Maths Ntse

8/16/2019 Maths Ntse

1/13

CAREER POINT : 128, Shakti Nagar, Kota-324009 (Raj.), Ph: 0744-2503892 Page # 1

1. If x = 9 + 4 5 and xy = 1, then 2x

1+ 2y

1 is -

(A) 81 (B) 322 (C) 97 (D) 2

2. The fraction

323

622 is -

(A)3

22(B) 1 (C)

3

32(D)

3

4

3. If N =15

2 –525

– 22 –3 , then N equals

(A) 1 (B) 1 –22 (C) 2

5

(D) 15

2

4. The value of

43

1

32

1

21

1---------- +

10099

1

is -

(A) Less than100

99(B) Equal to

100

99

(C) Greater than99

100(D) Equal to

99

100

5. The numerator of22

22

b –a –a

b –aa +

22

22

b –aa

b –a –a

is -

(A) a2 (B) b2 (C) a2 – b2 (D)2

22

b

b2 –a4

6. The ascending order of the surds 963 4,3,2 is -

(A) 369 2,3,4 (B) 639 3,2,4 (C) 963 4,3,2 (D) 396 2,4,3

7. If x =2

13 , then the value of 4x3 + 2x2 – 8x + 7 is -

(A) 10 (B) 8 (C) 6 (D) 4

8. On simplifying 2)999814(

1999815999813 , we get

(A) 1 (B) 2 (C) 3 (D) 4

SUMMER WORKSHOP (NTSE)

aily Practice Problem Sheet

Subject : Mathematics Topic : Number System DPPS. NO. – 01


2/13


9. The value ofqr

1

r

q

x

x

×

rp

1

p

r

x

x

×

pq

1

q

p

x

x

is equal to -

(A) r

1

q

1

p

1

X

(B) 0 (C) x pq+qr+rp

(D) 110. If ax = cq = b and cy = az = d, then

(A) xy = qz (B)y

x=

z

q(C) x + y = q + z (D) x – y = q – z

11. Number of two digit numbers having the property that they are perfectly divided by the sum of their digits with

quotient equal to 7, is

(A) 2 (B) 3 (C) 4 (D) 9

12. If a, a + 2 and a + 4 are prime numbers, then the number of possible solutions for a is :

(A) one (B) two (C) three (D) more than three

13. If least prime factor of a is 3 and the least prime factor of b is 7, the least prime factor of (a + b) is :(A) 2 (B) 3 (C) 5 (D) 11

14. The product of the HCF and LCM of the smallest prime number and the smallest composite number is :

(A) 2 (B) 4 (C) 6 (D) 8

15. When 1! + 2! + 3! + ... + 125! is divided by 7, what will be the remainder ?

(A) 3 (B) 5 (C) 1 (D) None of these

16. If HCF of 65 and 117 is expressed in the form of 65m + 117n, then the value of m and n respectively is

(A) 3, 2 (B) 3, – 1 (C) 2, – 1 (D) 2, – 3

17. Find out (A + B + C + D) such that AB x CB = DDD, where AB and CB are two-digit numbers and DDD is a three-digit

number.

(A) 21 (B) 19 (C) 17 (D) 18

18. If 27 = 123 and 31 = 133, than 15 = ?

(A) 13 (B) 31 (C) 11 (D) 33

19. How many zeros at the end of first 100 multiples of 10.

(A) 10 (B) 24 (C) 100 (D) 124

20. The last digit of (13 + 23 + 33 + ... 103)64 is :

(A) 2 (B) 5 (C) 9 (D) 0


3/13


1. L.C.M. of x3 + x2+ x + 1 and x3 – x2+ x – 1 is :

(A) x4+ 1 (B) x4 – 1 (C) x2+ 1 (D) x2 – 1

2. If (a-5)2 + (b-c)2 + (c-d)2 + (b+c+d-9)2 = 0, then the value of (a + b + c) (b + c + d) is :

(A) 0 (B) 11 (C) 33 (D) 99

3. If x +x

1 = 3, then the value of x6 + 6x

1 is :

(A) 927 (B) 114 (C) 364 (D) 322

4. If the zero of the polynomial f(x) = k 2x2 – 17x + k + 2(k > 0) are reciprocal of each other, then the value of k is

(A) 2 (B) – 1 (C) – 2 (D) 1

5. A farmer divides his herd of x cows among his 4 son's such that first son gets one-half of the herd, the second son gets

one fourth, the third son gets one-fifth and the fourth son gets 7 cows, then the value of x is :

(A) 100 (B) 140 (C) 160 (D) 180

6. If in 3 + 3 5 , x = 3 and y = 3 5 , then its rationalising factor is(A) x + y (B) x – y

(C) x5 + x4y + x3y2 + x2y3 + xy4 + y5 (D) x5 – x4y + x3y2 – x2y3 + xy4 – y5

7. If a, b, c are positive, thencb

ca

is :

(A) always smaller thanb

a(B) always greater than

b

a

(C) greater thanb

aonly if a > b (D) greater than

b

aonly if a < b

8. If the quotient of x4 – 11x3 + 44x2 – 76x + 48, when divided by (x2 – 7x + 12) is Ax2 + Bx + C, then the descending order of A,B,C is :(A) A,B,C (B) B,C,A (C) A,C,B (D) C,A,B

9. If a + b – c = 0, then the value o 2)cba( is :

(A) 2ab (B) 2bc (C) 4ab (D) 4ac

10. If13

37= 2 +

z

1y

1x

1

where x,y,z are natural numbers then values of x,y,z are

(A) 1,2,5 (B) 1,5,2 (C) 5,2,1 (D) 2,5,1

11. Ifa

1+

b

1+c

1=

cba

1

where (a+b+c) 0 and abc 0. What is the value of (a+b) (b+c) (c+a) ?

(A) 0 (B) 1 (C) –1 (D) 2



Subject : Mathematics Topic : Polynomial DPPS. NO. – 02


4/13


12. If, x +y

1= 1 and y –

z

1= 1, then the value of xyz is -

(A) 1 (B) –1 (C) 0 (D) –2

13. If x + y + z = 1, x² + y² + z² = 2 and x³ + y³ + z³ = 3 then the value of xyz is:

(A) 1/5 (B) 1/6 (C) 1/7 (D) 1/8

14. The polynomials ax3 + 3x2 – 3 and 2x3 – 5x + a when divided by (x – 4) leaves remainders R 1 & R

2 respectively then value

of ‘a’ if 2R 1 – R

2 = 0.

(A)127

18 – (B)

127

18(C)

127

17(D)

127

17 –

15. , , are zeros of cubic polynomial x3 – 12x2 + 44x + c. If , , are in A.P., find the value of c.

(A) –48 (B) 24 (C) 48 (D) – 24

16. If x2 – 4 is a factor of 2x3 + ax2 + bx + 12, where a and b are constant. Then the values of a and b are :

(A) – 3, 8 (B) 3, 8 (C) –3, – 8 (D) 3, – 8

17. If xy + yz + zx = 1, then the expression xy1

yx

+

yz1

zy

+

zx1

xz

is equal to

(A) zyx

1

(B) xyz

1(C) x + y + z (D) xyz

18. The value of (a + b)3 + (a – b)3 + 6a (a2 – b2) =

(A) 6a3 (B) 8a3 (C) 10a3 (D) 12a3

19. If 31

3

1

3

1

zyx = 0, then

(A) x3 + y3 + z3 = 0 (B) x + y + z = 27xyz

(C) (x + y + z)3 = 27 xyz (D) x3 + y3 + z3 = 27xyz

20. If, are the zero’s of polynomial

f(x) = x2 – p(x + 1) – c then ( + 1)( + 1) is equal

(A) c – 1 (B) 1 – c (C) c (D) 1 + c


5/13


1. If one root is 53 , then quadratic equation will be :

(A) x2 – 6x + 4 = 0 (B) x2 – 5x + 5 = 0 (C) x2 – 3x + 4 = 0 (D) None of these

2. If are the roots of x2 – 2x + 2 = 0 then 6 + 6 is:

(A) 20 (B) 22 (C) 0 (D) None of these

3. If the roots of the equation px2 + qx + r = 0 are in the ratio : m then :

(A) ( + m)2 pq = mr 2 (B) ( + m)2 pr = mq

(C) ( + m)2 pr = mq2 (D) none

4. If the equation x2 + x + = 0 has equal roots and one root of the equation x2 + x – 12 = 0 is 2, then ( , ) =(A) (4, 4) (B) (–4, 4) (C) (4, –4) (D) (–4, –4)

5. The value of ‘a’ for which one root of the quadratic equation (a2 – 5a + 3)x2 + (3a – 1)x + 2 = 0 is twice as large as the

other, is :

(A)

3

2(B) –

3

2(C)

3

1(D) –

3

1.

6. If the roots of the equation x2 – 8x + a2 – 6a = 0 are real and distinct, then find all possible values of a .

(A) –2 < a < 8 (B) 2 < a < 8 (C) –2 a 8 (D) 2 a 8

7. If x2 + ax + 10 = 0 and x2 + bx – 10 = 0 have a common root, then a2 – b2 is equal to

(A) 10 (B) 20 (C) 30 (D) 40

8. Out of a group of swans,2

7times the square root of the total number are playing on the shore of a tank. The two

remaining ones are playing, in deep water. What is the total number of swans?

(A) 16 (B) 15 (C) 17 (D) 18

9. Two friends tr ied solving a quadratic equat ion x2 + bx + c = 0. One started with the wrong value of b and got the

roots as 4 and 14; the other started with the wrong value of c and got the roots as 17 and – 2. Find the actual

roots.(A) 7, 8 (B) 28, 2 (C) 19, 2 (D) 13, 2

10. Number of integral values of ‘p’ for which the quadratic equation x2 – px + 1 = 0 has no real roots is :

(A) 2 (B) 3 (C) 5 (D) infinite

11. The roots of the quadratic equation (a + b 2c) x2 (2a b c) x + (a 2b + c) = 0 are(A) a + b + c & a b + c (B) 1/2 & a 2b + c

(C) a 2b + c & 1/(a + b 2c) (D) 1,c2ba

cb2a

12. If sec , cosec are the roots of quadratic equation, a x 2 + b x + c = 0, then:

(A) c2 + 2 a c = b2 (B) b2 a2 = 2 b c (C) a2 + 2 a b = c2 (D) b2 + c2 = 2 a c



Subject : Mathematics Topic : Quadratic Equation DPPS. NO. – 03


6/13


13. A fast train takes 3 hours less than a slow train for a journey of 900 km. If the speed of slow train is 15 km/ hr. less than

that the fast train. Find the sum of speeds of the two trains.

(A) 120 km/hr (B) 125 km/hr (C) 135 km/hr (D) 150 km/hr

14. If and are the root of ax2 + bx + c = 0, then the value of

ba

1

ba

1 is :

(A)bc

a(B)

ca

b(C)

ab

c (D) None of these

15. For what value of m does the equation x2 – x + m2 = 0 possess no real roots :

(A)

,2

1U

2

1, (B)

,2

1U1,

(C) 1,U2, (D) )1,(–U,1


7/13


1. The sum of the third and seventh terms of an A.P. is 6 and their product is 8, then common difference is :

(A) 1 (B) 2 (C) 2

1(D) 4

1

2. If a1, a

2..........., a

19 are the first 19 term of an AP and a

1 + a

8 + a

12 + a

19 = 224.

then

19

1i

ia is equal to :

(A) 896 (B) 1064 (C) 1120 (D) 1164

3. The sum of all two digit numbers each of which leaves remainder 3 when divided by 5 is

(A) 952 (B) 999 (C) 1064 (D) 1120

4. Let Sn

denote the sum of the first 'n' terms of an A.P. and S2n

= 3Sn

. Then, the ratio S3n

: Sn

is equal to :

(A) 4 : 1 (B) 6 : 1 (C) 8 : 1 (D) 10 : 1

5. The first term of an A.P. of consecutive integers is p2 + 1. The sum of (2 p + 1) terms of this series can be

expressed as :

(A) (2p + 1)(p2+p+1) (B) (2 p + 1) (p + 1)2 (C) (p + 1)3 (D) p3 +(p + 1)3

6. If 7 times the 7th term of an AP is equal to 11 times the 11th term, then 18th term in that AP is

(A) 143 (B) 0 (C) 1 (D) Cannot be determined

7. If the ratio of the sum of n terms of two A.P’s is (3n + 4) : (5n + 6), then the ratio of their 5 th term is

(A)31

21(B)

41

31(C)

51

31(D)

31

11

8 If in an A.P. , Sn = n2 p and S

m = m2 p, where S

r denotes the sum or r terms of the A.P., then S

p is equal to

(A)2

1 p3 (B) mnp (C) p3 (D) (m + n)p2

9. If sum of n terms of a sequence is given by Sn = 2n2 + 3n, find its 50 th term.

(A) 250 (B) 225 (C) 201 (D) 205

10. Sum of n terms of the series

321882 + .......... is :

(A)

2

)1n(n (B) 2n (n + 1) (C)

2

)1n(n (D) 1



Subject : Mathematics Topic : Progression DPPS. NO. – 04


8/13


11. Divide 600 biscuits among 5 boys so that their shares are in Arithmetic progression and the two smallest shares together

make one-seventh of what the other three boys get. What is the sum of the shares of the two boys who are getting lesser

number of biscuits, than the remaining three ?

(A) 75 (B) 85 (C) 185 (D) 90

12. Find the value of15

2 +

35

2+

63

2 +

99

2 + ....... +

9999

2

(A)33

8(B)

11

2(C)

303

98(D)

909

222

13. An A.P. consists of 21 terms. The sum of three terms in the middle is 129 and of the last three terms is 237. Then the A.P.

is :

(A) 3, 7, 11, 15 ......... (B) 2, 7, 12, ........ (C) 5, 9, 13, 15 ........ (D) none of these

14. The sum of first 20 terms of log2 + log4 + log8 + ......

(A) 20 log 2 (B) log 20 (C) 210 log 2 (D) log 2

15. If n is odd, then the sum of n terms of the series 1 – 2 + 3 – 4 + 5 – 6 + ....... .. will be

(A)2

n (B)

2

1n (C)

2

1n (D)

2

1n2

16. The sum of n terms of the series a, 3a, 5a,...is

(A) na (B) 2na (C) n2a (D) None


9/13


1. In aXYZ, LM || YZ and bisectors YN and ZN ofY &Z respectively meet at N on LM. Then YL + ZM =

(A) YZ (B) XY (C) XZ (D) LM

2. If D is any point on the side BC of aABC, then :

(A) AB + BC + CA > 2AD (B) AB + BC + CA < 2AD

(C) AB + BC + CA > 3 AD (D) None

3. In the given figure PQ II RS, QPR = 70º, ROT = 20º. Then, find the value of x.

Tx

20º

QP

O

R S

70º

(A) 20º (B) 70º (C) 110º (D) 50º

4. Two triangles ABC and PQR are similar, if BC : CA : AB = 1 : 2 : 3, thenPR

QR is :

(A)3

2(B)

2

1(C)

2

1(D)

3

2

5. ABC is a right-angle triangle, right angled at A. A circle is inscribed in it. The lengths of the two sides containing the

right angle are 6 cm and 8 cm, then radius of the circle is :

(A) 3 cm (B) 2 cm (C) 4 cm (D) 8 cm

6. In an isosceles ABC, if AC = BC and AB2 = 2 AC2, then C is equal to :(A) 45º (B) 60º (C) 30º (D) 90º

7. ABCD is a trapezium in which AB || CD and AB = 2CD. What is the ratio of the areas of triangles AOB and COD ?

D

A

O

C

B

(A) 2 : 1 (B) 4 : 1 (C) 3 : 1 (D) 3 : 2



Subject : Mathematics Topic : Lines and Angles & Triangles DPPS. NO. – 05


10/13


8. The sides of a right triangle are a and b and the hypotenuse is c. A perpendicular from the vertex divides c into

segments r and s, adjacent respectively to a and b. If a : b = 1 : 3, then the ratio of r to s is :

(A) 1 : 3 (B) 1 : 9 (C) 1 : 10 D) 3 : 10

9. In the accompanying figure CE and DE are equal chords of a circle with centre O. Arc AB is a quarter-circle with centre

O. Then the ratio of the area of triangle CED to the area of triangle AOB is :E

OC D

A B

(A) 2 : 1 (B) 3 : 1 (C) 2 : 1 (D) 3 : 1

10. In the given figure, AB and AC are produced to P and Q respectively. The bisectors of PBC and QCB intersect at

O. BOC is equal to

(A) 10º

(B) 30º

(C) 50º

P Q

A

B C

80º (

O(D) 60º

11. In the figure below BA || DC and EC = ED, the measureBED is -

50º

A

B C

D

E

(A) 80º (B) 50º (C) 100º (D) 105º

12. ABCDE is a regular pentagon. A star of five points ACEBDA is formed to join their alternate vertices. The sum of all five

vertex angles of this star is .......

D C

E B

A(A) Two right angle (B) Three right angle

(C) Four right angle (D) Five right angle

13. The sum of all the interior angles of n sided polygon is 2160º. Then this polygon can be divided into how many number

of triangles.

(A) 10 (B) 12 (C) 14 (D) 16


11/13


14. Given triangle PQR with RS bisectingR , PQ extended to D and n a right angle, then

R

PQ

Dp

m

n

q

S

d

(A)m =2

1( p –q) (B)m =

2

1( p +q)

(C)d =2

1(q + p) (D)d =

2

1m

15. In the given figure, x > y. Hence

(A) LM = LNL

M Nx y

(B) LM < LN

(C) LM > LN

(D) None of these

16. ABC is such that AB = 3 cm, BC = 2 cm and CA = 2.5 cm. IfDEF ~ ABC and EF = 4 cm, then perimeter ofDEF

is

(A) 7.5 cm (B) 15 cm (C) 22.5 cm (D) 30 cm

17. In an isosceles triangle ABC, AC = BC,BAC is bisected by AD where D lies on BC. It is found that AD = AB.

ThenACB equals

(A) 72° (B) 54° (C) 36° (D) 60°

18. In the figure BAC =ADC, thenCB

CA is

A

B CD

(A) CB CD (B)CA

CD(C) CD2 (D) CA2

19. In ABC right angled at C, AD is median. Then AB2 = ......

(A) AC2 – AD2 (B) AD2 – AC2 (C) 3AC2 – 4AD2 (D) 4AD2 – 3AC2


12/13


20. In ABC, AD, BE & CF are medians. Then 4(AD2 + BE2 + CF2) equals

A

E

CB

FO

D

(A) 3(OA2 + OB2 + OC2) (B) 3(OE2 + OF2 + OD2) (C) 3(AB2 + BC2 + AC2) (D) 3(AE2 + AF2 + AD2)

21. In DABC, BE AC and CF AB, then BC2 = ........

A

E

CB

F

(A) (AB BF) + (AC CE) (B) (AB AF) + (AC AE)

(C) (AB CF) + (AC BE) (D) AB + BC + AC

22. In an equilateral triangle ABC, if ADBC, then

(A) 2AB2 = 3AD2 (B) 4AB2 = 3AD2 (C) 3AB2 = 4AD2 (D) 3AB2 = 2AD2

23. ABCD is a parallelogram, M is the midpoint of DC. If AP = 65 and PM = 30 then the largest possible integral value of

AB is :

M

A B

CDP

(A) 124 (B) 120 (C) 119 (D) 118

24. In the diagram if ABC andPQR are equilateral. TheCXY equals

(A) 35º (B) 40º (C) 45º (D) 50º

25. Let XOY be a right angled triangle withXOY = 90º. Let M and N be the midpoints of legs OX and OY, respectively.

Given that XN = 19 and YM = 22, the length XY is equal to

(A) 24 (B) 26 (C) 28 (D) 34


13/13


DPPS - 1. Number System

Ques 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ans. B D A B D A A A D A C A A D B

Ques 16 17 18 19 20

Ans.C A D B B

DPPS - 2. Polynomials

Ques 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ans. B D D A B D D D C B A A B B A

Ques 16 17 18 19 20

Ans. C B B C B

Ques 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ans.

Ques 16 17 18 19 20

Ans.

DPPS - 3. Quadratic Equation

Ques 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ans. A C C A A C D A A B D A C B A

DPPS - 4. Progression

Ques 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Ans. C B B B A B C C C C A C A C C

Ques 16

Ans. C

DPPS - 5. Lines and Angles and Triangles

Ques 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ans. D A D D D D B B C C C A C B C

Ques 16 17 18 19 20 21 22 23 24 25

Ans. B C B D C A C C B B

ANSWER KEY


Daily Practice Problem Sheet

Documents

Maths Ntse