{糆 VýS$Æý‡$ÐéÆý‡…
™ðlË$VýS$ Ò$yìlĶæ$…
ѧéÅÆý‡$¦Ë$..
VýS×ìæ™èlÔ>{çÜ¢… ¼sŒæ »êÅ…MŠæ
MøçÜ… ^èl*yýl…yìl
www.sakshieducation.com
Ýë„ìS™ø E_™èl…
çܵÆý‡®Ä¶æ* Ð]lÆý‡®™ól ѧýlÅ
11&2&2016
ONLINE EDITION
www.sakshieducation.com/apbhavitha.aspx
MATHEMATICS
10/ 10 iï³H
Ë„ýSÅ Ý뫧lý ¯]lMýS$..
C…WÏ‹Ù Ò$yìlĶæ$…
★
^éç³tÆý‡Ï ÐéÈV> JMýS Ð]l*Æý‡$P {ç³Ô¶æ²Ë$, çÜÐ]l*«§é¯éË$
★ Ð]l¬QÅOÐðl$¯]l Æð‡…yýl$, ¯éË$VýS$ Ð]l*Æý‡$PË {ç³Ô¶æ²Ë$
★ ^éç³tÆý‡Ï ÐéÈV> ¼r$Ï
ѧéÅ iÑ™èl…ÌZ MîSËMýS OÐðl$Ë$Æ>Ƈ¬V> °Í^ól 糧ø ™èlÆý‡VýS† ç³È„ýSË$ ™èlÓÆý‡ÌZ
{´ëÆý‡…¿¶æ… M>¯]l$¯é²Æ‡¬. ѧéÅÆý‡$¦ÌZÏ ç³È„ýSË sñæ¯]lÛ¯Œl Ððl¬§ýlOÌñæ…¨. A…§ýlÇ §ýl–ïÙt
A™èlÅ«¨MýS {VóSyŠl ´ëƇ¬…rÏOò³¯ól! 10/10 iï³H Ë„ýSÅ…V> ѧéÅÆý‡$¦Ë$ ïÜÇĶæ$‹ÜV>
{í³ç³Æó‡çÙ¯Œl Mö¯]lÝëWçÜ$¢¯é²Æý‡$. AƇ¬™ól G…™èl ^èl¨Ñ¯é ÌZÌZç³Ë H§ø B…§øâ¶æ¯]l..
Ð]l$…_ {VóSyŠl Æ>ÐéË…sôæ C…M> H… ^ólĶæ*˯ól çÜ…§ólçßæ… ™èlÌñæ™èl$¢™èl$…¨.
D ¯ólç³£ýlÅ…ÌZ ѧéÅÆý‡$¦ËMýS$ Eç³Äñæ*VýSç³yólÌê ÐéÆ>°Mø çÜ»ñæjMýS$t òܵçÙÌŒæ™ø ¿¶æÑ™èl
Ò$ Ð]l¬…§ýl$MöÝù¢…¨.. C…§ýl$ÌZ ¿êVýS…V> D ÐéÆý‡… "Ð]l*Å£ýlÐðl$sìæMŠSÞ' {ç³™ólÅMýS…!
çÜ»ñæjMýS$t °ç³#×æ$Ë çÜ*^èl¯]lË$
1
{ç³çÜ$¢™èl… ѧéÅÆý‡$¦Ë$ ÇÑf¯ŒlMýS$
{´ë«§é¯]lÅÑ$ÐéÓÍ. ™éÐ]l¬
^èl§ýl$Ð]l#™èl$¯]l² ´ëu>Å…Ô>ÌZÏ°
Ð]l¬QÅOÐðl$¯]l A…Ô>˯]l$
´ëƇ¬…rÏ Æý‡*ç³…ÌZ
Æ>çÜ$MøÐ]lyýl… Ð]lËÏ _Ð]lÆøÏ MìSÓMŠS
ÇÑf¯ŒlMýS$ Eç³Äñæ*VýSç³yýl$™èl$…¨.
Ð]l¬QÅ…V> çÜ*{™éË$, ¿êÐ]l¯]lË$
CÑ$yìl E…yól VýS×ìæ™èl…ÌZ ©°Ð]lËÏ
{ç³Äñæ*f¯]l… E…r$…¨.
2
¸ëÆý‡$ÃÌêË$,
¥Æý‡ÐŒl$ÞOò³ AÐ]lV>çßæ¯]l
ò³…´÷…¨…^èl$Mö°,
°Æý‡…™èlÆý‡… {´ëMîSt‹Ü
^ólõÜ¢ Ë„>Å°MìS ™èlW¯]l
Ð]l*Æý‡$PË$
Ý뫨…^èlÐ]l^èl$a.
3
4
{´ëMîSt‹Ü Mø×æ…ÌZ çÜ…RêÅ
Ð]lÅÐ]lçܦ, °Æý‡*ç³MýS
Æó‡RêVýS×ìæ™èl…, Æó‡RêVýS×ìæVýS×ìæ™èl…,
„óS{™èlÑ$†, {†Mø×æÑ$†,
çÜ…¿êÐ]lÅ™èl, Ýë…QÅMýSÔ>ç܈…
A«§éÅĶæ*ËMýS$
{´ë«§é¯]lÅÑ$ÐéÓÍ.
ѧéÅÆý‡$¦Ë$ çÜÐ]l$çÜÅ Ý뫧ýl¯]l™ø´ër$
M>Æý‡×ê˯]l$ °Æý‡*í³…^èlVýSÍVóS,
M>Æý‡×ê˯]l$ Ð]lÅMîS¢MýSÇ…^ól O¯ðlç³#×êÅË™ø
´ër$ C™èlÆý‡ A…Ô>Ë™ø A¯]lÓĶæ$…,
A¯]l$çÜ…«§é¯]l… ^ólçÜ$MøÐ]lyýl… Ð]l…sìæ
O¯ðlç³#×êÅË$ AËÐ]lÆý‡$aMøÐéÍ. A糚yól
çÜ»ñæjMýS$tOò³ ç³r$t HÆý‡µyýl$™èl$…¨.
5
6
{糆 A«§éÅĶæ$…ÌZ°
Ð]l¬QÅOÐðl$¯]l
°Æý‡Ó^èl¯éË$,
çÜ*{™é˯]l$ JMýS^ør
Æ>çÜ$Mö°
ÒOÌñæ¯]l糚yýlÌêÏ ^èl§ýlÐéÍ.
{糆 A«§éÅĶæ$… _Ð]lÆøÏ
C^óla çÜÐ]l$çÜÅ˯]l$
™èlç³µ°çÜÇV> Ý뫧ýl¯]l
^ólĶæ*Í.
{V>‹œË$,
°Æ>Ã×ê™èlÃMýS
çÜÐ]l$çÜÅ˯]l$
Ý뫨…^ól…§ýl$MýS$
{´ëMîSt‹ÜMýS$ GMýS$PÐ]l
{´ë«§é¯]lÅ… CÐéÓÍ.
õ³ç³ÆŠ‡&1ÌZ {V>‹œ
B«§éÇ™èl
çÜÐ]l$çÜÅËOò³ {ç³™ólÅMýS
§ýl–íÙtò³sêtÍ.
A well defined collection of objects or ideas is known as a SET.
Set Theory is a comparatively new concept in Mathematics.
It was developed by Georg Cantor (1845-1918).
Cantor's work between 1874 and 1884 is the origin of Set Theory.
VýS$Æý‡$ÐéÆý‡… l íœ{ºÐ]lÇ l 11 l 2016
2
10th Class Special - Maths
Prepared by:
Y.Vanamraju,
School Assistant,
GHS, Nandikotkur.
PAPER - I
1. REAL NUMBERS
IMPORTANT QUESTIONS
1 Mark
1. ''LCM of two numbers, is a multiple of
HCF of that two numbers''. Is it true or
not? Justify your answer by giving an
example?
A. LCM of 4, 6 is 12 and HCF of 4, 6 is 2 ..
LCM 12 is a multiple of HCF 2.
2. Three bells toll at intervals of 9, 12, 15
minutes respectively. If they start tolling
together, after what time will they next toll
together?
A. Three bells will toll together at the time =
LCM of 9, 12, 15 = 180 minutes = 3 hours.
After 3 hours they will toll together.
3. From the given factor tree find x?
4
x
5
y
3
A. y = 5 × 3 = 15
x = 4 × y = 4 × 15 = 60
2 Marks
1. Simplify log 12 + 2 log 3 –3log 2 as logN.
Determine the value of N?
2. Explain why 7×11×13+13 and 7×6×5× 4
×3×2×1+5 are composite numbers?
4 Marks
1. Prove that √5 is an irrational?
 x+ y 1
2. If l og 
 = (log x + log y).
 3  2
x y
Then find the value of y + x ?
2. SETS
IMPORTANT QUESTIONS
1 Mark
1. Is the following sets are equal sets? or not?
Justify your answer?
A = {x:2<x<3, x∈N}
B = {x: 2<x<3, x∈Q}
A. A = {x:2<x<3, x∈N} = φ
(∵ there is no natural no. between 2 and 3.
B = {x: 2<x<3, x∈Q} = sets of rational
numbers between 2 and 3 are infinite.
∴ A and B are not equal sets.
2. Write all subsets of A = {1, 2, 3}?
A. Subsets of A = P(A) = [φ, {1}, {2}, {3},
{1, 2}, {2, 3}, {3, 1}, A}
3. From the venn Diagram find A–B?
B
A
3
1
2
4
5
7
µ
8
A. A–B = {3, 4}
2 Marks
1. A represents the set of 3 multiples and B
represents the set of 4 multiples. Which
set of multiples does A∩B represents?
2. P is the set of factors of 5, Q is the set of
factors of 25 and R is the set of factors of
Target - 10 grade points
125. Is it right to say that P⊂Q, Q⊂R and
R⊂P. Explain?
4 Marks
1. A = {x: x is a prime, x<10}
B = {x: x is an odd number, x<10}
C = {x: x is a factor of 24}
D = {x: x is a multiple of 3, x≤18}
Write the above sets in Roster form and
find A∪B, B∩C, C–D, D–A?
2. If A = {1, 2,3} and B = {3, 4, 5} then show
that A–B, A∩B, B–A are mutually disjoint
sets?
A.
2.
A.
3. POLYNOMIALS
IMPORTANT QUESTIONS
1 Mark
1. Find the quadratic polynomial 2, 3 are the
sum and product of its zeroes respectively?
A. α + β = 2, αβ =3
Required polynomial
= K{x2–(α + β)x+αβ}, K≠0
=K {x2–2x+3}, K≠0
2. If the length and breadth of a rectangle are
the zeroes of a polynomial x2–7x+12 then
find the area?
A. Let the zeroes are α, β.
Area of rectangle =
= lb = αβ =
c 12
=
= 12sq.units
a 1
3. From the figure find the sum of the zeroes?
–4 –3 –2 –1
1
2 3 4
A. zeroes are –2, 1. Their sum = –2+1= –1
2 Marks
1. Find the zeroes of x2–2x–8 and verify the
relation between the zeroes and coefficients?
2. If α and β are the zeroes of the polynomial f(x) = x2–5x+k such that α – β = 1, then
find the value of k?
4 Marks
1. Verify that 3, –1, –1/3 are the zeroes of the
cubic polynomial P(x) = 3x3–5x2–11x–3,
then verify the relationship between the
zeroes and coefficients?
2. Obtain all other zeroes of 3x4 + 6x3 – 2x2 –
10x – 5, if two of its zeroes are
5
5
,−
?
3
3
5 Marks
1. Draw the graph of polynomial x2 – 4x+5
and find the zeroes. Justify the answer?
4. PAIR OF LINEAR
EQUATIONS IN
TWO VARIABLES
IMPORTANT QUESTIONS
1 Mark
1. "The cost of 5 tables and 7 chairs is Rs.
6400." Represent this situation in a linear
3.
equation?
Let the cost of 1 table is Rs.x, the cost of 1
chair is Rs.y then 5x+7y = 6400.
Father's age is three times the sum of the
ages of his two children. After 5 years, his
age will be twice the sum of the ages of
two children. Find the age of father?
Let the present age of the father be x years
and the sum of the present ages of his two
children be y–years. Then according to the
question
x=3y →(1); x+5=2(y+5+5)
⇒ x–2y–15 = 0 →(2)
Solving (1) and (2) we get x=45, y=15
∴ Father age = 45 years.
Check whether the equations 2x–3y=8 and
7x+4y= –9 are consistent or inconsistent?
a
2 b
−3
1
1
A. From the given equations a = 7 , b = 4
2
2
a
b
1
1
Since a ≠ b
2
2
∴ The equations are consistent
2 Marks
1. The larger of 2 supplementary angles exceeds the smaller by 18o. Find the angles?
2. Find out the pair of linear equations
9x+3y+12=0 and 18x+6y+24=0 are intersect at a point, or parallel or coincident?
4 Marks
1. A fraction becomes 4/5 if 1 is added to
both numerator and denominator. If, however 5 is subtracted from both numerator
and denominator, the fraction becomes
1/2. What is the fraction?
2. Solve the following pairs of equations by
reducing them to a pair of linear equations
6x+3y = 6xy; 2x+4y=5xy?
5 Marks
1. Draw the graph of the equations 2x–y=5
and 3x+2y = 11. Find the solution of the
equations from the graph.
5. QUADRATIC EQUATIONS
IMPORTANT QUESTIONS
1 Mark
1. The roots of x2–4x+4=0 are equal or not?
Explain?
A. b2–4ac= (–4)2–4(1)(4) = 0
Since b2–4ac=0, the roots are equal
or
x2–4x+4=0 ⇒ (x–2)2=0, x=2, 2. The roots
are equal.
2. "The base of a parallelogram is 4 cm
longer than its altitude and its area is 48
sq. cm." Express this statement in a quadratic equation form?
A. Let altitude h = x, base = x+4 (given)
From the data (x+4) x=48 (∵ Area of parallelogram = bh)
⇒ x2+4x–48=0
3. The product of Ramu's age (in years) five
years ago with his age (in years) 9 years
later is 15. Find Ramu's present age?
A. Let the Ramu's present age = x years
Given (x–5)(x+9)=15 ⇒ x2+4x–60=0
x2+10x–6x–60=0
⇒ (x+10)(x–6)= 0, x= –10 or 6
∴ Ramu's present age = 6 years, (x≠–10)
2 Marks
1. Find the value of 'k' for the quadratic
equation 2x2+kx+3=0, so it has two equal
roots?
2. Find the dimensions of a rectangle field
whose perimeter is 200 meters and area is
2100 sq.m.?
4 Marks
1. A shop keeper buy a number of books for
Rs. 1200. If he had bought 10 more books
for the same amount, each book would
have cost Rs. 20 less. How many books
did he buy?
2. A train travels 360 km at a uniform speed.
If the speed had been 5km/h more it would
have taken 1 hour less for the same journey. Find the speed of the train?
6. PROGRESSIONS
IMPORTANT QUESTIONS
1 Mark
1. Check whether 62 is a term of the list of
numbers 4, 7, 10, .....?
A. Given list of numbers 4, 7, 10, ... is an A.P.
For this A.P. we have a = 4, d=3
Let we taken the nth term of A.P. is 62 i.e.
a+(n–1)d = 62 ⇒ 4+(n–1)3 = 62
n=
61
3
But n should be a positive integer....So, 62
is not a term of the given list of numbers.
2. In a G.P. 3rd term is 24 and 6th term is 192.
Find the common ratio?
A. In a G.P. a3=ar2=24, a6=ar5=192
a 6 ar 5 192
=
=
⇒ r3 = 8 ⇒ r = 2
a 3 ar 2 24
∴ Common ratio = r = 2
3. How many two-digit numbers are divisible by 8?
A. The list of two-digit numbers divisible by
8 is 16, 24, 32, ...96
In this A.P. a=16, d=8
tn = a+(n–1) d= 96 ⇒ 16+(n–1)8 = 96
n=11
∴ There are 11 two-digit numbers divisible by 8.
2 Marks
1. Determine the A.P. whose 3rd term is 5 and
the 7th term is 9?
You
should never
remain awake till
late night a day before
the examination as this
can create headache, the
next morning and
you might end
up in trouble.
Aryabhata the famous Indian mathematician gave formulas for the
sum of squares and cubes of natural numbers. His work was
"Arybhateeyam' (499 A.D.). He gave a formula for finding the sum of
"n terms' of an Arithmetic progression starting with any term.
2. A man repays a loan of Rs. 3250 by paying Rs. 20 in the first month and then
increases the payment by Rs. 15 every
month. How long will it take him to clear
the loan?
3. Which term of the G.P. 2, 2√2, 4, .. is 128?
4 Marks
1. Determine 32 into four parts which are in
A.P. such that the product of extremes is to
the product of means is 7:15?
2. If the Geometric progressions 162, 54, 18,
BD AB
4
6
=
⇒ =
DC AC
3 AC
⇒ AC = 4.5cm
∴
2. ∆ABC, ∆DEF are similar triangles and
their areas are respectively 81cm2 and 169
cm2. If EF =26 cm, then Find BC?
2
2
Area ∆ABC  BC 
81  BC 
=
=
 ⇒

Area ∆DEF  EF 
169  26 
A.
BC
81
9
9
=
= ⇒ BC = × 26 = 18c.m.
26
169 13
13
3. From the
figure find AD
A
7. COORDINATE
GEOMETRY
9
IMPORTANT QUESTIONS
1 Mark
1. Ganesh says that the distance between two
points A(4,2), B(8,6) is 4 units Do you
agree with Ganesh or not? Why?
A. Distance between
AB = (8 − 4) 2 + (6 − 2) 2 = 16 + 16
= 32 = 16 × 2 = 4 2 units .
But Ganesh says that AB = 4 units.
∴ I cannot agree with Ganesh statement.
2. Find the centre of the circle, whose vertices of a diameter are (–4, 3), (2, 5)?
A. Centre = midpoint of (–4, 3), (2, 5)
 −4 + 2 3 + 5 
=
,
 = (− 1, 4)
2 
 2
3. Find a relation between x and y such that
the point (x,y) is equidistant from the
points (3,0) and (0,4)?
A. Let P = (x, y), A=(3,0), B=(0,4)
Given PA=PB ⇒ PA2 = PB2
⇒ (x–3)2+(y–0)2 = (x–0)2+(y–4)2
⇒ x2–6x+9+y2 = x2+y2–8y+16
–6x+8y–7=0 ⇒ 6x–8y+7=0
Required conditon 6x–8y+7=0
2 Marks
1. Find the ratio in which the line segment
joining the points (–3,10) and (6,–8) is
divided by (–1, 6)?
2. Determine x so that 2 is the slope of the
line through P(2,5) and Q(x,3)?
4 Marks
1. If P(2,–1), Q(3,4), R(–2,3) and S(–3, –2)
be four points in a plane, show that PQRS
is a rhombus but not a square. Find the
area of the rhombus?
2. Find the points of trisection of the line segment joining the points(5,–6) and (–7,5)?
3. Find the area of the triangle formed by
joining the mid-points of the sides of the
triangle whose vertices are (0,–1) (2, 1),
(0, 3). Find the ratio of this area to the area
of the given triangle?
D
4
C
E
B
12
A. From the given figure DE = BC = 12
AE = AB – BE = 9–4=5
From Pythogoras theoreum
AD2 = AE2+ED2 = 52+122=169
AD = 169 =13
2 Marks
1. Can you show that sum of the squares of
the sides of a rhombus is equal to the sum
of the squares of its diagonals?
2. BL and CM are medians of a triangle
ABC, right angled at A. Prove that
4(BL2+CM2) = 5BC2?
4 Marks
1. AB, CD, PQ are perpendicular to BD.
AB = x, CD=y and PQ = z
1
1
1
Prove that x + y = z ?
C
A
P
x
y
z
B
D
Q
2. ABC is a right triangle at C. Let BC = a,
CA = b, AB = c and let 'P' be the length of
perpendicular from C on AB prove that
1
1
8. SIMILAR TRIANGLES
IMPORTANT QUESTIONS
1 Mark
1. "Basic Proportionality theorem is applicable only for right angled triangles". Can
you agree or not? Discuss?
A. No, I cannot agree, because Basic
3. State and Prove Basic Proportionality theorem?
4. State and Prove Pythagoras theorem?
5 Marks
1. Construct an Isosceles triangle whose base
is 9 cm and altitude is 5 cm. Then, draw
another triangle whose sides are 1½ times
the corresponding sides of the Isosceles
triangle?
9. TANGENTS AND
SECANTS TO A CIRCLE
1 Mark
1. If PA and PB are two tangents from a point
P to a circle with centre O and are inclined
to each other at an angle of 80o, then find
POA ?
A
40
40
O
B
L
2. Two Parallel lines touch the circle at
points A and B. If area of the circle is 25π
sq.cm then find the length of AB?
A
A. Area of circle
π r2 = 25π sq.cm
•
r = 5cm.
AB = 2r = 10 cm
B
3. Two concentric circles of radii 5 cm and 3
cm are drawn. Find the length of the chord
of the larger circle which touches the
smaller circle?
A. Length of chord
= 2 r12 − r22 (r1 > r2 )
= 2 (5) 2 − 3 2 = 8cm
A
0 r
r2 1
P
B
2 Marks
1. Prove that the lengths of two tangents
drawn from an external point to a circle
are equal?
2. Find the area of the shaded region in figure, if ABCD is a square of side 7 cm and
APD and BPC are semicircles?
22 

 use π =

7 

D
C
P
A
P
A. From the figure r=5 cm
h=12 cm
Slant height
10. MENSURATION
IMPORTANT QUESTIONS
1 Mark
1. How many balls, each of radius 1 cm, can
be made from a solid sphere of lead of
radius 2 cm?
Volume of solid sphere of lead
A. No. of balls =
Volume of small ball

11. TRIGONOMETRY
1
3
Volume of cone = πr 2 h
Ratio of their volumes
3
πr 2 h
= = 3 :1
1 2
πr h 1
3
IMPORTANT QUESTIONS
1 Mark
1. "The mutiplicative inverse of
1 − cos θ 1 + cos θ "
is
explain?
sin θ
sin θ
A. If a, b are multiplicative inverse to each
other then ab=1
here
1 − cos θ 1 + cos θ 1 − cos 2 θ sin 2 θ
.
=
=
=1
sin θ
sin θ
sin 2 θ
sin 2 θ
1 − cos θ
is multiplicative inverse to
sin θ
1 + cos θ
sin θ
∴
2. For what value of 'θ', the trigonometric
ratio sine, cosine are equal?
A. If θ=45o sinθ = sin 45o =
cosθ = cos 45o =
1
2
1
2
∴ sinθ = cosθ, if θ = 45o
3. In a ∆ABC, sin A =
3
then find the area of
5
the triangle?
3
5
From the figure
2. "A cylinder and cone have bases of equal
radii and are of equal heights. Then their
volumes are in the ration 3:1". Explain?
A. A cylinder and cone have equal radii r, and
equal height h.
Volume of cylinder = π r2h
=
22 
of radius 3.5 cm  π =  ?
7 

2. A right circular cylinder has base radius 14
cm and height 21 cm, then find volume of
the right circular cylinder?
3. If the volume of a cube is 1728 cm3, then
find the height of the cube?
4 Marks
1. A sphere, a cylinder and a cone are of the
same radius and same height. Find the
ratio of their curved surface areas?
2. A tent is in the form of a cylinder of diameter 4.2m and height 4m surmounted by a
cone of equal base and height 2.8 m. Find
the capacity of the tent and the cost of canvas for making the tent at Rs.100/Sq.m.
3. A 20 m deep well with diameter 7m. is
dug and the earth from digging is evenly
spread out to form a platform 22m by 14
m find the height of the platform?
A. In ∆ABC, sin A =
4
π(2) 3
=3
=8
4
π(1) 3
3
5
2 Marks
1. Find the total surface area of a hemisphere
B
4 Marks
1. A chord of a circle of radius 12 cm subtends an angle of 1200 at the centre. Find
the area of the corresponding minor segment of the circle (use π = 3.14 and √3 =
1.732)?
2. If a circle touches all the four sides of a
quadrilateral ABCD at points PQRS, then
show that AB+CD=BC+DA?
5 Marks
1. Draw a pair of tangents to a circle of
radius 5 cm which are inclined to each
other at an angle 60o?
12
L = r 2 + h2 = 52 + 122 = 13c.m.
7 cm
1
IMPORTANT QUESTIONS
PAPER - II
3. From the figure find the
slant height of circular cone?
1
1
APB = (80 o ) = 40 o
2
2
OAP = 90o (∵ OA ⊥ AP)
POA = 180o − (90o + 40o ) = 50o
A. OPA =
ii) p 2 = a 2 + b2
i) pc = ab
3
10th Class Special - Maths
Proporationality theorem is applicable for
any triangle.
2 2 2
, , ,.... have their nth term
... and
31 27 9
equal. Find the value of n?
VýS$Æý‡$ÐéÆý‡… l íœ{ºÐ]lÇ l 11 l 2016
The
syllabus has
been understood
effectively and the basics
of the topic are clear in
your mind, you will be able
to solve any given
problem with minimal efforts on
your part.
4
10th Class Special - Maths
A. From the given data
C
h
1
h
tan 30 =
⇒
=
30
3 30
o
5
3
B
A
AB2 = AC2 – BC2
AB2 =52 –32 =16
AB = 16 = 4
1
Area of ∆ABC = ×base×height
2
=
1
× 4 × 3 = 6sq.units
2
4. Express sin 75o+cos55o in terms of
trigonometric ratios of angles between 0o
and 45o?
A. Sin 75o + cos55o
= sin (90o–15o) + cos (90o–35o)
= cos 15o + sin 35o
2 Marks
1. Is it right to say that sin (A+B) = sinA+
SinB justify your answer?
2. If sin (A–B) =
1
1
, cos (A+B) = ,
2
2
0o<A+B ≤ 90o A>B, then find A and B?
3. If A, B and C are interior angles of a triangle ABC, then show that
C
 A + B
tan 
 = cot ?
2
 2 
4 Marks
1. Prove that
1 + cos θ
= cos ecθ + cot θ ?
1 − cos θ
2. In a right angle triangle ∆ABC, right angle
at B, if tan A = √3 then find the value of
i) sinA cosC + cosA sinC
ii) cos A cosC – sinA sinC
3. If secθ+ tan θ=k. then prove that
sin θ =
Carl Friedrich Gauss (1777-1855) the great German mathematician,
proposed a formula to find the Sum of first "n' terms in Arithmetic
Progression. He contributed significantly to many fields like number
theory, algebra, geophysics, optics etc.
VýS$Æý‡$ÐéÆý‡… l íœ{ºÐ]lÇ l 11 l 2016
k2 −1
?
k2 +1
12. APPLICATIONS OF
TRIGONOMETRY
h=
30 30 × 3 30 3
=
=
= 10 3m
3
3
3× 3
2 Marks
1. A circus artist is climbing a 20 m long
rope, which is tightly stretched and tied
from the top of a vertical pole in the
ground. Find the height of the pole if the
angle made by the rope with the ground
level is 30o?
2. The angle of elevation of a ladder leaning
against a wall is 60o and the foot of the
ladder is 9.5m away from the wall. Find
the length of the ladder?
4 Marks
1. A tree breaks due to storm and the broken
part bends so that the top of the tree touches the ground by making 300 angle with
the ground. The distance between the foot
of the tree and the top of the tree on the
ground is 6m. Find the height of the tree
before falling down?
2. Two pillars of equal height and on either
side of a road, which is 100 m wide. The
angles of elevation of the top of the pillars
are 60o and 30o at a point on the road
between the pillars. Find the position of
the point between the pillars and the
height of each pillar?
5 Marks
1. An aeroplane at an altitude of 200 meters
observes the angles of depression of opposite points on the two banks of a river to be
45o and 60o. Find the width of the river?
2. As observed from the top of a light house,
100m above sea level, an angle of depression of a ship, sailing directly towards it,
changes from 30o to 45o. Determine the
distance travelled by the ship during the
period of observation?
IMPORTANT QUESTIONS
1 Mark
1. The height of the pole and the length of its
shadow are equal. Then, what is the angle
of elevation of the sun rays with the
ground at that time?
A
h
θ
B
C
h
∴ θ= 45o
∴ The sun rays make 45o angle with the
ground at that time.
2. The angle of elevation of the top of a
tower from a point on the ground which is
30 m away from the foot of the tower is
30o. Find the height of the tower?
A
h
30o
30m
Number of outcomes favourable to E 1
=
Number of all possible outcomes
2
Die
E = getting an even number = [2, 4, 6]
h
tan θ = = 1 = tan 45 o
h
B
25 1
=
100 4
∴ P(E) =
3. From the figure find the probability of getting a ball of prime number?
14
13 12
1
2
15
11 10
3
4
9
5
8
6
7
A. From the figure number of all possible out
comes = 15
The numbers of outcomes favourable to
the event getting a ball of prime numbers
= 6. ({2, 3, 5, 7, 11, 13})
∴ Probability =
6 2
=
15 5
2 Marks
1. Harpreet tosses two different coins simulataneously. What is the probability that he
gets atleast one head?
2. One card is drawn from a well -shuffled
deck of 52 cards. Find the probability of
getting a face card?
3. A bag contains two black, three white, 5
yellow balls. One ball is drawn at random
from the bag. Find the probability that the
ball is drawn is not yellow?
4 Marks
1. Two dice are rolled simulataneously and
counts are added. Complete the table
given below?
Event sum
2 3 4 5 6 7 8 9 1011 12
on 2 dice
1
5
1
Probability
36
36
36
2. A box contains 100 discs which are numbered from 1 to 100. If one disc is drawn
at random from the box, find the probability that it bears
i) Two-digit numbers divisible by 5
ii) Factor of 100
iii) A perfect square number.
IMPORTANT QUESTIONS
1 Mark
1. Are the probabilities of getting a head
when the coin is tossed, and getting an
even number, when a die is thrown once,
equal or not? explain?
A. coin
E = the event "getting a head"
Probability P(E)
=
A. From the given data
C
13. PROBABILITY
The no. of favourable out comes to the
event E =25
P(E) =
3 1
=
6 2
∴ Probability in both situations are equal.
2. A kiddy bank contains twenty five Rs.1
coins, forty three Rs.2 coins, thirty two
Rs.5 coins. If it is equally likely that one
of the coins will fall out when the kiddy
bank is turned upside down. What is the
probability that the coin will be Rs.1?
A. The no. of possible out comes = 25 + 32 +
43 = 100
Let E denote the event ''the coin will be
Rs.1''
14. STATISTICS
IMPORTANT QUESTIONS
1 Mark
1. "Mean takes into consideration extreme
values of the data only" will you agree
with the statement? or not? why?
A. Not agree, because mean takes into
account all the observations, and lies
between the extremes.
2. Mean of observations 10, 15, 20, p, 30, 35,
40 is 25. Then find the value of P?
A. Mean ==
⇒ 25 =
2 Marks
1. Write the formula of median for a grouped
data. Explain the symbols in words?
2. Convert the given below distribution to a
'more than' type cumulative frequency distribution?
Class Interval
50-55
55-60
60-65
65-70
70-75
75-80
Frequency
2
8
12
24
38
16
4 Marks
1. A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes, and summarised
this in the table given below.
Find the mode of the data?
No. of Cars
0-10
10-20
20-30
30-40
40-50
50-60
60-70
70-80
Frequency
7
14
13
12
20
11
15
8
2. The following distribution show the daily
pocket allowance of children of a localilty.
The mean pocket allowance is Rs. 18.
Find the missing frequency f?
Daily Pocket
No. of
Allowance
Children
(in Rs.)
11-13
7
13-15
6
15-17
9
17-19
13
19-21
f
21-23
5
23-25
4
3. During the medical check-up of 35 students of a class, their weights were recorded as follows?
Weight
(in kg)
less than 38
less than 40
less than 42
less than 44
less than 46
less than 48
less than 50
less than 52
No.
of students
0
3
5
9
14
28
32
35
Draw a less than type ogive curve for the
given data. Hence obtain the median
weight from the graph?
Sum of observations
No.of observations
10 + 15 + 20 + p + 30 + 35 + 40
7
⇒ 25×7= p+150 ⇒ p=175−150= 25.
3. The heights of the students in a class are as
follows (in feets)
5.2, 5.6, 5.3, 4.9, 5.1, 5.2, 5.5, 5.3, 4.8, 5.7,
5.3, 4.8.
Find the mode of the data?
A. A mode is that value among the observations which occurs most frequently
∴ Mode of the given data = 5.3
Making brief
notes of the topics
always helps and you
should prepare brief notes
of whatever your teachers
share with you. These
help you prepare
effectively as and
when the exams
approach.
In a right angled triangle the square of the hypotenuse is equal to the
sum of the squares of the other two sides. Pythagoras (570 BC-495
BC), the great Greek mathematician announced it. More than 50
proofs are available for this theorem.
REAL NUMBERS
1. According to Euclid's Division Lemma,
given positive integers a and b, there exist
unique integers q and r satisfying a=bq+r
Here, the inequalities satisfied by r are
( )
1) 0 ≤ r < b
2) 0 < r < b
3) 0 < r ≤ b
4) 0 ≤ r ≤ b
2. The remainder when the square of any
prime number greater than 3 is divided by
6, is
( )
1) 1
2) 3
3) 2
4) 4
( )
3. log2 9 + log24 – log218=
1) 0
2) log24 3) log212 4) 2
Key: 1) 1; 2) 1; 3) 1
1. Set A = {F, L, W, O}. Which of the following is not a set builder form for set A
( )
1) {x : x is a letter from the word FOLLOW}
2) {x:x is a letter from the word FLOW}
3) {x: x is a letter from the word WOLF}
4) {x:x is a letter from the word SLOW}
2. From the figure which statement is true
( )
B
a
d
g
h b
µ
c
1) A∩B = {d, g}
2) A–B = {a, b, h}
3) B–A = {c, e, f}
4) All the above
3. Which statement is true
( )
1) A∩ φ = φ∩A = A
2) A∪φ = φ∪A=φ
3) A–B, A∩B, B–A are disjoint sets
4) None
Key: 1) 4; 2) 4; 3) 3
POLYNOMIALS
1. A cubic polynomial with the sum, sum of
the product of its zeroes taken two at a
time, and the product of its zeroes are 2,
–7 and –14 respectively, is
( )
2) x3–2x2+7x+14
1) x3+2x2+7x+14
4) x3–2x2–7x–14
3) x3–2x2–7x+14
2. Which of the following is not a quadratic
polynomial
( )
2) (x–2)2 +4
1) x2 + 5x+6
4) (x–3) (x+3) – (x2–5x)
3) 2x2 – 3
3. The graph of the equation y = ax2+bx+c is
an upward parabola, if
( )
1) a> 0 2) a<0
3) a= 0 4) a = –1
Key: 1) 3; 2) 4; 3) 1
PAIR OF LINEAR EQUATIONS
IN TWO VARIABLES
1. Which of the following is not a linear
equation
( )
1) x+y=6
2) x2–5x+6=0
3) y = 2x
4) x=0
2. If x = 1, then the value of y in the equation
4 3
+ = 5 is –––
x y
1) 1
2)
1
3
(
3) 3
)
4) –3
3. If the sum of the two digits of a two-digit
number is 12. The number obtained by
1) 27:20 2) 20:27 3) 9:4
4) 4:9
Key: 1) 4; 2) 2; 3) 2
Multiple Choice Questions
TRIGONOMETRY
1. From the figure, Sin θ=
interchanging the two digits exceeds the
given number by 18, then the number is
( )
1) 75
2) 57
3) 66
4) 48
4. If the pair of linear equations a1x+b1y+c1
= 0; a2x+b2y+c2=0 has a unique solution,
then
( )
a1 b1 c 1
1) a = b = c
2
2
2
a1 b1 c1
2) a = b ≠ c
2
2
2
a1 b1
3) a ≠ b
2
2
4) None
1
x
3) x + = 3
A
13
5
θ
B
(4,3), then the third vertex is
1) (13,9)
2) (9, 13)
3) (13,–9)
4) (–9, –13)
Key: 1) 2; 2) 3; 3) 1
1
=2
x2
4) (x+1)(x+2)(x+3)=0
Key: 1) 2; 2) 1; 3) 3
PROGRESSIONS
1. Which of the following in not an A.P.(
)
C
1) 5/12 2) 12/13 3) 5/13 4) 13/5
2. From the figure, tanQ-tan R =
( )
1. The standard form of a quadratic equation
is
( )
1) ax+b=0, a≠0
2) ax2+bx+c=0, a≠0
3) a2x+b2=0, a≠0
4) ax3+bx2+cx+d=0, a≠0
2. Which of the following is a quadratic
equation
( )
1) (x+1)2 = 2(x–3)
2) (x–2) (x+1) = (x–1) (x+3)
3) x2+3x+1 = (x–2)2
4) x3–1=0
3. Which of the following is a quadratic
equation
( )
2
2) x +
(
12
QUADRATIC EQUATIONS
1) x3–6x2+2x–1=0
e f
5
10th Class Special - Maths
Key: 1) 2; 2) 3; 3) 2; 4) 3
SETS
A
VýS$Æý‡$ÐéÆý‡… l íœ{ºÐ]lÇ l 11 l 2016
)
−5
−3
1) −3, , − 2, ,.......
2
2
2) 0.3, 0.33, 0.333, ......
3) 3, 12, 27, 48,......
4) p1 2p+1, 3p+2, 4p+3, ......
2. Which term of the A.P. 92, 88, 84, 80, –––
is 0
( )
1) 23
2) 32
3) 22
4) 24
3. If i) –1.0, –1.5, –2.0, –2.5, ... and
ii) –1, –3, –9, –27, ....
are two progressions, they which of them
is a Geometric progression
( )
1) i) only
2) ii only
3) i and ii both
4) None
Key: 1) 2; 2) 4; 3) 2
COORDINATE GEOMETRY
1. In a co-ordinate plane, if line segment AB
is parallel to X-axis, then which of the following is correct
( )
1) x co-ordinates of points A and B are
equal
2) y co-ordinate of point A and B are equal
3) x co-ordinate of point A and y co-ordinate of point B are equal
4) y co-ordinate of point A and x co-ordinate of point B are equal
2. The points which divide a line segment
into 3 equal parts are said to be the ( )
1) Midpoint
2) Centroid
3) Trisectional points 4) None
3. If two vertices of a triangle are (3, 6) and
(–4, –5) and the centroid of the triangle is
(
)
SIMILAR TRIANGLES
1. Observe the following
i) Any two similar figures are congruent
ii) Any two congruent figures are similar
iii) Which of the following is true
( )
1) Only (i) is true
2) Only (ii) is true
3) Both (i) and (ii) are true
4) Both (i) and (ii) are false
2. Basic proportionality theorem is also
known as
( )
1) Pythogorous theorem
2) Thales theorem
3) Fundamental theorem of Arithmetic
4) None
3. A man goes 24m due west and then 7m
due north. How far is he from the starting
point
( )
1) 31m 2) 17m
3) 25m 4) 26m
Key: 1) 2; 2) 2; 3) 3
TANGENTS AND
SECANTS TO A CIRCLE
1. The length of tangents from a point A to a
circle of radius 3cm is 4cm, then the distance between A and the center of the circle is
( )
1) 7cm 2) 6cm
3) 5cm 4) 4.5cm
2. In the figure, ∠BAC=
( )
A
B
600
600
C
1200
2)
3) 900 4) 300
1)
3. Number of circles passing through 3
collinear points in a plane is
( )
1) 3
2) 2
3) 1
4) 0
Key: 1) 3; 2) 4; 3) 3
MENSURATION
1. The radius of spherical baloon increases
from 8cm to 12 cm. The ratio of the surface area of the baloon in two cases is ( )
1) 2:3
2) 3:2
3) 8:27 4) 4:9
2. Volume of a cylinder is 88cm3 and the
radius is 2cm, then the height of the cylinder is ––– cm
( )
1) 8.5
2) 7
3) 6.4
4) None
3. The radii of the two cylinders are in the
ratio 2:3 and their heights are in the ratio
5:3. The ratio of their volumes is
( )
R
5
P
3
Q
1) 4/3
2) 3/4
3) 1
4) 7/12
3. Which of the following values is not possible with sin θ
( )
1) 3/4
2) 3/5
3) 4/5
4) 5/4
Key: 1) 3; 2) 4; 3) 4
PROBABILITY
1. If an event occurs surely, then its probability is
( )
1) 0
2) 1
3) 1/2
4) 3/4
2. A bag contains 8 red, 6 white and 4 black
balls. A ball is drawn at random from the
bag. The probability that the drawn ball is
neither white nor black is
( )
1) 4/9
2) 7/9
3) 5/9
4) 1/9
3. A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at
random from the box, the probability that
it bears a perfect square number is ( )
1) 8/90 2) 3/90
3) 1/10 4) 3/10
Key: 1) 2; 2) 1; 3) 3
STATISTICS
1. Which of the following is not a measure of
central tendency
( )
1) Mean
2) Median
3) Mode 4) Standard deviation
2. Mode is the value of the variable which
has
( )
1) Maximum frequency
2) minimum frequency
3) Mean frequency
4) middle most frequency
3. Which is the better measure of central tendency when individual observations are
not important
1) mode 2) median 3) mean 4) none.
Key: 1) 4; 2) 1; 3) 2
Do
not leave the
things to the end as
this can create unnecessary pressure on you
and can affect your
examination
results eventually.
Probability theory had its origin in the 16th century when an Italian physician
and mathematician J.Cardan wrote the first book on the subject, The Book
on Games of Chance. James Bernoulli, A.DeMoivre, and Pierre Simon
Laplace are among those who made significant contributions to this field.
VýS$Æý‡$ÐéÆý‡… l íœ{ºÐ]lÇ l 11 l 2016
10th Class Special - Maths
4. If α, β are the zeroes of the polynomial
PAPER - I
p(x)=
1. REAL NUMBERS
1. If m is some integer, then the square of
any positive integer is of the form____.
2. 7×11×13 + 13 is ____.
3. The exponent of 2 in the prime factorisation of 144, is ____.
4. If two positive integers m and n are
expressible in the form m= pq3 and n=
p3q2 where p, q are prime numbers, then
HCF (m,n) = ____.
5. The exponential form of log10 0.001 = –3
is ____.
6. The product of two irrational numbers is
____.
7. The rational number between √2 and √3 is
____.
8. The smallest number by which √27 should
be multiplied so as to get a rational number is ____.
9. log20162016 = ____.
10. The exponential form of log264=6 is ____.
ANSWERS
1) 3m or 3m+1; 2) a composite number;
3) 4; 4) pq2; 5) (10)–3 = 0.001; 6) sometimes a rational number, sometimes an
2. SETS
–3 –2 –1
3
4
5
6
7
9
8
10
5. If n (A) = 7, n(B)= 8, n (A∩B)= 5 then
n (A∪B) = ____.
6. If A = {5x/x∈N, x≤4} and
B = {5x/ x∈N, x≤4 } then A∩B = ____.
7. If A, B are disjoint sets then ____.
8. If A = {0, 2, 4}, then A∩ φ = ____.
9. If A⊂B then A∩B= ____.
10. If the union of two sets is one of the set
itself, then the relation between the two
sets is ____.
ANSWERS
4
; 2) 99o, 81o; 3) 57; 4) Coincident
3
x
1
1
lines; 5) 3 + 10 ; 6) x = , y = 7) 2; 8) 6;
2
3
1)
9) x=0, y=0; 10) Inconsistent equations
5. QUADRATIC EQUATIONS
1) 2; 2) 2b = a+c; 3) 25; 4) 2; 5) 960;
6) 2; 7) 3072; 8) –1; 9) 17; 10) 1; 11) 128
1. The roots of the quadratic equation
x2 − 8 1
= are ____.
x 2 + 20 2
–4 –3 –2 –1 0
1 2 3 4
x
9. The quadratic polynomial, whose zeros
are 2 and 3 is ____.
10. If one root of the polynomial f(x) =
5x2+13x+k is reciprocal of the other, then
the value of k is ____.
ANSWERS
−3
−3
; 5) 2 ; 6) 3; 7) –1,
7
2; 8) –2, 1; 9) x2–5x+6; 10) 5
4. PAIR OF LINEAR
EQUATIONS IN
TWO VARIABLES
1. If the line y = px–2 passes through the
point (3, 2), then the value of P is ____.
2. If the larger of two supplementary angles
exceeds the smaller by 18 degrees, then
the angles are ____.
3. The sum of the two digits of a two-digit
number is 12. The number obtained by
interchanging the two digits exceeds the
given number by 18, then the number is
____.
4. If the pair of lines 2x+y+5=0 and 4x+2y
+10=0 represent ____ lines
5. The age of a son is one-third of the age of
his father. If the present age of father is x
years, then the age of the son after 10
years is ____.
6. The solution of the pair of equations
2 3
5 4
+ = 13; + = − 2 is ____.
x
y
x
2. If the product of two consecutive natural
numbers is 72, then the natural numbers
are ____.
3. The roots of x2–2x–(r2–1)=0 are ____.
4. ____ is the conditon that one root of the
quadratic equation ax2+bx+c=0 is reciprocal of the other.
5. The product of the roots of the quadratic
equation x2–5x+6=0 is ____.
6. If α, β are the roots of the quadratic equation √2x2+7x+5√2=0, then αβ = ____.
7. If one root of the equation 4x2–2x+(λ–4)
=0 be the reciprocal of the other, then λ =
____.
8. The sum of a number and its reciprocal is
5/2 Represent this situation as ____.
9. From the figure the quadratic equation has
____.
Y
X
0
10. From the figure the roots of the quadratic
equation are ____.
7. From the graph the value of K is ____.
Y
2
y=
x+
X
0
8. ____ is the value of 'k', the pair of equations 3x+4y+2=0 and 9x+12y+k=0 represent coincident lines.
9. Solution for the equations √3x+√5y=0 and
√7x+√11y=0 is ____.
10. A pair of linear equations in two variables
7. COORDINATE
GEOMETRY
1. The distance of the point (3, 4) from Xaxis is ____.
2. The distance between (x1,y1) and (x2, y2)
is ____.
3. The triangle with vertices (–2, 1), (2,–2)
and (5,2) is ____.
4. The co-ordinates of the point, dividing the
join of the points (5,0) and (0,4) in the
ratio 2:3 internally are____.
5. The co-ordinates of the centroid of the triangle whose vertices are (8,–5), (–4,7) and
(11, 13) are ____.
6. If two vertices of a triangle are (3, 6) and
(–4, –5) and the centroid of the triangle is
(4,3), then the third vertex is ____.
7. If (1,2), (4, y), (x,6) and (3, 5) are the vertices of a parallelogram taken in order,
then the values of x and y are ____.
8. 'Heron's formula to find the area of a triangle is ____.
9. From the figure, if area of ∆ABC=5sq.
units, then the area of given parallelogram
is ____ sq.units.
D
C
2
1
0
–3–2–1
123
ANSWERS
1) ±6; 2) 8, 9; 3) 1–r, 1+r; 4) a = c; 5) 6;
1
x
6) 5; 7) 8; 8) x + =
5
; 9) No roots;
2
10) –2, 1
y
3. POLYNOMIALS
1. The degree of a constant polynomial is
____.
2. The number of zeros that the polynomial
f(x) = (x–2)2 +4 can have is ____.
3. If –1 is a zero of the polynomial f(x) =
x2–7x–8 then the other zero is ____.
ANSWERS
6. The common ratio of the G.P. 3, 6, 12, 24,
... is ____.
7. In a G.P. 3rd term is 24 and 6th term is 192,
then 10th term is ____.
8. The 100th term of 1, –1, 1, –1, .... is ____.
9. In a garden there are 32 rose flowers in
first row and 29 flowers in 2nd row, 26
flowers in 3rd row, then ____ rose flowers
are there in the 6th row
10. The common difference of an Arithmetic
progression, whose 3rd term is 5 and 7th
term is 9, is ____.
11. ____ three-digit numbers are divisible by
7.
ANSWERS
5
y=
+k
2x
1) {1, 2, 3, 4, 5}; 2) 8; 3) {4}; 4) 5; 5) 10;
6) {5}; 7) A∩B = φ; 8) φ; 9) A; 10) one
set is a subset of other set
are 2x–y=4 and 4x–2y=6. This pair of
equations are ____.
1 2 3
8. The following is the graph of the polynomial. The zeroes of the polynomial from
the graph is ____.
1) 0; 2) 2; 3) 8; 4)
1. Roster form of {x: x∈N,1≤ x≤5} is ____.
2. Cardinal no.of a set {x:x is a factor of 42}
is ____.
3. Roster form of {x:x∈N, x2+x–20=0}is
____.
4. From the figure, cardinal number of set A
____.
µ
1 A 2 B
is equal to
____.
5. If the product of two zeroes of the polynomial f(x) = 2x3 +6x2 – 4x +9 is 3, then its
third zero is = ____.
6. ____ is the coefficient of the first term of
the quotient when 3x3+x2+2x+5 is divided
by 1+2x+x2
7. In the given figure the zeroes of the polynomial f(x) are ____.
f(x)
2
1
3
; 8) √3; 9) 1; 10) 26 = 64
2
irrational; 7)
1 1
then +
α β
4x2+3x+7,
ax 2+bx+
c=0
6
6. PROGRESSIONS
1. The common difference of the A.P. –4, –2,
0, 2, ..... is____.
2. If a, b, c are in A.P., then ____.
3. The number of odd numbers between 0
and 50 is ____.
4. If the sum of first k terms of an A.P. is
3k2–k and its common difference is 6 then
the first term is ____.
5. The sum of first 15 multiples of 8 is
____.
A
B
10. If a straight line passing through the points
P (x1,y1), Q(x2, y2) is making an angle 'θ'
with positive X-axis, then the slope of the
straight line is ____.
11. The area of a trinagle whole vertices are
(0,0), (3,0) and (0,4) is ____.
12. If the slope of a line joining the points
P(2,5) and Q(x,3) is 2, then x = ____.
13. ____ is the condition that A,B,C are the
successive points of a line.
ANSWERS
1) 4; 2)
(x 2 − x) 2 + (y 2 − y1 ) 2
; 3) Right an-
8
gled isosceles; 4)  3,  ; 5) (5,5); 6) (13, 9);
 5
7) x=6, y=3; 8) s(s − a)(s − b)(s − c) ; 9) 10;
y 2 − y1
10) x − x ; 11) 6 sq.units; 12) 1;
2
1
13) AB+BC=AC;
Hipparchus, a Greek mathematician established the relationships
between the sides and angles of a triangle. The first trigonometric
table was apparently compiled by Hipparchus, who is now consequently known as "the father of trigonometry'.
PAPER - II
8. SIMILAR TRIANGLES
1. If ∆ABC and ∆DEF are similar such that
2AB=DE and BC=8cm, then EF=____.
2. In triangles ABC and DEF, ∠A = ∠E =
40o, AB:ED =AC:EF and ∠F= 65o, then
∠B = ____.
3. Sides of two similar triangles are in the
ratio 4:9. Areas of these triangles are in the
ratio ____.
4. If D, E, F are the mid-points of sides BC,
CA and AB respectively of ∆ABC, then
the ratio of the areas of triangles DEF and
ABC is ____.
5. In an equilateral triangle ABC, if AD⊥BC,
then ____.
6. If ∆ABC is an isoscles triangle and D is a
point on BC such that AD ⊥BC, then____.
7. The areas of two similar triangles are 121
cm2 and 64 cm2 respectively. If the median of the first triangle is 12.1cm, then the
corresponding median of the other triangle
is ____.
8. In the figure ∆ACB ∼ ∆APQ. If AB = 6
cm, BC = 8 cm and PQ = 4 cm then AQ =
____ cm
B
A
P
Q
C
9. In the adjacent figure, AC = 13 cm then
the length of the median BD = ____ cm
A
D
B
C
10. Vertical stick 20m long casts a shadow
10m long on the ground. At the same time,
a tower casts a shadow 50 m long on the
ground. The height of the tower is____.
11. In the figure, AD bisects ∠A. AB = 6cm
BD = 8 cm, DC = 6 cm. Then the value of
AC is ____.
A
B
D
C
ANSWERS
1) 16cm; 2) 75o; 3) 16:81; 4) 1:4; 5) 3
AB2 = 4 AD2; 6) AB2– AD2 = BD.DC; 7)
8.8 cm; 8) 3 cm; 9) 6.5 cm; 10) 100m; 11)
4.5 cm;
VýS$Æý‡$ÐéÆý‡… l íœ{ºÐ]lÇ l 11 l 2016
circle touching it at T and 'O' is the centre
of the circle, then ∠OPT+∠POT is ____.
7. From the figure, find the length of the
chord AB if PA = 6 cm and ∠PAB = 60o
A
5. If tan x = sin 45°. cos 45° + sin 30° then x
equals ____.
6. The value of 1+tan 5o. cot 85o is equal to
____.
7. In any triangle ABC, the value of
 B+C
sin 
 is ____.
 2 
60o
P
8. (sec A + tan A) (1–sinA) is equal to ____.
B
8. The number of pair of tangent can be
drawn to a circle, which are parallel to
each other are ____.
9. Circles touch internally P is external point
PM, PN, PA are tangents PM = 6 cm then
M
PN = ____.
N
9. If sin A =
24
then cot A = ____.
25
its 1, 2, 3, 4, 5, 6, 7, 8, 9 then probabilities
that it is odd, is ____.
5. A letter is chosen at random from the
English alphabet. The probabilities that
the letter chosen is a vowel is ____.
6. A bag contains 4 red, 5 black and 3 yellow
balls. A ball is taken out of the bag at random. The probability that the ball taken
out is not of red colour is ____.
7. A jar contains 27 marbles. Five of them
ANSWERS
4
; 5) 45°;
3
A
7
6) sec2 5o; 7) cos ; 8) cos A; 9)
2
24
1) sin 60° ; 2) 45°; 3) 2 ; 4)
•
P
A
ANSWERS
2
1) 2; 2) 2; 3) 360o; 4) 8 cm; 5) 102 sq cm ;
3
6) 90o; 7) 6 cm; 8) Infinite; 9) 6
10. MENSURATION
1. The area of the square that can be
inscribed in a circle of radius 8cm is ____
(in cm2)
2. The surface areas of two spheres are in the
ratio 1:4. then,the ratio of their volumes is
____.
3. If each side of a cube is doubled then its
volume becomes ____ times.
4. Diagonal of a cuboid is ____.
5. The ratio of the volume of a cube to that of
the sphere which will exactly fit inside the
cube is ____.
6. Eight solid spheres of the same size are
made by melting a solid metallic cylinder
of base diameter 6cm and height 32 cm.
The diameter of each sphere is ____.
7. ____ lead balls of radius 2 cm can be
made from a ball of radius 4 cm ____.
8. If a sphere, a cylinder and cone are of the
same radius and same height, then the
ratio of their curved surface is ____.
9. Surface area of a solid hemisphere whose
radius is 7cm is ____ cm2
10. In the picture, Height of cone is ____.
9. TANGENTS AND
SECANTS TO A CIRCLE
1. The height of the tower is 100cm. When
the angle of elevation of sun is 30o, then
shadow of the tower is ____.
2. A pole 6m high casts a shadow 2√3 m long
on the ground, then Sun's elevation is___.
3. If the height and length of the shadow of a
man are the same, then the angle of elevation of the sun is ____.
4. The angle of elevation of the top of a
tower, whose height is 100m, at a point
whose distance from the base of the tower
is 100m is ____.
5. The tops of two poles of height 20m and
14 m are connected by a wire. If the wire
makes an angle of 30o with horizontal,
then the length of the wire is ____.
6. If two towers of height h1 and h2 subtend
angles of 60o and 30o respectively at the
midpoint of the line joining their feet, then
h1 : h2 is ____.
7. From adjacent figure, h = ____.
8. A ladder 'x' meters long is laid against a
wall making an angle 'θ' with the ground.
If we want to directly find the distance
between the foot of the ladder and the foot
of the wall, ____ trigonometrical ratio
should be considered?
9. Two persons A and B observe the top of a
pole at an angle of elevation α and β
respectively. If α > β, then ____.
10. The angle of elevation of the Sun is 45o.
Then the length of the shadow of a 12 m
high tree is ____.
ANSWERS
5
1) 100 √3 m; 2) 60o; 3) 45o; 4) 45o;
5) 12m; 6) 3:1; 7) 8√3 mts; 8) cos θ; 9) A
is nearer to the pole than B; 10) 12m
2
2
2
1) 128; 2) 1 : 8; 3) 8; 4) l + b + h ;
5) 6 : π ; 6) 6cm; 7) 8; 8) 4 : 4 : √5;
9) 144π; 10) 13cm
11. TRIGONOMETRY
1. The value of
12. APPLICATIONS OF
TRIGONOMETRY
12
ANSWERS
1. ____ tangent lines can be drawn to a circle
from a point outside the circle?
2. A circle may have ____ parallel tangents
3. Sum of the central angles in a circle is___.
4. If two concentric circles are radius 5 cm
and 3 cm are drawn, then the length of the
chord of the larger circle which touches
the smaller circle is ____.
5. If the length of the minute hand of a clock
is 14 cm, then find the area swept by the
minute hand in 10 minutes ____.
6. If PT is tangent drawn from a point P to a
7
10th Class Special - Maths
2 tan 30 0
= ____.
1 + tan 2 300
2. If secθ= cosecθ, then the value of θ is ___.
3. sin 45° + cos 45° = ____.
4. If A+B= 90°, sin A= 3/4, then secB is ___.
13. PROBABILITY
1. If the probability of an event is 0.65, then
the probability of not happening of that
event is ____.
2. Two coins are tossed simultaneously. The
probability of getting a head on only one
of the two coins is ____.
3. The probability of getting a prime number
in a single throw of a die is ____.
4. If a digit is chosen at random from the dig-
are green and other are blue. If a marble is
drawn at random from the jar, the probability that it is green is 2/3. The number of
blue marbles in the jar is ____.
8. 12 defective pens are accidentally mixed
with 132 good ones. It is not possible to
just look at a pen and tell whether or not it
is defective. One pen is taken out at random from this lot. The probability that the
pen takenout is a good one is ____.
9. One card is drawn from a well shuffled
deck of 52 cards. The probability of getting a face card is ____.
ANSWERS
1
; 3)
2
2
11
6) ; 7) 9; 8) ; 9)
3
12
1) 0.35; 2)
5
1
5
; 4) ; 5)
;
26
2
9
12
52
14. STATISTICS
1. The width of the class interval 40-50 is
____.
2. The most frequently used measure of central tendency is ____.
3. The sum of the deviations of the variate
values 3, 4, 6, 7, 8, 14 from their mean
____.
4. Arun scored 36 marks in English, 44
marks in Hindi, 75 marks in Maths and x
marks in Science. If he has scored an average of 50 marks, the value of x is ____.
5. ____ cannot be determined graphically
6. The mean of first n odd natural numbers is
____.
7. The wickets taken over by a bowler in 10
cricket matches are as follows 2, 6, 4, 5, 0,
2, 1, 3, 2, 3, then the mode of this data is
____.
8. Cumulative frequency curves are called as
____ curves.
9. Data having two modes is called ____data
10. From the figure median = ____.
ANSWERS
1) 10; 2) Mean; 3) 0; 4) 45; 5) Mean;
6) n; 7) 2; 8) Ogive; 9) Bimodal; 10) 40