Maths IJSO Stage-1

I J S O ( S TAG E - I ) DAILY PRACTICE PROBLEMS SESSION-2012-13 IIT-JEE | AIPMT | AIEEE | OLYMPIADS | KVPY | NTSE Subject : Mathematics 1.

Topic : Number System

DPP No. 01

If 0 < a < b < c < d where (a, b, c, d) are positive integers, which among the following has the least value ? (A)

ab cd

(B)

ad bc

(C)

cd ab

(D)

2.

Three boys P, Q, R agree to divide a bag of marbles as follows : P takes one more than half of the marbles; Q takes a third of the remaining marbles; R takes the marbles left out now in the bag. The original number of marbles found at the beginning in the bag must be (A) a multiple of 6 (B) one more than a multiple of 6 (C) Two more than a multiple of 6 (D) three more than a multiple of 6.

3.

If a – 1 = b + 2 = c – 3 = d + 4 then the largest among a, b, c, d is : (A) a (B) b (C) c

bd ac

(D) d

4.

4ab5 is a four digit number divisible by 55 where a,b are unknown digits. Then b –a is : (A) 1 (B) 4 (C) 5 (D) 0

5.

(a – 1)2 + (b – 2)2 + (c – 3)2 + (d – 4)2 = 0.Then a  b  c  d + 1 is : (A) 02 (B) 102 (C) 52

(D) 12 + 22 + 32 + 42 + 1

6.

Given that a,b,c and d are natural numbers and that a = bcd, b = cda, d = abc then (a + b + c + d )2 is : (A) 16 (B) 8 (C) 2 (D) 1

7.

The number of 3 digit numbers which end in 7 and are divisible by 11 is (A) 2 (B) 4 (C) 6

(D) 8

8.

The product of two numbers is 27. 33. 55. 73. Then the sum of these two numbers may be divisible by : (A) 16 (B) 9 (C) 25 (D) 49

9.

Let a,b,c,d be positive integers where a + b + c = 53, b + c + d = 51, c + d + a = 57, d + a + b = 58. Then the greatest and the smalllest numbers among a,b,c,d are respectively : (A) b and d (B) a and c (C) c and a (D) d and b

10.

It is given that 5 (A) 10

3 1  b = 19 (where the two fractions are mixed fractions); then a + b = a 2 (B) 12 (C) 9 (D) 15

11.

a and b are two primes of the form p and p +1 and M = aa + bb ; N = ab + ba then : (A) M and N are composite (B) M is a prime but N is composite (C) M and N are primes (D) M is composite, N is prime

12.

What is the value of x if log3 x  log9 x  log27 x  log81 x  (A) 9

(B) 27

(C) 81

25 ? 4 (D) None of these

1 1

1

13.

The pages of a book are numbered 1 through n. When the page numbers of the book were added, one of the page numbers was mistakenly added twice resulting in the incorrect sum 1998. What was the number of the page that was added twice ? (A) 45

14.

(B) 54

(B) 3

If a =

(B) 4 –3

 3  2

(A) 4 17.

(C) 1

(D) 0

Find the last digit in the finite decimal representation of the number 1/52003 ? (A) 2

16.

(D) 51

If x and y are natural numbers, find the number pairs (x, y) for which x2 – y2 = 31. (A) 4

15.

(C) 50

and B =

(C) 8 –3

 3 – 2

(B) 3

(D) 6

, find the value of (a + 1)–1 + (b + 1)–1 (C) 1

(D) 0

Find the smallest positive number from the numbers below 10 – 3

11 , 3 11 – 10, 18 – 5

13 ,

51 – 10 26 , 10 26 – 51. (A) 51 – 10 26 (B) 10 26 – 51 18.

(D) 10 + 3

11

(B) 11

(C) 4

(D) 5

How many different four digit numbers are there in the octal (Base 8) system, expressed in octal system ? (A) 3584

20.

11

Find the number of two digit numbers divisible by the product of the digits. (A) 7

19.

(C) 10 – 3

(B) 2058

(C) 6000

(D) 7000

A hundred and twenty digit number is formed by writing the first x natural number in front of each other as 12345678910111213..... Find the remainder when this number is divided by 8. : (A) 6

(B) 7

(C) 2

(D) 0

2 2

2

I J S O ( S TAG E - I ) DAILY PRACTICE PROBLEMS SESSION-2012-13 IIT-JEE | AIPMT | AIEEE | OLYMPIADS | KVPY | NTSE Subject : Mathematics 1.

Topic : Geometry - 1

DPP No. 02

In ABC , BE and CF are medians. BE= 9cm, CF = 12 cm. If BE is perpendicular to CF, find the area of ABC in sq. cms. (A) 72

2.

(B) 24

(C) 144

(D) Cannot be determined

In ABC, BC = 20 , medians BE = 18 and median CF = 24 (E, F are midpoints of AC, AB respectively). Find the area of ABC. (A) 244 sq. units

3.

(C) 144 sq. units

(D) None of these

In ABC , the median from A is perpendicular to the median from B. If BC = 7 and AC = 6 find AB. 15 units

(A) 4.

(B) 288 sq. units

(B) 17 unit

(C) 13 unit

(D) None of these

Let ABC be an acute angled triangle and CD be the altitude through C. If AB = 8 and CD = 6, find the distance between the mid-points of AD and BC. (A) 1 unit

5.

(B) 2 unit

(C) 5 unit

In a triangle ABC, medians AD and BE are drawn. If AD = 4, DAB =

(D) 10 unit

  and ABE = , then the area 6 3

of the ABC is : (A)

6.

8 3

(C)

32 3 3

(D)

64 . 3

(B) 7 : 2

(C) 3 : 7

(D) 7 : 3

Let ABC be an equilateral triangle and AD be the altitude through A. Then (A) AD2 = 3BD2

8.

16 3

In a triangle ABC, if a : b : c = 3 : 7 : 8, then R : r is equal to (A) 2 : 7

7.

(B)

(B) AD2 = 5BD2

(C) AB2 + AC2 = BC2

(D) AD2 = 2BD2

In triangle ABC with angle A = 90 . The bisectors of the angle B and C meet at P. The distance from P to the hypotenuse is 4 2 . Find the distance AP.. (A) 4

9.

(D) None of these

(B) 9

(C)

48 5

(D) None of these

In a ABC, AC > AB. The bisector of A meets BC at E, then : (A) CE > BE

11.

(C) 8

Let ABC be a triangle with AB = AC = 6, if the circumradius of the triangle is 5 then find BC (A) 10

10.

(B) 6

(B) CE = BE

(C) CE < BE

(D) None of these

ABC is an isosceles triangle with mA = 20° and AB = AC. D and E are points on AB and AC such that AD = AE. I is the midpoint of the segment DE. If BD = ID, then the angle of IBC are : (A) 110°, 35°, 35°

(B) 100°, 40°,40°

(C) 80°,50°,50°

(D) 90°,45°,45°

1 1

1

12.

A, B, C and D are points on a line. E is a point outside this line. Given that AE = BE = AB = BC and CE = CD, we find that the measure of DEA is :

(A) 90° 13.

14.

(B) 15°

In triangle ABC, BD bisects angle B. If mA =

(D) either I or II but not both

(B) 105°

(C) 20°

(D) 50°

2 mB and mB = 3mC, then mBDC is : 3

(C) 90°

(D) 120°

In the adjacent figure, DE bisects BDA and AD bisects BAC. The measure of BED is :

(A) 85° 17.

(D) 150°

In the adjoining figure find the size of ACE, given AD = DB and DE = DC.

(A) 75° 16.

(C) 120°

In an isosceles acute angled triangle one angle is 50°. I. The othere two angles are 65° and 65° II. The other two angles are 50° and 80° Then which one of the above ststements can be true ? (A) I only (B) II only (C) I and II both

(A) 45° 15.

(B) 105°

(B) 95°

(C) 75°

(D) 105°

ABC is an equilateral triangle (shown in figure). D is some point on BC. IF DE = 3 and DF = 7 find the length of altitude form A to BC.

(A) 5

(B) 7

(C) 10

(D) 12 2 2

2

18.

Given triangle PQR with RS bisecting R, PQ extended to D and n a right angle, then : R

m

P

19.

n q

p

d Q

S

(A) m =

1 (p – q) 2

(B) m =

(C) d =

1 (q + p) 2

(D) d =

D

1 (p + q) 2 1 m 2

In a right triangle ABC, AD = AE and CF = CE as show. If DEF = x degrees then the value of x equals : A

E X D B

(A) 30º

(B) 45º

F

C

(C) 60º

(D) 75º

20.

In the diagram if ABC and PQR are equilateral. The CXY equals :

21.

(A) 35º (B) 40º (C) 45º (D) 50º Find the number of isosceles traingles having integral sides and perimeter less than or equal to 100 is : (A) 24 (B) 20 (C) 27 (D) None of these

3 3

3

I J S O ( S TAG E - I ) DAILY PRACTICE PROBLEMS SESSION-2012-13 IIT-JEE | AIPMT | AIEEE | OLYMPIADS | KVPY | NTSE Subject : Mathematics 1.

Topic : Geometry - 2

DPP No. 03

PQRS is a trapezium, in which PQ is parallel to RS, and PQ = 3 RS. The diagonal of the trapezium intersect each other at X, then the ratio of PXQ and RXS is : (A) 6 : 1

2.

(B) 3 :1

(C) 9 :1

(D) 7 : 1

Two circles APQC and PBDQ intersect each other at the points P and Q and APB and CQD are two parallel straight lines. Then only one of the following statements is always true. Which one is it ?

3.

(A) ABDC is a cyclic quadrilateral

(B) AC is parallel to BD

(C) ABDC is a rectangle

(D) ACQ is a right angle

A triangle ABC with an obtuse angle B is inscribed in a circle. The altitude AD of the triangle is tangent to the circle. The altitude AD of the triangle is tangent to the circle. The side BC has length 4 cm. Find the area of the triangle ABC (in cm2). (A) 48

4.

(B) 60

(C) 72

(D) 36

The length of the common chord of two circles of radii 15 cm and 20 cm, whose centres are 25 cm apart is : (A) 24 cm

5.

(B) 25 cm

(C) 15 cm

(D) 20 cm

Chords AB and PQ meet at K and are perpendicular to one another. If AK = 4, KB = 6 and PK = 2, then the area of the circle is : (A) 25sq. units

6.

(B) 20sq. units

(C) 100sq. units

(D) 50sq. units

In quadrilateral ABCD , diagonals AC and BD meet at O. If AOB, DOC and BOC have areas 3, 10 and 2 respectively, find the area of AOD. (A) 10 sq. units

7.

(C) 15 sq. units

(D) 12 sq. units

If P is a point inside a rectangle ABCD such that PA = 3, PB = 4 and PC = = 5 find PD. (A) 3 2 units

8.

(B) 20 sq. units

(B) 2 3 units

(C)

5 units

(D) None of these

In a trapezium ABCD , AB II CD ; the measure of angle D is twice that of B. If AD = a and CD = b, find the length of AB. (A) a + b

9.

(B) 2ab

(C) a – b

(D) None of these

ABCD is a quadrilateral and P,Q are the midpoints of AB , CD respectively. If AQ, DP intersect at X and BQ , CP at Y, then ar AXD + arBYC =

10.

(A) Area of quad.PXQY

(B) Area of quad.PDCB

(C) Area of quad.PQCB

(D) None of these

In a right triangle ABC the incircle touches the hypotenuse AC at D , If AD = 10 and DC = 3 then the inradius is : (A) 1

11.

(B) 2

(C) 4

(D) 6

Let P be any point in the interior of the rectangle ABCD which of the following sets of numbers can form the areas of the four triangles PAB, PBC, PCD, PDA. (A) 10, 9, 12, 5

(B) 21, 15, 6, 12

(C) 10, 9, 8, 6

(D) 12, 8, 7, 5

1 1

1

12.

In a triangle ABC , AB = AC = 20 cm, D and E are points on AB and AC rspectively such that AD = AE = 12 cm , If F is the intersection point of BE and CD and area of ADFE is 24 cm2 . Find the area of the triangle BFC in sq. cm (A) 24

13.

(B) 20

(C)

40 3

(D) None of these

RST is an angle in the minor segment of a circles of centre O, then the angle (in degrees) RST less the angle ORT is : (A) 90

14.

(C) 60

(D) None of these

Triangle ABC is divided into four regions with areas as shown in the diagram. Find x.

(A) 15.

(B) 45

1900 67

(B)

54 11

(C)

1998 67

(D)

544 11

A circle is inscribed in a rhombus, one of whose angles is 60°. Find the ratio of area of the rhombus to the area of the inscribed circle. (A) 7 :  3

16.

(B) 8 :  3

(C) 4 :  3

(D) None of these

In the figure, CD is the diameter of a semicircle CBED with centre O, and AB = OD. If EOD = 60º, then BAC is : E B A

(A) 15º 17.

(B) 20º

C

D

O

(C) 30º

(D) 45º

Semi-circle C1 is drawn with a line segment PQ as its diameter with centre at R. Semicircles C2 and C3 are drawn with PR and QR as its diameter respectively, both C2 and C3 lying inside C1. A full circle C4 is drawn in such a way that it is tangent to all the three semicircles C1, C2 and C3. C4 lies inside C1 and outside both C2 and C3. The radius C4 is :

1 1 1 1 PQ PQ (B) PQ (C) (D) PQ 2 3 6 4 ABCD is a parallelogram and P is any point within it. If the area of the parallelogram ABCD is 20 sq. units, (A)

18.

then what is the sum of the areas of the DPAB and DPCD ? (A) 5 sq. units 19.

(B) 10 sq. units

(C) 12 sq. units

(D) Cannot be determined

A circle passes through the vertex A of an equilateral triangle ABC and is tangent to BC at its midpoint. Find the ratio in which the circle divides each of the sides AB and AC. (A) 1 : 1

20.

(B) 3 : 2

(C) 3 : 1

(D) 2 : 1

In any quadrilateral ABCD, the diagonal AC and BD intersect at a point X. If E, F, G and H are the midpoints of AX, BX, CX and DX respectively, then what is the ratio of (EF + FG + GH + GE) : (AD + DC + CB + BA)? (A)

1 2

(B)

3 2

(C)

3 4

(D) Data insufficient 2 2

2

IJ S O (S TAG E -I ) DAILY PRACTICE PROBLEMS SESSION-2012-13 IIT-JEE | AIPMT | AIEEE | OLYMPIADS | KVPY | NTSE Subject : Mathematics 1.

Topic : Mensuration

DPP No. 04

Find the area of the largest square which can be inscribed in a right angled triangle with legs 4 and 8. (A)

8 3

(B)

7 3

(C) 4

(D) None of these

2.

In a right angled triangle, if the square of the hypotenuse is twice the product of the other two sides, then which of the following is true . (A) triangle is isoscels (B) Triangle is equilateral(C) Triangle is scalene (D) None of these

3.

Find the ratio of the area of the equilateral triangle inscribed in a circle to that of a regular hexagon inscribed in the same circle. (A) 1 : 2 (B) 1 : 4 (C) 2 : 3 (D) None of these

4.

If the altitudes of a triangle are in the ratio 2 : 3 : 4, then the lengths of the corresponding sides are in the ratio : (A) 2 : 3 : 4 (B) 6 : 4 : 3 (C) 3 : 2 : 4 (D) 3 : 2 : 1

5.

In case of a right circular cylinder the radius of base and height are in the ratio 2 : 3. Therefore, the ratio of lateral surface area to the total surface area is : (A) 5 : 3 (B) 3 : 5 (C) 2 : 5 (D) 2 : 3

6.

ABC is an isosceles right triangle with area P. The radius of the circle that passes through the point A, B and C is : (A)

7.

P

(B)

P 2

(C)

P 2

(D)

2P

The diagonals of a rhombus are 12 and 24. The radius of the circle inscribed in the rhombus, is :

12 (A)

6 (B)

5

5

(C) 6 5 (D) not possible as a circle in a rhombus can not be inscribed 8.

A round pencil has length 8 units when unstreched and diameter 1/4. It is sharpened perfectly so that it remains 8 units long with 7 units section still cylindrical and remaining 1 unit giving a conical tip. Volume of the pencil now is : (A)

9.

11 96

(B)

37 192

(C)

7 64

(D) None

Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is m times the area of the square. The ratio of the area of the other small right triangle to the area of the square is : (A)

1 4m

(B)

1 2m  1

(C) m

(D)

1 8m 2

1 1

1

10.

In this figure, AOB is a quarter circle of radius 10 and PQRO is a rectangle of perimeter 26. The perimeter of the shaded region is : B

Q

R

O

(A) 13 + 5

P

(B) 17 + 5

(C) 7 + 10

A

(D) 7 + 5

11.

Three parallel lines 1, 2 and 3 are drawn through the vertices A, B and C of a square ABCD. If the distance between 1 and 2 is 7 and between 2 and 3 is 12, then the area of the square ABCD is : (A) 193 (B) 169 (C) 196 (D) 225

12.

Each of the congruent circles shown is externally tangent to other circles and/or to the side(s) of the rectangle as shown. If each circle has circumference 16, then the length of a diagonal of the rectangle, is :

(A) 80 13.

(B) 40

(C) 20

(D) 15

ABCD is a rectangle and lines DX, DY and XY are drawn as shown. Area of AXD is 5, Area of BXY is 4 and area of CYD is 3. If the area of DXY can be expressed as X

A

x where x  N then x is equal to

B Y C

D

(A) 72 14.

(C) 84

(D) 96

A sphere is inscribed in a cone of radius . 3 3 and slant height 6 3 . The radius of the sphere, is :

3 3 2 ABCD is a rectangle with AB = 12 cm and BC = 7 cm. Point E is on AD with DE = 2 cm. Point P is on AB. How far to the right of point. A should point P be placed so that the shaded area comprises exactly 40% of the area of the rectangle ? (A) 0

15.

(B) 75

(B) 3 3

(C) 6 3

D

(D)

C

E

A (A) 8

(B) 8.4

B

P (C) 8.2

(D) 8.6

2 2

2

16.

A lead ball of radius 24 cm is melted down and recast into smaller balls of radius 6 cm. Assuming that no metal is lost in this process, number of complete smaller balls that can be made, is : (A) 4 (B) 16 (C) 36 (D) 64

17.

The sides of a triangle are in the ratio 4 : 6 : 11. Which of the following words best described the triangle? (A) obtuse (B) isosceles (C) acute (D) impossible

18.

The volume of a cube (in cubic cm) plus three times the total length of its edges (in cms) is equal to twice its surface area (in sq. cm). The length of its diagonal is : (A) 6

19.

(C) 3 6

(D) 6 6

A square with side length 1 is inscribed in a semicircle such that one side of the square is on the diameter of the semicircle. The perimeter of the semicircle is : (A)  5

20.

(B) 6 3

(B)

 5 2

(C)

  5   1 2  

(D)

5   1 2

A semicircle is drawn with AB as its diameter. From C, a point on AB, a line perpendicular to AB is drawn meeting the circumference of the semicircle at D. Given that AC = 2 cm and CD = 6 cm, the area of the semicircle is : (A) 32 (B) 50 (C) 40 (D) None of these

3 3

3

I J S O ( S TAG E - I ) DAILY PRACTICE PROBLEMS SESSION-2012-13 IIT-JEE | AIPMT | AIEEE | OLYMPIADS | KVPY | NTSE Subject : Mathematics

1.

2.

Topic : Algebra -1

 1 7  1  1 a, b, c are real numbers such that a +   =   ; b +   = 4 ; c +   =1, find abc. b 3 c       a (A) 0 (B) 1 (C) 2 (D) 3

If a + b + c = 0, find

b2  c 2  a2 b 2 – ca (B) 1

(A) 0 3.

4.

x 1 x –1

If

x 1– x –1

If

5 3

37 =2+ 13

(A) (1, 5, 2)

7.

(D) 3

(B)

1 x

4 3

(C) 2

(D) None of these

(D) 3

where x, y, z are positive integers find x, y, z.

1 1 z

(B) (1, 5, 4)

(C) (1, 3, 2)

For x2 + 2x + 5 to be a factor of x4 + px2 + q, find the values of p and q. (A) p = 6 , q = 24 (B) p = 6 , q = 25 (C) p = 24 , q = 6

(D) (1, 3, 4)

(D) None of these

1 1 1 If 2x = 4y = 8z and xyz = 288, then 2x  4 y  8z  (A)

8.

(C) 2

= 3 find x.

y

6.

.

If x + y = 5xy, y + z = 6yz, z + x = 7zx find the value of x + y + z. (A) 0 (B) 1 (C) 2

(A) 5.

DPP No. 05

7 96

(B)

11 96

(C)

13 96

If a + b + c = 3, a2 + b2 + c2 = 13 and a3 + b3 + c3 = 27, then

(A) 3

(B) –3

(C)

(D)

15 96

(D)

1 3

1 1 1    a b c

1 3

9.

Given that ax + by = 4, ax2 + by2 = 2 and ax3 + by3 = – 3. Find the value of (2x – 1) (2y – 1). (A) 4 (B) 3 (C) 5 (D) Cannot be determined

10.

The graph of the equation y = 2x2 + 4x + 3 has its lowest point at : (A) (– 1, 9) (B) (1, 9) (C) (– 1, 1)

(D) (0, 3)

11.

If the common points of the graphs of y = x2 – 6x + 2 and x + 2y = 4 are A and B find the coefficients of the midpoint of AB (A) A =

11 5 ,B= 4 8

(B) A = –

1 5 ,B=– 3 8

(C) A =

11 5 ,B= 4 8

(D) A =

1 5 ,B= 3 8

12.

A quiz has 20 questions with seven points awarded for each correct answer; two points deducted for each wrong answer and zero mark given for each question omitted. Ram scores 87 points. How many questions did he omit ? (A) 2 (B) 5 (C) 7 (D) 9

13.

Two candles of same length are lighted at 12 noon. The first is consumed in 6 hours and trhe second in 4 hours. Assumung that each candle burns at a constant rate, in how many hours after being lighted, was the first candle twice the length of the second ? (A) 3 p.m. (B) 2 p.m. (C) 1.30 p.m. (D) 2.30 p.m.

14.

If

a b c   , then each fraction is equal to : bc c a ab

(A)

15.

1 2

(B) 1

If a2 = by + cz, b2 = cz + ax, c2 = ax + by, then

17.

(D) – 2

x b z   =? ax by cz

abc xyz If 2x – 3y + 4z = 0 & x + 2y – 5z = 0, then x : y : z is : (A) 2 : 3 :4 (B) 1 : 1 : 1 (C) 1 : 2 : 1 (A) abc

16.

(C) 2

(B) 1

(C)

Given a number of the form a + b number p + q

(A)

a 2  2b 2 2

2 2

(a  4b )

a 2  2b 2 2

xyz abc

(D) 3 : 4: 2

2 , where a, b are rational numbers with a  or b  0, there exists a

2 , with p, q rational number, such that (a + b

(B)

(D)

a  2b

2

(C)

2 2 2 )(p + q 2 ) = 1. The value of (p – 2q ), is

1 2

a  2b

2

(D)

1 2

a  4b 2

18.

The number of triplets (a, b, c) of the positive integer, which satisfy simultaneous equation ab + bc = 44 & ac + bc = 23 (A) 1 (B) 2 (C) 4 (D) Infinite solutions

19.

If y varies jointly as x and the square of z and inversely as the cube of w, the factor by which y increases when z is doubled and w is divided by 3, if x remains the same, is : (A)

20.

n

4 27

(B)

27 4

(C)

1 108

(D) 108

n

x 3  y 3 is divisible by x + y if :

(A) (C)

n is any integer n  n is an even positive integer

(B) (D)

n is an odd positive integer n is a rational number

I J S O ( S TAG E - I ) DAILY PRACTICE PROBLEMS SESSION-2012-13 IIT-JEE | AIPMT | AIEEE | OLYMPIADS | KVPY | NTSE Subject : Mathematics 1.

Topic : Algebra - 2

DPP No. 06

 and  are two numbers such that  +  = 6,  –  = 8. Then trace the equation whose roots are  and . (A) x2 – 6x – 7

2.

(B) x2 – 7x – 6

(C) x2 + 6x + 7

(D) None of these

If  and  are the roots of the equation x2 – 9x + 5 = 0. then find the equation whose whose roots are      and  . 2   2   

3.

(A) 10x2 – 122 x – 61 = 0

(B) x2 – 122x – 61 = 0

(C) 20x2 – 122x – 61 = 0

(D) None of these

For the natural number n, Sn = 1 – 2 + 3 – 4 + ..... + (–1)n-1n; the value of S17 + S33 + S50 is : (A) 0

4.

(B) 1

(C) –1

(D) 2

If ,  are the roots of x 2 + x + 1 = 0 and ,  are the root of x 2 + 3x + 1 = 0, then

                = (A) 2 5.

(B) 4

(D) 8

If X + Y + Z = 30, (X,Y, Z > 0), then the value of (X – 2) (Y – 3) (Z – 4) will be : (A)  1000

6.

(C) 6

(B)  800

(C)  500

(D)  343

For what values of 'a' , the equations 1998x2 + ax + 8991 = 0 , and 8991x2 + ax +1998 = 0 have a common root (A) ± 10989

7.

r p

(B) –

r p

(C)

1 p

b a  is : a b

(D)

1 r

(B) a + 1 = 0

(C) a = 0

(D) a + b + 1 = 0

The ratio of the sum of n terms of two AP's is (3n – 13) : (5n + 21). Find the ratio of their 24th terms. (A) 1 : 2

10.

(D) None of these

If x2 + ax + b = 0 and x2 + bx + a = 0, a  b , have a common root 'a' then which of the following is true ? (A) a + b = 1

9.

(C) 0

If the roots of the equation px2 + rx + r = 0 are in the ratio a : b, then value of

(A)

8.

(B) ± 1

(B) 1 : 3

(C) 2 : 3

(D) 2 : 5

A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying odd places, then the common ratio will be (A) 2

(B) 3

(C) 4

(D) 5

1 11.

If a, b, c are in A.P., then (A) A.P.

12.

1

a b (B) G.P.

,

a c

1 ,

b c (C) H.P.

are in : (D) None of these

Suppose a, b, c are in A.P. and a2, b2, c 2 are in G.P. if a < b < c and a + b + c =

3 , then the value 2

of a is :

1 (A)

13.

1

2 2

(B)

2 3

(C)

1 1 – 3 2

(D)

1 1 – 2 2

In the quadratic equation ax 2 + bx + c = 0,  = b2 – 4ac and  + , 2 +  2, 3 +  3, are in G.P. where  are the root of ax 2 + bx + c = 0, then (A)   0

(B) b  0

(C) c  0

(D)   0

14.

The number of terms in an AP is even. The sums of the odd and even numbered terms are 24 and 30 respectively, if the last term exceeds the first term by 10.5, then the number of terms in the AP is : (A) 6 (B) 8 (C) 10 (D) 12

15.

Let a, b, c be the sides of a triangle (No two of them are equal) and  R. If the roots of the equation x2 + 2(a + b + c)x + 3 (ab + bc + ca) = 0 are real, then : (A)  <

16.

4 3

(B)  >

5 3

1 5 (C)   ,  3 3

4 5 (D)   ,  3 3

If roots of the equation x2 – 10ax – 11b = 0 are c and d and those of x2 – 10cx – 11d = 0 are a and b, then find the value of a + b + c + d. (where a, b, c, d are all distinct numbers) (A) 1000

17.

(B) 1200

(C) 1210

(D) 1250

A circle with area A1 is contained in the interior of a large circle with area A1 + A2. If the radius of the larger circle is 3 and A1, A2, A1 + A2 are in AP, then the radius of the smaller circle is : (A)

2

3 2

(B) 1

(C)

(D)

3

3

18.

If b1, b2, b3.......belongs to A.P. such that b1 + b4 + b7 + ..... + b28 = 220, then the value of b1 + b2 + b3 .......+.........+ b28 equals : (A) 616 (B) 308 (C) 2200 (D) 1232

19.

If 2 – (A)

20.

3 is a root of the quadratic equation x2 + 2 ( 3  1) x + 3 – 2 3 = 0, then the second root is : 3 –2

(B)

3

(C) 2 +

3

(D) –

3

If a is a root, repeated twice, of the quadratic equation (a – d) x2 + ax + (a + d) = 0 then

d2 a2

equal to : (A) sin290º

(B) cos260º

(C) sin245º

(D) cos230º

has the value

MATHEMATICS _IJSO STAGE-I : DPP-1

Number system :

Que s. Ans. Que s. Ans.

1 A 11 C

2 C 12 D

3 C 13 A

4 A 14 C

5 C 15 C

6 A 16 C

7 D 17 A

8 A 18 D

9 C 19

10 A 20

7 A 17 C

8 C 18 C

9 C 19 B

10 C 20 B

7 A 17 B

8 A 18 B

9 A 19 C

10 B 20 A

7 A 17 D

8 A 18 B

9 A 19 C

10 B 20 D

7 B 17 C

8 C 18 B

9 A 19 D

10 C 20 A

7 B 17 D

8 D 18 A

9 A 19 D

10 C 20 D

MATHEMATICS _IJSO STAGE-I : DPP-2

Geometry : I

Que s. Ans. Que s. Ans.

1 A 11 B

2 B 12 B

3 B 13 C

4 C 14 C

5 C 15 B

6 B 16 A

MATHEMATICS _IJSO STAGE-I : DPP-3

Geometry : II

Que s. Ans. Que s. Ans.

1 C 11 B

2 B 12 C

3 A 13 A

4 A 14 C

5 D 15 B

6 C 16 B

MATHEMATICS _IJSO STAGE-I : DPP-4

Mensuration :

Que s. Ans. Que s. Ans.

1 A 11 A

2 A 12 A

3 A 13 C

4 B 14 B

5 C 15

6 A 16 D

MATHEMATICS _IJSO STAGE-I : DPP-5

Algebra : I

Que s. Ans. Que s. Ans.

1 B 11 C

2 C 12 B

3 A 13 A

4 A 14 A

5 A 15 B

6 B 16 C

MATHEMATICS _IJSO STAGE-I : DPP-6

Algebra : II

Que s. Ans. Que s. Ans.

1 A 11 A

2 C 12 D

3 B 13 C

4 D 14 C

5 15 A

6 A 16 C