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]
