iit_rama test papers

SCHOLASTIC APTITUDE TEST - 1995.

MATHEMATICS

Time: Two Hours Max. Marks: 60(8.30 AM - 10.30 AM)

Answers must be written either in English or the medium of instruction of the candidate in high school.

There will be no negative marking. There are FIVE parts. Answer all questions of a part at one place. Use of calculators or graph papers is not permitted.

PART A

This part contains five statements. State whether the statements are true or false. Give reasons. (5 x 1 = 5 Marks)

1. The product of three positive integers is equal to the product of their L.C.M and G.C.D.

2. 443 is prime number 3. (a b)x2 +(b c)x +(c a) = 0 and (c a)x2 +(b c)x +(a b) = 0 have a common

solution for all real numbers a,b,c. 4. Every odd number can be expressed as a difference of two squares.

5. The graph of cuts the x-axis exactly once.

PART B

This part contains FIVE questions. Each question has only one correct answer. Indicate your choice by marking with the correct letter. Show your working in brief. (5 x 2 = 10 Marks)

1. If a2+b2+c2 = D where a and b are consecutive positive integers and c = ab, then is (A) always an even integer (B) sometimes an odd integer and sometimes not (C) always an odd integer (D) sometimes rational, sometimes not (E) always irrational

2. 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 (A) one (B) two (C) zero (D) zero or infinite (E) infinite

3. In any triangle ABC, a and b are the sides of triangle. Given that ( where S is the area of the triangle), then

A) B)

C) D) E) None of these

3. If a0, a1, .............., a50 are respectively the coefficients of in

the expansion of , then is

A) odd B) even C) Zero D) None of these

4. The smallest value of for all real values of x is

(A) 16.25 (B) 16 (C) 15 (D) 8 (E) None of these

PART C

This part contains FIVE incomplete statements. Complete the statements by filling the blanks. Write only the answers in your answer sheet strictly in the order in which the questions appear. (5 x 2 = 10 Marks)

1. The value of a R for which the equation (1+a2)x2 +2(x-a)(1+ax)+1 = 0 has no real roots is _______________________________.

2. The greater of the two numbers 3114 and 1718 is _______________. 3. 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)

4. 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 , , and

then the 100 th term is _______________________. 5. 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 ____________________.

PART D

Attempt any FIVE questions. (5 x 5 = 25 Marks)

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

2. 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 kI, where k is a real number and I the identity matrix .

3. Determine all 2 x 2 matrices A such that A2=I, where I = . 4. Find an explicit formula ( in terms of n N ) for f(n), if f(1)=1 and f(n)=f(n 1) + 2n

1 5. 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. 6. Let ABC be any triangle. Construct parallelograms ABDE and ACFG on the outside

D ABC. Let P be any point where the lines DE and FG intersect. Construct a parallelogram BCHI on the outside D ABC such that PA and BI are equal and parallel. Show that Area of ABDE + Area of ACFG = Area of BCHI.

7. 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 .

PART E

Attempt any ONE question (1 x 10 = 10 Marks)

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

2. Let x be any real number. Let and

. Prove that . 3. Let . Show that there exists an infinite sequence

such that for all positive integers n.

SCHOLASTIC APTITUDE TEST 1996

MATHEMATICS


NOTE:-

Answers must be written in English or the medium of instruction of the candidate in High school.

There are nine questions in this paper. Answer to each question should begin on a fresh page, and rough work must be enclosed to answer book.

Attempt all questions. There is no negative marking. Answer all parts of a question at one place. Use of calculators, slide rule, graph paper and logarithmic, trigonometric and

statistical tables is not permitted

PART-A

I. This section contains (7) questions. Each question has one or more than one correct answer(s). Indicate the answer(s) by A,B,C,D with brief reasoning. Order of the questions must be maintained. (7 x 2 =14 MARKS)

i. Let n be a product of four consecutive positive integers then

A) n+1 is always a perfect square B) n is never a perfect square C) n is always divisible by 24 D) n+12 is always a perfect square

ii. log(x+3)+log(x-1)=log(x2-2x-3) is satisfied for

A) All real values of x B) No real value of x C) All real values of x except x=0 D) No real value of x except x=0 E) All real values of x except x=1

iii. |sinx|tanx > |sinx|cotx then

A) B) C) D) None of these

iv. Which of the two functions are equivalent

A)

B)

C)

D)

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

A) Angles of the traingle are in ratio 1:1:2

B) Angles of the traingle are in the ratio 1:2:3

C) The traingle is right angle traingle

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

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

A) n=4 B) n=10 C) n=12 D) n=50

vii. The relation on the X={1,2,3,4}

A) not a function from X X B) an one to one function from X X

C) an onto function of X X D) R is a transitive relation on X

II. This section contains (6) in-complete statements. complete the statements by filling the blanks. Write your answers in your answer book in the order in which the statements are given below. (6 x 2=12 MARKS)

i. If for some angle q and for some real number x it is given that sinq = x2-2x+2 then x should be equal to _______________ and q should be equal to ________________ degrees

ii. If a, b, c, d are four distinct integers between 1 to 10 such that for a traingle 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 __________________

iii. In a cylinder, AB is a diameter of the base circle and C is the centre of the top circle.

Given that the volume of the cylinder is cm3. = 300, the perimeter of the traingle ABC is _____________ and the curved surface area of the cone with vertex C (having the same base) is ______________

iv. If the sum of the intercepts of the line with coordinate is 5 and the area of the traingle formed by the line and coordinate axes is 3 then intercepts are ______________________

v. In the figure given below

= = 900 AP = 7.5 cm, AB = 10 cm, BC = 26 cm, Then CD is ________

vi. Then a, b are __________________

III. This section contains (8) statements in GROUP-A and (4) questions in GROUP-B. Answer questions in GROUP-B on the basis of the statements made in GROUP-A. (4 x 1 = 4 MARKS)

GROUP-A

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

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

ii. B lies on the perpendicular bisector of AC

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

iv. There is a point D such that ABCD is a rhombus

v. There is a point D such that ABCD is a rectangle

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

vii. AB is a tangent to the circle through B with centre C

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

GROUP-B

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

2. Which of the above statements mean AB = BC

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

4. Which of the above statements infer ABCD is cyclic quadrilateral

PART-B

Attempt all questions Justify your answers with mathematical arguments

IV. 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 . Find the first fraction . (5)

V. Let be the roots of the quadratic equation x2+ax+b=0 and , be the roots of the equation

x2-ax+b-2=0. Given that and = 24. Find the value of the coefficient a. (5)

VI. Find the number of points with integral coordinates in the cartesian plane satisfying the inequalities

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

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

VIII. 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)

IX. Let the sequence be defined by

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

Express un as a polynomial in n.

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


MATHEMATICS


NOTE:-

1. Answers must be written in English or the medium of instruction of the candidate in High school.

2. Answer to each question should begin on a fresh page, and rough work must be enclosed with answer book.

3. Attempt all questions. 4. There is no negative marking. 5. Answer all parts of a question at one place. 6. Use of calculators, slide rule, graph paper and logarithmic, trigonometric and


Note:- All answers to questions in Section-A, Section-B, Section-C must be supported by mathematical arguments unless explicitly exempted.

SECTION-A

I. This section has Five questions. Each question is provided with five alternative answers. Only one of them is the correct answer. Indicate the correct answer by A,B,C,D,E. Order of the questions must be maintained. (5 x 2 =10 MARKS)

1. n1, n2, .., n1997 are integers, not necessarily distinct.

X = ;

Y =

A) (-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 = and ASB = . Then

A) is acute and nothing can be said about

B) is obtuse and nothing can be said about

C) nothing can be said about and is acute

D) nothing can be said about and 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. Ni 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 D) N0=12, N1=20, N2=19 E) none of these

5 Consider the decimal 0.111010100010, where the digit in the nth 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

SECTION-B

II. This section has Five questions. In each question a blank is left. (5 x 2 =10 MARKS)

Fill in the blank.

1. k is a given real constant. f(x) is a real quadratic in x such that f(x+k) = f(-x). The coefficient of x2 is 1. Then f(x) is of the form ___________________

2. A is a 2x2 real matrix such that A2 = 0 and none of its elements is a zero. Then A is of the form __________________________

3. 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) = ___________.

4. p is a natural number such that no term in the expansion of is independent of x. The p is of the form ______________________.

5. The circle C1 has centre at (2,3) and radius 3. The circle C2 has centre at (-1,4) and radius 5. The number of common tangents they possess is ________________

SECTION-C

III. This section has Five questions. Each question has a short answer. Elegant solutions will be rewarded. (5 x 2 =10 MARKS)

1. Sketch the graph of the region 0 < x < y < 1 on plain paper and describe it in words.

(No verbal argument is required)

2. Find the value of x3+y3+z3-2xyz, if

3. The remainder when x5+kx2 is divided by (x-1) (x-2) (x-3) contains no term in x2. Find k, without performing division.

4. if x 0. Determine the function f(x) and constants a,b

such that

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

SECTION-D

IV. This section has six questions. These questions are long answer questions. Elegant solutions will be rewarded (6 x 5 =30 MARKS)

1. If x3+px+q = (x- ) (x- ) (x- ), then rewrite the polynomial ( x+ + ) ( x+ + ) ( x+ + ), in terms free of , , . 2. If x+y+z = 15, xy+yz+zx = 72, prove that 3 x 7. 3. If the sides and the area of a triangle have integral measures prove that its perimeter must have even integral measure.

4. 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).

5. Two circles 6cm, 18cm in diameter are externally tangent. Compute the area bounded by the circles and an external common tangent.

6. From any point P within a triangle ABC, perpendiculars PA , PB , 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.

SCHOLATIC APTITUDE TEST 1998

MATHEMATICS

Time: Two Hours (8.30 AM - 10.30 AM) Max. Marks: 60

NOTE:-


Attempt all questions. Answer all the questions in the booklets provided for the purpose. No pages should

be removed from the booklets. There is no negative marking. Answer all questions of section I at one place. Same applies to section II. The

remaining questions can be answered in any order. Answers to sections I and II must be supported by mathematical reasoning. Use of calculators, slide rule, graph paper and logarithmic, trigonometric and


Section I

This section has 5 questions. Each question is provided with five alternative answers. Only one of them is correct. Indicate the correct answer by Aor B or C or D or E. Order of the questions must be maintained. (5x2=10 Marks)

1. p is the smallest positive such that every positive integer greater than p can be written as a sum of two composite numbers. Then A) p=3 B) p=6 C) p=10 D) p=11 E) None of these

2. ABC is a triangle such that ma > a/2, where ma = length of the median through A and a = length of BC. Then A) such a triangle does not exist B) A is acute C) A is obtuse D) A is right E) None of these

3. A vegetable shop keeps only four weights, one each of 1 kg, kg, 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 weighings possible and W the total weight of all possible weighings, then

A) n=4,W= kg B) n=16, W=30kg C) n=15,W=15 kg D) n=15, W=7.5 kg E) None of these

4. are the roots of the polynomial

and

Then

is independent of A) All ai B) All ai, i 10 C) None of ai D) All ai ,i even E) All ai ,i odd

5. 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 , 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

A) 2S B) S/2 C) S D) (p / 2) S E) None of these

Section II

This section has 5 questions. Each question is in the form of a statement with a blank. Fill the blank so that the statement is true. Maintain the order of the questions. (5x2=10 Marks)

1. 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=________

2. 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 ADB= radians

3. is a bijection from a 3-element set into a 3-element set. It is given that exactly one of the following statements A,B,C is true and the remaining

two are false: A: B: C: . Then = .

4. Given that x is real, the solution set of is .

5. In the standard expansion of , the number of terms appearing with -ve sign is .

Section III

This section has 5 questions. The solutions are short and methods, easily suggested. Very long and tedious solutions may not get full marks. (5x2=10 Marks)

1. Determine all positive integers n such that n+100 and n+168 are both perfect squares. 2. Determine with proof whether integers x and y can be found such that x+y and x2+y2

are consecutive integers. 3. The first term of an arithmetical progression is Log a and the second term is Log b.

Express the sum to n terms as a logarithm. 4. p,q,r,s are positive real numbers. Prove that (p2+p+1)(q2+q+1)(r2+r+1)(s2+s+1)

81pqrs 5. 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).

Section IV

This section has 6 questions. The solutions involve either slightly longer computations or subtler approaches. Even incomplete solutions may get partial marks. (6x5=30 Marks)

1. If x+y+z=0, then prove that (x2+xy+y2)3 + (y2+yz+z2)3 + (z2+zx+x2)3 = 3(x2+xy+y2) (y2+yz+z2) (z2+zx+x2)

2. Simplify till it is obtained equal to 3. 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).

4. 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.

5. 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 orthocenters 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

Time: Two Hours Max. Marks: 60






section I

This section has 5 questions. Each question is provided with five alternative answers. Only one of them is correct. Indicate the correct answer by A or B or C or D or E. Order of the questions must be maintained. (5x2=10 Marks)

1. Let R be the set of all real numbers. The number of functions satisfying

the relation is

A) Infinite B) One C) Two D) Zero

E) None of the above

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

A) there is no such B) there are infinitely many such B) there are precisely nine such D) there are precisely seven such E) there are precisely ten such

3. f(n)=2f(n1)+1 for all positive integers n. Then

A) B)

C) D)

E) None of these

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

A) any triangle B) no triangle

C) a right angled triangle D) an isosceles triangle

E) an obtuse angled triangle

5. If f(x)=x3+ax+b is divisible by (x-1)2 , then the remainder obtained when f(x) is divided by x+2 is

A) 1 B) 0 C) 3 D) 1

E) None of these

Section II


6. is a given line segment, H, K are points on it such that BH=HK=KC. P is a variable point such that (i) has the constant measure of radians. (ii) has counter clockwise orientation Then the locus of the centroid of is the arc of the circle bounded by the chord

with angle in the segment ____________ radians.

7. The coefficient of in is __________ 8. 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 tens place is not a 9. Then the smallest positive integer p such that 11 divides the sum of digits of (n+p) is _____________

9. The number of positive integers less than one million (106) in which the digits 5, 6, 7, 8, 9, 0 do not appear is ______________

10. The roots of the polynomial are all positive and are denoted by i, for i=1, 2, 3, .., n. Then the roots of the polynomial

are, in terms of i, _________.

Section III


11. Given any integer p, prove that integers m and n can be found such that p=3m+5n. 12. 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.

13. If a, b, c are all positive and c 1, then prove that 14. Find the remainder obtained when x1999 is divided by x21.

15. Remove the modulus :

Section IV

This section has 6 questions. The solutions involve either slightly longer computations or subtler approaches. Even incomplete solutions may get partial marks. (6x5=30 Marks)

16. Solve the following system of 1999 equations in 1999 unknowns : x1+x2+x3=0, x2+x3+x4=0., x1997+x1998+x1999=0, x1998+x1999+x1=0, x1999+x1+x2=0

17. 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.

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

, prove that

for any valid base of the logarithms.

19. Let XYZ denote the area of triangle XYZ. ABC is a triangle. E, F are points on and respectively. and 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).

20. Prove that 80 divides 21. 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

Time: Two Hours Max. Marks: 60






Section I

This section has 5 questions. Each question is provided with five alternative answers. Only one of them is correct. Indicate the correct answer by A or B or C or D or E. Order of the questions must be maintained. (5x2=10 Marks)

1. A1A2..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

A) There is no such P B) There is exactly one such P

C) Either A or B D) The locus of P is a straight line

E) The locus of P is a circle

2. Consider the following expressions in a,b,c.

3. The expressions which are not perfect squares among E1,E2,E3 and E4 are

A) E1,E3 B) E2,E3 C) E2,E4 D) E1,E4 E) E3,E4

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

A) The hypothesis is not sufficient to evaluate n

B) n=3 C) n=6 D) n=9 E) None of these

5. There are n rods, . 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

A) 0 B) n C) n! D) nn E) 2n-1

6. Integers from 1 to 2000 are written. A single operation consists of cancelling any two of the numbers and replacing them by their product. After 1999 such operations A) No number will be left B) Precisely 2 numbers will be left C) More than two numbers may be left D) Only one number will be left and it is 2000! E) Only one number will be left, but it cannot be uniquely determined

Section II


6. All possible 2000-gons are inscribed in a circle of perimeter p. Let X= an

inscribed polygon has a side of length x. . Then the greatest member of X is ____________ .

7. 8. Triangle ABCThe altitude through A has length h. is right angled at A. Then

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

______________ and .

10. The units digit of is _________.

Section III


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

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

13. If real numbers x,y,z satisfy , then prove that x=y=z

14. Write down the solutions of ,if x >0 and and

denotes the integral part of

15. Given and a point D in its interior. The problem is to find E on and

F on such that D is the mid point of line segment . A student suggests the following method of construction: Join BD and extend it to B such that BD =

DB. Draw parallel to through B. Take for E the point where this parallel

cuts . Join ED and produce it to meet .Take for F this point of intersection.

Discuss the validity or otherwise of this method of construction.

Section IV

This section has 5 questions. The solutions involve either slightly longer computations or subtler approaches. Even incomplete solutions may get partial marks.

16. Consider the equation in positive integers with .

a) Prove that . b) Rule out the possibility that one of is even and the other is odd. c) Rule out the possibility that both are odd. d) Prove that is a multiple of 4. e) Obtain all the solutions. (10 Marks )

17. 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 . ( 5 Marks ) 18. Manipulate the equality

until the equlity is obtained - 5 Marks

6. 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 )

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

at , , then evaluate ( 5 Marks )


MATHEMATICS

Time: Two Hours(8.30 AM - 10.30 AM) Max. Marks: 60

NOTE:-

1. Answers must be written in English or the medium of instruction of the candidate in High school.

2. Answer to each question should begin on a fresh page, and rough work must be enclosed with answer book.

3. Attempt all questions. 4. There is no negative marking. 5. Answer all parts of a question at one place. 6. Use of calculators, slide rule, graph paper and logarithmic,

trigonometric and statistical tables is not permitted

Note:- All answers to questions in Section-A, Section-B, Section-C must be supported by mathematical arguments unless explicitly exempted.

SECTION-A

I. This section has Five questions. Each question is provided with five alternative answers. Only one of them is the correct answer. Indicate the

correct answer by A,B,C,D,E. Order of the questions must be maintained. (5x2 =10 MARKS)

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

A) 2001 B) 2002 C) 2003 D) 2004 E) 2005

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

A) m B) 2002m C) 2003m D) mn E) None

3. 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

A) 2000 units B) 2001 units C) 2002 units D)2003 units E) 1999 units

4. Let S be any expression of the form ek2k, where each ek is varied independently between 1 and 1. The number of expressions S such that S=0 is

A) 0 B) 1 C) 2 D)22001 units E)None

5. 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

A. always B. only if a1 = b1, a2 = b2,., a2001 = b2001 C. only if b1, b2,., b2001 is a cyclic permutation of a1, a2,., a2001 D. only if the center of the polygon P1 lies in its interior E. none of these

SECTION-B

II. This section has Five questions. In each question a blank is left. Fill in the blank. (5x2 =10 MARKS)

1. 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 = _________.

2. 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 ________.

3. The smallest positive integer k such that (2000)(2001)k is a perfect cube is____________.

4. ABCD is a rectangle with AB = 16 units and BC = 12 units. F is a point on

and E is a point on such that AFCE is a rhombus. Then EF measures _________units.

5. n is a positive integer not exceeding 9. The list of all n such that 10 divides nn n is

SECTION-C

III. This section has Five questions. Each question has a short answer. Elegant solutions will be rewarded. (5x2 =10 MARKS)

1. Given a set of n rays in a plane, we mean by a reversal the operation of reversing precisely one ray and obtaining a new set of n rays. Starting from 2001 rays and performing one million reversals, is it possible to reverse all the rays?

2. If x + y+ z = a, prove that x2 + y2 + z2 . 3. Given the locations 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.

4. If for every x > 0 there exists an integer k(x) such that a0 + a1x + a2x2 + .. + anxn xk(x) - 1 , then find the value of a0 + a1 + + an. 5. We know that we can triangulate any convex polygonal region. Can we parallelogramulate a convex region bounded by a 2001-gon?

SECTION-D

IV. This section has Five questions. These questions are long answer questions. Elegant solutions will be rewarded (5x6 =30 MARKS)

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

2. The product of two of the roots of x4 11x3 + kx2 + 269x 2001 is 69. Find k.

3. 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.

4. If (1+2x)(1+2y) = 3, xy 0, then show that = .

5. a) 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 is a constant.

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


MATHEMATICS Time: Two Hours (8.30 AM 10.30 AM) Max.Marks:60

NOTE:-

1. Attempt all questions. 2. Answer to each question should begin on a fresh page, and rough work must be

enclosed with answer book. 3. There is no negative marking. 4. Answer all parts of a question at one place. 5. Use of calculators, slide rule, graph paper and logarithmic, trigonometric and

statistical tables is not permitted.

Note:- All answers to questions in Section-A, Section-B, Section-C must be supported by mathematical arguments. In each of these sections order of the parts must be maintained.

SECTION-A

I. This section has Five questions. Each question is provided with five alternative answers. Only one of them is the correct answer. Indicate the correct answer by A,B,C,D,E. (5 2=10 MARKS) 1. Let m be the l.c.m. of 32002-1 and 32002+1. Then the last digit of m is

A) 0 B) 4 C) 5 D) 8 E) none

Hint: Read off g.c.d. and hence determine the l.c.m.

2. 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

3. f,g,h:R R are defined by f(x) = x-1, g(x) = if x -1 h(x) =

if x -1, x -2 -2 if x = -1 -2 if x = -1 -3 if x = -2 Then f(x) + g(x) 2.h(x) is

A) 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

4. A, B are two distinct points on a circle with center C1 and radius r1. 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.

A) S happens if C1C2 > r2

B) If S happens then C1C2 > r2

C) If S happens then it is possible that C1C2 = r2

D) If S happens then it is necessary that C1C2 < r2

E) none of these

5. 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 points l m where the line segment l is a side of T1 and the line segment m is a side of T2, whenever l m is a unique point. These points of intersection determine a convex polygon, which is a

A) triangle B) quadrilateral C) pentagon D) hexagon E) septagon

SECTION-B

II. This section has Five questions. In each question a blank is left. Fill in the blank. (5 2=10 MARKS)

1. The number of integral solutions of the equation x4-y4 = 2002 is _____

2. A,B,C,D are distinct and concyclic. is perpendicular to , is perpendicular

to . intersects in N. The nonobtuse angle enclosed between and is _____ in degrees

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

4. If x is a real number Int.x denotes the greatest integer less than or equal to x.

Int. = __________

5. O is the origin and Pi is a point on the curve x2 + y2 = i2 for any natural number i.

= OPj, where j = _________

SECTION-C

III. This section has Five questions. Each question has a short answer. Elegant solutions will be rewarded. (5 2=10 MARKS) 1. If m is the right most nonzero digit of (n!)4 where n is a positive integer greater than 1, determine the possible values of m.

2. Sketch the graph of the function f(x) = , .

3. 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.

4. Do there exist positive integers p, q, r such that ?

5. 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

SECTION-D

IV. This section has Five questions. These questions are long answer questions. Elegant solutions will be rewarded. (5 6=30 MARKS) 1. If m and n are natural numbers such that

Prove that 2002 divides m.

2. 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.

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

4. 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. If a,b,c, are real numbers such that the sum of any two of them is greater than the third

and =3abc, then prove that a=b=c.

6. 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.

SCHOLASTIC APTITUDE TEST 1995.

PHYSICS

Time: ONE Hour Max. Marks: 60


There will be no negative marking. There are FOUR Sections. Answer all questions of a section at one place. Use of calculators or graph papers is not permitted. Marks alloted for individual sections are as follows. All questions in an

individual section carry equal marks.

Section I 10 Marks Section II 20 Marks Section III 15 Marks Section IV 15 Marks

SECTION I

1. Velocity of light in centimeter per nanosecond is _______________________ . 2. Phase of vibrating particles varies linearly and continuously when a wave travels in a

medium. The linear distance between particles differing in phase by p when a wave of frequency 2 kHz travels at 350 m/s is a medium is __________________.

3. Density and youngs modulus of a certain hypothetical material are 5 gm/cm3 and 2.0 x 1010 Pa. Velocity of sound wave in this material is ______________.

4. Field strength at a certain distance from a charge Q is E. Field strength at one fourth of this distance from a charge Q/4 is _______________ .

5. Farad x Ohm expressed in the simplest form equals _________________. 6. A conductor carries a current vertically upwards. Magnetic field due to this current to

the East of the conductor would be directed ____________________________. 7. An electron is moving southward in a region of space having a magnetic field directed

vertically upwards. It experiences a force towards _________________ . 8. The change that takes place in the nucleus when a beta particle is emitted is

________________. 9. Two Deutrons combine to form a Helium nucleus. The masses of Deuterium and

Helium atoms are 2.0141 u and 4.0026 u respectively. The energy yield in MeV per reaction is ________________.

10. Electromagnetic radiations of wavelength 5 nm fall in the _________________ region.

SECTION II

1. A man moves 100 m 30 East of south then another 100 m 60 North of East. He is at a distance of _________________ from the starting point.

2. Displaced air exerts a vertcally upward force equal to its own weight on a body. Force exerted by the displaced air on a man of weight 60 kg is nearly __________________ . ( Average density of a human being is nearly same as that of water and density of air is 1.3 gm/l. ).

3. A mirror produces a magnified image of a linear object placed 30 cm away from it 30 cm away from the object. If the object is moved towards the mirror by 5 cm the image moves by ____________.

4. With velocity of light in vaccum chosen as a standard the refractive indicies of water and glycerine are 4/3 and 8/5 respectively. If velocity of light in water is chosen as standard refractive index of glycereine would be ___________________ .

5. A source of sound of frequency 5 kHz is moving at 20 m/s. The wavelength of waves in front of the source is _______________________ .( Velocity of sound in air is 350 m/s ).

6. Two wires of circular cross section are made of equal amounts of same material and have their radii in the ratio 1:2. The ratio of their electrical resistances is ___________________.

7. Current between junctions A and B is _______________ .

8. ABCDEFGH is a wireframe of twelve conductors forming the edges of a cube. Resistance between any two adjacent corners is R. If the connection between A,E and C,G are removed the effective resistance between B and D would be ________________.

9. A battery is rated 12 V and has a capacity of 12 Ah. ( 1 Ah means the battery can drive one ampere of current for one hour or half ampere for two hours and so on..... ). The energy it can deliver is _______________________.

10. An electric lamp is rated 40 W at 200 V. If the lamp is operated at 150 V its power would be __________________.

SECTION III

The amount of heat Q flowing along the length of a rod in a time t is given by the expression

Where q 2 and q 1 are temperatures of hot and cold ends, A and L are area of cross section and length of the rod, and K the coefficient of thermal conductivity ( a constant for the given material ) of the rod. It is assumed that there is no heat loss from the sides of the rod.

1. A rod is 50 cm long and has a cross sectional area of 10 mm2. If coefficient of thermal conductivity of the material of the conductor is 400 W/mK, the rate of heat flow along the rod is _____________________.

2. Temperature gradient is the rate at which the temperature varies with the length of the rod and is a constant for a rod carrying heat in steady state. If the temperature gradient of a rod is 5 K/mm, the heat that flows in one minute through a rod of cross sectional area 2 cm2 and coefficient of thermal conductivity 250 W/mK is equal to ________________.

3. When two rods are placed end to end the rate of heat flow in them must be same. If two rods of equal areas of cross section and having their lengths and coefficients of thermal conductivities in the ratio 1:2 and 2:1 respectively have their free ends maintained at 350 K and 300 K respectively, the temperature of the junction will be _______________.

4. The amount of heat that escapes from a house through a 3 m x 2 m glass window pane of thickness 5 mm for a temperature difference of 10 C in one hour is ____________________. ( Coefficient of thermal conductivity of glass is 0.008 W/cmK )

5. One end of rod of cross sectional area 5 cm2, length 25 cm, coefficient of thermal conductivity 210 W/mK is maintained at 100 C and the other end is connected to beaker of water containing melting ice. The amount of ice that melts in a minute is _______________. ( Latent heat of fusion of ice is 80 cal/gm )

SECTION IV

A body floats when the weight of the liquid it displaces equals its own weight. Questions in the following section are based on this principle.

1. A block of ice (density 900 kg/m3) is floating in a beaker of liquid (density 1200 kg/m3). The fraction of its volume seen above the liquid is ____________________ .

2. A body of density 1200 kg/m3 just floats in a homogenous mixture of two liquids of densities 1000 kg/m3 and 1500 kg/m3. The ratio by volumes in which the liquids are mixed is ____________________ .

3. A cubical block of Aluminum has a cavity inside. If it floats in water with 7/8 of its volume submerged in water. The ratio of the volume of the material to the ratio of its external volume is __________________ .

4. A cubical block C floats as shown with one fourth of its volume submerged in liquid A and the remaining three fourths in liquid B as shown. If density of liquid A is 600 kg/m3and that of B is 400 kg/m3, density of the material of the block is _____________________.

5. A test tube containing some lead shot floats vertically in a beaker of water. A few lead shots are removed from the tube and dropped in to the beaker. The level of water in the beaker _______________ .( Assume that the test tube continues to float vertically after the lead shots are removed from it ).


PHYSICS

Time: ONE Hour (10.45 AM - 11.45 AM) Max. Marks: 60


There will be no negative marking. There are THREE Sections. Answer all questions of a section at one place. Use of calculators or graph papers is not permitted. Marks allotted for individual sections are given below. All questions in an

individual section carry equal marks.

Section A 10 Marks Section B 20 Marks Section C 30 Marks

SECTION A

1. A large amount of ice at 0 C is added to a beaker of water at 30 C. Some of the ice melts and the rest floats on water in the beaker. The temperature of the water in the beaker is .

2. The angle of minimum deviation of an equiangular prism is equal to the angle of the prism. The refractive index of the material of the prism is .

3. A wave of frequency 170 Hz travels through a narrow pipe closed at one end where it is reflected. The reflected wave and the incident wave interfere forming a stationary wave in the pipe. If the nearest antinode in the pipe is 50 cm from the closed end, the velocity of the wave is .

4. For compressions and rarefactions to form in a medium the direction of vibration of the particles and the direction of propagation of the wave must be .

5. When a person speaks in to a microphone the vibrations of its diaphragm are free/ resonant/forced . ( Mention the correct answer in your response sheet )

6. A current of 3.2 mA is flowing through a wire. The number of electrons crossing any section of the conductor in a minute would be .

7. A long wire has electronic current flowing through it from north to south. The north pole of a compass needle placed above the wire would deflect towards .

8. A filament lamp is rated 100 W at 200 V. The filament is cut in to two equal parts and one of them is used at 250 V. Its power would be .

9. A hypothetical radioactive nucleus emits a proton and a deuteron in succession. The mass number and charge number of the product nucleus are reduced by and respectively.

10. When aluminum is bombarded with alpha particles, the resulting nucleus becomes radioactive and emits particles.

11. With x axis westward and y axis southward the coordinates of two places are 150 m, 200 m and 450 m, 600 m. The westward distance between the places is . The southward distance between the places is . The distance between the places is .

12. A solid is heated by a source supplying heat at a constant rate. The temperature at first rises from 20 C to 80 C in 5 minutes and then remains steady for 60 minutes before it starts rising again. The ratio of specific heat of the solid to its latent heat of fusion is .

13. A concave mirror is kept some distance away from a wall. A twice magnified image is formed on the wall when an object is placed between the mirror and the wall at a distance of 30 cm from the wall. The focal length of the mirror is .

14. In the expression v, E and D denote velocity, Elasticity and density. Then a and b are equal to and respectively.

15. Charge stored by a condenser is 5 mC when a potential difference of 2 V is maintained between its plates and the plates have a material of dielectric constant 5 between them. Charge on the plates when the potential difference is doubled and the material between the plates is replaced by material of dielectric constant 2 would be .

16. A cell drives a certain charge through a circuit in a certain time. Due to a change in the resistance of the circuit in which this cell is a part the same charge takes twice greater time to be driven. Work done in the first and second cases by the cell are in the ratio .

17. A charged liquid is flowing through a pipe of cross sectional area a. Unit volume of this liquid has a charge q. If the velocity of the flow is v, current through any section of the pipe is .

18. Two wires are made of equal mass of same material and have their cross sections in the ratio 3:4. Their resistances will be in the ratio .

19. To raise the temperature of 10 liters of water through 20 C a 1000 W heater would require minutes.

20. A length of wire is made in to circular coil with certain number of turns. For a certain current through it the magnetic field at its center is X. If another coil made of same length of wire has twice greater number of turns, magnetic field at its center for same current as the first coil would be .

SECTION C

21. The distance of a particle moving along a straight line from another stationary point on the same line is given by l = 2+ 4t + 5 t2 with l in meters and t in seconds. The distance travelled by the body between 4 th and 6 th seconds will be .

22. Two cars are travelling on a straight road with the one car 200 m from the other at the same speed. At a certain moment brakes are applied to the rear car and it decelerates uniformly travelling 20 m before coming to rest. The distance between the cars at the moment the rear car comes to rest is .

23. Two points A and B are located on the floor of a room 30 cm and 20 cm away from a wall. The distance between the points parallel to the wall is 50 cm. An insect is at A moving on the floor at 5 cms1. It has to first reach a point C on the wall and then reach B such that AC and BC make equal angles with normal to the wall at C The time taken for this motion would be .

24. Two particles are going round concentric circular paths making 30 RPM and 60 RPM. the time interval between their being at the minimum and maximum distances is .

25. A wooden block of certain volume is floating in water with a metal block of one eighth its volume hanging from below it. The volume of the submerged portion of the wooden block is one fourth its total volume. When the metal block is placed on top of the wooden block, the volume of the submerged portion of the wooden block is three fourths its total volume. The ratio of densities of wood and metal is .

26. Intensity of light at a point is proportional to square of the distance of the point from a source of light. A source of light is at a height of 30 cm above the center of a circular table of radius 40 cm. The ratio of intensities at the center and the edge of the table is .

27. Two points on a thin circular ring are located such that the radial lines joining them to the center have a certain angle between them. When this angle is 90 the electrical resistance offered by the ring for a current entering one point and leaving the other is R. The resistance when the angle is 60 will be .

28. An electron moving vertically upward in a horizontal magnetic field experiences a force directed towards east. The force experienced by an electron travelling eastward in this magnetic field will be directed towards .

29. Radii of circular paths of electrons round the nuclei are proportional to the square of quantum numbers. If square of time taken to go round circular path once by an electron is proportional to cubes of the radius, The ratio of frequencies of revolution in orbits of quantum numbers 1 and 4 is .

30. Mass of a radioactive substance at a certain moment is 16 gm and its half life is 2 hrs. The mass of the substance that disintegrates in 1 hour is greater than / equal to / less than 4 gm. ( Mention the correct answer in your response sheet and justify your answer)


PHYSICS

Time: ONE Hour (10.45 AM - 11.45 AM) Max. Marks: 60


There will be no negative marking. There are THREE sections. Answer all questions of a section at one place. The relevant working in arriving at an answer has to be shown wherever

required. However if the answer can be arrived at without detailed working, write only the answer

Use of calculators or graph papers is not permitted.

Marks allotted for individual sections are given below. All questions in an individual section carry equal marks.


Useful Physical constants:

i. acceleration due to gravity = 10 ms2 ii. Charge of electron e = 1.6 x 10-19 C

iii. 1/4p e o = 9 x 109 Nm2C2 iv. Density of wood = 0.5 gm/cc

SECTION A

1. A wire carries a current vertically downwards. The magnetic field to the East of the wire is directed towards .

2. A source of sound is moving at 36 kmph straight towards an observer moving away from it at 36 kmph and along the same straight line. The frequency of the source is 400 Hz. Frequency of the note received by the observer is .

3. If 0.16 mA of current is flowing through a wire, the number of electrons crossing any section of the wire in a minute is .

4. A hypothetical radioactive nucleus emits a proton and a positron. Its mass number decreases by and its charge number decreases by .

5. Mass equivalent of energy of 186.2 MeV is amu. 6. In electrolysis of copper sulphate solution, an alternating current source is

used. The mass of copper deposited on the cathode plate is . 7. In wavelength of a wave in a certain medium is 20 cm and its velocity is 350

m/s. Its wavelength in a medium where the velocity is 1400 m/s would be equal to .

8. The energy consumed by a 20 W lamp used for 25 hrs is kWh

9. If a cell of emf 6 V drives a current of 2 amp through a circuit for 10 minutes, the amount of work done by it is .

10. 20 mJ of work is done in moving one micro Coloumb of charge between two points. The potential difference between these points is .

SECTION B

1. Light travels at v1 in a medium and at v2 in another medium. The ratio of refractive indices of the media is .

2. The closest distance between particles one at an extreme position and another at the equilibrium position is 2 m. If the frequency of this wave is 2.5 kHz, the velocity of the wave is .

3. At 1 atmosphere pressure and a certain temperature the velocity of sound is 350 m/s. At 2 atmosphere pressure and same temperature the velocity would be .

4. Two stones are released from rest one from a height of 9 m and another from some unknown height above the ground. One of the stones hits the ground 1 s after the other strikes the ground. The height from which the second stone is released can have values of .

5. Light falling on a sphere made of a photoelectric material causes it to lose electrons. If a sphere initially uncharged has acquired a charge of 3.2 m C, the number of electrons it has lost is .

6. Energy stored by a capacitor is given by the expression . If a 5 m F capacitor charged to 200 V, the energy it possesses is equal to .

7. 2 kgf/cm2 = N/m2.( kgf = g N ) 8. Momentum is mv and kinetic energy is mv2/2. A body of mass 4 kg has a

kinetic energy of 32 J. Its momentum is . 9. A solid object A of mass 200 gm object placed in contact with a source

supplying heat at a constant rate undergoes a rise in temperature of 20 C in 40 s. Another solid object B of mass 300 gm needs 20 s for a 5 C rise in temperature. The ratio of specific heats of A and B is .

10. Mass of a radio active substance at a certain moment is 16 gm. The mass of it that disintegrates in its third half-life period is .

SECTION C

In some of the following questions information ( A new rule, a new equation) required to answer the questions is provided. Read the information provided carefully and answer the following question. The questions can be answered by proper appreciation of the information provided.

1. Resultant R of two parallel forces P and Q acting at A and B is equal to P+Q and the point C where it acts can be located from the relation P(AC)=Q(BC). ( Refer to the figure shown below )

The resultant of two parallel forces is 40 N and acts 10 cm from the force of 10 N. The other force acts at a distance of from the 10 N force.

2. A stone projected in to the air from the ground level with a velocity v directed at q above the horizontal hits the ground at a horizontal

distance of from the point of projection. The maximum height attained by it during the flight is given by the

expression .

A projectile hurled at 45 attains a maximum height of 25 m. The horizontal distance it travels before it strikes the ground is .

3. A positively charged particle travelling vertically upwards experiences a northward force due to magnetic field directed at right angles to it. A negatively charged particle travelling northward in the same magnetic field will experience a force.

4. A long uniform wooden stick of cylindrical cross section is weighted at the bottom such that a length of 20 cm of it protrudes out of water as it

floats vertically. If a length of 10 cm is sawed off from the stick and it is floated again, the length that protrudes above the water surface would be .

5. A lens forms a twice magnified image of an object kept 30 cm away from it on oneside on a screen kept on the other side. Without disturbing the object, the lens is moved through 5 cm towards the object, the screen must be moved by to obtain a sharp image.

6. Between any two adjacent junctions the resistance of a wire frame shown is same. If the resistance between B and E is R, resistance between A and B is .

7. An insect is located at the edge of a fan blade of length 40 cm. It starts walking toward the centre of the fan. As the insect moves 10 cm from its initial position, the blade has turns through a right angle. The distance between initial and final positions is .

8. Two cars are travelling towards a junction along two different intersecting roads. One of them is 200 m away and is moving uniformly at 10 m/s. The other located 100 m away starts from rest and accelerates uniformly at 2 m/s2 towards the junction. The interval between the moments when the cars pass the intersection is .

9. A wire is of square cross section of side a and length L and has a resistance R. Another wire made of same material and of twice greater length has the same volume as the first. The electrical resistance of the second wire is .

10. A ring with a uniformly distributed charge (Q) of radius R repels a point charge (q) placed on its axis with a force F given by the

expression where x is the distance of q from the centre of the ring.

A point charge of 5 m C is located on the axis of a ring of radius 60 cm carrying a charge of 40 m C. The distance between any point on the ring and the point charge is 100 cm. The force experienced by the point charge is .

SCHOLASTIC APTITUDE PHYSICS 1998

PHYSICS



Answer all the questions in the booklets provided for the purpose. There will be no negative marking. There are THREE sections. Answer all questions of a section at one place. The relevant working in arriving at an answer has to be shown wherever required. However if the

answer can be arrived at without detailed working, write only the answer Use of calculators or graph papers is not permitted. Marks allotted for individual sections are given below. All questions in an individual section carry

equal marks


Useful Physical constants:

I. Plancks constant = 6.6 x 10-34 Js II. Velocity of Light c = 3 x 108 m/s

III. Electron Volt = 1.6 x 10-19 J IV. Relative density of cork = 0.25

SECTION A

1. What temperature change on the Kelvin scale is equivalent to a 10 degree change on the Celsius scale? A) 283 K B) 273 K C) 18 K D) 10 K E) 0

2. A light year is a unit of A) acceleration B) distance C) speed D) time E) velocity

3. The critical angle of a material is the angle of incidence for which the angle of refraction is: A) 0 B) 30 C) 45 D) 90 E) 180

4. A plane mirror produces an image that is: A) real, inverted, and larger than the object. B) real, upright, and the same size as the object. C) real, upright, and smaller than the object. D) virtual, upright, and the same size as the object.

5. An object is located 0.20 m from a converging lens which has a focal length of 0.15 m. Relative to the object, the image formed by the lens will be: A) real, erect, and smaller. B) real, inverted, and smaller. C) real, inverted, and larger. D) virtual, erect, and larger.

6. If the frequency of sound is doubled, the wavelength: A) halves and the speed remains unchanged. B) doubles and the speed remains unchanged. C) is unchanged and the speed doubles. D) is unchanged and the speed halves.

7. When metal rod 1 is placed in contact with metal rod 2, thermal energy flows from 1 to 2. A possible explanation is that 1 has a higher A) specific heat B) heat capacity C) mass D) temperature E) Density

8. When the velocity of a moving object is doubled, another physical quantity associated with the object also gets doubled. This quantity is A) acceleration B)kinetic energy C) mass D) momentum E) Potential Energy

9. Which of the following could NOT be used to indicate a temperature change? A change in: A) color of a metal rod. B) length of a liquid column. C) electrical resistance. D)mass of one mole of gas at constant pressure.

10. Two charges repel each other with a force F when they are a certain distance apart. The charges are halved and the distance between them is also halved. The force between the charges now is

A) F/16 B) F/4 C) F/8 D)F E) F/2

SECTION B

11. A mass m of helium gas is in a container of constant volume V. It is initially at pressure p and absolute temperature T. Additional helium is added, bringing the total mass of helium gas to 3m. After this addition, the temperature is found to be 2T. What is the gas pressure?

A) 2 /3 p B) 3 /2 p C) 2 p D) 3 p E) 6 p

12. Which would be the most comfortable temperature for your bath water?

A) 0o C B) 40 K C) 110o C D) 310 K E) 560 K

13. A wire of length 50 cm has a mass of 20 gm. If its radius is halved by stretching, its new mass per unit length will be

A) 0.4 gm/cm B) 0.2 kg/m C) 2 kg/m D) 0.1 gm/cm E) 0.2 gm/cm

14. A string is tied around a cork of mass m and the free end of the string is attached to the bottom of a water-filled container with the cork totally immersed in the water. The tension in the string is:

A)0 B) mg C) 2 mg D) 3 mg E) 4 mg

15. A long straight wire conductor is placed below a compass. When the conventional current in the conductor is from south to north and large, the N pole of the compass:

A) remains undeflected. B) points to the south. C) points to the west. D) points to the east. E) has its polarity reversed.

16. A beam of light passes from medium 1 to medium 2 to medium 3 as shown in the accompanying figure. What is true about the respective indices of refraction (n1, n2, and n3)? A) n1 > n2 > n3 B)n1 > n3 > n2 C) n2 > n3 > n1 D)n2 > n1 > n3 E) n3 > n1 > n2

17. A positively charged conductor attracts a second object. Which of the following statements could be true? I. The second object is a conductor with negative net charge. II. The second object is a conductor with zero net charge. III. The second object is an insulator with zero net charge.

A) I only B) II only C) III only D) I & II only E) I, II & III 18. A ray of light is

directed toward a point P on a boundary ( represented by the dotted line ) as shown to the right. Which segment best represents the refracted ray?

A) PA B) PB C) PC D) PD E) PE

19. A book is placed on a table. Let FEB be force exerted by earth on the book, FTB be force exerted by table on the book, FBT be force exerted by book on the table and FBE be force exerted by book on the earth. Among these forces, the action, reaction pairs are

A) FEB , FTB B) FBE , FTB C) FEB , FBE D)FEB , FBT E)FBE , FBT

20. You are given three 1.0 W resistors. Which of the following equivalent resistances CANNOT be produced using all three resistors?

A)1 /3 W B) 2 /3 W C) 1.0 W D) 1.5 W E) 3.0 W

SECTION C

21. Two cars are travelling in the same direction at 60 kmph and are 30 km apart. A third car travelling at 30 kmph crosses one of the cars at a certain moment. Find the time elapsed before it crosses the second car. Solve the problem if the first two cars are travelling in the opposite directions.

22. A point object is located at a distance of 30 cm on the principal axis of a convex lens of focal length 20 cm. If the lens is moved away from the object by 10 cm, find the distance through which the image moves.

23. From amongst resistances of 2 W ,3 W ,6 W and 7 W , two resistors are picked and combined first in series and then in parallel. Their equivalent resistances are found to be 2 W and 9 W . Find the parallel equivalent of the remaining two resitances.

24. Which of the following is NOT possible for the images formed by the lens in the accompanying figure? A) real and inverted B) real and smaller in size C) real and larger in size D) virtual and erect E) virtual and smaller in size

25. A cyclist covers a certain distance in 50 minutes riding his bicycle at a certain speed

and in 40 minutes riding at a faster speed. If he rides his bicyle changing his speed from lower to higher speed every ten minutes how long would he take to cover the distance. Consider both the cases that arise.

26. A horse running along a 5 m wide circular track takes p seconds more to travel along the outer edge than the time taken to travel the inner edge. Find the speed of the horse.

27. In a certain process initial energy of the system is -13.6 eV and final energy is - 3.4 eV. The enegy loss is due to the emission of a quantum of light by the system. If the energy of the quantum of light is given by E=hn , where h is the Plancks constant and n is the frequency of the light, Find the wavelength of light emittted.

28. A metallic sheet of length L, width W and some thickness is melted and formed in to a hollow cylinder. If the inner and outer radii of the cylinder are a and b respectively, and the length of the cylinder is same as the width of the sheet, Find the thickness of the sheet.

29. Two convex lenses of focal lengths 20 cm and 30 cm are placed 50 cm apart coaxially. (i) A point object is located at a distance of 20 cm from the lens of shorter focal length. The rays of light from this object pass through the first lens and then through the second. Find the distance where the rays of light converge from the second lens. (ii) If a parallel beam of light were to be incident on the first lens and the rays refract a second time after first converging, find the distance where the rays converge after second refraction

30. In the circuit shown adjacent junctions are connected by equal resistances. The effective resistance between junctions A and G when the resistances between junctions C,E and D,F are removed is R. The removed resistances are reconnected and the resistances between junctions B,D and F,H are now removed. Find the new effective resistance between A and G.


PHYSICS



Answer all the questions in the booklets provided for the purpose. There will be no negative marking. The relevant working in arriving at an answer has to be shown wherever required. However if the

answer can be arrived at without detailed working, write only the answer

1. At a certain moment of time a particle moving along a straight line is located at a distance of 5 m from a point and at a subsequent moment it is at a distance of 10 m from the same point. The minimum and maximum distances the particle could have moved are .

2. A particle is going round a circular path with a constant speed. The angle between its acceleration and velocity vectors is .

3. An object placed on an equal arm balance requires 12 kg to balance it. When placed on a spring scale, the scale reads 120 N. Everything (balance, scale, set of masses, and the object) is now transported to the moon where the gravitational force is one-sixth that on Earth. The new readings of the balance and the spring scale are respectively.

4. A body is thrown up from a certain height and another thrown down from the same point with the same speed at the same time. If they hit the ground with a time gap of 4 s, the speed of projection is .

5. On applying brakes a car slows down from v1 to v2 as it travels a certain distance and from v2 to v3 as it travels an equal distance. the relation between the velocities is .

6. A ball of mass m is thrown vertically upward. Air resistance is not negligible. Assume the force of air resistance has magnitude proportional to the velocity, and direction opposite to the velocity's. At the highest point, the ball's acceleration is (greater than /less than /equal to) acceleration due to gravity.

7. A converging lens with focal length +10 cm would produce a twice magnified image of an object, if the object is placed at a distance of from it.

8. Two small identical conducting spheres are separated by a distance much larger than their diameter. They are initially given charges of -2.00 C and +4.00 C, and are found to exert a force on each other of magnitude 1 N. Without changing their position, they are connected by a conducting wire. When the wire is removed, the magnitude of the force between them is .

9. Three 60 W light bulbs are mistakenly wired in series and connected to a 120 V power supply. Assume the light bulbs are rated for single connection to 120 V. With the mistaken connection, the power dissipated by each bulb is .

10. In the circuit shown the resistance R1, R2, R3, R4 and R5 are 80 , 20 60 25 and 20 respectively. A current of 0.10 A flows through the 25 resistor shown in the diagram. The current through the 80 resistor is .

11. A wire of a certain radius has a resistance R. Another wire of same length, same material has one end of same radius and the other end of twice greater radius, the radius increasing linearly from the narrower end. If the resistance R' of the second wire is less than nR where n is an integer, n would be .

12. Angular momentum of a particle about a point is defined as the product of its mass, velocity and perpendicular distance from the given point. A particle of mass m is moving in the xy plane parallel to x- axis at a distance d from it with a speed v. Its angular momentum about the origin would be

13. An alloy consists of a metal A of density 3.1 gm/cc and a metal B of density 8.8 gm/cc mixed in the ratio 2:1. A hollow ball made of this material has an external volume of 80 cc and a mass of 100 g. The volume of the cavity is .

14. A body is dropped from a height of 20 m above the ground. If gravity disappears 1 second after it starts falling , the time it takes to hit the ground is ( Assume acceleration due to gravity = 10 ms2 ) .

15. The diagram to the right shows the velocity-time graph for two masses A and B that collided elastically. Consider the following statements. I. A and B moved in the same direction after the collision. II. The velocities of A and B were equal at the mid-time of the collision III. The mass of A was greater than the mass of B. The true statements from these are .

16. Potential difference between A and B is and the current flowing from C to D is .

17. Parallel beam of light parallel to the principal axis of a lens falls on a lens of focal length 50 cm. After passing through the first lens they pass through another coaxially mounted convex lens of focal length 50 cm placed 150 cm from the first lens. After passing through the second lens, the rays converge at a distance of __________ from the first lens.

18. If velocity of light in medium A is 2.25 108 m/s. Refractive index of medium B with respect to medium C is 1.2. Velocity of light in a medium C is 1.5 108 m/s. Refractive index of A with respect to B is . ( You should not use the value of velocity of light in vaccum or air ).

19. Particles A and B are free to move along parallel paths. B is moving at a constant velocity of 10 m/s and A starts moving at the moment B passes it. If A accelerates at 1 m/s2 for 5 s and then travels at a constant velocity, the distance between A and B 10 s after B passes A is .

20. Dust particles are uniformly distributed in space with a mass per unit volume . A circular plate of area A is moving in this space at v. The dust particles that come in to contact with the plate stick to it. The mass of dust that collects on the plate


PHYSICS



Answer all the questions in the booklets provided for the purpose. There will be no negative marking. The relevant working in arriving at an answer has to be shown wherever

required. However if the answer can be arrived at without detailed working, write only the answer

Use of calculators or graph papers is not permitted. All questions carry equal marks

1. A body floats with half its volume submerged in a liquid A. The same body floats with three fourth of its volume submerged in a liquid B. Equal masses of these two liquids are mixed to form a homogeneous mixture and the body is floated in this mixture again. The fraction of the volume of the body that would be seen above the surface of the mixture is .

2. A line of cyclists separated by 50 m are travelling along a road at a constant speed of 4 m/s. Another cyclist is riding his bicycle at a constant speed of 6 m/s. The time taken by him to cross 10 cyclists is if he is travelling in the same direction as the others and if he is travelling in a direction opposite the others.

3. A particle starting from rest accelerates at 2 m/s2 for some time and then travels uniformly and passes a point 175 m from the starting point 20 s after start. The velocities of the particle at the moment of passing the 20th m and 30th m marks are and .

4. A body is dropped from a certain height. 2 seconds later another body is projected vertically upward from the same point at 20 m/s. The distance between the first and second bodies 5 seconds after the projection of the second body is . ( acceleration due to gravity = 10 ms2 )

5. A boy drops stones from the roof of a building on to the ground 9 m below. The first stone released by him strikes the ground at the moment he has just released the fifth stone. The second, third and fourth stones are at heights of , and above the ground at this moment. ( acceleration due to gravity = 10 ms2 )

6. A pendulum has a time period of 2 s at a place where acceleration due to gravity is 9.83 m/s2. To have the same time period of 2 seconds at a place where acceleration due to gravity is 9.78 m/s2, it's length must be .

7. A beaker with negligible water equivalent contains some water of mass m. Some mass of ice at 0 c is added to this water. The mass of ice added being equal to half the mass of water in the beaker and all the ice added just melts. If half of this mass of ice were to be added, the final temperature would have been .

8. Suppose gravitational force between two masses were to be given by

where k is some constant. Two equal masses attract each other with a certain force when the distance is d. If each of the masses is doubled, the distance between them must be increased times for the force to remain the same as earlier.

9. If end correction is neglected it is possible to obtain resonance for lengths of air column which are in the ratio 1:3:5:7: .. ( This ratio is an approximation ). A meter long tube with its lower end closed is held vertically. A vibrating tuning fork of frequency 480 Hz is held at the mouth of the tube and water is slowly poured in to the tube. Shortest and longest lengths of the water column for which resonance is obtained are and . ( Velocity of sound in air is 345.6 m/s)

10. A candle flame and a screen are 180 cm apart. A lens is placed between them. For a certain position of the lens a twice magnified clear image is obtained on the screen. The lens is now moved away from the candle through 30 cm. To obtain another clear image on the screen, the distance through which the screen has to be moved is and the new magnification of the image is .

11. A ray of light incident normally on one face of an equiangular prism, gets internally reflected from the second face and emerges through the base of the prism. The angle of deviation of the ray is . ( Draw a neat ray diagram. )

12. 110 Particles are chasing each other on a circular path of radius 7 m. The speed of any particle is 2 m/s. Number of particles crossing any radial line every second is . If each of the particles carries a charge of 4 microColoumb, the average current in the path is .

13. All the resistances shown are equal. With resistance between A and D removed and the resistance between different points is measured. If the resistance between points B and C is R, the resistance between points A and B would be

14. A potential difference applied to the ends of a wire causes a certain current to flow through it and a certain amount of heat to develop in it. The wire is stretched to obtain a new thin wire of half the original thickness. If the same potential difference is applied to the ends of the new wire, the heat developed _______________ times and the current flowing __________ times.

15. The current in a wire of resistance 10 Ohm varies as shown. The current remains constant for one second and in the next second it is half of the value in the previous second. If the initial value of the current is 8 A, the heat produced in the wire in 5 seconds is joule, and the charge that flows across any section of the wire in this time is .

16. A house uses 5 lamps of 40 W each for 5 hours a day and an electrical heater of 1000 W for 1 hour a day. The energy consumed in a month of 30 days in kWh is . The mass ( in milligram ) that is equivalent to this energy is .

17. Magnetic field produced by a circular coil of radius R and having N turns carrying

a current 'i'is given by . The magnetic field is perpendicular to the plane of the coil and is directed out of the plane if the current is anticlockwise. Two concentric coils made of wires o

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