**RE-231**

**SEE 2075(2019)**

**Optional Mathematics**

**Time: 3hrs Full Marks: 100**

**Answer all the questions**

**.**

**To Download The Question Paper, Click Here.**

**Group ‘A’ [8x (2+ 2) = 32]**

a. If

**then find the value of**

*f(x) = 2x - 1*

*ff (-1**).*

b. If a factor of a polynomial

**find the value of**

*p(x) = 2x*is^{3 }- 6x^{2 }- 5m - 2*x - 2*,

*m.*2.

a. Calculate the arithmetic mean and geometric mean between

**and**

*4*

*16.*b. Write the definition of inverse matrix. In which condition the inverse matrix cannot be defined? Write it.

3.

a. If find the matrix of

**.**

*C*b. Write down the formula to find the angle between the lines

**and**

*y = m*_{1}x + c_{1}**Also write the condition of perpendicularity of these lines.**

*y = m*_{2}x + c_{2}.4.

a. If the lines represented by

**are coincident, find the value of**

*16x*^{2 }– kxy + 9y^{2 }= 0

*k.*b. The end points of a diameter of a circle are

**and**

*(O, 6)***Find the equation of the circle.**

*(8, O).*5.

a. Find the value of

*sin165*

^{0}^{ }without using a trigonometric table or a calculator.

b. If , prove that .

6.

a. Prove that:

b. Solve:

7.

a. If unit and the angle between is

**, find the length .**

*30*^{0}a. Find the coordinates of the image of a point

*A (3, -5**)*after reflection on the line

**followed by the rotation through**

*x + y = 0*

*+90*^{0}^{ }about the origin.

b. Find the transformation represented by the matrix

**Group ‘B’ [17x4= 68**

**]**

**and**

*g(x)= 2x -3***then find**

*fog(x) = 6x -11*,

*f*^{-1}(x).10.

**Solve:**

*2x*^{3}- 9x^{2}+ 7x + 6 = O.11. The product of the first five terms of a geometric series is

**. If the third term of the geometric series is equal to the tenth term of an arithmetic series, find the sum of the first**

*243***terms of the arithmetic series.**

*19*12. Solve graphically the quadratic equation

*x*^{2 }- 3x = 10.13. Solve by matrix method:

14. Find the equation of any one straight line passing through the point

**and making an angle of**

*(4, -1)***with the line**

*45*^{0}^{ }

*2x -3y = 5.*15. If an angle between the pair of lines represented by the equation

**is**

*2x*^{2 }+ + 3y^{2 }= O

*45*^{0}*,*then find the positive value of k and also find the separate equations of the lines.

16. In the given figure, the circle

**with centre**

*A***passes through the centre**

*X***of the circle**

*Y*

*B.**If the equation of circle*

**is**

*B*

*x*^{2 }+ Y^{2 }-4x + 6y - 12*and the co-ordinates of*

**are**

*X***then find the equation of the circle**

*(-4, 5),*

*A.*

17. Without using the calculator or table, find the value of:

*sin 100*^{0}.sin120^{0}.sin140^{0}.sin160^{0}18. If

**and**

*P, Q***are the angles of a**

*R*

*Î”PQR**,*prove that .

19. Solve:

20. From the roof and foot of a house, the angles of depression and elevation of the top of a tree are

*60*^{0}*and*

^{ }

*30*^{0}*respectively. If the height of the tree is*

^{ }**, find the height of the house.**

*15ft*21. Prove by vector method that the circumference angle

**of a semicircle with a diameter**

*ACB***is a right-angle.**

*AB*22. A triangle

**with vertices**

*ABC***and**

*A (2, 3), B (2, 6)***is translated by and the image so obtained is enlarged by**

*C (3, 4)***Write the 3 co-ordinates of the vertices of the images so formed and represent**

*E [(O, O), 2].***and its both images in the same graph.**

*AABC*23. A line segment

**joining the points**

*AB*

*A (4, 1)**and*

*B (7, 5**)*is transformed to the line segment

**joining the points**

*AB'***and**

*A'(-4, 1)***Find the**

*B'(-7, 5).***matrix that represents this transformation.**

*2x2*24. Find the mean deviation and its coefficient from median of the data given below:

*10, 50, 60, 40, 30, 2*25. Calculate the coefficient of variation from the data given below.

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