Kendriya Vidyalya Sangathan New Delhi

Kendriya Vidyalya Sangathan
New Delhi
Workshop for high order
Thinking Skills ( HOTS )
Questions as per new pattern of CBSE
( effective from 2008-2009)
Director :Mr. S.K.Vohra
Asst. Director :- Mrs. Rashmi Gupta
Participants:- Sh. H.N. Singh, Smt. Praveen Gulati,
Sh. S.K. Rai, Sh. Balraj Singh,
Smt. Harjeet Bakshi, Smt. Smita Saha,
Smt Sapna Ratwaya, Smt. Nisha Sobti,
Smt. Sujata Anand, Smt. Sonia Baveja
1
Aims And Objectives Of Workshop
Dear teachers,
You are aware that CBSE has introduced High Order Thinking Skills questions in
Board papers w.e.f. 2008.
As enough exposure, practice and drill is required to be given to the students for
attempting these questions which are based on higher mental ability, a workshop
has been conducted on the subject for preparing the Question-Bank on “HOTS”.
Hope that our Students and Math’s teacher will be benefited and prepare well for
ensuing board Exams – 2009.
With Best Regards
S.K.Vohra
2
REAL NUMBERS
(Questions of 1 mark)
Q1.
What is the maximum no. of factors of a prime number?
Q2
Given HCF of (16, 100) = 4. find L.C.M of (16, 100).
Q3 Write a rational no. between √2 and √3 .
Q4.
Write if 343/28 is a terminating or non-terminating repeating decimal without doing actual
division.
Q5.
Tell whether the prime factorization of 15 is 1X 3 X 5 or not.
Q6
If x and y are two irrational numbers then tell whether x + y is always irrational or not.
Q7.
What is the L.C.M of x and y if y is a multiple of x?
Q8.
Write the sum of exponents of prime factors of 98.
Q9
State if (√2 - √3)( √2 + √3) is rational or irrational.
Q10
Express 0.03 as a rational number in the form of p/q.
(Questions of 2/3 marks)
Q1
Explain why 7 X 13 X13 + 13 and 7 X 6 X 5 X 4 X 3 X 2 X 1 + 5 are composite numbers.
Q2
Find the missing number
3
34
34
2
2
Q3.
Show that one and only one out of n, n+4, n+8,n+12 and n+16 is divisible by 5 where n is any
positive integer.
Q4
Show that the sum and product of two irrational numbers 7 + √5 and 7 - √5 are rational numbers.
Q5.
Use Euclid’s division lemma to find the H.C.F of 615 and 154.
Answers
1)
2
2)
400
3)
4) Non terminating & repeating decimal 5) Not, 15=3X5
6) Not, x+y may be rational
7) y
8) 1+2=3
4
9)rational
10)
1
10
3
2
7
Polynomials
1-mark questions
1. The value of quadratic polynomial f (x)=2x2- 3x- 2 at x =-2 is ……
2. If the product of zeroes of the polynomial ax2–6x-6 is 4, find the value of a.
3. Find the zeroes of the polynomial x2-1.
1
4. The sum and product of the zeroes of a quadratic polynomial are – and –3respectively. What is
2
the quadratic polynomial?
5. Find the number of zeroes of y=p(x) from the graph
5
6.
7.
8.
9.
Find the zeroes of the polynomial f(x)=43x2+ 5x-23
2x2-3x+5 is a polynomial. True or false. Justify
What is the zeroes of the polynomial ax+b=0,a0
Give examples of polynomials f(x), g(x) and r(x) which justify the division algorithm f(x) =g(x)
q(x)+r(x) and (i) deg r(x)=0 (ii) deg f(x) =deg g(x)=2 (iii) deg q(x) =deg r(x)=1
10. Write a polynomial whose zeroes are 2 and -2
2/3 marks Questions
1. Obtain all the zeroes of the polynomial x2 +7x+10 and verify the relationship between the zeroes
and its coefficients.
2. If two zeroes of the polynomial of (x)=x4-6x3 -26x2+138x-35 are 2  3 find other zeroes.
3. If  and B are the zeroes of the quadratic polynomial f (x) =x2 +2x+1, then find  and 1/.
4. If  and  are the zeroes of the polynomial f(x) =x2 –px+q such that α 2 +2.
5. If  and  are the zeroes of the polynomial f(x)=x2 –5x+k such that -=1,find the value of k.
6. Check whether 2x3+1 is a factor of 2x5+10x4+6x3+2x2+5x+1.
7. Obtain all the zeroes of the polynomial f(x)=3x4+6x3-2x2-10x-5 if two of its zeroes are
5
and 3
5
.
3
8. If the polynomial f(x)=x4-6x3+16x2-25x+10 is divided by another polynomial x2-2x+k, the
remainder comes out to be x+a, find k and a
9. Find the polynomial of least degree which should be subtracted from the polynomial x4+2x34x2+6x-3 so that it is exactly divisible by x2-x+1
10. Divide 3x2-x3-3x+5 by x-1-x2 and verify the division algorithm
Polynomials(Answer)
1 mark
1)12
2.)
-3/2
3) +1
4)2x2+x-6
5) zeros=3
6)
-2/3,3/4
7) false
8) –b/a
9) f(x)=g(x)Xq(x)+r(x), deg r(x)=0
x 2-1=(x+1)(x-1)+0
2/3 marks
1. -5,-2
2. 7,-5
3. –2
6
10) x2-1
4. p2-2q
5. k=6
6. a=1, b=+ 2, Not a factor
7. –1,-1
8. k=5,a=-5
9. 2x-2
10. Q-(x-2), R=3
Linear Equations
(Questions of 1 Mark)
Q1. Does the point(1,-2) lie on the line whose equation is 3x-y-5=0?
Q2. For what value of ‘K’ the pair of equations x-ky+4=0 and 2x-4y-8=0 is
inconsistent?
Q3. Which axis is the graph of equation x=0?
Q4. How many solutions of the equation 5x-4y+11=0 are possible?
Q5. What is the value of ‘K’ for which the graph of the equations 2x-3y=9
and kx-9y=18 are parallel lines?
Q6. Find the value of ‘a’ for which the system of equations 3x+2y-4=0 and
ax-y-3=0 will represent intersecting lines?
Q. Write a linear equation in two variables which is consistent to equation
5(x-y)=3.
Q8. What is the solution of 2 x -5 y=0 and 2. 3 x-7 y=0.
Q9. Find the co-ordinates of the point where the line 2x-3y=6 meets x-axis
and y-axis.
(Questions of 2/3 Marks)
Q1. In a cyclic quad. ABCD;
A  (2 x  4) 0 , B  ( y  3) 0
C  (2 y  10)0 and D=(4x-5)0
Find the four angles in degree.
Q2. Solve for x and y:
103x+51y=617,
97x+49y=583.
Q3. The sum of two numbers is 35 and their difference is 13 find the
Numbers.
Q4. For what value of’K’ will the following equations has no solution.
3x+y=1,
(2K-1)x+(K-1)y=2K+1.
7
Q5. Aftab tells his daughter, “Seven years ago I was 7 times as old as you
were then, also 3 years from now I shall be 3 times as old as you will be.
Represent the situation algebraically.
ANSWERS
Answers of One mark Question
1. yes
4. Infinite solution
2. k=2
5.k=6
7. 2x+3y=7 or more others
8. x=0, y=0
Answers of 2/3 marks Question
1.  A =700, .  B =530, .  C =1100, .  D=127 0
2. x= 5, y=2
3. x=24, y=11
4. k=2
5. x-7y+42=0
x-3y-6=0
8
3. y-axis
3
6.a 
2
9.(3,0),(0,-2)
Quadratic Equation
1 mark
1. Find the sum & product of the roots of the equation x2- 3 =0
2. Divide 51 into two parts whose product is378.
3. Determine K, so that the equation x2-4x+k=0 has no real roots.
4. Find the equation whose roots are 5+ 2 and 5- 2
5. For the given equation 3 x2-2 2 x-2 3 =0
6. Find the value of a and b such that x=1, and x= -2 are solutions of the quad. Equation x2+ax+b=0
7. Find the value of K for which the following quad. Equation have equal roots
(a). (k-4) x2+2(k-4) x+4=0
(b). kx (x-2) +6=0
7. For what value of ‘P’, the following equations have real roots
(b) Px2r +4x +1=0
(c) 4x2+8x-P=0
(d) 2x2+Px +8=0
8.
If  and  are the roots of the equation x2 – 4x –5=0. Find -1 + -1
2 3
9. If one root of a quad. Equation is
, then what is the other root?
2
2/3 MARKS
1. Solve the following by factorization method
(a) 2x2+ax-a2=0,
a R
(b) 4x2-4ax+(a2-b 2)=0
(a,b  R)
2. Using quadratic formula, solve the following equation for x
abx2 + (b2-ac) x-bc=0
3. Solve for x, x+
(a,b,c  R)
1
1
= 25
.
x
25
4. If one root of the quadratic equation 2x2 +kx -6 =0 is 2, find the value of k also find the other root
5. If -5 is a root of 2x2 + px-15=0 & the quad. Equation f (x2+x)+k = 0 has equal root find the value
of k
6. If the roots of the equation (b-c) x2 +(c-a) x+ (a-b)=0 are equal then prove that
2b= a+c
7. Find two consecutive multiple of three whose product is 270
8. Two no. differ by 4 and their product is 192, find the numbers..
9
9. Is the following situation possible ?if so determine their present ages
The sum of the ages of two friend are 20 years. 4 years ago, the product of their age in years was
48.
10. If the list price of a toy is reduced by Rs 2 a person can by 2 toys more for Rs360.find original
price of the toy.
Answers
1 Marks Questions
(1) sum = 0 , Product = - 13
(2) 42,9
(3) K > 4
(4) X2 – 10x + 23 = 0
(5)
(6)
(7)
(8)
(9)
 6
3
4 , 4
3
3
(a) P  4 (b) P  8 (c) P  8 ,  8
4
5
2 3
2
6 ,
2/3 Marks Questions
a b
(1) (a) x = -a , a 2 (b)
2

b
c
(2) X = a , x = b
(3) 25 , 1
25
(4) 2 , 3
2
(5) P=7 , k = 7
4
(6)
(7) 15,18 ; -15, -18
(8) 12,16
(9) Not possible
(10)
Rs.20
10
ARITHMETIC PROGRESSION
1 MARK
1)
Which term of the sequence 4,9,14...is 124?
2)
Find the 10th term from the end of A.P
3, 8, 13, 18...253
3) For what value of K, the number x,2x+k,3x+6,are three consecutive terms of A.P
4) How many numbers of two digits are divisible by 8?
5) Find the middle term of A.P
1,8,15,...,505
6) Write next term of A.P ∫8, ∫18, ∫32....
7) What is the common difference of an A.P in which a23-a18=45
8) Find first three terms of an A.P whose n th term is -5 + 2x
9) In the given A.P, find the missing terms 0,_,-8,-12,_.
10) Find the sum of all odd integers between 1 and 100 which are not multiples of 4
2/3 MARKS
Q.1 How many terms of A.P 18, 16, 14...Should be taken so that their sum is zero?
Q.2 If the 10th term of an A.P is 47 and first term is 2, find the sum of the first 15 terms
Q.3 Solve the equation-2+5+8...+x=155
Q-4 Which term of A.P 121,117,113....is the first negative term?
Q-5 How many multiples of 4 lie between 10 and 250?
Q-6 If the sum of n terms of an A.P is n2 +2x, find the A.P and the 20th term.
Q-7 For what values of n, nth term of the series 3, 10, 17...and 63,65,67...are equal?
11
Q-8 If seven times the seventh term of an A.P is equal to 11 times the eleventh term, show that 18th term
of an A.P is zero.
Q-9 If 9 th term of an A.P is 0, prove that 29th term is double of the 19th term
Q-10 Determine the A.P whose zero term is 16 and when 5th term is subtracted from 7th ,we get 12
Answers
1 mark
a. n=25
b. 209
c. k=3
d. n=11
e. middle term=73
f. 52,62,72
g. d=9
h. –3,-1,1
i. 0,-4,-8,-12,-16
j. 2500
2/3 marks
1.
19
2.
255
3.
29
4.
32nd term
5.
60
6.
3,5,7,9
T20=41
12
7.
n=13
8.
a+17d=0
T18=0
10.
4,10,16,22
TRIGONOMETRY
Q1.
If θ = 45°, find the value of sec² θ.
Q2
Evaluate: cos 60°cos 45° - sin60° sin 45°.
Q3
Find the value of
tan15°.tan25°.tan30°.tan65°.tan85°
Q4
If θ is a positive acute angle such that sec θ = cosec60°, then find the value of 2 cos² θ-1.
Q5
Find the value of sin65° - cos25° without using tables.
Q6
Can cos θ =
Q7
If sec 5A = cosec(A - 36°), find the value of A.
Q8
If 2 sin
Q9
If A, B and C are interior angles of ∆ ABC, then prove that cos
Q10
Find the value of 9 sec² A – 9 tan²A.
5
be possible?
4
x
– 1 = 0, find the value of x.
2
(Question of 2/3 marks)
Q1
Prove that
Sin 6 θ + cos 6 θ = 1 – 3 sin² θcos² θ.
Q2
13
From the figure find the value of sin x and cos y.
BC
A
= sin .
2
2
C
3 cm
x
D
4 cm
A
y
B
12 cm
Q3
If 5 tan θ – 4 = θ, then find the value of
5 sin θ – 4 cos θ
5 sin θ + 4 cos θ
Q4
In ∆ ABC, c = 90°, tan A =
Q5.
If XAC = 45°, find the value of x and y in the figure
Answers of 1 mark Questions
1 3
1. 2
2.
2 2
1
4.
5. 0
2
7. 210
8.90 0
14
1
and tan B = √3. Prove that sinA .cosB + cos A .sin B =1.
3
3.
1
3
6.No
10. -9
Answers of 2/3 mark Questions
4
5
12
cos y=
13
2. sinx=
3. 0
5. x=20 2 cm
y=20 m
CO-ORDINATE GEOMETRY
(Question of 1 Marks)
1. Find the distance between the prints A(10 Cos ) and B(0,10 Sin)
2. Find the area of ∆ABC where A (2,3), B (-2,1) and C (3,-2)
3. Find the co-ordinates of the point which divides the line-segment joining the point (1,3) and (2,7)
in the ratio 3:4
4. Find the area of the triangle formed by the points O (0,0), A(a,0) and B(0,h).
5. AB is the diameter of a circle whose centre is (2,-3). If the co-ordinates of A.B are (1,4), then
find the co-ordinates of A.
6. Find the distance of the point (3,-4) from y-axis
If A(3,2) , B(-2,1) are two vertices of ∆ABC whose centroid G has the co-ordinates (5/3,
1/3). Find the co-ordinates of the third vertex C.
7. Find the co-ordinates of vertex B of the equilateral triangle OAB in the fig.
B
y
0
A
y’
15
x
(Question of 2/3 Marks)
1. Find the ratio in which the line-segment joining the points (-3,-4) and (1,-2) is divided by y-axis
2. Find the point of trisection of the line-segment joining the points (-3,4) and (1,-2).
3. In the figure BOA is a right triangle and C is the midpoint of hypt AB.Show that it is equidistant
from the vertices O,A and B.
y
(0,2 l)
C
x’
A (2a, 0)
x
y’
4. If the area of a triangle formed by (x, 2x), (-2, 6) and (3,1) is 5 square units, then find the value
of x.
5. If the centroid of the triangle formed by the points (a,b), (b,c) and (c,a) at the origin, then find the
value of a3+b3+c3
ANSWERS
Answers of one mark
1) 10
ab
2
7) (4,-2)
4)
10 33
,
7 7
2) 11sq. units
3)
5) (3,-10)
6) 3 units
8) (a, 3 a)
Answers of 2/3 marks
1) 3:1
4) x=2
16
5
1
2) (- ,2) ,(- ,0)
3
3
5) 3abc
3)
a 2  b 2  AC  BC  OC
Triangles
1 Mark Questions
Q1. In fig, if  A=  CDE, AB=9cm, AD=7cm, CD=8cm, CE=10cm. Find DE
if  CAB CED.
C
8
10
D
E
D
7
B
A
9
Q2.Find ‘y’ if ABC   PQR
P
A
9
Y
12
10
Q
B
R
C
7
Q3. ADE ˜
 ABC , if DE=4cm, BC=8cm and ar( ADE)=25sq cm. Find the area of ABC.
Q4. If D and E are respectively the points on the sides AB and AC of a ABC such that AD=6cm,
BD=9cm, AE=8cm, EC=12cm, Then show that DE||BC.
Q5.
17
Q
A
80
40
40
80
60
60
C
B
P
R
Write symbolic representation for above similarity.
Q6. If ar ABC =
Q7. Find
1
AB
ar DEF Find
4
DE
AE
if D is mid point of AB and DE||BC.
CE
A
D
E
B
C
Q8.  ABC is a right triangle, right angled at  B, BD  AC then  ADB  _____________
A
D
C
B
Q9. In  ABC, if 2AB2=AC2, Find  B.
Q10. In fig  B=900, if AB 2+BC2+CD2=AD2
D
C
18
Find  ACD.
A
B
2/3 Mark Questions
Q1. If  PQR ˜ ABC and PQ=AB show that  PQR   ABC.
Q2. In fig.
QR QT
=
and  1=  2 Show that
QS PR
PQS ˜  TQR
T
P
∟1
2
Q
R
S
Q3.In fig
PS PT
=
and  PST=  PRQ. Prove that PQR is an isosceles triangle.
SQ TR
P
S
T
R
Q
19
Q4. In fig ,AC||BD. Is
A
AE DE
=
. Justify your answer.
CE BE
C
E
B
Q5. In fig, DE|| AC and DF ||AE. Prove that
BF BE
=
FE EC
A
D
B
C
F
E
Q6. In fig, considering  BEP and  CPD. Prove that BP X P D= EP X PC
A
E
D
P
B
C
Q7. In fig, E is a point on side CB produced of an isosceles triangle ABC with AB=AC.
A
20
F
B
D
C
If AD  BC and EF  AC, Prove that  ABD ˜  ECF.
Q8. In PQR, M is a point on QR such that PM  QR and PM2 =QM.MR. Show that PQR is a right
triangle.
Q9. In Fig, ACB=900 and CD  AB. Prove that BC2/AC2=
BD
.
AD
C
A
B
D
Q10. Two poles of height ‘a’ meters and ‘b’ meters (a<b) are ‘p’ meters apart. Prove that the height of
the point of intersection of the lines joining the top of each pole of the foot of the other pole is given by
ab
(
)m.
a b
Triangles
1 Marks Questions
21
(1)
(2)
(3)
(4)
(5)
6 cm
2.5
100
ABC  RQP
1
(6)
2
(7) 1
(8) ABC
(9) 90
(10)
90
Circles
1 Mark Questions
1.
In the fig.1 PA and PB are tangents to the circle with centre O drawn from an external point P.
CD is a third tangent touching the circle at Q. If PB=10cm Q =2 cm. What is the length of PC?
2. The length of tangent from point A at a distance at 5 cm. from the centre of the circle is 4 cm. What
will be the radius of the circle?
3.
22
Two tangents TP and TQ are drawn from an external point ‘T’ to a circle with centre O as shown
in fig they are inclined to each other at an angle of 1000 then what is the value of  POQ?
4. How many tangents can be drawn to a circle from a point outside the circle?
2/3 MARKS
5.In fig, a circle touches all the four sides of a quadrilateral ABCD whose sides AB = 8 cm., BC = 9
cm. and CD = 6 cm. find AD.
6. What is the distance between two parallel tangents of a circle of the radius 4 cm.?
7. How many tangents can be drawn to a circle from a point inside the circle.
8. TP and TQ are tangents to the circle with centre O if  TOQ = 500 find  OTQ.
23
9. If O is the centre of two concentric circles of radius 5 cm. and 13 cm. if AB is chord of larger
circle which touches the smaller circle. Find the length of chord AB.
10.In the adjoining figure (fig. 5) a circle is inscribed in a quadrilateral ABCD in which  B= 90 0.If
AD= 23cm, AB =29cm, and DS =5cm. find the radius of the circle.
Circles(Answer)
(1) 8cm
(2) 3 cm
(3) 80
(4) Two
(5) 5 cm
(6) 8 cm
(7) None
(8) 40
(9) 24 cm
(10)
11 cm
24
CONSTRUCTIONS
Q1. Construct a triangle with sides 4cm, 5cm and 6 cm and then another triangle whose sides are
8
of
5
the corresponding sides of the first triangle.
Q2. Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of
300.
Q3. Draw a line segment of length 8 cm and divide it in the ratio 3:2. measure the two lengths.
Q4. Draw a circle of radius 5 cm. From a point 11 cm away from its center. Construct the pair of
tangents to the circle.
Q5. Let ABC be a right triangle in which AB= 6 cm and BC is 8 cm,  B=900, BD is perpendicular
from B on AC. The circle through B, C, D is drawn. Construct the tangents from this circle.
25
Area related to Circle
(1 Mark Questions)
1. A wire in the shape of square of perimeter 88 cm. is bent so as to form a circular ring. Find the
radius of the ring.
2. What is the perimeter of a semi circular protractor whose diameter is 14cm?
3. In fig. 6 OACB is a quadrant of a circle of radius 7 cm. What is its perimeter?
4. In fig.7, which is a sector of a circle of radius 10.5 cm. Find the perimeter of the sector (٨
22/7)
=
5. What is the ratio of the areas of a circle and an equilateral triangle whose diameter and a side are
respectively equal.
6. The length of a minute hand of a wall clock is 7 cm. What is the area swept by it is 30 minutes.
7. What is the radius of the circle whose circumference and area are numerically equal?
8. Find the area of a sector whose radius is ‘r’ and central angle is complete angle.
9. What is the perimeter of a sector of radius ‘r’ and of central angle Q (in degrees).
10. In fig.8 ABPC is a quadrant of radius ‘r’. Find the area of the shaded region.
26
2/3 Marks Questions
1.
Find the perimeter of adjoining figure (fig. 9), if a semicircular arc is drawn on side BC as
diameter.
2.
In the given figure (fig. 10) AB=BC=CD and AD=42 cm. find the area of shaded figure.
3.
27
In the given figure (fig.11) OABC is a rhombus, three of whose vertices lie on a circle with
centre O. If the area of the rhombus is 32√3 cm2. Find the radius of the circle.
4.
Find the area of shaded portion in the given figure (fig.12).
5. A wire is looped in the form of a circle of radius 28cm. if is reverted into a square form.
Determine the area of the square so formed.
6.
In the following figure(fig. 13) chord AB is equal to the radius of the circle. Find area of shaded
region if radius is 7 cm.
7. If the perimeter of a sector of a circle of radius 5.2 cm. is 16.4 cm. find the area of the sector.
8. The short and long hands of a clock are 4cm. & 6cm. long respectively. Find the sum of distance
22
travelled by their tips in two days. (Take  =
)
7
9.
28
∆ ABC is a right-angled triangle,  B= 900, AB= 28cm, BC= 21cm. With AC as diameter
semicircle is drawn and with BC as radius a quarter circle is drawn. Find the area of the shaded
portion. (Fig. 14)
10.
Above figure is a running track with semicircular ends of diameter 7 cm. Find the length of the track
as per given dimensions.
Area related to Circle
1 Marks Questions
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
29
14 cm
36 cm
25 cm
32 cm
22
7 3
12.8 cm
2
 r2
PROBABLITY
1)
A die is thrown once. Find the probability of getting multiple of 2 or 3.
2)
Find the probability of getting 53 Fridays in a leap year.
3)
Write a sample space when two coins are tossed simultaneously?
4)
A bag contain 4 red, 5 black and 6 white balls & ball is drawn from the bag at random.
Find the probability that ball drawn is
a)
b) Red or White
5)
In a single throw of two dice, find the probability of not getting the same number on the
two dice.
6)
The probability that it will rain tomorrow is 0.85. What is the probability that it will not
rain tomorrow ?
7)
What is the probability of
a)
8)
9)
12)
Impossible event
Red and a queen
b)
Either a king or jack
Exactly one head
b)
No head
15 cards numbered 1,2,3………15 are put in a box and mixed thoroughly & card is
drawn at random from the box. Find the probability that the card drawn is
a)
11)
b)
Two coins are tossed simultaneously once. What is the probability of getting
a)
10)
A sure event
A card is drawn at random from a well shuffled pack of playing cards.
Find the probability that the card drawn is
a)
30
Not black
neither prime nor odd
b) a number which is a perfect square.
A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random
from the box, find the probability that it bears.
a)
not divisible by 5 or 10.
b)
two digit number.
Two coins are tossed simultaneously once. Find the probability of getting
a)
at least one head and one tail.
b)
at most 2 heads.
13)
In a single throw of two dice, find probability of getting
a)
b)
14)
A bag contains 12 balls out of which x are black. If one ball is drawn at random from the
box, what is the probability that it will be a black ball. If more black balls are put in the
box the probability of drawing a black ball is now double of what it was before. Find x
15)
Two customers Shyam and Ekta are visiting a particular shop in the same week (Tuesday
to Saturday) each is equally likely to visit the shop on any day as on another day. What is
the probability that both will visit the shop on
i)
ii)
iii)
16)
17)
a black face card
a queen of diamond
a spade
a black card
Savita and Hamida are friends what is the probability that both will have
i)
ii)
20)
She will buy it
She will not buy it
All the three face cards of spade are removed from a well shuffled pack of 52 cards &
card is drawn from the remaining pack. Find the probability of getting
a)
b)
c)
d)
19)
the jack of hearts
a face card.
A lot consists of 144 ball pens of which 20 are defective and the others are good. Nuvi
will buy a pen if it is good, but will not buy it if it is defective. The shopkeeper draws one
pen at random and gives it to him. What is the probability that
i)
ii)
18)
the same day?
consecutive day?
Different day?
One card is drawn from well-shuffled pack of 52 cards. Find the probability of getting
a)
b)
the same birthday?
different birthday?
A letter is chosen at random from the letters of the word “ASSASSI NATION” and the
probabilities that the letter chosen is a
i)
31
doubles
a total of 11
vowel
ii)
consonant
ANSWERS
(1) S = {1,2,3,4,5,6}  n(S) = 6
E = {2,3,4,6}  n(E) = 4  P(E) =
4
2
=
6
3
(2) A leap year consist of 52 weeks and 2 extra days.
 n(S) = 7 , n(E) = 2 , P(E) =
2
7
(3) S = { HH,HT,TH,TT}
(4) 4R
5B
10
2
=
15
3
10
2
(b) p( R or W ) =
=
15
3
(a) P( B ) =
6W
(5) n(S) = 36
n(E) = 30
30
5
P(E) =
=
16
6
(6) P(E) = 0.85
P(E) = 1 – P(E) = 1 – P(E) = 1 - .085 = 0.15
(7) (a) P(S) = 1 (b) P(2) = 0
(8) (a)
4
1
8
2
28
7
=
(b)
=
(c)
=
52 13
52 13
52 13
(9) S = { HH,HT,TH,TT}
3
1
(a)
(b)
4
4
32
(10)
(a) E= {4,6,8,10,12,14}  n(E) = 6; n(S) = 15
6 2
P(E) =
=
15 5
(b) E = {1,4,9}  n(E) = 3
3
1
P(E) =
=
15 5
(11)
(a) n(E) = divisible by 5 or 10 = 18
18
1
P(E) =
=
90
5
1
4
P( E ) = 1 – =
5
5
81
9
=
90 10
(12) S= {HH,HT,TH,TT}
(a) E= {HT,TH}  n(E)=2
2
1
n(S) = 4 , P(E) =
=
4
2
4
(b) P(E) = =1
4
6 1
2
1
(13) (a) P(E) =
=
(b) P(E) =
=
36 6
36 18
(b) P(E) =
(14) xb+6
x
x6
(1)
(2) P(E) =
12
18
x  6 2x
=
 x + 6 = 3x  2x = 6  x = 3
18
12
(15) S= { Tue, Wed, Thur, Fri, Sat} n(S) = 25
(1)
(16) (a)
5
1
8
20 4
=
(2)
(3)
=
25 5
25
25 5
1
12 3
(b)
=
(Jack, King & Queen)
52
52 13
(17) 20 Defected & 124 Non-Defected
124
31
20
5
(1)
=
(2)
=
144 36
144 36
(18) (1)
9
49
(19) (1)
1
365
(2)
1
10
23
(3)
(4)
49
49
49
(2)
364
365
(20) A,A,I,A,I,O = 6’s Vowels
6
7
(1)
(2)
13
13
33
S,S,S,S,N,T,N = 7’S Constants
Surface Area & Volumes
Take  = 22/7, unless stated otherwise.
Q-1
What is the surface area of a cube whose volume is 64 cm3?
Q-2 A wooden solid sphere of radius r cm is divided into two equal parts. What is the whole surface
area of the two parts?
Q-3 If the curved surface area of a right circular cylinder is 1760 cm2 and its radius is 21 cm, then
what is its height?
Q-4 Two cubes each of volume 64 cm3 are joined face to face. What is the surface area of the
resulting cuboid ?
Q-5
How many balls, each of radius 1 cm, can be made from a solid sphere of lead of radius 8 cm ?
HOTS
Q-6 The diameter of a metallic sphere is 6 cm. It is melted and drawn into a wire having diameter of
the cross-section as 0.2 cm. Find the length of the wire?
Q-7
What is the height of a cone whose base area and volume are numerically equal?
Q-8 A rectangular piece of paper of dimensions 60 cm X 88 cm is rolled to form a hollow circular
cylinder of height 60 cm. What is the radius of the cylinder?
Q-9 In the adjoining figure, the bottom of the glass has a hemispherical raised portion. The glass is
fitted with orange juice. How much juice a person will get?
34
Q-10 Two right circular cones X & Y are made, X having three times the radius of Y and Y having half
the volume of X. Calculate the rates of X & Y?
2/3 marks Questions
Q-1 The curved surface area of a cone is 550 cm2. Find the volume of the cone given that its base
diameter is 14 cm.
Q-2 A square field and an equilateral triangular park have equal perimeters. If the cost of ploughing
the field at the rate of Rs. 5 per m2 is Rs. 720, find the cost of maintaining the park at the rate of Rs. 10
per m2.
Q-3
An iron solid sphere of radius 3 cm is melted and recast into small spherical balls of radius 1
cm each. Assuming that there is no wastage in the process, find the number of small spherical balls
made from the given sphere?
Q-4
A rectangular water tank of base 11 cm X 6 cm contain water up to height of 5 m. If the water in
the tank is transferred to a cylindrical tank of radius 3.5 m, find the height of water level to be raised in
this tank?
Q-5 The base radii of two circular cones of the same height are in the ratio 3:5. Find the ratio of their
volumes.
Q-6 Circumference of the edge of hemispherical bowl is 132 cm. Find the capacity of the bowl.(  =
22/7)
Q-7 In the given figure, a cone of radius 10 cm is divided into two parts by drawing a plane through
the mid-points of its axis, parallel to its base. Compare the volume of the two parts?
Q-8 Find the volume of the largest right circular cone that can be cut out of a cube whose radius is 9 cm.
Q-9
50 circular plates, each of radius 10.5 cm and thickness 1.6 cm, are placed one above the other
to form a solid circular cylinder. Find the curved surface are and volume of the cylinder so formed?
Q-10 the radii of the internal and external surfaces of a metallic spherical shell are 3 cm and 5 cm
2
respectively. It is melted and recast into a solid right circular cylinder of height 10 cm. Find the
3
diameter of the base of the cylinder.
35
Answers
1) 16 cm2
2) 3 r2 cm 2
3)13.3 cm
4) 160 cm2
5) 512
6)9 m
7) 3 units
8) 14 cm
9) 396 cm3
10) H/h=1/18
2/3 marks questions
1) 1334.66 cm 3
2) Rs. 4428.80
3) 27
4) 8.57m
5) 9:25
6) 19104 cm3
7)1:7
8)190.93 cm3
9) (5280 cm2, 27720 cm3)
10) r=3.5 cm, d=7 cm
STATISTICS(HOTS)
Q1. The given distribution shows the number of runs scored by some top batsman of the world in one
day international cricket matches.
RUNS SCORED
3000-4000
4000-5000
5000-6000
6000-7000
7000-8000
8000-9000
9000-10000
10000-11000
NUMBER OF
BATSMAN
4
18
9
7
6
3
1
1
Find the mode of the data.
Q2. Give the empirical relation between median, mode and mean.
Q3. If the median of the distribution given below is 28.5. Find the value of ‘x’ and ‘y’.
C.I
0-10
36
FREQUENCY
5
10-20
20-30
30-40
40-50
50-60
x
20
15
y
5
Q4. Find the median of the following frequency distribution
MARKS
0100
FREQUENCY 2
100- 200200 300
5
9
300400
12
400500
17
500600
20
600700
15
700800
9
800900
7
9001000
4
Q5. The median of the following data is 20.75. Find the missing frequencies ‘x’ and ‘y’ . If the total
frequency is 100.
CI
0-5
FREQUENCY 7
5-10
10
10-15
x
15-20
13
20-25
y
25-30
10
Q6. If the mean of the following distribution is 6, find the value of’p’.
X
2
4
6
10
Y
3
2
3
1
Statistics (Hots)
(1) 4608.7 runs
(2) Mode = 3 median – 2 Mean
 x 8
(3) 

 y 8
(4) 525
(5) y = 20 , x = 17
(6) P = 7
Statistics (Hots)
(1)
(2)
(3)
(4)
(5)
37
5 cm
.
.
.
30-35
14
P+6
2
35-40
9
SAMPLE PAPER-1
CLASS-X
Gereral Instructions:
1. All questions are compulsory.
2. Section A consists of 10 questions of 1 mark each.
Section B consists of 5 questions of 2 marks each.
Section C consists of 10 questions of 3 marks each.
Section D consists of 5 questions of 6 marks each.
3. There is no overall choice however internal choice has been provided.
SECTION-A
1. State fundamental theorem of arithmatic.
2. One card is drawn at random from a well shuffled deck of 52 cards.
Find the probability of getting a king of red suit.
3. If kx+y=7 and 3x-2y=11 represent inte rsecting lines then find the
value of K.
4.Find the 18th term of A.P.
2,3 2,5 2,........
o
5. If sec5A=cosec(A-36 ),where 5A is an acute angle. Find the value of A.
6. If mode of the following data is 15. Find K.
k
10,15,17,15,  6,17, 20
2
7. In the given fig DE is parallel to BC.
AD 2
If
 and AC=18 cm. Find AE.
BD 3
8. Find X in the give n fig. if PT is tangent to the circle.
38
9.If  and  are zeros of 2x 2  5( x  2), then find the product of  and  .
22
10. Find the perimeter of a quadrant of circle of radius 7cm( = ).
2
SECTION-B
11. Find 31 term of an AP whose 11 term is 38 and 16 term is 73.
12. Find the ratio in which the y-axis divides the join of (5,-6) and(-1,-4).
Also find the point of intersection.
13. Evaluate without using trignometric table
tan 50  sec50
 cos 40 cos ec50
cot 40  cos ec50
OR
If sec 4 A  cos ec( A  20 ), where 4A is an acute angle,
find the value of A.
14. If AB,AC and PQ are tangents in the given fig. and AB=5 cm.
Find the perimeter of APQ.

39
15. If the mean of the following data is 18.
find the missing frequency P.
Xi
fi
10
5
15
10
20
p
25
8
SE C T IO N -C
2
16. Find the zeros of polynom ial p(x)= 2 x  3 x  2 2 and
verify the relationship betw een the zeros and co-efficients.
OR
3
2
O n dividing x -3x + x+2 by the polynom ial g(x), the quotient
and rem ainder w ere x-2 and -2x+ 4 respectively. Find g(x).
17. Find the value of K for w hich (k-3)x+3y=K , kx+ky=12 w ill
have infinitely m any solutions.
18. T he hypotenuse of a right triangle is 1 m less than tw ice the
shortest side. If third side is 1 m m ore than the shortest side.
Find the sides of the triangle.
40
19.Construct a triangle of sides 4cm, 5cm & 6cm, and then a triangle
2
similar to it whose sides are of corresponding sides of first triangle.
3
20. Prove that:
(1+cotA+tanA)(sinA-cosA)=sinAtanA-cotAcosA
or
1  cos 
(cosec -cot ) 2 =
1  cos 
21.The mid points of the sides of a triangle are (3,4), (4,6) and(5,7).
Find the co-ordinates of the vertices of the triangle.
22. How many coins 1.75 cm in diameter and 2mm thick must be melted
to form a cuboid 11cm X 10 cmX 7cm.
23.Prove that 5 -3 2 is an irrational number.
24. In a trapezium ABCD, AB  CD and CD=2 AB Cuts AD in F and BC in E, such that
BC 3
 .
EC 4
Diagonal BD intersect EF at G.
Prove that 7EF=10AB.
25. Prove that the points(a,0),(0,b) and (1,1) are collinear if
1 1
  1.
a b
SECTION-D
26.Prove that the ratio of areas of two similar triangles is equal to the ratio
of squares of their corresponding sides using the above theorem do the
following:
The area of two similar triangles ABC and PQR are in the ratio 9:16,. If BC=4.5 cm.
Find the length of QR.
OR
State and prove Pythagoras theorem. Using the above theorem, Prove the following.
In the given figure PQR is a right triangle right angled at Q. If QS=SR, show that PR2=4PS2-3PQ2.
41
27. A sailor can row a boat 8 km downstream and return back to the starting
point in 1 hr. 40min. If speed of stream is 2 km/hr. Find the speed of boat in
still water.
28. An aeroplane when 3000m high, passes vertically above another
aeroplane at an instant when the angles of elevation of the two
aeroplanes from the same point on the ground are 600 and 450 respectively.
Find the vertical distance between two aeroplanes.
29. A solid in the form of right circular cylinder, with hemisphere at one
end and a cone at other end. the radius of the common base is 3.5 cm and
the heights of the cylindrical & conical portion are 10cm amd 6 cm respectively.
Find the total surface area of the (use  =
42
22
)
7
30. Calculate Medians of the following data:
Marks obtained
Less than 20
Less than 30
Less than 40
Less than 50
Less than 60
Less than 70
Less than 80
Less than 90
Less than 100
No. of students
0
4
16
30
46
66
82
92
100
Compartment 2008, SET-1
General Instructions:
(1) All questions are compulsory.
(2) The questions paper consist of 30 questions divided into four sections – A,B,C and D. Section A
comprises of ten questions of 1 mark each, Section B comprises of five questions of 2 marks
each, Section C comprises of ten questions of 3 marks each and Section D comprises of five
questions of 6 marks each.
(3) All questions in Section A are to be answered in one word, one sentence or as per the exact
requirement of the questions.
(4) There is no overall choice. However, an internal choice has been provided in one question of 2
marks each, three questions of 3 mark each and two questions of 6 marks each. You have to
attempt only one of the alternatives in all such questions.
(5) In questions on construction, the drawings should be neat and exactly as per the given
measurement.
(6) Use of calculators is not permitted.
SECTION A
1. Write the HCF of the smallest composite number and the smallest prime number.(HOT)
43
2. The graph of y = f(x) is given in Fig 1. what is the number of zeroes of f(x)?
3. For what value of k, are the roots of the quadratic equation 3x2 + 2kx +27 =0 real and equal?
4. find the next term of the A.P..
 2,  8, 18,……………..
1
cos ec 2  sec2 
5. Given that tan  
, what is the value of
?
cos ec 2  sec2 
3
6. The perimeter of two similar triangles ABC and LMN are 60 cm and 48 cm respectively. If LM
= 8 cm, then what is the length of AB?
7. From a point P, the length of the tangent to a circle is 15 cm and distance of P from the center of
the circle is 17 cm. Then what is the radius of the circle?
8. A chord of a circle of radius 14 cm subtends 60 at the center. Find the area of the major sector.
9. Cards bearing of numbers 3 to 20 are placed in a bag and mixed
thoroughly. A card is taken out from the bag at random. What is the probability that the number
on the card taken out is an even number?
10. What measure of central tendency is obtained graphically as the xcoordinate of the point of intersection of the two ogives for this data ?
SECTION-B
11. Find a quadratic polynomial whose zeros are 1 and –3. Verify the relation between the
coefficients and zeros of the polynomial.
12. If A, B and C are interior angles of a triangle ABC, then show that
BC
A
sin
 cos (HOT)
2
2
13. For what value of p, are the point (-3,9),(2,p) and (4,-5) collinear ?
14. In Figure2, DE || BC and
AD 3
= . If AC = 4.8 cm, find the length of AE.
DB 5
15. A bag contains 5 red , 4 blue and 3 green balls. A ball is taken out of the bag at random. Find the
probability that the selected ball is (i) or red colour (ii) not of green colour.
Or
A card is drawn at random from a well shuffled deck of playing cards. Find the probability of
drawing a (1) face card (2) card which is neither a king nor a red card.
Section C
16. Show that 2+ 3 is an irrational number.(HOT)
OR
Show that only one of the numbers n, n+2 and n+4 is divisible by 3.
44
17. Find all the zeroes of 2x4 –9x3+5x2 +3x –1 , If two of its zeroes are 2+ 3 and 2- 3.
18. Solve the following system of equations for x and y:
OR
For what of a and b does the following pair of linear equations have an infinite number of solutions?
2x+3y =7
a(x+y)-b(x-y) =3a + b – 2
(HOT)
19. The first and the last term of an A.P. are 4 and 81 respectively. I(f the common difference is 7, how
many terms are there in the A.P. and what is their sum?
20. Prove that :
21. Show that the points (3,2), (0,5), (-3,2) and (0, -1) are the vertices of a square.
22. Find the area of the quadrilateral whose vertices are A(0,0), B(6,0), C(4,3) and D(0,3).
23. Prove that the angle between the two tangents to a circle drawn from an external point, is
supplementary to the angle subtended by the line segment joining the points of contract at the centre.
24. Construct a ABC in which BC = 5cm, CA=6 cm and AB = 7 cm. Construct a A’BC’ similar to BC ,
each of whose sides are 7/5 times the corresponding sides of ABC.
25. In fig. 3, ABCD is a Quadrant of a circle of radius 14 cm and a semicircle BED is drawn with BD as
diameter. Find the area of the shaded region.(HOT)
OR
Find the area of the shaded region in fig 4. , If PR = 24 cm, PQ= 7cm and O is the center of the circle.
Section D
26. An aeroplane left 30 minutes later than its scheduled time, and in order to reach its destination 1500
km away in time , it has to increase its speed by 250 km/hour usual speed. Determine its usual speed.
27. a person standing on the bank of a river observes that the angle of elevation of the top of the tower
standing on the opposite bank is 600. When he moves 40 m away from the width of the river.( use  3=
1.732)
OR
As observed from the top of a lighthouse, 100 m high above sea level, the angle of depression of a ship
sailing directly towards it, changes from 300 to 600, Determine the distance travelled by the ship sailing
45
directly towards it, changes from 300 to 600. Determine the distance traveled by the ship during the
period of observation.( use  3= 1.732)
28. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their
corresponding sides.
Using the above , do the following :
If D is a point on the side AB of ABC such that AD: DB = 3:2 and the areas of ABC and BDE.
OR
Prove that the lengths of the two tangents drawn from an external point to a circle are equal.
Using the above, do the following:
Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which
is tangent to the smaller circle.
29. A woolen article was made by scooping out a hemisphere from each end of a solid cylinder. If the
height of the cylinder is 20 cm and radius of the base is 3.5 cm . find the total surface area of the article.
30. Find the mean, mode and median for the following data:
CLASS
46
FREQUENCY
0-10
8
10-20
16
20-30
36
30-40
34
40-50
6
TOTAL
100
47
48
49
50
51
52
53