º without using a calculator or table.

b. Prove that: tan 55º – tan 35º = 2 tan 20º

6. a. If sin x = 0.6 , find the value of sin.

b. Solve: 4cos2 – 1 = tan0º (0º , 180º )

7. a. If $$\overrightarrow a$$ = 3$$\overrightarrow i$$ + k$$\overrightarrow j$$ and $$\overrightarrow b$$ = 4$$\overrightarrow i$$ - 2$$\overrightarrow j$$ are perpendicular to each other, then what is the value of k?

b. If |$$\overrightarrow a$$| = 16 ,  $$\overrightarrow a$$. $$\overrightarrow b$$ = 24$$\sqrt{3}$$ and angle between  $$\overrightarrow a$$ and  $$\overrightarrow b$$ is 30º, find |$$\overrightarrow b$$|.

8. a. If r1 is the reflection on the line x = 4 and r2 is the rotation through +90º about the origin, then find the image of the point A(3, – 2) under the combined transformation r1.r2.

b. Which transformation is represented by the matrix  $$\begin{bmatrix}0 & (-1)\\ (-1)&0\\ \end{bmatrix}$$ ? Find the image of P(3, –4) by using this matrix.

Group 'B' [17×4=68]

9. If f(x) =  and f(x) = f–1(x), find the values of x.

10. Solve: y3 – 21y – 20 = 0.

11. The sum of three numbers in AP is 45. When the numbers are increased by 5, 15 and 40 respectively, the resulting numbers are in GP. Find the numbers.

12. Solve graphically: x2 + 2x – 3 = 0.

13. Solve by matrix method:  $$\frac{3}{y}$$ - $$\frac{1}{x}$$  =  $$\frac{2}{xy}$$

14. Find the equations of straight lines passing through (2, 3) and making an angle of 45º with the line x – 3y – 2 = 0.

15. Find the angle between two straight lines represented by the equation ax2 + 2hxy + hy2 = 0. When are these lines perpendicular? What happens when h2 = ab?

16. Find the equation of a circle with radius 4 units, whose centre lies on the line 13x + 4y = 32 and which touches the line 3x + 4y + 28 = 0.

17. Prove that: sin 20º sin 40º sin60º sin80º = 3/16.

18. If A, B and C are angles of a triangle then prove that: cos2A + cos2B + cos2C = 1 – 2 cosA cosB cosC

19. Solve: (0º , 360º)

$$\sqrt{3}$$ sin + cos = 1

20. A 1.5m tall boy is standing at some distance from a 30m tall building. The angle of elevation from his eyes to the top of the building increases from 30º to 60º as he walks towards the building. Find the distance he walked towards the building.

21.Prove by vector method that in a right-angled triangle, the midpoint of the hypotenuse is equidistant from the vertices of the triangle.

22. A(2, 1), B(4, 3) and C(3, 5) are the vertices of a triangle ABC. If r1 represents the reflection in X-axis and r2 the reflection in the line x = 5, find the image of triangle ABC under the combined transformation r2.r1 and draw both object and its image on the same graph paper.

23. Find a 2×2 transformation matrix that transforms a unit square ABCD to a parallelogram A'B'C'D' with vertices A'(0, 0), B'(3, 0), C'(4, 1) and D'(1, 1).

24. Find the mean deviation from median and its coefficient from the given data.

 Wages 200 300 250 350 400 No. of Workers 5 12 10 8 5

25. Find the standard deviation from the given data.

 Class 0-10 10-20 20-30 30-40 40-50 Frequency 5 4 4 6 1

**Good Luck**