1. 	(a) 	Find the equation of the line which passes through (-4,2)and (2,4).    [15x3=45 Marks] 

	(b) 	Find the equation of the circle whose radius is r=4 and whose centre is (0,0). 

	(c) 	Find the co-ordinates of the vertex and the focus of the parabola y2=4(x+y). 

	(d) 	Find the foci and vertices of the hyperbola 9x2-y2-36x+8y-5=0. 

	(e) 	Evaluate:


		d/dx(3x2+7) 


	(f) 	Fill in the blanks:
		]
		(i) |2x+3y|<= -------
		(ii)|2 x y| = -------


	(g) 	Evaluate ?(3x2+14)dx 

	(h) 	State for each, whether the following is true:

		(i) 7 € [-8,5]
		(ii)6 € [1,16] 


	(i) 	Solve the following equations:

		3x+5y=13
		2x+9y=20 


	(j) 	If U=[1,2,3,4,5,6,7,8,9,10]

		  A=[1,3,5] B=[2,4,6,8]


	Find AUB. 
	

	(k) 	Give the functions

		f(x)=4x2+3
		Find value of f(x)for each
		(i)x=3 and (ii)x=7 


	(l) 	Obtain 3+5i/2+i in the form of a+ib,a,b € R. 


	(m) 	Solve the equation 2x2+5x+6=0 


	(n) 	Show that for sets A,B and C:If A is subset of B and B is subset of C then A is subset of C. 


	(o) 	Find the equation of the ellipse whose principle axis lies along the axis of co-ordinates, focus is (2,0)and length of latus rectum is 6. 





2. 	(a) 	Evaluate: ?(3sinx+11)dx.    [3+4+3=10 Marks] 

	(b) 	Evaluate:?05(2x2+6)dx 

	(c) 	Find the area of the smaller region lying above the x-axis and included between the circle x2+y2=2x and the parabola y2=x. 



3. 	(a) 	Find the equation of the circle having radius 4 and as centre the point (3,4).     [3+4+3=10 Marks] 

	(b) 	Find the equation of the parabola whose vertex is (-2,2) and focus is (-6,6). 

	(c) 	Find the eccentricity, the foci and equations of the directrix of the ellipse 9x2+4y2=36 


4. 	(a) 	Draw the curve for the following equation:    [3+4+3=10 Marks]

		x2+4y2+4xy=0 

	(b) 	Draw the curve for the following equation:

		3x+7y=8 

	(c) 	Find the value of the determinant:
		
		|1 1 1|

		|x x2 x3|

		|y y2 y3| 



5. 	(a) 	For any two sets A and B in a universal set U.Prove that:    [3+4+3=10 Marks]

		(i) (AnB)' = A'U B'
		(ii)(AUB)' = A'n B' 


	(b) 	Apply De Moivoe's theorem, prove that 

		(i) cos2? = cos2? - sin2?.
		(ii)sin2? = 2 sin? cos?. 


	(c) 	Solve x2+7x+10=0 


6. 	(a) 	Find the equation of the plane passing through (1,2,0) and the line
		
		xcosa + ycosß +zcos?=1,x+y=z.    [4+3+3=10 Marks] 


	(b) 	Find the angle of intersection of the spheres

		x2+y2+z2-2x+2y-4z+2=0 and x2+y2+z2=4 

	(c) 	Find the equation of the tangent plane at the point (3,4,-1) to the cone 2yz-3zx-2xy=0. 



6. 	(a) 	Fill in the blanks : [3 marks] 

`		1.	By commutativity of '.' in real numbers, we get x.y = ........... where x and y ae real numbers.
		2.	By associative property of '.' in real numbers, we get (x.y).z = ......... for real numbers x,y and z.
		3.	For real numbers x, y and z, then using transitivity of '>' in R, we get if x > y and y > z then........ . 



	(b) 	For real numbers x and y, tell for each of the following, whether it is true or False : [3 marks] 

		1.	|x+y| always equals |x|+|y|
		2.	|x.y| always equals |x|.|y|
		3.	|x-y| always equals |x|-|y|,


	where |x| = absolute value of x. 
	

	(c) 	In each of the following, if f : R ~ {0}-> R is a function and is defined as : [3 marks] 
		
		1.	f(x)=5x, then tell wheter f is 1-1 or not and why.
		2.	f(x)=2x4, then tell whether is 1-1 or not and why.


	f(x) = 2/x2, then tell whether f is onto or not andy why.


	(d) 	Given : f : R -> R and g : R -> R are two functions such that f(x) = 2x3 and g(x) = 7x+5, then find fog and gof. [3 marks] 

	(e) 	Find dy/dx for each of the following : [3 marks] 

		1.	y=3 sin x
		2.	y=17+5x
		3.	y=x6


	(f) 	Evaluate each of the following : [3 marks] 
			
		1.	? (3+x4) dx
		2.	? sin x dx
		3.	? 7 dx
	
	
	(g) 	Evaluate each of the following : [3 marks] 
	
		1.	2?3 (7+8x) dx
		2.	1?2 e5x dx


.	0?p/2 cos x dx

	(h) 	Solve the following system of linear equations : [3 marks] 


		•	5x + 4y = 14
		•	3x + 7y = 13


	(i) 	Find the value of the determinant : [3 marks] 
			
			|2 3  6| 
			|4 1 12|
			|3 2  9|  


	(j) 	Find the arithmetic mean of the following numbers : 8, 15, 10, 12, 6. [3 marks] 
		

	(k) 	Find the geometric mean of the following numbers : 2, 4, 8, 64. [3 marks] 


	(l) 	For each of the following, tell whether it is true or false, where A, B and C are sets and U, n denote respectively set union and set intersection : [3 marks] 

		•	A U B always equals B n A
		•	(A U B) U C = A U (B U C)
		•	A n ? = A, whether ? denotes empty set.


	(m) 	Draw a Venn diagram for sets A and B with universal set U such that A is a subset of B. [3 marks] 


7. 	(a) 	State the following properties/laws of real number : [4 marks] 

		1.	Associative property of '+' in real numbers
		2.	Distributivity of '.' over '+' in R
		3.	Archimedean property
		4.	Monotone property of '+' in R
	

	(b) 	Draw a graph for each of the following functions : [4 marks] 

		1.	f : R -> R such that f(x) = 7 for all x in R
		2.	f : R -> R such that f(x) = 2x + 3 for alll x in R


	(c) 	Define each of the following concepts and give an example for each : [4 marks] 
	
		•	Odd function
		•	composition of two functions


8. 	(a) 	Evaluate the following : [4 marks] 

		1.	0?p (x - sin x) dx
		2.	? ((2 / (3(1+x2))) dx


	(b) 	For each of the following function, find whether te function is monotonically increasing or monotonically decreasing or neigher , on given interval : [4 marks] 

		1.	f(x) = x2 - 1 on [0,2]
		2.	f(x) = cos x on [0,p/2]


	(c) 	Prove the following inequality : ex > 1 + x2 / 2 + x 3 / 6. [2 marks] 

	(d) 	Find the area of the region bounded by x = 0, x = 3, and y = 3. [2 marks] 


9. 	(a) 	Do as directed : [4 marks] 
	
		1.	Describe the following set by listing method : { x|x is a divisor of 36}
		2.	Describe the following by property method : {2, 4, 6, 8, ...}
		3.	Show the following for any set A, ? sub set A, where ? denotes empty set.


	(b) 	Obtain conjugate of each of the following complex numbers : [3 marks] 

		1.	3 + 5i
		2.	8i3.	12


	(c) 	Explain the following with suitable exmple : [5 marks] 
		
		•	Proof by counter-example
		•	Proof by contradiction


10. 	(a) 	Solve the following : [6 marks] 
		
		•	x + 2y + 3z = 10
		•	2x + y + 2z = 10
		•	3x + 4y + z = 18


	(b) 	Find the value of the following determinant : [3 marks] 
		
		|1 2 3|
		|1 2 4|
		|3 1 2| 
	

	(c) 	Explain each of the following concepts with one suitable example for each : [3 marks] 

		•	Harmonic Mean
		•	Arithmethic Mean
		•	Geometric Mean


6. 	(a) 	Find the mid point of the straight line joining the line segment A(2,3) and B(-5,7). [2 marks] 

	(b) 	Find the equation of the straight line parallel to the line 2y + 3x + 1 = 0 and passing through the point (0,0). [3 marks] 

	(c) 	Find the equation of a straight line in three-dimensional space joining the points (-1,0,1) and (2,1,4). [3 marks] 

	(d) 	Let A(0,2,6), B(3,-4,7), C(6,3,2) and D(5,1,4) be four points in three-dimensional space. Find the projection of the line AB on CD. [4 marks] 

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