Mathematics NTSE Stage-1

NTSE STAGE-1 DAILY PRACTICE PROBLEMS SESSION-2012-13 SUBJECT : MATHEMATICS

DPP : 01

Class – X

TOPIC : NUMBER SYSTEM 1.

(A) 81

2.



2  3 

6

2

 3  

5 2

The value of

(A) a2

7.

(C) 97

(C)

99 100 100 99 a  a2 – b 2 a – a2 – b2

+

a – a2 – b2 a  a2 – b 2

(B) b2

4 , 6 3, 3 2

If x = (A) 10

2 3 3

(D)

4 3

3 – 2 2 , then N equals

–

2

5 2

(C)

(D)

(B)

9

(B) Equal to

99 100

(D) Equal to

100 99

5 1

is -

(C) a2 – b2

The ascending order of the surds 9

(D) 2

1 1 1 1    ---------- + is 1 2 2  3 3  4 99  100

The numerator of

(A)

is -

(B) 2 2 – 1

(C) Greater than

6.

y2

is -

5 –2

5 1

(A) Less than

5.

1 +

(B) 1

(A) 1

4.



2 

2 2 3

If N =

x

2

(B) 322

The fraction

(A)

3.

1

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

3

(D)

4a 2 – 2b 2 b2

2, 6 3 , 9 4 is -

4 , 3 2, 6 3

(C)

3

2, 6 3 , 9 4

(D)

6

3 1 , then the value of 4x3 + 2x2 – 8x + 7 is 2 (B) 8

(C) 6

(D) 4

3 , 9 4, 3 2

8.

On simplifying

999813  999815  1 (999814 )2

(A) 1

(B) 2 1

9.

X

1 1 1   p q r

(D) 4

1

(B) 0

(C) xpq+qr+rp

(D) 1

(C) x + y = q + z

(D) x – y = q – z

If ax = cq = b and cy = az = d, then (A) xy = qz

11.

1

(C) 3

 x q  qr  x r  rp  xp  pq The value of  r  ×  p  ×  q  is equal to x  x  x 

(A) 10.

, we get

(B)

x q = y z

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

12.

(C) 4

(D) 9

If a, a + 2 and a + 4 are prime numbers, then the number of possible solutions for a is : (A) one

13.

(B) 3

(B) two

(C) three

(D) more than three

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

16.

(C) 1

(D) None of these

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

17.

(B) 5

(B) 3, – 1

(C) 2, – 1

(D) 2, – 3

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

18.

(D) 18

(B) 31

(C) 11

(D) 33

How many zeros at the end of first 100 multiples of 10. (A) 10

20.

(C) 17

If 27 = 123 and 31 = 133, than 15 = ? (A) 13

19.

(B) 19

(B) 24

(C) 100

(D) 124

(C) 9

(D) 0

The last digit of (13 + 23 + 33 + ... 103)64 is : (A) 2

(B) 5

NTSE STAGE-1 DAILY PRACTICE PROBLEMS SESSION-2012-13 DPP : 02

SUBJECT : MATHEMATICS

Class – X

TOPIC : ALGEBRA - 1 (POLYNOMIAL & LINEAR EQUATION IN TWO VARIABLE) 1.

2.

3.

The product of all the solutions of x4 – 16 = 0, is (A) –16 (B) – 4 (C) 4

(D) 16

1 1 = a + b and x – = a – b, then x x (A) ab = 1 (B) a = b

(D) a + b = 0

If x +

The sum of the x-coordinates of the intersection of x2 + y2 = 36 and x2 – y2 = 6 is(A) –2 21

4.

5.

6.

(C) ab = 2

(B) 0

The value of (a + b)3 + (a – b)3 + 6a (a2 – b2) = (A) 6a3 (B) 8a3 1

1

1 2 a

+

1 2 b

(D) 2 21

(C) 10a3

(D) 12a3

1

If x 3  y 3  z 3 = 0, then (A) x3 + y3 + z3 = 0 (C) (x + y + z)3 = 27 xyz

If

(C) 2 15

–

1 2 c

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

= 0, then the value of (a + b – c)2 is -

(A) 2ab

(B) 2bc

(C) 4ab

(D)4ac

7.

If 2x3 + ax2 + bx – 6 has (x – 1)as a factor and leaves a remainder 2 when divided by (x – 2), find the values of ‘a’ and ‘b’. (A) a = –8, b = 12 (B) a = 8, b = –12 (C) a = –4, b = 10 (D) a = 4, b = –10

8.

The expression

bx(a 2 x 2  2a 2 y 2  b 2 y 2 )  ay(a 2 x 2  2b 2 x 2  b 2 y 2 )

(A) a(x + y)

9.

(B) bx + ay

(C) ax + by

If a + b + c = 0 and none of them is zero, then the value of (A) – 2

10.

(ax  by )2

(B) 0

(C) 2

is equal to (D) b(x + y)

a4  b4  c 4 2 2

a b  b 2 c 2  c 2a 2 (D) 4

is :

In solving an equation ax – b = 0, where 'a' and 'b' are coprime natural numbers, julie made a mistake in 7 8 as its solution whereas Rashmi made a mistake in reading 'a' and obtained 3 5 as its solution. The correct solution differs from the mean of these solution by

reading 'b' and obtained

(A) 11.

7 10

(B)

59 30

If xy + yz + zx = 1, then the expression 1

(A) x  y  z

1

(B) xyz

(C)

8 3

(D)

59 15

xy yz zx 1  xy + 1  yz + 1  zx is equal to

(C) x + y + z

(D) xyz

12.

If (x + a) is a factor of x2 + px + q and x2 + mx + n then the value of a is : (A)

m–p n–q

(B)

n–q m–p

(C)

nq mp

(D)

mp nq

13.

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

14.

6 men and 10 boys can finish a piece of work in 15 days, while 4 men and 12 boys can finish it in 18 days. Find the time taken by 1 man above and that by 1 boy alone to finish the work. (A) 480 days (B) 440 days (C) 420 (D) 400 days

15.

2 (i) If x 

1 x2

(A) ± 12 16.

(C) ± 19

(D) none of these

Two candles of equal length start burning at the same instant. One of the candles burns in 5 hrs and the other in 4 hrs. By the time one candle is 2 times the length of the other. The candles have already burnt for : (A) 2

17.

 7 , then the value of x 3  1 is: x3 (B) ± 18

1 hrs. 2

(B) 3

1 hrs. 2

(C) 3

1 hrs. 9

(D) 3

1 hrs. 3

Find the value of x 3  y 3  12xy  64 , when x + y = – 4. (A) 0

(B) 1

(C) 4

(D) 64

18.

If quotient = 3x2 – 2x + 1, remainder = 2x – 5 and divisor = x + 2, then the dividend is : (A) 3x3 – 4x2 + x – 3 (B) 3x3 – 4x2 – x + 3 (C) 3x3 + 4x2 – x + 3 (D) 3x3 + 4x2 – x – 3

19.

The polynomials ax3 + 3x2 – 3 and 2x3 – 5x + a when divided by (x – 4) leaves remainders R1 & R2 respectively then value of ‘a’ if 2R1 – R2 = 0. (A) –

18 127

(B)

18 127

(C)

17 127

(D) –

17 127

20.

A quadratic polynomial is exactly divisible by (x + 1) & (x + 2) and leaves the remainder 4 after division by (x + 3) then that polynomial is : (A) x2 + 6x + 4 (B) 2x2 + 6x + 4 (C) 2x2 + 6x – 4 (D) x2 + 6x – 4

21.

The zeros of the quadratic polynomial x2 + 99x + 127 are : (A) Both positive (B) Both negative (C) One positive and one negative (D) Both equal

22.

If one zero of 2x2 – 3x + k is reciprocal to the other, then the value of k is : 2 3 (A) 2 (B)  (C)  (D) – 3 3 2 , ,  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

23.

24.

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

25.

If 4x +3y = 120, find how many positive integer solutions are possible ? (A) 8 (B) 9 (C) infinite (D) can’t be detremined

NTSE STAGE-1 DAILY PRACTICE PROBLEMS SESSION-2012-13 DPP : 03

SUBJECT : MATHEMATICS

Class – X

TOPIC : ALGEBRA – 2 (QUADRATIC EQUATION, ARITHMATIC PROGRESSION) 1.

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

2.

Sum of first 24 terms of the AP a1, a2, a3 ........, if it is known that a1 + a5 + a10 + a15 + a20 + a24 = 225 is – (A) 450

(B) 900

(C) 1350

(D) None of these

3.

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

4.

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

5.

The sum of first 20 terms of log2 + log4 + log8 + ...... (A) 20 log 2 (B) log 20 (C) 210 log 2

(D) log 2

6.

If n is odd, then the sum of n terms of the series 1 – 2 + 3 – 4 + 5 – 6 + ......... will be n n 1 n 1 2n  1 (A)  (B) (C) (D) 2 2 2 2

7.

The sum of n terms of the series a, 3a, 5a,...is (A) na (B) 2na

(C) n2a

(D) None

8.

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

9.

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

21 31

(B)

31 41

(C)

31 51

(D)

11 31

15

10.

The value of

 (2k  3) is k 1

(A) 390 11.

(C) 210

(D) 420

If in an A.P. , Sn = n2p and Sm = m2p, where Sr denotes the sum or r terms of the A.P., then Sp is equal to (A)

12.

(B) 195

1 3 p 2

(B) mnp

(C) p3

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) 2,200

(D) (m + n)p2

(D) 1,232

13.

If a2(b + c), b2(c + a), c2 (a + b) are in A.P., then either a,b,c are in A.P. or (A) ab + bc + ca = 0 (B) a + b + c = 0 (C) a – b – c = 0 (D) a – b + c = 0

14.

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)

15.

3 2

2 (B) 1

(C)

3

(D)

3

For which value of k will the equations x 2 – kx – 21 = 0 and x 2 – 3kx + 35 = 0 have one common root. (A) ± 1

(B) ± 2

(C) ± 4

(D) none of these

16.

Let a1, a2 ..........and b1, b2 ......... be the arithmetic progressions such that a1 = 25, b1 = 75 and a100 + b100 = 100. The sum of the first one hundred terms of the progressions (a1 + b1), (a2 + b2), ..... (A) 0 (B) 100 (C) 10000 (D) 5,05,000

17.

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 If the difference of the roots of equation x2 – bx + c = 0 be 1 then (A) b2 – 4c – 1 = 0 (B) b2 – 4c = 0 (C) b2 – 4c + 1 = 0 (D) b2 + 4c – 1 = 0

18.

19.

If the roots of quadratic equation x2 – 2ax – cx + 2ab + 2ac –b2 – bc = 0 are equal then find relation between a, b, c :

c ab 5 (B) 2a – c = 2b (C) 2a + c = 2b (D) 2a – 2b + c = 0 2 What is the condition for one root of the quadratic equation ax 2 + bx + c = 0 to be twice the other ? (A) b2 = 4ac (B) 2b2 = 9ac (C) c 2 = 4a + b2 (D) c 2 = 9a – b2 (A)

20.

21.

If  and  are the roots of the equation x2 – a(x + 1) – b = 0, then ( + 1)( + 1) = (A) b (B) –b (C) 1 – b (D) b – 1

22.

Let p and q be the roots of the quadratic equation x 2 – ( – 2) x –  – 1 = 0. What is the minimum possible value of p2 + q2? (A) 0 (B) 3 (C) 4 (D) 5

23.

Evaluate 6  6  6  6  ...... (A) 6

24.

The equation (A) 3 roots

(B) 0

x  2x  5  x  2 x  3x  6 x  4

(C) 1

(D) 3

(C) 1 roots

(D) no roots

has

(B) 2 roots

25.

If x 2 + ax + 10 = 0 and x 2 + bx – 10 = 0 have a common root, then a2 – b2 is equal to (A) 10 (B) 20 (C) 30 (D) 40

26.

If sin a and cos a are the roots of the equation 4x2 – kx – 1 = 0 (k > 0) then the value of k is : (A) 2 2

(B) 4

(C) 2

(D) 4 2

27.

The quadratic equation a x2 + bx + c = 0 has real roots  and . If a, b, c real and of the same sign, then (A)  and  are both positive (B)  and  are both negative (C)  and  are of opposite sign (D) nothing can be said about the signs of  and  as the information is insufficient.

28.

Number of solutions of the equation x 4  16 = x2 – 4, is : (A) 0 (B) 1 (C) 2

(D) 4

NTSE STAGE-1 DAILY PRACTICE PROBLEMS SESSION-2012-13 DPP : 04

SUBJECT : MATHEMATICS

Class – X

TOPIC : LINES AND ANGLES, TRIANGLES 1.

In a XYZ, LM || YZ and bisectors YN and ZN of Y & Z respectively meet at N on LM. Then YL + ZM = (A) YZ

2.

3.

(B) XY

(C) XZ

(D) LM

If D is any point on the side BC of a ABC, then : (A) AB + BC + CA > 2AD

(B) AB + BC + CA < 2AD

(C) AB + BC + CA > 3 AD

(D) None

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

T

x

R

S

20º

O

(A) 20º 4.

5.

(B) 70º

(C) 110º

(D) 50º

QR Two triangles ABC and PQR are similar, if BC : CA : AB = 1 : 2 : 3, then is : PR 1 2 2 1 (A) (B) (C) (D) 3 3 2 2

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

6.

(C) 4 cm

(D) 8 cm

In an isoscelesABC, if AC = BC and AB2 = 2 AC2, then C is equal to : (A) 45º

7.

(B) 2 cm

(B) 60º

(C) 30º

(D) 90º

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

C O

A

(A) 2 : 1 8.

(B) 4 : 1

B

(C) 3 : 1

(D) 3 : 2

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

O

C

D

A

B

(A)

(

10.

(B) 3 : 1 (C) 2 : 1 (D) 3 : 1 2 :1 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 (A) 10º 80º

(B) 30º B

(C) 50º

C

(D) 60º O

P

11.

In the figure below BA || DC and EC = ED, the measure BED is A

B

(A) 80º 12.

Q

50º

E

C

D

(B) 50º

(C) 100º

(D) 105º

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 (C) Four right angle

(B) Three 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

14.

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

m

n q

p P

S

d Q

(A) m =

1 (p – q) 2

(B) m =

(C) d =

1 (q + p) 2

(D) d =

D

1 (p + q) 2 1 m 2

15.

In the given figure, x > y. Hence L

(A) LM = LN (B) LM < LN M

(C) LM > LN

N x

y

(D) None of these 16.

ABC is such that AB = 3 cm, BC = 2 cm and CA = 2.5 cm. If DEF ~ ABC and EF = 4 cm, then perimeter of DEF 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. Then ACB equals

(A) 72° 18.

(B) 54°

In the figure BAC = ADC, then

(C) 36°

(D) 60°

(C) CD2

(D) CA2

CA is CB

A

B (A) CB  CD 19. 20.

(B)

C

D

CD CA

In ABC right angled at C, AD is median. Then AB2 = ...... (A) AC2 – AD2 (B) AD2 – AC2 (C) 3AC2 – 4AD2

(D) 4AD2 – 3AC2

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

A

F B

O E C

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

F

(A) (AB  BF) + (AC  CE) (C) (AB  CF) + (AC  BE)

B

E C (B) (AB  AF) + (AC  AE) (D) AB + BC + AC

22.

23.

In an equilateral triangle ABC, if AD  BC, then (A) 2AB2 = 3AD2 (B) 4AB2 = 3AD2

(C) 3AB2 = 4AD2

(D) 3AB2 = 2AD2

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 : A

B

P D

(A) 124 24.

C

M

(C) 119

(D) 118

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

(A) 35º 25.

(B) 120

(B) 40º

(C) 45º

(D) 50º

Let XOY be a right angled triangle with XOY = 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

NTSE STAGE-1 DAILY PRACTICE PROBLEMS SESSION-2012-13 DPP : 05

SUBJECT : MATHEMATICS

Class – X

TOPIC : CIRCLES & QUADRILATERALS 1.

Two circle touch each other externally at C and AB is a common tangent to the circles. Then, ACB = (A) 60º (B) 45º (C) 30º (D) 90º

2.

ABC is a right angled triangle, right angled at B such that BC = 6 cm and AB = 8 cm. A circle with centre O is inscribed in ABC. The radius of the circle is : (A) 1 cm (B) 2 cm (C) 3 cm (D) 4 cm PQ is a tangent to a circle with centre O at the point P. If OPQ is an isosceles triangle, then  OQP is equal to : (A) 30° (B) 45° (C) 60° (D) 90°

3.

4.

5.

If two tangents inclined at an angle 60º are drawn to a circle of radius 3 cm, then length of each tangent is equal to : 3 3 cm (A) (B) 6 cm (C) 3 cm (D) 3 3 cm 2 The length of a chord of a circle is equal to the radius of the circle. The angle which this chord subtends on the longer segment of the circle is equal to : (A) 30o

6.

(B) 45o

(C) 60o

(D) 90o

In the given figure, AB = BC = CD, If BAC = 25º, then value of AED is : C

(A) 50º

D

B

(B) 60º

25º

(C) 65º (D) 75º 7.

A

E

A, B and C are three points on the circle whose centre is O. If BAC = x, CBO = BCO = y,, BOC = t, reflex BOC = z, then :

(A) x + y = 90°

(B) x – y = 90°

(C) t + 2y° = 90°

(D) None of these

8.

In a circle of radius 17 cm. two parallel chords are drawn on opposite side of a diameter, the distance between the chords is 23 cm. If the length of one chord is 16 cm, then the length of the other is : (A) 15 cm (B) 23 cm (C) 30 cm (D) 34 cm

9.

Two circles of radii 20 cm and 37 cm intersect at A and B. If O1 and O2 are their centres and AB = 24 cm, then the distance O1O2 is equal to (A) 44 cm (B) 51 cm (C) 40 ½ cm (D) 45 cm

10.

If a regular polygon of sides ‘n’ is circumscribed about a given circle of radius R then the length of side of the polygon is (A) 2R tan

 n

(B) 2R sin

 n

(C) 2R cot

 n

(D) 2R sec

 n

11.

The common tangents to the circle  and  with centres P and Q meet the line joining P and Q at ‘O’ as shown. Given that the length OP = 28 cm and the diameters of  and  are in the ratio 4 : 3. The radius of the circle  is

(A) 2 cm

(B) 3 cm

(C) 4 cm

(D) 5 cm

12.

Two circles of radius 1 and 4 units are tangent to each other externally. The length of part of the common tangent line between the points of tangency is closest to which the following ? (A) 4.8 (B) 4.5 (C) 3.9 (D) 3.6

13.

In the given figure, O is the centre of a circle and BD is a diameter. AB and AC are tangents touching the circle at B & C respectively. If BAC = 700 then OBC is : (A) 300 (B) 350 (C) 400 (D) 450

14.

In the figure, O is the centre, OAB = 20°, OCB = 55°. Then AOC

O A

20°

B

55°

C

(A) 75° 15.

(C) 110°

(D) 40°

BC is the diameter of a semicircle. The sides AB and AC of a triangle ABC meet the semicircle in P and Q respectively. PQ subtends 140° at the centre of the semi-circle. Then A is

(A) 10° 16.

(B) 70°

(B) 20°

(C) 30°

(D) 40°

two circles of radii 10 cm and 8 cm intersect each other and the length of common chord is 12 cm. The distance between their centres is. A O

D

O'

B

(A)

7 cm

(B) 3 7 cm

(C) 4 7 cm

(D) (8 + 2 7 ) cm

17.

In the figure, O is the centre of circle. If AOD = 140 and CAB = 50°, then EDB is C D 50° 140°

A

(A) 140° 18.

(B) 90°

(C) 120°

C

50°

(A) 110°

20°

B

O A (B) 70°

(C) 160°

(D) 40°

In the figure, O is the centre of the circle, then XOZ is

Y

Z

X

(A) 2  X 20.

(D) 50°

In the figure, AB is diameter of a circle with centre O and CD || BA. If BAC = 20°, CAD = 50°, then ADC is D

19.

E

B

O

O

(B) 2  Y

(C) 2  Z

(D) 2[ XYZ +  YXZ]

In the given diagram shown below, if PB = 8 cm, AB = 4 cm, PD = 6 cm, then CD = ? A

4 cm

B

8 cm P

C

(A) 21.

16 cm 3

?

(B) 10 cm

D

6 cm

(C) 6 cm

(D) 7 cm

In given figure, PQRS is a ||9m. The sides of this ||9m are given by SP = (2x – 4) cm, PQ = (3y + 5) cm, 1  QR =  x  8  cm and RS = (y + 12) cm, then the perimeter of the ||9m PQRS is 2 

1 x + 8 cm. 2

(A) 55 cm

(B) 60 cm

(C) 65 cm

(D) None

22.

The diagonals of a rectangle ABCD meet at O. If BOC = 44°, then OAD is : (A) 68º (B) 44º (C) 54º (D) None of these

23.

Sides BA and DC of a quadrilateral ABCD are produced as shown in fig., then a + b is equal to : (A) x – y (B) x + y (C) x + 2y (D) None of these

24.

In fig. ABCD is a parallelogram. P and Q are mid points of the sides AB and CD, respectively. Then PRQS is : (A) Parallelogram (B) Trapezium (C) Rectangle (D) None of these

25.

In the figure, ABCD is a parallelogram and PBQR is a rectangle. R

Q C

D

A

B

P

If AP : PB = 1 : 2 = PD : DR, what is the ratio of the area of ABCD to the area of PBQR ? (A) 1 : 2 26.

(B) 2 : 1

(C) 1 : 1

(D) 2 : 3

A square board side 10 centimeters, standing vertically, is tilted to the left so that the bottom-right corner is raised 6 centimeters from the ground.

6 cm

By what distance is the top-left corner lowered from its original position ? (A) 1 cm (B) 2 cm (C) 3 cm (D) 0.5 cm 27.

ABCD is a trapezium in which AB || CD. If ADC = 2ABC, AD = a cm and CD = b cm, then the length (in cm) of AB is : 2 2 a (A) + 2b (B) a + b (C) a+b (D) a + b 3 3 2

28.

E is the midpoint of diagonal BD of a parallelogram ABCD. If the point E is joined to a point F on DA such 1 that DF = DA, then the ratio of the area of DEF to the area of quadrilateral ABEF is : 3 (A) 1 : 3 (B) 1: 4 (C) 1 : 5 (D) 2 : 5

29.

In figure, if ar(ABC) = 28 cm2 then ar (AEDF) =

(A) 21 cm2 30.

(B) 18 cm2

(C) 16 cm2

(D) 14 cm2

In the figure, ABCD is a square and P, Q, R, S are the midpoints of the sides. D

R

S

A

C

Q

P

B

What is the area of the shaded part, as a fraction of the area of the whole square ? (A)

1 3

(B)

1 4

(C)

1 5

(D)

1 6

NTSE STAGE-1 DAILY PRACTICE PROBLEMS SESSION-2012-13 DPP : 06

SUBJECT : MATHEMATICS

Class – X

TOPIC : MENSURATION 1.

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.

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

3.

The hypotenuse of a right angled triangle is 10 cm and the radius of its inscribed circle is 1cm. Therefore, perimeter of the triangle is : (A) 22 cm (B) 24 cm (C) 26 cm (D) 30 cm

4.

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

12 (A)

5 6

(B)

5

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

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

6.

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 A

X

x where x  N then x is equal to

B Y

D

(A) 72 7.

(B) 75

C

(C) 84

In a right angled triangle ABC, right angled at C, a + b = of the triangle. (A) 1 sq.unit

(B) 2 sq.unit

(D) 96 17 unit and c = 3 unit, then find the area

(C) 3 sq.unit

(D) 4 sq.unit

8.

An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is 12 dm2, then find the difference of their areas . (A) 3dm2 (B) 4dm2 (C) 5dm2 (D) 6dm2

9.

Area of the four walls of a room is 108 m2. If the height and length of the room are in the ratio of 2 : 5 and the height and breadth in the ratio 4 : 5, then find the area of the floor of the room. (A) 45 m2 (B) 50 m2 (C) 55 m2 (D)60 m2

10.

Water flows into a tank 150 metres long and 100 metres broad through a pipe whose cross-section is 2 dm by 1.5 dm at the speed of 15 km per hour. In what time, will the water be 3 metres deep ? (A) 100hr (B) 10hr (C) 50 hr (D)25 hr

11.

Two circles seen in the figure are concentric. Chord AB of the larger circle is tangent to the smaller circle and its length is equal to 16. The area of the shaded region, is : (A) 32  (B) 64  (C) 32  (D) 162 – 16 

12.

A right circular cone is divided into 3 portions A, B and C by planes parallel to the base as shows in the figure. The height of each portion is 1 unit, Calculate.

A

1

B

1

C

1

(a) the ratio of the volume of A to the volume of B. (b) the ratio of the volume of B to that C (c) the ratio of the area of the curved surface of B to that of C. (A) 1 : 7, 7 : 19, 5 : 3 (B) 1 : 7, 19 : 7, 5 : 3 (C) 1 : 7, 19 : 7, 3 : 5 (D) 1 : 7, 7 : 19, 3 : 5 13.

The radii of three cylindrical jars of equal height are in the ratio 1 : 2 : 3. Second jar is full of water which is first poured into the first jar. After filling the first jar, water is poured into the third jar. Which of the following statements is ture ? (A) Third jar is half filled (B) Third jar is one third filled (C) Third jar is two thirds filled (D) Third jar is four ninths filled.

14.

27 metal balls each of radius r are melted together to form one big sphere of radius R. Then the ratio of surface area of the big sphere to that of a ball is : (A)

15.

27 : 1

(B)

3 :1

(C) 3 : 1

(D) 9 : 1

A conical vessel, with internal radius of the base 8 cm and height 42 cm, is full of water. This water is poured in a right circular cylindrical vessel the radius of whose base is 16 cm. what is the height, in cm, of the water level in the cylindrical vessel? (A)

8 5

(B)

7 3

(C)

8 3

(D)

7 2

16.

If the area (in m2) of the square inscribed in a semicircle is 2, then the area (in m2) of the square inscribed in the entire circle is : (A) 4 (B) 5 (C) 6 (D) 7

17.

In the figure, F is taken on side AD of the square ABCD. CE is drawn perpendicular to CF, meeting AB extended to point E. If the area of CEF = 200 cm2 and area of square ABCD = 256 cm2, then the length ( in cm) of BE is : (A) 10 (B) 11 (C) 12 (D) 16

18.

A point is taken anywhere inside an equilateral triangle. From this point perpendiculars are drawn to each side of the triangle. If 's' be the sum of these perpendiculars and 2a be the length of the side of the triangle, then (A) s >

19.

3a

(B) s =

3a

3a

(D) s = 3 a

If in the figure, each circle is of radius 2 cm, then the width AD of the rectangle ABCD is : D

C

A

(A) 10 cm 20.

(C) s <

B

(B) 12 cm

(C) 4 ( 3  1) cm

(D) 4 ( 3  1) cm

An altitude h of a triangle is increased by a length m. How much must be taken from the corresponding base b so that the area of the new triangle is half of the original triangle ? (A)

bm hm

(B)

bm 2(h  m)

(C)

b(2m  h) 2(h  m)

(D)

b(m  h) 2m  h

MATHEMATICS _X-NTSE STAGE -I : DPP-1

NUMBER SYSTEM : 1

Que s. Ans. Que s. Ans.

1 B 11 C

2 D 12 A

3 D 13 A

4 B 14 D

5 A 15 B

6 A 16 C

7 A 17 A

8 A 18 D

9 D 19 D

10 A 20 B

7 A 17 A

8 B 18 D

9 C 19 B

10 A 20 B

7 C 17 B 27 B

8 B 18 A 28 A

9 C 19 B

10 B 20 B

7 B 17 C

8 C 18 B

9 C 19 D

10 C 20 C

7 B 17 D 27 B

8 C 18 A

9 B 19 D

10 A 20 C

7 B 17 C

8 D 18 B

9 A 19 D

10 A 20 C

MATHEMATICS _X-NTSE STAGE -I : DPP-2

Algebra - 1 (Polynomial & Linear Equation in two variable)

Ques. 1 Ans. B Ques. 11 Ans. B Ques. 21 Ans. B

2 A 12 B 22 A

3 C 13 C 23 A

4 B 14 A 24 B

5 C 15 B 25

6 C 16 D

B

MATHEMATICS _X-NTSE STAGE -I : DPP-3

Algebra – 2 (Quadratic equation, Arithmatic Progression)

Ques. Ans. Ques. Ans. Ques. Ans.

1 A 11 C 21 C

2 B 12 A 22 D

3 B 13 A 23 D

4 A 14 D 24 C

5 C 15 C 25 D

6 C 16 C 26 A

MATHEMATICS _X-NTSE STAGE -I : DPP-4

GEOMETRY : 1 (LINES ANGLE & TRIANGLE)

Que s. Ans. Que s. Ans. Que s. Ans.

1 D 11 C 21 A

2 A 12 A 22 C

3 D 13 C 23 C

4 B 14 B 24 B

5 D 15 C 25 B

6 B 16 B

MATHEMATICS _X-NTSE STAGE -I : DPP-5

GEOMETRY 2 (CIRCLE & QUADRILATERAL)

Que s. Ans. Que s. Ans. Que s. Ans.

1 D 11 B 21 A

2 B 12 C 22 A

3 B 13 B 23 B

4 D 14 B 24 A

5 A 15 B 25 A

6 D 16 D 26 B

MATHEMATICS _X-NTSE STAGE -I : DPP-6

MENSURATION

Que s. Ans. Que s. Ans.

1 A 11 B

2 A 12 D

3 A 13 B

4 A 14 D

5 A 15 D

6 C 16 B