SAT 2010

MATHEMATICS

2 marks questions

1). Let l1,l2 be any two parallel lines and B,C be any two points on l1 and A1,A2.........A2010 be points on l2 . If ∆i denotes the area of the triangle AiBC and if ∑_(i=l)^2010▒〖∆i=2010〗, Then the area of ∆ A2010BC is

A) 1 B)1/2 C) 2 D) 2010 E)1005

2). Let {an} be a sequence of integers such that a1 =1, am+n =am+an+mn for all positive integers m and n. Then a12 is

A) 6 B) 70 C) 78 D)76 E) 72

3). In a triangle ABC, a, b ,c denote the lengths of the sides BC, CA, AB.If D is the midpoint of the side BC and AD is perpendicular to AC, then

A) 3b2 = a2-c2 B) 3a2 = b2-3c2 C) b2 = a2-c2 D) a2 + b2 = 5c2 E) none of these

4). If is an integer then which of the following is true ?

A) An integer of the form 4k+1 can always be put in the form 2k-1

B) An integer of the form 4k+3 can always be put in the form 2k+1

C) An integer of the form 2k-1 can always be put in the form 4k+1

D) An integer of the form 2k-1 can always be put in the form 4k+3

E) An integer of the form 2k+1 can always be put in the form 4k+3

5). The number of elements in {(a,b,c) / a=b, (a-c)2 = 0, a+b+c = 0, a, c are real numbers} is

A) 0 B) 1 C) 6 D) 3 E) infinitely many

6). The no. of solutions of the equation xy(x+y)=2010 where x and y denote positive prime numbers, is--------------------

7) The number of elements in the set {nϵN/n3-8n2+20n-13 is a prime number} is------------------------

8) The solution set of the equation √(x2-4x+4) + (x-2)=0 is -------------------------------

9). Given any two diameters of a circle the convex quadrilateral formed by joining the extremities of the diameters is always a rectangle. True/False--------------------------------

10). If P=32010 + 3-2010,Q=32010- 3-2010 then P2- Q2= ------------------------------------------------------------------------

11). Solve the equation log2010(2009x)= log2009 (2010 x ).

12). In a quadrilateral ABCD, AB = 3, BC =4, CD = 5, ∠ABC = ∠BCD = 1200 . Find the area of the quadrilateral

13). I was trying to solve 4/(x-2)>5. While writing the question I mistakenly wrote a digit other than 5 and solved the inequality and got 2<x<4. What digit did I write possibly?

14). In a right angled triangle what is the relation between the square of the altitude on to the hypotenuse and the product of the segments of the hypotenuse?

15). Is is possible to find two functions f and g such that the domain of f is not finite, the domain of g is finite, go f is defined? Justify your answer.

4 marks questions

1). If the last digits (unit places) of the products1.2.,2.3,3.4,…,n(n+1) are added, the result is 2010. How many products are used?

2). Show that four divides any perfect square or leaves a remainder 1. Also show that nine divides cube of any integer or leaves 1 or 8 as remainder.

3). Let AB be a line segment of length 26. Let C and D be located on the line segment AB such that AC=1 and AD=8. Let E and F be the points on one of the semi circles with diameter AB for which EC and FD are perpendicular to AB. Find the length of the line segment EF.

4). In each of the following cases give an example of a system of two linear equations in two variables x and y .

i) A system having exactly one solution ii) A system having no solution iii) A system having infinitely many solutions.

5). Using Mathematical Induction Prove that 32n+ 7 is divisible by 8,∀ n∈N.

MATHEMATICS 2009

2 Marks

log_(1/2009)2009 =

〖2009〗^(1/2009) b) – 2009 c) 〖(1/2009)〗^2009 d) – 1 e) 1

Let a, b, c be the lengths of a triangle ABC. Let ∝,β,γ be positive integers and p = p=βa+γb+αc,q=γa+αb+βc and r=αa+βb+γc. Then p, q, r are lengths of the sides of a triangle

Only if α,β,γ are all distinct

Only if α+β>γ,β+γ>α,γ+α>β

Only if α>β>γ

Only if α>a,β>b,γ>c

For All values of α,β,γ

3). Let f: R → R be defined by f(x) = x2 for all x in R. let A = {4,3,0,-1}. Then

a) f-1(A) is not defined as f-1 does not exist

b) for any function f: P→Q, if T is a subset of Q, then f-1( T ) is defined and here

f-1(A) = {-2,2,√3,- √(3,) 0}

f-1 ( A ) is not defined as ‘-1’ has no pre-image under the function f

f-1 ( A ) = f ( A ) e) none of these

4). If 1216451 * 0408832000 is equal to 19 !, where * denotes a digit, then the missing digit represented by * is

a) 0 b)4 c)2 d)5 e) 7

5) A polynomial f( x ) is said to be reducible if it can be written as f( x ) = g( x )h( x ) where deg(g( x )) > 1 and deg(h( x )) > 1 otherwise it is said to be irreducible.

If f(x) is a polynomial of degree 2009, Choose the correct statement from the following

There are infinitely many f(x) that are irreducible

There are exactly 2009 f(x) that are irreducible

F(x)=x2009 +1 is an irreducible polynomial

There exists no f(x) that is irreducible

None of these

6) the number of positive integers less than 2009 that have exactly three prime factors, not necessarily distinct, such that each prime factor is greater than 10 is_______________

7) if the odd natural numbers are arranged as shown in the figure, then the extremities f the row containing the number 2009 are ___________

1

3 5

7 9 11

13 15 17 19

21 23 …. …. …..

8) The number of triangles having a fixed base of ‘a’ units and area 2009 square units is ____________

9) The coefficient of x is the expansion of (√7 x4 + 13/4 x3+ x2+ 2x-1)2009 is____________

10) There are four circles in the figure. Two of them have radius 2 and the smallest one has radius 1. The radius of the largest circle, if the area of the dashed part is equal to the area of the dotted part, is__________________

11) How many positive integers less than 2009 are divisible by any one perfect square greater than 168 ?

12) The square ABCD of side ‘a’ and the square BEFG of side ‘b’ are drawn next to each other as shown in the figure. Express in terms of a and b the area of the quadrilateral formed by the mid points of the line segments AB, BE, FC and DG.

13) Find out the unequal pairs of sets, if any, among the following sets.

a) ( A – B ) ∪ ( B – A ) b) ( A ∪ B ) – ( A ∩ B ) c) ( A ∪ B ) ∩ ( A ∩ B)’

d) ( A ∩ B’) ∪ ( B ∩ A’) e) (A ∪ B) ∩ ( A’ ∪ B’)

14) If ABC is a triangle with AB,BC measuring p and q units respectively, find the maximum area possible for triangle ABC.

15) Two real numbers x and y are varying such that their sum is 2009. Find the values of x and y such that their product is maximum.

4 marks

Solve 2x2y – 3x2 + 4y = 2009 for x and y in positive integers.

The lower part of the house ( see the figure ) in the circle is a square, and the top is an equilateral triangle. Find a relation between the length of the side of the house and the radius of the circle

A function f: N→Z is defined by f(n) = {█(n/2,if n is even@-(n-1)/2 ,if n is odd)┤ Is f a bijection ?

Solve the system of equations ax = 2y = b and 2x – y =3 for all real values of a and b.

Is it possible to insert exactly 125 Harmonic Means between (-1)/49 and 1/2009 ? if so, what is the 6th HM?

MATHEMATICS 2008

2 marks questions

1). Let A={1,2,3,……,2008},B={1,2,3,….,1004}.[ ]

A). There can be infinitely many functions from A to B

B) There cannot be an onto function from A to B

C) There can be at least one one-one function from A to B

D) There can be infinitely many onto functions from A to B

E) None of these

2). The number of equiangular octagons fixing 6 consecutive sides is [ ]

A) Infinitely many B) Exactly 8 C) At most 8 D) 0 E) None of these

3). All the numbers between 1947 and 2008 are written, including 1947 and 2008. From the list, all the multiples of 3 and 5 are stuck off. The sum of the remaining numbers is [ ]

A) 41517 B) 73137 C) 73138 D) 65247 E) 65248

4). A square ABCD is inscribed in a quarter circle where B is on the circumference of the circle and D is the center of the circle. The length of diagonal AC of the square, if the circle’s radius is 5, is [ ]

A) 〖5 〗^(π/4) B) 5^(π/2) C) 5 D) 5^√2 E) The length cannot be determined

5). 50 x 50 x 50 x ……(where there are a hundred 50s) is how many times 1000 x 100 x 100 x.(where there are fifty 100s)? [ ]

A) 25 x 25 x 25 x …..(where there are fifty 25s)

B) 4 x 4 x 4 x……(where there are fifty 4s)

C)2 x 2 x 2 x…. (where there are fifty 2s)

D) 1 time E) None of these

6). a, b are positive integers. A is the set of all divisors of ‘a’ except for ‘a’ . B is the set of all divisors of ‘b’ . If A=B then which of the following is a wrong statement?

A) a ≥b B) a is a multiple of b C) b is a not a multiple of a D)a/b is a prime number

E) none of these

is ---------------------------------------------

8). The number of points P strictly lying inside an equilateral triangle ABC such that the sum of the perpendicular distances from P to the three sides of the triangle is minimum is ----------------

9). Positive integers a,b are such that both are relatively prime and less than or equal to 2008, a2+ b2 is a perfect square and that b has the same digits as a in the reverse order. The number of such ordered pairs (a,b) is------------------------------------------

10). Let ABCD be a square E, F, G, H be the mid points of the sides ¯AB, ¯BC, ¯CD, ¯(DA ) respectively. Let P, Q, R, S be the points of intersection of the line segments ¯AF, ¯BG, ¯CH, DE inside the square. The ratio of the areas ∆ PQRS : ∆ ABCD is

11). The number of elements in the set {(a,b,c) ∶a,b,c are three consecutive integers in some order, a+b+c=abc} is

12). The sum of all positive integers for which the quotient and remainder are equal if the number is divided by 2008 is

13). Is y a real function of x in the equation y+2008 = x2 + 2 ?

14). The people living on street ‘S’ of Y-City all decide to buy new house numbers so they line up at the only Hardware store in order of their addresses: 1,2,3,… If the store has 100 of each digit, what is the first address that won’t be able to buy the digits for its house number ?

14). Let ABCD be a quadrilateral such that AB is perpendicular to BC, AD is perpendicular to BD and AB=BC, BD=a, AD=c, CD=x. find x in terms of a and c.

15). For what pair wise different positive integers is the value of a/(a+1)+b/(b+1)+c/(c+1)+d/(d+1) an integer ?

16). Side AB of rectangle ABCD is 2 units long and side AD is 3 units. E is a point on the line AC such that C is the mid point of the line segment AE. What is the length of line segment BE ?

17). How many number of integers are there between 2008 and 2,00,82,008 including 2008 and 2,00,82,008 such that the sum of the digits in the square is 42 ?

4 marks questions

1). Let S={1,2,3,…2008}. find the number of elements in the set {(A+,B): AUB=S}

2). A square is drawn in side a triangle with sides 3, 4 and 5 such that one corner of the square touches the side 3 of the triangle, another corner touches the side 4 of the triangle, and the base of the square rests on the longest side of the triangle. What is the side of the square ?

3). State and prove the test of divisibility of a positive integer ‘a’ by 11.

4). A square cake 6” x 6” and 3” tall was cut into four pieces of equal volume as shown in the figure. Determine how far in from the side of the cake the cut should be made ? (i.e., x = ?)

CAKE

5). Solve the following simultaneous equations for a and b :

a√a+ b√b=183,

a√b+ b√a=182

6). f and g are two real variable real valued functions defined by

f(x)= {█(x+2 ,if x≥0@2x-3,if x <0)┤ , g(x)= {█(3x+7,if x≥2@x-1,if-2<x<2@x+1,if x ≤ -2)┤ find gof(x)

MATHEMATICS 2007

2 marks

1). Real numbers x1, x2,….x2007 are chosen such that (x1,x2) , (x2,x3)….,(x2006, x2007) are all points of the graph of y= 1/(x-1).

A) such a choice is possible for all real x1≠1

B) In every choice x1,x2,….x2007 are all distinct

C) There are infinitely many choices in which all x 1’ are equal

D) There exists a choice in which the product x1,x3,x2007 = 0.

E) None of these

2). The consecutive sides of an equiangular hexagon measure x,y,2,2006,3,2007 units

A) The hypothesis never takes place

B) The greatest side measures 2007 units and the smallest 2

C) The greatest side measures 2007 units, but the smallest x

D) The greatest side measures y units, but the smallest 2

E) The smallest side measures x units, but the greatest y

3). ABCD is a convex quadrilateral

A) A circle can always be circumscribed to it

B) A circle can never be circumscribed to it

C) A circle can always be inscribed in it

D) A circle can never be inscribed in it

E) None of these

4). A lattice point in a plane is one both of whose coordinates are integers . let O be (√(2,) 1) and p any given lattice point. Then the number of lattice points Q, distinct from P , such that OP = OQ is

A) 0 B) 1 C) not necessarily 0, not necessarily 1, but either 0 or 1

D) infinitely many E) none of these

5). i) f (x,y) is polynomial f0(x)y2006+f2(x)y2005+ …+f2006(x)y+f2007(x) , where each f1(x)

Is a polynomial in x with real coefficients, and ii) (x-∝) is a factor of f(x,y), where a is real number. Then

there exists a f(x,y) such that f(x)=x2+1 for some i∈{ 0,1,2,…2007}

there exists a f(x,y) such that fi(x)=x+1 for some i∈{ 0,1,2,…2007}

There exists a f(x, y) such that fi(x - ∝)2006 + 1 for some i ∈{0,1,2,…2007}

There exists a f(x, y) such that f0(x) = x2 – (2α+1)x+(α2+α)

None of these

6). (b – c)(x – a)(y – a) + (c – a )(x – b)(y – b) + (a – b )(x – c )(y – c ) is

A) independent of x, but not of y B) independent of y, but not of x

C) independent of both x and y D) independent neither of x , nor of y

E) independent of x only if not independent of y

7). For the purpose of this question, a square is considered a kind of rectangle. Given the rectangle with vertices (0, 0) , (0, 223),(9, 223), (9, 0), divided into 2007 unit squares by horizontal and vertical lines. By cutting off a rectangle, we mean making cuts along horizontal and (or) vertical lines to produce a smaller rectangle. Let m be the smallest positive integer such that a rectangle of area ‘m’ cannot be cut off from the given rectangle. Then m =

8). A line has an acute angled inclination and does not pass through the origin. If it makes intercepts a and b on x-,y- axes respectively then |ab|/(ab )= ________________

9) if k is a positive integer, let Dk denote the ultimate sum of digits of k. T-hat is , if k is a digit, then DK=k. if not, take the sum of digits of k. if this sum is not a single digit, take the sum of its digits. Continue this process until you obtain a single digit number. By Dk we mean this single digit number {DP / p is a positive multiple of 2007} = ___________, in roster form.

10) The digits of a positive integer m can be rearranged to form the positive integer n such that m+n is the 2007-digited number, each digit of which is 9. The number of such positive integers m is---------------------------

11) (AB) ⃗and (CD) ⃗ are chords of a circle such that (BA) ⃗ and (DC ) ⃗ intersect in a point E outside the circle. F is a point on the minor arc BD such that ∠FAB = 220, ∠ FCD = 180 . then ∠ AEC + ∠ AFC =

12) the quadratic ax2+bx+a=0 has a positive coincident root ∝ = -----

13) explain a way of subdividing a 102 X 102 square into 2007 non-overlapping squares of integral side.

14) ABC is a triangle. Explain how you inscribe a rhombus BDEF in the triangle such that D ∈ ¯BC ; E ∈ ¯CA and F ∈ ¯(AB.)

15) Equilateral triangle ∆ABC has centroid G, A1,B1 , C1 are points on ¯AG,¯BG,¯CG such that (AıBı) ⃗, (BıCı) ⃗,(CıBı) ⃗ are respectively parallel to (AB) ⃡,(BC) ⃡,(CA) ⃡. If the distance between (BC) ⃡ and (BıCı) ⃡ is one-sixth of the altitude of ∆ABC , determine the ratio of areas (∆ Aı Bı Cı)/∆ABC .

15) P(x) is a polynomial in x with real coefficients. Given that the polynomial p2(x) + ( 9x – 2007)2 has a real root α , determine α and also the multiplicity of α.

16) find the homogeneous function of 2nd degree in x,y, which shall vanish when x = y and also when x=4, y=3 and have value 2 when x = 2, y = 1.

17) If 3yz ÷2y+z+1=0 and 3zx+2z+x+1=0, then 3xy+2x+y+1=0.

4 marks

x3 is the 753rd AM of 2007 AM’s inserted between x1 and x2 ,,y3 is the 753rd AM of 2007 AM’s inserted between y1 and y2 show that A(x1,y1),P(x3 ,y3),B(x2,y2) are collinear. Determine also the ratio AP : PB.

Lines I and m intersect in 0. Explain how you will construct a triangle OPQ such that P∈I,Q∈m, ¯OP and ¯OQ are equal in length and ¯PQ is of length 'a^'.

i) ∠ABC=1200; ii) ∆ABC is equilateral, iii) B and D are on opposite sides of (AC.) ⃡ prove that a) (BD ) ⃡ bisects ∠ABC and b) ¯BD is in length equal to the sum of lengths of ¯AB and ¯(BC.)

a1 , a2,…….,a2007 are 1,2,……,2007 in some order. If x is the greatest of 1.a1 , 2.a2,…..,2007. a2007, prove that x≥(1004)2.

Prove that for all integers n≥2,2n-1(3n+ 4n)>7n.

Resolve x8+y8 into real quadratic factors.

MATHEMATICS 2006

2 MARKS

A triangle has integral sides. Its perimeter is 2006. One of the sides is sixteen times another side. The number of such triangles is

0 b) 1 c) 2 d) 3 e) 4

Given such that ∆ABC is an arbitrary triangle. the number of points P in the interior of the triangle such that ∠APB= ∠BPC= ∠CPA is

0 b. 1 c. 2 d. not necessarily 0, not necessarily 1, but either 0 or 1

e. not necessarily 1, not necessarily 2, but either 1 or 2

3) The number of rhombi that can be inscribed in a parallelogram such that each side of the parallelogram contains precisely one vertex of the rhombus is

a. 0 b. 1 c. 0 or 1 d. 0 or 2 e. infinitely many

4) In this question a polynomial means a polynomial with real coefficients. P(x) is a polynomial of degree 2006 and P(x)≥0 for all real x.

a. P(x) must have at least one complex root

b. P(x) cannot have 2006 distinct real roots

c. P(x;) is a product of 1003 quadratics, none of which has linear factors.

d. P(x) cannot have a polynomial factor of the form (x-k)2005

e. none of these

5) If 0<a ≤x ≤b ,0<a ≤y ≤b and x/b= (2x+y)/(x+a+b) ,then

a). a=x=y=b b). a=x <y=b c). a=y <x=b d). a=y ≤x=b

e) none of these

6) p,q,r are given real numbers. Consider the polynomials (q-r)(x-q)(x-r)+ (r-p)(x-r)(x-p)+(p-q)(x-p)(x-q)+ k for all real k. then

a). there exists a k such that the degree of the polynomial is 2

b). there exists a k such that the degree of the polynomial is 1

c). there exists no k such that the degree of the polynomial is 0

d). there exists no k such that the degree of the polynomial is undefined

e). none of these

A lattice point is a point both of whose c0-ordinates are integers. A is (0,0) B IS (4012,6018). If n is the number of lattice points lying on ¯AB, THEN n = ________________

Two circles of equal radius r are inscribed in a semicircular sector of radius ‘I’ such that they touch each other externally. Then r = _________________

In a triangle ABC,D,E are points on ¯AC ¯BC respectively such that ¯DE // ¯AB and DE = 5. If AB = 8 and ¯AE bisects ∠DEB, then the length of ¯CE=__________________________________

1+ 1/(1+2)+ 1/(1+2+3)+⋯+ 1/(1+2+3+⋯+2006)= _______________________________

The characteristic of log_3 2006 is ____________________

If the positive integer m is such that m/2006 is closest to √2 then m = __________________

A(0,0), B(4012, 0), C(4012, 6018), D(0, 6018) determine a rectangle which is the union of 4012 x 6018 unit squares. Determine the number of these unit squares which have a non-empty intersection with ¯AC

P is a point not lying on a line l. Explain how you construct an equilateral triangle PQR such that ¯QR ⊂l .

Let E be the midpoint of the side ¯BC of ∆ABC and let F be chosen in ¯AC so that AC = 3FC determine the ratio of the areas, ∆ABEF∶ ∆EFC .

Does the equation 2006 = (x+y)2+ 3x+y have a solution in non-negative integers ?

Without removing brackets at any stage factorize 2y(y+z)-(x+y)(x+z)

Factorize the polynomial in x∶(t2+1)x3-(t2-1)x2-(t2+1)x+(t2-1)

4 marks

Treating the equalities log102 = 0.3010 and log103 = 0.4771 as exact, prove that the sum to 30 terms of the series 2+6+18+54+………exceeds 2 x 1014

Factorize a5 (b-c)+ b5(c-a) + c5(a-b)

Prove that (n+1) is either itself a prime or a product of primes for every positive integer n.

A triangle ABC is turned about the vertex A in the plane of the triangle into the position AB’C’ if ¯AC BISECTS ¯BB ,prove that ¯AB bisects ¯CC .

P is a point in the plane of parallel lines l,m. explain how you locate a point Q on l and a point R on m such that ¯PQ and ¯PR are equal in length and perpendicular to each other. discuss only the case when P lies between l and m.

A,B,C are respectively (1002,1005),(2004,2001) and (1003,1000) and P0 is (1,4). Points Pi , i ∈N are such that

Is the midpoint of ¯(P_0 P_1 ) ,¯(P_3 P_4 ) , ¯(P_6 P_7 ) ,_____________

Is the midpoint of ¯(P_1 P_2 ) ,¯(P_4 P_5 ) , ¯(P_7 P_8 ) ,_____________

Is the midpoint of ¯(P_2 P_3 ) ,¯(P_5 P_6 ) , ¯(P_8 P_9 ) ,_____________

Estimate the ratio of lengths (P_2005 P_2006)/(P_1 P_2 ).

MATHEMATICS 2005

2 marks

C is a circle and P is a point exterior to it. Several lines are drawn through P such that each line has nonempty intersection with C . If 2005 points of intersection are formed, then among the lines.

There need not be even one tangent

There may be two tangents

There may be three tangents

There has to be precisely one tangent

None of these

For I = 1,2,…..,2005, ai is a positive integer.

a n , an+1, an+2 is a G.P. if n is odd.

a n , an+1 , an+2 is an A.P. if n is even.

There is no such sequence

a_n=1 for all i∈{1,2,…..,2005}

a_n/a_1 is the square of a rational number,if n is odd

there are only a finite number of such sequenes

None of these

P is an integer point of a circle C whose centre is O AND O≠P. C_1 is the circle with ¯OP as diameter. Q is a point on the circumference of C_1 such that Q ∉{O,P}. (PQ) ⃡ intersects C in A and B. then AQ/QB is

Independent of P and Q

Independent neither of P nor of Q

Independent of P but not of Q

Independent of Q but not of P

Dependent on the radius of C

n is a positive integer. The number of its positive integral divisors is 2005. Then

It cannot be the 4-th power of a positive integer

It cannot be the 3-rd power of a positive integer

It cannot be the 12-th power of a positive integer

It cannot be the 6-th power of a positive integer

None of these

D, G are points on the side ¯AB of ∆ABC. E and F are points on the sides ¯AC and ¯BC respectively such that ¯DE// ¯BC//, ¯EF// ¯AB and ¯FG//¯CA . then D,E,F,G are the consecutive vertices of a quadrilateral

Always

Only if AD/AB> 1/2

Only if AD/AB= 1/2

Only if AD/AB< 1/2

None of these.

Let S be the set of points common to the lines ax + by=q. let S’ be the set of points common to the lines dx + by=p’ and cx+ay=q’. then :

If S is empty then S’is empty.

If S is infinite so is S’

If S consists of only one element, then S’ is empty

If S consists of only one element, then S’ is infinite

If S consists of only one element, so does S’.

The number of positive integers n such that 2005 is a divisor of n2+n+1 is ___________________

I1, I2 ,…..I2005 are arcs of circles. Ik is part of a circle with radius rk and subtends an angle θk radians at the centre. C is the circle with radius r1+r2+⋯,+r2005. The arc of C whose length is the sum of the lengths of I1,I2….,I2005 subtends at the centre of c an angle of __________________________ radians.

The greatest positive integer k such that xk – 1 is a divisor both of x2005 -1 is ____________________

The distance between the parallel sides ¯AB and ¯CD of a trapezium is 12 units. AB=24 units, CD=15 units. E is the midpoint of ¯AB. O is the point of intersection of ¯DE with ¯AC . the area of the quadrilateral EBCO is _________________________

The set of all natural numbers n such that n ≥ 2 and 2005 is the sum of n consecutive natural numbers is _______________

Consider the smallest multiple of 2005 such that its digits are the lengths of the sides of a pentagon. The number of unit sides in the pentagon is ___________________

C is in the interior of ∠AOB. locate a point P on (OA) ⃗ and Q on (OB) ⃗ such that C is the midpoint of (PQ) ⃗.

P(x) is a polynomial with integral coefficients. P(x) = 4010 for 5 different integral values of x. prove that there is no integer x such that P(x) = 2005.

Determine the radius of the circle inscribed in a rhombus whose diagonals measure 10 units and 24 units.

There are 15 sets of lines in a plane, one consisting of 77 parallel lines, another 5 parallel lines, another 4 parallel lines and another 3 parallel lines. The remaining 11 sets consist of one line each. If no two lines are coincident, no three of them are concurrent and lines belonging to different sets intersect, determine the points of intersection.

In two triangles ∆ABC and ∆DEF,AB=DE,AC=DF,∠ACB,∠DFE are of equal measure. is it necessary that ∠ABC and ∠DEF are of equal measure ? discuss.

In the adjoining figure ∆ OAB=∆OPQ in the indicated order of correspondence. (QP) ⃗ meets ¯AB in X. Prove that (OX) ⃡ bisects ∠AXP.

4 MARKS

Factorize 2(a-b)2+(b-c)(c-a).

In a ∆ABC,∠ABC=7^0 . Explain how you locate all points P in the plane of the triangle such that ∠APC=〖112〗^0.

x≠0.if u_1=0 and if u_(n+1)=(1-x) u_n+nx for all natural numbers n, then prove by induction that u_n=1/x{nx-1+(1-x)n} for all natural numbers n.

ABC are three non collinear points. Explain how you will draw a circle with centre at C such that parallel tangents can be drawn from A and B.

Given positive integers a_0,a_1,a_2…..a_2005. it is known that a_1>a_0,a_2=3a_1-2a_0,a_3=3a_2-2a_1,…..,a_2005=3a_2004-2a_2003.prove that a_2005>2^2004

¯AB is a diameter of a circle π_1,π_2the two half-planes determined by (AB) ⃡ .P_1,P_2,…..P_2004 are points on ¯AB such that 〖AP〗_1=P_1 P_2=P_2 P_3=⋯….=P_2003 P_2004=P_2004 B.for every r∈{1,2,3…,2004} draw a hook, which is the figure formed by the semicircle in π1 on ¯AP as diameter and the semicircle in π2 on ¯PB as diameter. Prove that these hooks divide the circle into 2005 regions of equal area.

MATHIMATICS 2004

2 MARKS

A lattice points is a point (x, y)in the plane such that x and y are integers. The number of rectangles with corners at lattice points, sides parallel to the axes, and center at the origin which have precisely 2004 lattice points on all of its sides put together is

0 b) 501 c) 499 d)500 e)2004

The number of positive integers which divide 2004 to leave a remainder of 24 is

a) 36 b) 20 c) 22 d) 21 e) 34

ABCD is a parallelogram. C’D’ is parallel to AB, with D’ and C’ in the interior of ¯AD and ¯BC respectively. then

AC < AC’ is possible if ∠BCA is acute

AC < AC’ is possible if∠BAC is right

AC < AC’ whenever ∠BAC is obtuse

AC ≤ AC’ whenever ∠BAC is obtuse

None of these

a, b, c, d are real constants, x a real variable. Real number p and q exist such that p(ax+b)+q(cx+d)=ex+f

for every pair of real’s (e. f)

for every pair of real’s (e, f) if they exist for one pair of real’s (e, f)

for every pair of real’s (e, f) if they exist uniquely for one pair of real’s (e, f)

for no pair of real’s (e, f), if they do not exist for some pair of reals (e, f)

none of these

By a chord of the curve y = x3 we mean any line joining two distinct points on it the number of chords which have slope -1 is

Infinite b) 0 c) 1 d) 2 e) none of these.

The descending A.P. of 4 distinct positive integers with greatest possible last term and sum 2004 is____________________

The radii of circle C_1,C_2……C_2004 are respectively r1,r2……….r2004. if r1 = 1 and ri = ri-1+1 for i = 2, 3,……..,2004, then r2004 = _______________________

f(n) = (4^n-4^(-n))/(4^n+4^(-n) ) for every integer n. p and q are integers such that p > q . the sign of f(p) – f(q) is ______________________

M1 is the initial point of a ray in a plane. Mi , for I ∈{2,3,….,2004},are points on the ray such that M1M2=M2M3=M3M4=………..=M2003M2004 . M1 is (a,b) and M2004 is(c, d). if sx and sy are respectively the sum of all x-coordinates and the sum of all y-coordinates of Mi ¬ for i∈{1,2,3,…..,2004},then (sx , sy) =____________________

The number of 2-element sets of non unit positive integers such that their g.c.d is 1 and 1.c.m is 2004 is__________________

Print mistake

(AB) ⃡ and (BC) ⃡ are distinct lines. P is a point in the plane ABC and P ≠B explain how to draw a line through such that if the line interests (AB) ⃡ in Q and (BC) ⃡ in R,then BQ=BR.

Prove that there is no polynomial f(x) with integral coefficients such that f(1)=2001 and f(3)=2004.

∆ABC is right angled at B. A square is constructed on ¯AC on the side of (AC) ⃡ bisects ∠ABC.

4 MARKS

Evaluate ∑_(n=1)^2004▒〖〖(-1)〗^n [√(n],)〗 where [x] denotes the integral part of x.

ABCD is aline segment trisected by the points B, C ; P is any point on the circle whose diameter ¯BC. If the angles APB and CPD are respectively α,β,evaluate tanα.tanβ.

Solve the system of equations in positive integers x, y, z:

x^2-y^2+z^2=2004,x+y-z=48,xy-yz-zx=125

a and b are positive reals and ¯AB a line segment in a plane. For how many distinct points C in the plane will it happen that for triangle ABC, the median and the altitude through C have lengths a and b respectively ?

prove by induction that 4( 1/2.3 )+8 ( 2/3.4 )+16 ( 3/4.5 )+⋯to n terms=2^(n+2)/(n+2)-2

6 marks

find x if x+y+z+t=1

x+3y+9z+27t=81

x+4y+16z+64t=256

x+167y+〖167〗^2 z+〖167〗^3 t=〖167〗^4

Hint : Avoid successive elimination of variables.

The perimeter of a triangle is 2004 . One side of the triangle is 21 times the other. The shortest side is of integral length. Solve for lengths of the sides of the triangle in every possible case.

MATHEMATICS 2003

2 MARKS

C is the circle with radius r. C_1,C_2,C_3,….C_2003 are unit circles placed along the circumference of C touching C externally. Also the pairs C_1,C_2;C_2,C_3,……;C_2002,C_2003;C_2003,C_1 touch. Then r is equal to

Cosec (π/2003) b) Sec(π/2003) c) Cosec(π/2003)-1 d) Sec(π/2003)-1

e)none of these

2) If n is a natural number, then 〖(2003+1/2)^n+〖(2004+1/2)^n is a positive integer〗 〗

a) when n is even b) when n is odd c) only when n=117 or n=119

d) only when n=1 or n=3 e) none of these

3) The line segment ¯AB is completely external to a fixed circle S . A variable circle C through A,B moves such that it intersects S in distinct point P and Q. In one position of C, ¯AB is parallel to ¯PQ. Then

a) there is precisely one more position of C such that ¯AB is parallel to ¯PQ

b) ¯AB is parallel to ¯PQ for infinitely many positions of C and (AB) ⃡ intersects (PQ) ⃡ for infinitely many positions of C

c) ¯AB is parallel to ¯PQ for all positions of C

d) the hypothesis is wrong because there cannot be any position of C such that ¯AB is parallel to ¯PQ

e) None of these

4) f(x) is a polynomial of degree 2003. g(x)=f(x+1)-f(x). then g(x) is polynomial of degree m, where

a)m=2003 b)m=2002 c) m cannot be uniquely determined, but 0 ≤m ≤2002 d) m may be undefined in some cases e) None of these

5) ∆PQR is the midpoint triangle of ∆ ABC.Then the both the triangles

a) have the same circum centre b) have the same orthocenter c) have the same centroid

d) are such that the circum centre of the smaller triangle is the incentre of the bigger triangle.

e) None of these

(m,n) is a pair of positive integers such that i) m<n, ii) their god is 2003, iii)m,n are both 4-digit numbers. The number of such ordered pairs is __________________

In a triangle ABC, AB=12, BC=18, CA=25. A semicircle is inscribed in ∆ ABC such that the diameter of the semicircle lies on ¯AC. If O is the centre of the circle, then the length AO=______________________

If x is the recurring decimal 0.0 ̇ 3 7 ̇ then x^(1/3) is the recurring decimal __________________

The diagonals of a quadrilateral ABCD intersect in the point O. AO=OC. P is the midpoint of BD. Then ∆APB+∆APD-∆BPC-∆DPC = _______________________

If x is any real number, then Int x stands for the unique integer n satisfying x-1<n≤x and Ff x stands for (x – Int x). if m is a positive integer, then Frm/√2+Fr m/(2+√2)=_____________________________

Solve in positive integers m, n, the equation 2^(3m+1)+3^2n+5^m+5^n=2003

Given location P,Q,R, of the points of contact of the sides of a triangle ABC with its in circle, describe the procedure to construct triangle ABC.

The solution set of the in equation (〖ax〗^2+bx+c)/(x^2-x+1) ≤0 is { x /x is real and-1 ≤x ≤2}. determine a:b:c.

ABCD is a cyclic quadrilateral. ¯BD bisects ¯AC. AB= 10, AD=12, DC=11. Determine BC.

4 MARKS

To each of the first two of the 4 numbers 1, 19, 203, 2003, is added x and to each of the last two y. The numbers form a G.P. fill all such ordered pairs (x, y) of real numbers

Show that the 4 points A, Q, X, R are concyclic in the following figure

Factorize a(b^2+c^2-a^2 )+ b(c^2+a^2-b^2 )+ c(a^2+b^2-c^2 )-2abc.

6 marks

π is a plane . O is the origin. K>0. f_k: π→π is called a k-stretch of the plane if f_k (O)=O and for every A ∈π ,A≠O,f_k (A) is B where B ∈ ¯(OA ) and OB=k.OA. P,Q are two points on the unit circle with origin as centre and PQ=1. The plane is subjected alternately to a 1/2- stretch and a 3-stretch,starting from the former. P and Q are transformed to P_2003,Q_2003 respectively at the end of the 〖2003〗^rd stretch. Determine the distance P_2003 Q_2003.

f(x)={√(x+2√(x-1)) +√(x-2√(x-1)) ┤ }^2

Determine the domain of f(x), in other words, determine from the graph the area of the region enclosed between the lines y=0, x=1 and x=3 and the curve y=f(x).

8 marks

a and b are both 4-digit numbers a>b and one is obtained from the other by reversing the digits. Determine b if (a+b)/5=(b-1)/2 .

MATHEMATICS 2002

2 MARKS

Let m be the l.c.m of 3^2002-1 and 3^2002+1. then the last digit of m is

a) 0 b) 4 c) 5 d) 8 e) none of these

hint : read of g.c.d and hence determine the l.c.m

In every cyclic quadrilateral ABCD with AB parallel to CD

a) AD =BC, AC≠BD b) AD≠BC, AC≠BD c) AD≠BC, AC=BD

d) AD=BC, AC=BD e) None of these

f,g,h: R→R are defined by f(x) = x-1, g(x) =(x^2-1)/(x+1) if x ≠ -1 h(x)=(x^3+〖2x〗^2-x-2)/(x^2+3x+2) if x ≠-1, x≠-2-2 if x=-1-3 if x=-2 then f(x)+g(x)-2.h(x) is

not a polynomial b) a polynomial whose degree is undefined c) a polynomial of degree 0 d) a polynomial of degree 1 e) none of these

A,B are two distinct points on a circle with center C1 and radius r1.¯AB also is a chord of a different circle with center C2 and radius R2.S denotes the statement: The arc AB divides the first circle into two parts of equal area.

S happens if C1C2>r2

If S happens then C1C2>r2

If S happens then it is possible that C1C2=r2

If S happens then it is necessary that C1C2<r2

None of these

Triangle T1 has vertices at (-3, 0), (2, 0) and (0, 4). Triangle T2 has vertices at (-3, 3),

(3, 5) and (1, -2). Plot all 1 ∩m where the line segment 1 is a side of T1 and the line segment m is a side of T2 , where 1 ∩m is a unique point. These points of intersection determine a convex polygon, which is

Triangle b) quadrilateral c) pentagon d) hexagon e) septagon

The number of integral solutions of the equation x^4-y^4 = 2002 is ____________________

A,B,C,D are distinct and concyclic . ¯DL is a perpendicular to ¯BC,¯DM is perpendicular to ¯AC.¯LM intersects ¯AB in N. the no obtuse angle enclosed between ¯AB and ¯DN is _____________ in degrees

Let m be the 2002-digit number each digit of which is 6. The remainder obtained when m is divided by 2002 is______________________________________________

If x is a real number Int.x denotes the greatest integer less than or equal to x. Int.√(5-(Int.-2002.2002)) = _________________________________

O is the origin and Pi is a point on the curve x^2+y^2=i^2 for any natural number i.

∑_(i=1)^2002▒〖OP〗_i = OPj , where j =____________________________

If m is the right most nonzero digit (n!)4 where n is a positive integer greater than 1, determines the possible values of m.

Sketch the graph of the function f(x)=√(x^2-4x+4) , 1≤x ≤2 .

∆ABC is right angled at A. AB = 60, AC = 80, BC = 100 units. D is a point between B and C such that the triangles ADB and ADC have equal perimeters. Determine the length of AD.

do there exist positive integers p,q,r such that p/q+q/r+r/p=1 ?

k is a constant, (xi , 0) is the mid point of ( i-k, 0) and (i+k, 0) for i = 1,2…………,2002 then find

∏_(1≤i≤j≤2002)▒〖x_i x_j 〗

6 MARKS

If m and n are natural numbers such that m/n=1/3+1/5+1/17+1/19+1/23+1/1979+1/1983+1/1985+1/1997+1/1999 prove that 2002 divides m.

∆ABC is right angled at A. prove that the three mid points of the sides and the foot of the altitude through A are concyclic.

Determine all possible finite series in GP with first term 1, common ratio an integer greater than 1 and sum 2002.

A convex pentagon ABCDE has the property that the area of each of the five triangles ABC, BCD, CDE, DEA and EAB is unity. Prove that the area of the pentagon is (5+√5)/2

If a,b,c are real number such that the sum of any two of them is greater than the third and ∑▒〖{c(a^2+b^2-c^2 )}〗=3 abc,then prove that a=b=c.

Explain how you will construct a triangle of area equal to the area of a given convex quadrilateral. Use only rough sketches and give proof.

MATHEMATICS 2001

2 MARKS

∆ABC has integral sides AB, BC measuring 2001 units and 1002 units respectively. The number of such triangles is

2001 b) 2002 c) 2003 d) 2004 e) 2005

Let m = 2001!, n = 2002 x 2003 x 2004. The LCM of m and n is

m b) 2002m c) 2003m d) mn e) none

The sides of a right-angled triangle have lengths, which are integers in arithmetic progression. There exists such a triangle with smallest side having length of

2000 units b) 2001 units c) 2002 units d) 2003 units e)1999 units

Let S be any expression of the form ∑_(k=0)^2001 e_k 2^k , where each e_k is varied independently between 1 and-1. the number of expressions S such that S=0 is

Let a1 , a2 , ………..,a2001 be the lengths of the consecutive sides of a convex polygon P1, which is cyclic. Let b1 , b2 , …………, b2001 be a permutation of the numbers a1 , a2 ,…………….,a2001 . let P2 be a polygon with consecutive sides having lengths b1 , b2 ,……….,b2001 . then P2 is cyclic

Always

Only if a1 =b1 , a2 =b2 , ……,a2001=b2001

Only if b1, b2,……,b2001 is a cyclic permutation of a1 , a2,……a2001

Only if the center of the polygon P1 lies in its interior

None of these

n is the smallest positive integer such that (2001+n) is the sum of the cubes of the first m natural numbers. Then m =___________________, n = __________________

An equilateral triangle is circumscribed to and a square is inscribed in a circle of radius r. the area of the triangle is T and the area of the square is S. then T/S=______________

The smallest positive integer k such that (2000) (2001) k is a perfect cube is _____________

ABCD is a rectangle with AB = 16 units and BC =12 units. F is a point on ¯AB and E is a point on ¯CD such that AFCE is rhombus. Then EF measures ________units.

n is a positive integer not exceeding 9. The list of all n such that 10 divides n2 –n is _______________________

Given the location P, Q, R of the midpoints of AB, BC, CA of a triangle ABC, explain how you will construct triangle ABC. Use only rough sketches and give proof.

If for every x> 0 there exists an integer k(x) such that |a_0+a_1 x+a_2 x^2+⋯…..+a_n x^├ n┤| ≤ |x^(k(x))-1| ┤, then find the value of a0+ a1+…..+an

We know that we can triangulate any convex polygonal region. Can we ‘parallelogram late’ a convex region bounded by a 2001-gon?

6 marks

Solve in positive integers the cubic x3-(x+1)2=2001.

The product of two of the roots of x4-11m3+kx2+269x-2001 is -69 find k.

a) AB,BC,CD are the consecutive sides of a cyclic polygon of equal angles. Prove that ∆BCA ≅ ∆BCD. b) deduce that if a cyclic (2n + 1)-gon has all its angles equal, then all its sides must be equal. c) prove by means of an example that a cyclic 2n-gon need not be equilateral if it is equiangular.

If (1+2x)(1+2y)=3 , xy ≠ 0 , then show that 〖1+8x〗^3/(x(1-x^3))=〖1+8y〗^3/(y(1-y^3))

a) ¯AB is the common chord of two intersecting circles. A line through A terminates on one circle at P and on the other at Q. then prove that BP/BQ is a constant.

If two chords of a circle bisect each other prove that the chords must be diameters.

MATHEMATICS 2000

2MARKS

A1¬A2,……,A2000 is a polygon of 2000 sides. P is a point in the plane of the polygon which is equidistant from all the vertices of the polygon. Then

There is no such P

There is exactly on such P

Either A or B

The locus of P is a straight line

The locus of P is a circle.

Consider the following expressions in a,b,c

E1=a2+b2+c2+2ab+2bc+2ca E2 =a2+b2+c2+2ab-2bc-2ca

E3=a2+b2+c2-2ab-2bc-2ca E4 =a2+b2+c2+2ab+2bc-2ca

The expression which are not perfect squares among E1,E2,E3 and E4 are

E1,E3 b) E2,E3 c) E2,E4 d) E1,E4 e) E3,E4

Three angles of a convex polygon measure each π/3 radians. Let n be the number of sides of the polygon. Then

The hypothesis is not sufficient to evaluate n

n=3 c) n=6 d) n=9 e) none of these

there are n rods, n ≥ 3 . the ith rod has length 2i units. The number of closed convex polygonal frames that can be made by joining the rods end-to-end is

0 b) n c) n! d) nn e) 2n-1

Integers from 1 t0 2000 are written. A single operation consists of cancelling any two of the numbers and replacing them by their product. After 1999 such operations

No number will be left

Precisely 2 numbers will be left

More thatn two numbers may be left

Only one number will be left and it is 2000!

Only one number will be left, but it cannot be uniquely determined

All possible 2000-gons are inscribed in a circle of perimeter p. let X = {├ x┤| an inscribed polygon has a side of length x. } then the greatest member of X is_______________

Triangle ABC the altitude through A has length h. is right angled at A. then

1/b^2 +1/c^2 -1/h^2 =__________________

p is a prime, n a positive integer and n+p=2000. LCM of n and p is 21879. Then p =_______________ n=__________________

The unit digit of 〖777〗^777 is __________________________

Does there exist a right angled triangle with integral sides such that the hypotenuse measures 2000 units of length ?

If a, b, c are not all equal and (a + b + c)>0, determine the sign of a3 + b3 + c3 – 3abc.

If real numbers x, y, z satisfy x^2/y^2 +y^2/z^2 + z^2/x^2 =x/y+y/z+z/x , then prove that x=y=z

Write down the solutions (x, y) of x^[siny ] =1 ,if x>0 and 0≤y≤π/2 and [siny ] denotes the integral part of sin. Y

Given ∠ABC and a point D in its interior. The problem is to find E on □(→┬BA ) and F on □(→┬BC ) such that D is the mid point of line segment ¯EF . A student suggests the following method of construction: join BD and extend it to B’ such that BD = DB’ . draw parallel to □(↔┬BC ) through B’. take for E the point where this parallel cuts □(→┬BA ) join ED and produce it to meet □(→┬BC ) . take for F this point of intersection. Discuss the validity or otherwise of this method of construction.

Consider the equation in positive integers x2+y2 = 2000 with x<y.

Prove that 31< y< 45

Rule out the possibility that one of x,y is even and the other is odd.

Rule out the possibility that both x, y are odd.

Prove that y is a multiple of 4.

Obtain all the solutions. (10 marks)

ABCD is a trapezium with AB and CD as parallel sides. The diagonals intersect at O. the area of the triangle ABO is p and that of the triangle CDO is q. prove that the area of the trapezium is (√p+√q )^2. (5 marks)

Manipulate the equality

a2b2(bc – a2) + b2c2 (ca – b2) + c2a2(ab – c2) = a2b2(b2-ac)+b2c2(c2-ab) + c2a2(a2-bc)

until the equality (a2+b2)(b2+c2)(c2 +a2) = abc(a+b)(b+c)(c+a) is obtained (5 marks)

If a triangle and a convex quadrilateral are drawn on the same base and no part of the quadrilateral is outside the triangle, show that the perimeter of the triangle is greater than the perimeter of the quadrilateral ( 5 marks)

If a line parallel to , but not identical with, x- axis cuts the graph of the curve

y= (x-1)/((x-2)(x-3)) at x=a,x=b,then evaluate (a-1)(b-1) (5 marks)

MATHEMATICS 1999

MARKS

Let R be the set of all real numbers. The number of functions f∶R→R satisfying the relation (x-1)^2 f(x)=0 is

Infinite b) One c) Two d) Zero e) None of these

α is a number such that the exterior angle of a regular polygon measures 10α degrees. Then

There is no such α

There are infinitely many such α

There are precisely nine such α

There are precisely seven such α

There are precisely ten such α

f(n)=2f(n-1)+1 for all positive integers n. then

f(n)=〖2^n〗^(-1) [f(1)+1]

f(n)+1=〖2^n〗^(-1) f(1)

f(n)+1=〖2^n〗^(-1) [f(1)+1]

f(n)+1=2^n [f(1)+1]

None of these

For any triangle let S and I denote the circum centre and the incentre respectively. Then SI is perpendicular to a side of

Any triangle

No triangle

A right angled triangle

An isosceles triangle

An obtuse angled triangle

if f(x)=x^3+ax+b is divisible by (x-1)^2 , then the remainder obtained when f(x) is divided by x+2 is

1 b) 0 c) 3 d) -1 e) None of these

¯BC is a given line segment, H, K are points on it such that BH=HK=KC. P is a variable point such that (i) BPC has the constant measure of α radians.

(ii) ∆BPC has counter clockwise orientation

Then the locus of the centroid of ∆BPC is the arc of the circle bounded by the chord ¯HK with angle in thesegment ________________ radians.

The coefficient of 〖x^n〗^(+1) in (a0+a1x+a2 x2+……+an xn)2 is _________________

n is a natural number such that i) the sum of its digits is divisible by 11 ii)its units place is non-zero iii) its ten place is not a 9.

Then the smallest positive integer p such that 11 divides the sum of digits of (n+p) is _________________

The number of positive integers less than one million (〖10〗^6) in which the digits 5,6,7,8,9,0 do not appear is __________________________

The root of the polynomial a_0 x^n+a_1 x^(n-1)+⋯..+a_n are, in terms of α for i=1,2,3……,n. then the roots of the polynomial

a_0 (2x^2-3)^n+2a_1 〖(2x^2-3)^n〗^(-1)+2^2 a_2 〖(2x^2-3)^n〗^(-2)+⋯……………+2^n a^n are all positive and are denoted by αI ________________________________

Given any integer p, prove that integers m and n can be found such that p=3m+5n.

E is the midpoint of side BC of a rectangle ABCD and F the midpoint of CD. The area of ∆AEF is 3 square units. Find the area of the rectangle.

If a, b, c are all positive c 1, then prove that a^log〖c^b 〗 =b^log〖c^a 〗

Find the remainder obtained when x^1999 is divided by x^2-1.

Remove the modulus |〖119〗^99-〖99〗^19 |

5 marks

Solve the following system of 1999 equations in 1999 unknowns :

x_1+x_2+x_3=0,x_2+x_3+x_4=0….,x_1997+x_1998+x_1999=0,

x_1998+x_1999+x_1=0 ,x_1999+x_1+x_2=0

Given base angles and the perimeter of a triangle, explain the method of construction of the triangle and justify the method by a proof. Use only rough sketches in your work.

If x and y are positive numbers connected by the relation.

y=(ax^2)/((x-a)(x-b)(x-c))+bx/((x-b)(x-c))+c/(x-c)+1

log〖y=3 logx-|x-a┤〗 |-log┤ |x-b┤ |- log┤ |x-c| for any valid base of the logarithms.

Let ∆ XYZ denote the area of triangle XYZ. ABC is a triangle. E,F are points on ¯AB and ¯AC respectively.¯CE and ¯BF intersect in O.if ∆ EOB=4,∆ COF=8,∆ BOC=13, develop a method to estimate ∆ ABC (you may leave the solution at a stage where the rest is mechanical computation).

Prove that 80 divides |〖19〗^99-〖99〗^19 |

ABCD is a convex quadrilateral. Circles with AB, BC, CD, DA as a diameters are drawn. Prove that the quadrilateral is completely covered by the circles. That is, prove that there is no point inside the quadrilateral which is outside every circle.

MATHEMATICS 1998

2 MARKS

p is the smallest positive such that every positive such that every positive integer greater than p can be written as a sum of two composite numbers. Then

p=3 b)p=6 c)p=10 d)p=11 e)None of these

A B C is a triangle such that m_a>a/2,where m_a=length of the median through A and a=length of BC.then

Such a triangle does not exist

D A is acute

D A is obtuse

D A is right

None of these

A vegetable shop keep only four weights, one each of 1 kg, 1⁄2 kg,1⁄4 kg,1⁄8 kg. A newly appointed shop assistant claims that he has so far taken once and only once each weighing possible with the available weights. If n is the number of all weighing possible and W the total weight of all possible weighings, then

N=4,W=17⁄8 kg b) n=16,W=30kg c) n=15,W=15 kg d) n=15,W=7.5 kg e) None of these

x_1,x_2 ,…,x_1997 are the roots of the polynomial

x^1997+a_1 x^1996+a_2 x^1995+⋯………………….+a_1996 x+1 and

y_1=x_1 x_2…x_10,y_2=x_2 x_3…..x_11,…..y_1997=x_1997 x_1…….x_9 then

(y_1 y_2……..y_1997 )^(〖(19〗^98 )) is independent of

All a_(i ) b) All a_i,i^1 10 c) None of a_i d) All a_i,i even e) All a_1 ,i odd

Triangle ABC is isosceles, right angled at B and has area S. A circle is constructed with B as center and BA as radius. A semicircle is constructed externally on ¯AC , that is, on the side of AC which is opposite to that of B. then the area of the crescent or sickle formed between the circle and the semicircle is

2S b) S/2 c) S d) (p/O ̈ 2) S e) None of these

Numbers 1, 2, 3,……….,1998 are written in the natural order. Numbers in odd places are stricken off to obtain a new sequence. Numbers in odd places are stricken off from this sequence to obtain another sequence and so on, until only one term a is left. Then a = _________________

S is the circumcircle of an equilateral triangle ABC. A point D on S is such that C and D lie on opposite sides of AB. Then D ADB =______________________ radians

f∶{x,y,z}→{a,b,c} is bijection from a 3-element set into a 3-element set. It is given that f(x)=a B: f(y) a C:f(z) b then f^(-1) (a)=________________________

Given that x is real , the solution set of √(x+1)+√(x-1)=1 is___________________

In the standard expansion of

(∑_(i=0)^28▒〖(-1)^i a_i 〗)^2,

the number of terms appearing with –ve sign is ______________________________

Determine all positive integers n such that n+100 and n+168 are both perfect squares.

Determine with proof whether integers x and y can be found such that x+y and x^2+y^2 are consecutive integers.

The first term of an arithmetical progression is log〖a 〗 andthe second term is log〖b.〗 expression the sum to n terms as a logarithm.

p ,q ,r ,s are positive real numbers. Prove that (p^2+p+1)(q^2+q+1(r^2+r+1) (s^2+s+1)^3 81 pqrs

With the aid of a rough sketch describe how you will draw a direct common tangent to two circles having different radii. (No formal proof is required).

5 marks

If x+y+z=0,then prove that

(x^2+xy+y^2 )^3+(y^2+yz+z^2 )^3+(z^2+zx+x^2 )^3=3(x^2+xy+y^2 )(y^2+yz+z^2 )(z^2+zx+x^2)

Simplify (2x+3)^3/((2x+3)^4+4)-((x+1))/(4(x+1)^2+1) till it is obtained equal to ((x+2))/(4(x+2)^2+1)

a, b, c are distinct and p(x) is a polynomial in x, which leaves remainders a, b, c on division by x-a, x-b, x-c, respectively. Find the remainder obtained on division of p(x) by (x-a)(x-b)(x-c).

A point A is taken outside a circle of radius R. Two secants are drawn from this point: One passes through the center, the other at a distance of R/2 from the center. Find the area of the circular region enclosed between the two secants.

A right angled triangle has legs a, b, a>b. The right angle is bisected splitting the original triangle into two smaller triangles. Find the distance between the

orthocenter’s of the smaller triangles using the co-ordinate geometry methods or otherwise.

6) If two sides and the enclosed median of a triangle are respectively equal to two sides and the enclosed median of another triangle, then prove that the two triangles are congruent.

MATHEMATICS 1997

2MARKS

n_1,n_2,……,n_1997 are integers,not necessarily distinct.

X=(-1)^n 1+(-1)^n 2+⋯..+(-1)^n 998 ;

Y=(-1)^n 999+(-1)^n 1000+⋯…..+(-1)^n 1997

(-1)^X=1,(-1)^Y=1 b) (-1)^X=1,(-1)^Y=-1

c) (-1)^X=-1,(-1)^Y=1 d) (-1)^X=-1,(-1)^Y=-1 e) none of these

2) I and S are the incentre and circumcentre of ABC. Let AIB=α and ASB=β Then

a) α is acute and nothing can be said about β b) α is obtuse and nothing can be said about β c) Nothing can be said about α and β is acute d) nothing can be said about α and β is obtuse e) none of these

3)Treating all similar polygons as indistinct, the number of convex decagons such that four of its angles are acute is

a) 0 b)infinitely many c)1 d)10 e)none of these

4) The first hundred natural numbers are written down. N_i denotes the number of times the digit i appears. Then

a) N0=12, N1=20, N2=20 b) N0=11, N1=20, N2=20

c) N0=11, N1=21, N2=20 c) N0=12, N1=20, N2=19

e) None of these

5) Consider the decimal 0.111010100010……….where the digit in the n^th decimal place is 1 if either n=1 or n is a prime. Then the decimal

a) terminates b) has only finite number of zeros c) recurs

d) denotes a rational number e) denotes an irrational number

k is a given real constant. f(x) is a real quadratic in x such that f(x+k)=f(-x). the coefficient of x^2 is 1. Then f(x) is of the form ____________________

A is a 2 x 2 real matrix such that A^2=0 and none of its elements is a zero. Then A is of the form_________________________

If X is a finite set, let P(X) denote the set of all subsets of X and let n(X) denote the number of elements in X. if for two finite sets A, B, n(P(A)) = n(P(B)) + 15 then n(B)=__________________ , n(A) = ___________________

p is a natural number such that no term in the expansion of [x/y+y/x]^p is independent of x. the p is of the form________________________________

The circle C1 has centre at (2, 3) and radius 3. The circle C2 has centre at ( -1, 4) and radius _____________________________

The number of common tangents they possess is _________________________________

Sketch the graph of the region 0 < x < y < 1 on plain paper and describe it in words. (no verbal argument is required)

Find the value x^3+y^3+z^3-2xyz,if

x=√7-√5,y=√5-√3,z=√3-√7.

The remainder when x^5+kx^2 is divided by (x-1) (x-2) (x-3) contains no term in x^2. Find k, without performing division.

g(x)= log_4〖x^2/4〗-2 log_4〖4x^4 〗 if x 0. Determine the functions f(x) and constants a, b such that g(x)=a+b log_4〖f(x)〗

How do you construct a tangent to a circle through a given external point ? Justify your answer.

5 MARKS

If x^3+px+q=(x-)(x-β)(x-γ), then rewrite the polynomial (β γ x+β+γ) (γ α x+γ+α) (α β x+α+β), in terms free of α,β,γ.

If x+y+z=15, xy+yz+zx=72 , prove that 3 x 7.

If the sides and the area of a triangle have integral measures prove that its perimeter must have even integral measure.

m and n are positive integers such that (i) m<n , (ii) they are not relatively prime (iii) their product is 13013. Find their g.c.d. and hence determine all such ordered pairs (m, n).

two circles 6 cm, 8 cm in diameter are externally tangent. Compute the area bounded by the circles and an external common tangent.

From any point P within a triangle ABC, perpendiculars PA' , PC' are dropped on the sides BC , CA , AB ; and circles are described about PA' B ' , PB' C' , PC' A'

Show that the area of the triangle formed by joining the centres of these circles is one-fourth of the area of ABC.

MATHIMATICS 1996

2 MARKS

Let n be a product of four consecutive positive integers then

n+1 is always a perfect square

d

n is always divisible by 24

n+12 is always a perfect square

log(x+3)+log(x-1)=log(x^2-2x-3) is satisfied for

All real values of x

No real value of x

All real values of x except=0

No real value of x except x=0

All real values of x except x=1

|sinx|^tanx>|sinx|^cotx then (x ≠n π)¦(x≠(2n-1)π/2) }n ∈N

a) 0 < x < π/2 b) 0 < x < π/4 c) π/4<x< π/2 d) None of these

Which of the two functions are equivalent

x^2-1=√x and x^2-1+√(1-x)=√x+√(1-x)

x^2+x=0 and (x^3+x)/x=0

x^2+1=√x and x^2+1+√(1-x)=√x+√(1-x)

log_(1/(8 cos^2x ))sinx =1/2 and sinx=1/√(8 cos^2x )

In a triangle ABC, the sides are in the ratio 1:1: it follows that [ ]

Angles of the triangle are in ration 1:1:2

Angles of the triangle are in the ratio 1:2:3

The triangle is right angle triangle

The circum circle of the D ABC has its centre on one of the sides

There are exactly 4 prime numbers between n and 2n. Then possible value of n is

n = 4 b)n=10 c)n=12 d)n=50

the relation R = { (1, 1)(2, 3)(3, 4)(4, 1) on the X = {1, 2, 3, 4}

not a function from X → X

An one to one function from X→ X

An onto function of X→ X

R is a transitive relation on X

If for some angle q and for some real number x it is given that sin q =x^2-2x+2 then x should be equal to _________________________ and q should be equal to _________________ degrees

If a, b, c, d are four distinct integers between 1 to 10 such that for a triangle whose sides are in the ratios a:b:c, the altitudes are d:c:b. then greatest of these four integers is ____________________ and the smallest is __________________

10) In a cylinder, B is a diameter of the base circle and C is the centre of the top circle. Given that the volume of the cylinder is √(9&3π_(cm^3 ) ) . CAB =〖30〗^0 , the perimeter of the triangle ABC is ___________________ and the curved surface area of the cone with vertex C (having the same base)

Is ________________________________

11)if the sum of the intercepts of the line with coordinate is 5 and the area of the triangle formed by the line and coordinate axes is 3 then intercepts are _________________________

12)In figure given below

A = D = 〖90〗^0 AP = 7.5 cm, AB = 10 cm, BC = 26 cm, Then CD is __________________________

13 ([1-[a/b]^(-2) ] a^2)/((√a-√b)^2+2√ab)=1 and a+b=5 then a, b are _______________________

1 mark

Group –A

A, B and C are three distinct points in a plane.

There is no point D equidistant from A, B, C

B lies on the perpendicular bisector of AC

Among the three lengths AB, BC, AC the largest is the sum of the other two

There is a point D such that ABCD is a rhombus

There is a point D such that ABCD is a rectangle

In triangle ABC, the angular bisector B is perpendicular to the side AC

AB is a to the circle through B with centre C

AC is a diameter of the circle passing through A, B, C.

Group-B

Which of the above statements mean that A, B, C are collinear

Which of the above statements mean AB = BC

Which of the above statements mean that AB is perpendicular to BC

Which of the above statements infer ABCD is cyclic quadrilateral

PART-B

Attempt all questions

Justify your answers with mathematical arguments

The numerator of a fraction is less than its denominator by 2. When 1 is added to both the numerator and denominator we get another fraction. The sum of these two fractions is 19/15 find the first fraction. (5 marks)

Let α β be the roots of the quadratic equation x^2+ax+b=0 and γ ,δ be the roots of the equation

x^2-am+b-2=0 given that 1/α+1/β+1/γ+1/δ=5/12 and α β γ δ=24.

find the value of the coefficient a. (5 marks)

Find the number of points with integral coordinates in the Cartesian plane satisfying the inequalities

|x| 100; |y| 100;|x-y| 100 (5 marks)

Perpendiculars are drawn from the vertex of the obtuse angle of a rhombus to its sides. The length of each perpendicular is equal to ‘a’ units. The distance between their feet being equal to ‘b’ units. Determine the area of the rhombus. (5 marks)

Let the sequence 〖{u〗_n} be defined by

u1 =5 and the relation un+1-un =3+4(n-1)n ∈ N

express un as a polynomial in n:

The remainder R, when x100 is divided by x2-3x+2, is a polynomial . find x. (3marks)

Given an ABC of magnitude less than 1800 and O is a point inside it. Drawn a straight line through O which forms a triangle with minimum perimeter. (Describe the construction and give proof). (5 marks)

MATHEMATICS 1995

2 MARKS

If a^2+b^2+c^2=D where a and b are consecutive positive integers and c =ab, then √D is

Always an even integer

Sometimes an odd integer and sometimes not

Always an odd integer

Sometimes rational, sometimes not

Always irrational

A triangle ABC is to be constructed. Given side a ( opposite A); B and hc , the altitude drawn from C on to AB. If N is the number of non-congruent triangles then N is

One b) Two c) zero d) zero or infinite e)infinite

In any triangle ABC, a and b are the sides of triangle. Given that S =1/2 ab SinC(Where S is the area of the triangle) , then

S (a^2+b^2)/4 b) S (a^2+b^2)/4 c) S (a^2+b^2-ab)/2 d) S (a^2+b^2-ab)/2 e)none

If a0 ,a1,……………,a50 are respectively the coefficients of x^0 ,x^1 ,……,x^50 in the expansion of (1+x+x2)25 then a1+a2+…………+a50 is

Odd

Even

Zero

None of these

The smallest value of x^2+8x for all real values of x is

-16.25

-16

-15

-8

None of these

The value of aI ̂R for which the equation (1+a^2 ) x^2+2(x-a)(1+ax)+1=0 has no real roots is_________________________

The greater of the two numbers 〖31〗^14 and 〖17〗^18 is ______________________

A and B are known verticies of a triangle. C is the unknown vertex. P is a known point in the plane of the triangle ABC. Of the two following hypotheses,

Hypothesis I -P is the incentre of D ABC

Hypothesis II -P is the circumcentre of D ABC.

The hypothesis which will enable C to be uniquely determined is _________I/II________(strike off the inapplicable)

If x = 0.101001000100001………… is the decimal expansion of positive x, and in the series expansion of x the first, second and third terms are 1/10 ,1/1000 and 1/1000000 then the 〖100〗^th term is __________________

If ABC is a triangle with AB = 7, BC = 9 and CA = n where n is a positive integer, then possible two values of n are__________________

Solve the equations xy+yz+zx=12-x2 =15-y2 = 20-z2

A is 2 x 2 matrix such that AB = BA whenever B is a 2 x 2 matrix. Prove that a must be of the form kl, where k is a real number and I the identity matrix [■(1&0@0&1)]

Determine all 2 x 2 matrices A such that A^2=I, where I = [■(1&0@0&1)]

Find an explicit formula (in terms of nI ̂N) for f(n), if f(1)=1 and f(n)=f(n-1)+2n-1

Let ABCD be a square. Let P , Q , R , S be respectively points on AB,BC,CD,DA such that PR and QS intersect at right angles. Show that PR = QS.

Let ABC be any triangle. Construct parallelograms ABDE and ACFG on the outside D ABC outside D ABC such that PA and BI are equal and parallel. Show that area of ABDE + Area of ACFG = Area of BCHI.

A belt wraps around two pulleys which are mounted with their centres s apart. If the radius of one pulley is R and the radius of the other is (R+r), show that the length of the belt is π(2R+r)+2√(s^2-r^2 )

10 MARKS

Deduce pythogaras theorem from the result of problem 6 of part D.

Let x be any real number. Let AX={y / y ∈ 0<y(y+1) (x+1)^2} and

Bx ={y/y∈0<y(y-1)≤x^2} prove that A_x⊂ B_x

Let x_1=x_2=2 show that there exists an infinite sequence x_1,x_2,x_3,x_4,…… such that x_1 x_2……x_n=x_1+x_2+⋯……+x_n for all positive integers n.

MATHEMATICS 1994

2 MARKS

If x={{11},11,2},3,2,1} and

p=(x/x∈X and x ∈y for some y∈X}

Q=(S_x 〖IS〗_x={x} for each x such that x∈X and X ∈y for some y ∈X}. write does PUQ.

If a and b are odd positive integers and a^3-b^3 is divisible by 2^n (n a positive integer),prove that a-b is divisible by 2^n

Prove that n(n + 1) (2n + 1) is divisible by 6, where n is an integer.

For subsets A. B of a set X, defines A’ B as A’B = (A ∩B)∪[(X-A)∩(X-B) ]. Mark in Venn’s X-(A *B), X – B. . A*(X-B)

In solving a certain problem which reduces to a quadratic expression on student makes a mistake only in the constant term and obtain 8 and 2 for the root Another student makes a mistake in the coefficient of first degree term and obtains -9 and -1 for the roots. Find the 4uadratic expression if coefficient! Of x’ is 1.

In a number system written to a particular base the following equality holds good: 10,000 = 7727 + 51. Find the base.

If A = {1,2,3,4,5,6} and B = {1,2,3} find the number of surjection’s from A to B.

If log〖█(@2) 〗 sinθ+log█(@2) cos〖θ=-2〗 find he value θ satisfying 0 θ 〖360〗^0.

Represent the set S = {(x , y) |x|+|y| 1} as a region in the Cartesian plane.

If 2 sin +3cos =5 can 3sin +2cos be evaluated ? if so , evaluate it.

If P(x)=x^4+ax^3+bx^2+cx+d is a polynomial such that P(1) = P(2) =P(3) =0, compute the value of P(4) = P(O).

Determine all natural numbers n such that (n+1)^2/((n+7)) an integer.

If f is defined for all real x , f(a +b) = f(ab) for all real a , b and f(-1/2) = -1/2, evaluate f(-311/515).

ABCD is a trapezium with AD and BC parallel . if AB = y. AD = x |▁C= α,| ▁A= 2 , evaluate the length of BC

4 marks

A man on his way to dinner shortly after p.m. observes that the hand on his watch form an angle of 〖110〗^0. Returning before 7 p.m. he notices that again the hands of his watch form an angle of 〖110〗^0. Find the number of minutes he had been out for dinner.

The median of a trapezoid cuts the trapezoid into two regions whose areas are in the ration 1:2 compute the ratio of the smaller base of the trapezoid to its larger base. (note : by the median of a trapezoid we mean the line joining the midpoints of the 2 non-parallel sides or the trapezoid).

A student entered a four digit positive number in his calculator and then performed the following procedure ; he multiplied it by 123 and he extracted the 4th root of the product. He repeated the results themselves were uncharging. If the uncharging number is N compute (I) closet integer to N, (ii) the integral part of N.

Show that two circles of radius 13 may be drawn through the point (0, 8) to touch the X-axis and find the length of their common chord.

Prove that in any arithmetic progression all a1, a2, a3,_____________________

We have S =1/(√(a_1 )+√(a_2 ))+1/(√(a_2 )+√(a_3 ))+⋯……………+1/√(a_(n-1)+√(a_n )) =(n-1)/(√(a_1 )+√(a_n ))

5 marks

Prove that 1^99+2^99+3^99+4^99+5^99 is divisible by 5.

If 1/a+1/b+1/c=1/(a+b+c) then 1/a^n +1/b^n +1/c^n =1/(a^n+b^n+c^n ) if n is an odd positive integer

Given a circle and a point inside it (not the centre) prove that the shortest chord through the point is the ne bisected by the point. Given construction for such a chord.

A point within an equilateral triangle. Whose perimeter is 30 mtrs. is 2 mtrs from one side and 3 mtrs from another side, find its distance from the 3rd side.

MATHEMATICS 1993

1 MARK

Enunciate the principle of Componendo and Dividendo.

Sate the ‘ Alternate Segment Theorem ‘.

Write the negation of the statement : it is raining and all the people are carrying umbrellas.

Write down, preferably y inspection, the solution of the equation ;

x-ab/ a+b+x-ac/ a+c+x-bc/ b+c= a+b+c.

Can the bisectors of two angles of a triangle be perpendicular why? Or why not?

Can the lateral side of an isosceles tnangle be equal to half its base ? why or why not?

Can the following graph be the graph of a function ? why of why not ?

If the answer to (7) is yes, can it be the graph of an injection ? why ? or why not? (a one – one function).

Solve for x, if: i) x is 5-digit number ii) x is divisible by 11 and iii) x has 1 in its units, hundreds and ten thousands and ten thousands place.

If cos〖x=3〗/ √10, O < x < /2. Compute log█(@10) sin〖x 〗+log█(@10) cosx+log█(@10) tan〖x .〗

2 marks

A circle touches the y-axis at A(0, 9) and cuts the x-axis at B (3, 0). Find the other point one the x-axis through which the circle passes.

Slove |x^2+y^2-41|+|y-x-1|=O,if x^2+y^2>41,x-y -1

The numbers 1, 2, 3, 4, 5 are divid into two sets. Prove that one of the sets should contain two numbers and their diference. Do not enumerate all possible divisions.

If Sinx+Sin^2 x=1,evaluate Cos^2+Cos^4 x.

Divide the unit square into 9 equals squares by means of two pairs of lines parallel to the sides. Then how many squares are loft? What will be the length of the sides of squares.

A rectangle is inscribed under the y = sin x between x=0 and x = as shown in the figures. If the base ABa of rectangle has length 2 /3, find the area of the rectangle.

Sketch the region x+37 12 O x 8, 0 y 3.

If in a polynomial, the coefficients are alternatively and negative, prove that it can not have negative roots.

The angle of a plane quadrilateral are in A.P. and the difference of the greatest and the least is a right angle. Find the number of degrees in each angle.

A train left station A for C via B. the speed of the train in the section from A to B was as required, but it fell off by 25% between B to C. on return trip, the required speed was maintained between C and B but decreased by 25% between B and A. how long did it take for the train to cover from A to C if we know that the same time was speed on the A – B section as on the B – C section and that on the A – C section the train spent 5/12 of an hour less than what is spent on the return trip ?

marks

if A is a finite set, let n(A) denote the number of element in A. if A and B are finite sets, A B and n(A) = n(B) show that n(AB) < n(A).

If n is a positive integer greater than 2, show that , in c right angled triangle, (Hypotenuse)n > (Base)n+(Height)n

Prove that x^3+px+q=O if x=[(-q)/2+(q^2/4+p^3/27)^(1/2) ]^(1/3)+[(-q)/2-(q^2/4+p^3/27)^(1/2) ]^(1/3)

AB is a diameter of a circle, CD chord half of AB, and AB is parallel to CD. AC meets the tangent at B in E. prove that AE=2 AB.

Let n=2^(p-1) where 2^(p-1) is a prime. find the number of divisors of n and prove that their sum is 2n. 2^p-1.

In the following figure AB CD –{C} is the graph of y =f(x) where f(x) is a function. Obtain a formula for f(x).

ABC is an equilateral triangle and p its centroid. Q is any point in the plane of the triangle. Choosing a covenant coordinate system for the plane , find the ration of: QA^2+QB^2+QC-BC^2 to QP^2.

The sides of a quadrilateral are 3, 3, 4 and 8 (in some order) Two of its angles have equal sines but unequal cosines. Two quadrilateral cannot be inscribed in a circle. Compute the area of the quadrilateral .

Prove that log█(@10)^n k log〖█(@10)^2 〗 where n is a natural number and k the number of distinct primes that divide n.

If Sec A + Tan A = a, find sin A in terms of a.

MATHEMATICS 1992

2 MAKRS

Let N be the set of all natural numbers. Then the number of ordered pairs (a. x) in N x N shuch that ax = a + 4xis.

1 b) 2 c) 3 d) 4

a,b,c are three numbers such that :

abc 0, b) a+b+c = abc c)(a+b) (b+c) (c+a) 0, d) a+b/l-ab+b+c/l-bc+c+a/l-ca = k abc. Then :

K is a constant B)the value of k depends on the values of a. b,c C) k =0 D)k=l B)=-1

If the lines l1 and l2 intersect on the A – side of m as shown in the fig; then :

y > p b) z>1800 c)x+p<y+z d)x+y<p+y e) none of these

in the diagram AB and C are the equal sides of an isosceles triangle ABC in which is inscribed an equilateral triangle DEF. x. y,z are as shown in the figure then

y = x +z/2 b) y=x-z/2 c) x = y-z/2 d) x = y + Z/2 e) None of these

if the quadratic ax^2+bx+c=0, a 0, a, b, c are integers, has natural numbers as its roots , then :

ac can be expressed as the sum of squares of two natural numbers.

a divides b and c

divides c and a

c divides a and b

none of these

“Three-circles have a, b, c as the intrathein a natural numbers and a b c . If the centres of the circles form a right-angledtriangle, the :[]

a, b, c must all be odd

a must be odd, b and c must be even

a must be given b and c must be odd

a, b, c must all be even

None of these

If three real numbers a, b, c none of which is zero, are related by a^2=b^2+C^2-2abc√(1-a^2 ) ,b^2=c^2+a^2-2ca √(1-b^2 ) ,c^2=a^2+b^2-2ab √(1-c^2 ) then

a=c√(1-b^2 )+√(1-c^2 )

b=a√(1-c^2 )+c√(1-a^2 )

c=b√(1-a^2 )+a√(1-b^2 )

a^2+b^2+c^2 /2abc = √(1-a^2 )+√(1-b^2 ) /b = √(1-c^2 )/c

None of these

The base of a triangle is of length b, and altitude h. A rectangle of height x is inscribed in the triangle such that the base of the rectangle forms parts of the base of the triangle. Then the area of the rectangle is : []

hx/b (b-x) b) bx/h (h-x) c) bx/h (h-2x) d) x(b-x) e) x(h-x)

given two numbers a and b m and p are to be so chosen that the system (3m- 5p+b) x+(8m-3p-a) y = 1, and (2m-3p+b) x+ (4m-p)y = 2 has several distinct solutions. Then :[]

Such a choice is impossible

m can be chosen in infinite many ways

p can be chosen in infinitely many ways

m = 5b-14a/64, p = 3b+2a/16

None of these

Let a, b, c be three distinct and non zero members. Let P,Q,R be the points(a^2,ab),(b^2,ab),(ca,cb). then a point S can be found such that :

P , Q , R , S are collinear

P , Q , R , S form a parallelogram (in some order)

P , Q , R , S form a rectangle if a, b, c are in G.P.

P , Q , R , S form a square if a , b, c, are in G.P.

None of these

1 mark questions

x and y are two distinct positive numbers and 2A= X2+y2,B = xy, C = B2/A. then , the arrangement of A , B , C in the ascending order is

A circle is inscribed in a square of side 2a touching all its sides. The points of contact are joined to form another square. Another circle is drawn touching all its sides. The points of contact are again joined to form another square. The process is continued. The limit to the sum of the areas of the squares is

If log█(3@10)=a,log█(2@10)=b,then log█(6@5)=

The chord of the larger of two concentric circles is tangent to the smaller circle and has length a, the area of the annulus is

ABC is a triangle. The medians wand BE intersect in M. N is the midpoint of AM. Then the area of MNE/ the area of ABO =

TRUE OR FALSE

If x^2=x+1,then x^3=2x+1.

if|x-2|<3,then it cannot be true that-1 x x 5.

For all real p,p^2+p+1/p^2 +1<3/2.

If a, b, c are all distinct, then the lines x +ay = b+c, x+by=c+a, x+cy = a + b are concurrent.

√(1-x)+√(x-1) is constant on its domain of definition.

5 MARKS

If children in a school are formed into columns, eight a breast, then one row remains incomplete. If they are arranged seven abreast, then there will be two rows more and all rows complete. If they are again arranged five abreast there will be another seven rows, but one of them incomplete. How many children seven rows, but one of them incomplete. How many children are there in the school?

Making use of the identity cos2 =cos2 - sin2 or otherwise, evaluate Tan2 /2n if sin + cos =1/5.

If n is a natural number, express n5 -3n3 + 4 n as the produt of a consecutive integers.

Sketch the region bounded by [ ( x , y) / |x+y|+|x-y|=├ 4┤|] and find it area.

If a + b + c = 0 , considering the expansion of (a+b)5 , show that:

5 ac^3+10a^2 c^2+10a^3 c+5a^4=5bc^3+10b^2 c^2+10b^2 c+5b^4

The parallel sides of a trapezium are 30 and 20 units and the nonparallel sides are 13 and 15 unit. Find its area.

MATHEMATICS 1991

2 MARKS

Log (x-a) > x ∈ R^+

a ∈ R then a can be

a^x>0 b) a<0 c) a=0 d) Absurd inequality

a √b= √(a^2 ) b a m b ∈ R ∈ = Belong

true for all values of a, b

true for all a R and some b R

true for some a R and all b R

true for some a r and some b R

if sin^6x+cos^4〖X=1 〗 then

sin x = o and cos x = 1

cos x = ¼ √22 and sin x = 1/6 √2

sin x = 1 and cos x = o

sin x = 6√2/3 and cos x = 4 √1/3

4 – 10 = 9 – 15…….(i) 1- 10 + 25/4 = 9 - `5 +25/4………\(ii), (22 – 5/2)2 = (3 – 5/2)2 ….(iii) 2 – 5/2 = 3 – 5/2 …….(iv), 2 = 3 ……….. (v) then we conclude

(i) =>(ii)

(ii) => (iii)

(iii)=>(iv)

(iv) => (v)

None of these

Y=f(x)=(1990-x^3 )^(1/3) g(y)=(i990-y^3 )^(1/3)

fogofog ……………….. fog is equal to

f(x) b) g(x) c) fog d) gof e) x

(a +1/a)2 is greater than

0 b) 2 c) 4 d) 8 e) None of these

If a = bc…..(i), a, b, c R, loga =logb + log c…………….(ii), Ch = Characteristic Mt = Mantesia, Ch of a = ch of b = ch of c (iii) Mt a = Mt b + Man ‘ C………..(iv)

(i) = >(ii)

(ii)=>(iii)

(i) =>(iv)

(ii) => (iv)

None of these

In the figure a pentagon has four of its sides cm 7, 8, 10, a and AE =cm x and x N greatest possible value of x is

16 cm b) 22 cm c) 24 cm d) 28 cm

√ +√ +√r = 0 then

Cos ( + ) = cos ( + r) = cos (r + )

Sin ( + ) = sin ( + r) = sin (r + )

1/( + + r) = 1/ +1/ +1/r

( + +r)n = n + n + rn n N

In the diagram two circles pass through each other’s centre in the radius of each circle is unity . the peremeter of shaded part is

4 /3 b) 4 /2 c) 8 /3 d)7 /2 e) None of these

1 MARK

(x+1)(x^2+1)(x^4+1)……..(x^2n+1)=_______________________

Least radius of the family of circles passing through two given points is ___________

The base of a ABC is a = 6 CM A = 1000 and having greatest area is ____________

The number of positive integral solutions of 12(x+y) = xy is_________________

If a triangle is circumscribed by a circle of unit radius. The ratio between the area and the perimeter of the triangle is ______________________________

ax^2+2hxy+by^2 as a product of three matrices is ________________________

ax^2+bx+13=0 and bx^2+ax+13=0 have a common root then a + b =____________________________________

x√^x= √xx then x =___________________________________

MARKS

Prove that numbers 49,4489,444889,………………… Obtained by inserting 48 in to the middle of the preceding number are squares of integers

M is a set of 2 x 2 matrices, and D is a set of 2 x 2 Determinants

M = { a / [■(x&y@z&r)]x,y,z,r ∈R}

D = { a /[■(x&y@z&r)] x,y,z,r, R}

f: M D f ([■(x&y@z&r)] )=|■(x&y@z&r)| then prove that f Is not (objective)

Determine the quadrants in which is point M (x,y) can be situated

Xy>0 ii) xy<0 iii) x – y =0 iv) x + y =0 v) x-y>0 vi) x+ y<0 express each case by separate diagram

Under what conditions the equation sec2 =4xy/(x+y)2 is valid.

Find the number of rectangles whose area, length and breadth are digits (excluding zero) and find the dimensions of the rectangle of greatest area.

Coordinates of 0 (0, 0) A (75, 0) and B (0, 100) are the vertices of a triangle, M is the midpoint of OB. Draw a perpendicular from M to AB which meets AB in P. find the coordinates of P.

Forty students took part in a mathematics test. They had to solve one problem in Algebra one in geometry and one in trigonometry. The results are tabulated below.

Problems solved No.of students having solved problems

In Algebra 20

In Geometry 18

In Trigonometry 18

In Algebra and Geometry 7

In Algebra and Trigonometry 8

In Geometry and Trigonometry 9

It is also know that three students did not solve a single problem. How many students solved all the three problems ? how many have solved two problems?

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