Chapterwise Additional questions to be practiced other than Text book questions.

1. STUDY MATERIAL FOR CLASS 10
PREPARED BY
G R AHMED
SESSION: 2020-21
NUMBER SYSTEM
Q. “The product of two consecutive positive integers is divisible by 2”. Is this statement true or false? Give
reasons.
Q. Write the number of zeroes in the end of a number whose prime factorization is 22 × 53 ×32 ×17.
Q. “The product of three consecutive positive integers is divisible by 6”. Is this statement true or false?
Justify your answer.
Q. Find after how many places of decimal the decimal form of the number
27
233254
will terminate.
𝟓
Q. If remainder of (𝟓𝐦+𝟏) (𝟓𝐦+𝟑) (𝟓𝐦+𝟒)
is a natural number then find it?( Hint: take m=1 and find
the
remainder)
Q. What are the two numbers lying between 60 and 70, each of which exactly divides 224-1?(Ans: 63,65)
Q. If HCF of 65 and 117 is expressible in the form 65n – 117, and then find the value of n.
Q. On a morning walk, three persons step out together and their steps measure 30 cm, 36 cm and 40 cm
respectively. What is the minimum distance each should walk so that each can cover the same distance in
complete steps?
Q. Use Euclid’s division algorithm to find HCF of 441, 567 and 693.
Q. Find a rational and irrational number between √2 𝑎𝑛𝑑 √3.
Q. Prove that √3+√5is irrational
5
Q. Prove that 2+√3
is irrational number. Given that √3 is irrational number.
Q. Prove that 2 + 5√3 is an irrational number, given that √3is an irrationalnumber.
Q. Prove that √p+√qis irrational, where p and q are primes.
Q. Show that 12n cannot end with the digit 0 or 5 for any natural number n.
Q. Explain why 3 × 5 × 7 + 7 is a composite number.
Q. Find total number of factors of 2000.
Q. Can two numbers have 18 as their HCF and 380 as their LCM? Give reasons.
Q. If two positive integers a and b are written as a=x3y2 and b=xy3 ; x,y are prime numbers,then find lcm
(a,b) .
Q. When n is divided by 6 the remainder is 4 find remainder when 2n is 6 divisible by?(Hint: n=6q +4,
2n=12q +8 =6(2q +1) +2= 6m +2, remainder=2)

2. Q. A positive integer n when divided by 9,gives 7 as remainder.what will be the remainder when (3n-1)
divided by 9?
Q.Arrange in ascending order:
√𝟓, 𝟑
√𝟏𝟏, and 2 𝟔
√𝟑
Q. Write the denominator of rational number 257
in the form of 2m× 5n, where m, n are non-negative5000
integers. Hence, write its decimal expansion, without actual division.
Q. On a morning walk, three persons step off together and their steps measure 40cm, 42 cm and 45cm
respectively. What is the minimum distance each should? Walk, so that each can cover the same distance in
complete steps?
1Q. If the H C F of 657 and 963 is expressible in the form of 657x + 963× (−15) find x.
2Q. Express the GCD of 48 and 18 as a linear combination.
Q. Prove that one of every three consecutive integers is divisible by 3.
Q. Find the largest possible positive integer that will divide 398, 436, and 542 leaving remainder 7, 11, 15
respectively.
Q. Find the least number that is divisible by all numbers between 1 and 10 (both inclusive).
Q. If d is the HCF of 30, 72, find the value of x & y satisfying d = 30x + 72y.
(Ans:5, -2 (Not unique)
Q. If d is the HCF of 30, 72, find the value of x & y satisfying d = 30x + 72y.
(Ans:5, -2 (Not unique)
Q. Show that the product of 3 consecutive positive integers is divisible by 6.
Q. Show that for odd positive integer to be a perfect square, it should be of the form 8k +1.
Q. Show that the square of any positive integer is either of the form 4q or 4q + 1 for some integer q.
Q. Show that the cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3 for some integer m.
Q. Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.
Q. Show that the square of any positive integer cannot be of the form (6m + 2), or (6m + 5) for any integer
m.
Q. Show that the square of any odd integer is of the form (4q + 1) for some integer q.
Q. Show that every odd positive integer is in the form of (4q+1) or (4q+3) where q is an integer.
Q. If n is an odd integer, and then show that n2 – 1 is divisible by 8
Q. Prove that, if x and y, both are odd positive integers, then (x2 + y2) is even but not divisible by 4.
Q. Show that the cube of a positive integer of the form (6q + r) where q is an integer and r = 0, 1, 2, 3, 4 and
5 which is also of the form (6m + r).
Q. Prove that one and only one out of n, (n + 2) and (n + 4) is divisible by 3, where n is any positive integer.
Q. Prove that one of any three consecutive positive integers must be divisible by 3.
Q. For any positive integer n, prove that n3 – n is divisible by 6.
Q. Show that one and only one out of n, (n + 4), (n + 8), (n + 12), (n + 16) is divisible by 5, where n is any
positive integer.

3. Q. If a and b are positive integers. Show that √2 always lies between 𝑎
and 𝑎−2𝑏
𝑏 𝑎+𝑏
Q. Prove that √𝑛− 1 + √𝑛+ 1 is irrational.
Q. Let a, b, c, and p be rational numbers such that p is not a perfect cube and a +b1/3 +cp2/3 =0 then show
that a=b=c (HOTS)
Q. Find 100th root of 𝟏𝟎(𝟏𝟎 𝟏𝟎).(hint: 100th root mean=
𝟏𝟎𝟎
√ 𝟏𝟎(𝟏𝟎 𝟏𝟎) =(𝟏𝟎 𝟏𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎)1/100=𝟏𝟎(𝟎 𝟖)
POLYNOMIAL:
Q. Find the value for K for which x4 + 10x3 + 25x2 + 15x + K exactly divisible by x + 7.(ans:-91)
Q. What must be added to the polynomial p(x)= x4 + 2x3 – 2x2 + x –1 so that the resulting polynomial is
exactly divisible by x2+2x-3.
Q.What must be subtracted from 8x4 + 14x3 – 2x2 + 7x –8 so that the resulting polynomial is exactly
divisible by 4x2+3x-2.
Q. Check whether g(x) is a factor of p(x) by dividing polynomial p(x) by polynomial g(x),
where p(x) = x5 – 4x3 + x2 + 3x + 1, g(x) = x3 – 3x + 1
Q. Find all zeros of the polynomial 3x³+10x²-9x-4. If one its zero is 1
Q. For what value of k, is the polynomial f(x) = 3x4 – 9x3 + x2 + 15x + k completely divisible by 3x2 – 5 ?
3 3
Q. Find the zeroes of the quadratic polynomial 7y2 – 11
y - 2
and verify the relationship between the zeroes
and the coefficients.
Q. If two zeros of the polynomial f(x) = x4 - 6x3 - 26x2 + 138x – 35 are 2± √3.Find the other zeros. (Ans:7, -
5)
3 3
Q. Obtain all the zeros of the polynomial p(x) = 3x4 - 15x3 + 17x2 +5x -6 if two zeroes are -1
and 1
.
(Ans:3,2)
Q. Find the Quadratic polynomial whose sum and product of zeros are √2+ 1 and
1
√2 +1
Q. If a,b are the zeros of the polynomial 2x2 – 4x + 5 find the value of a) a2 + b 2 b)(a - b)2 (Ans: a) -1 , b)
–6)
Q. On dividing the polynomial 4x4 - 5x3 - 39x2 - 46x – 2 by the polynomial g(x) the quotient is x2 - 3x – 5
and the remainder is -5x + 8.Find the polynomial g(x). (Ans:4 x2+7x+2)
Q. If the squared difference of the zeros of the quadratic polynomial x2 + px + 45 is Equal to 144, find the
value of p. (Ans:± 18).
Q. If p and q are the zeros of the polynomial f(x) =2x²-6x+3, find the value of (p3+q3) -3pq (p2 +q2) +3pq
(p+q)
Q. If p(x) a polynomial and p(a)p(b)<0 then find number of zeroes of the polynomial between a and b.
Q. What will be the value of p and q for x2 + 2x + 5 to be a factor of x4 + px2 + q?
Q. If a,b are the zeros of a Quadratic polynomial such that a + b = 24, a - b = 8. Find a quadratic polynomial
having a and b as its zeros. (Ans: x2– 24x + 128)
Q. If a & b are the zeroes of the polynomial 2x2 -4x + 5, then find the value of
i). a2 + b2 ii). 1/ a + 1/ b iii). (a -b)2 iv). 1
+ 1
v). a3+ b3
a2 𝑏2
Q.Give examples of polynomials p(x), g(x), q(x) and r(x) which satisfy the division algorithm.
a. deg p(x) =deg q(x) b. deg q(x) = deg r(x) c. deg q(x) = 0.
Q.If the ratios of the polynomial ax3+3bx2+3cx+d are in AP, Prove that 2b3-3abc+ a2d=0
Q. Find the number of zeros of the polynomial from the graph given.

4. Q. Find the value of k such that the polynomial x2-(k+6)x+2(2k-1) has sum of its zeros equal to half of their
product.
Q. If one zero of the polynomial 3x2 - 8x +2k+1 is seven times the other, find the zeros and the value of k.
(Ans k= 2/3)
PAIR OF LINEAR EQUATION IN TWO VARIABLES:
Q. At a certain time in a deer park, the number of heads and the number of legs of deer and human visitors
were counted and it was found there were 39 heads & 132 legs. Find the number of deer and human visitors
in the park. (Ans:27,12)
Q.Graphically, solve the following pair of equations 2x + y = 6 and 2x – y + 2 = 0. Find the ratio of the
areas of the two triangles formed by the lines representing these equations with equations with the x–axis
and the lines with the y–axis.
Q. Determine, graphically, the vertices of the triangle formed by the lines y = x, 3y = x, x + y = 8.
Q. Draw the graphs of the equations x = 3, x = 5 and 2x – y – 4 = 0. Also find the area of the
quadrilateral formed by the lines and the x–axis.
SOLVE:
i) 𝑥+𝑦−8
= 𝑥+2𝑦−14
=3𝑥+𝑦−12
2 3 11
ii) 7(y + 3) -2(x + 2) = 14,
iii) (a+2b)x + (2a- b)y = 2,
4 (y - 2) + 3(x -3) = 2
(a - 2b)x + (2a +b)y = 3
𝑎2 𝑏2
iii) 𝑥
+ 𝑦
= a+b, 𝑥
+ 𝑦
=2
𝑎 𝑏
iv) 2x + 3y= 17,
v) 41x + 53y = 135,
2x+2 -3y+1 =5
53x +41y =147
vi) (a-b)x +(a+b)y = a2 - 2ab – b2 , (a+b)(x+y) = a2+b2
Q. Find the value of p and q for which the system of equations represent coincident lines
2x +3y = 7, (p+q+1)x +(p+2q+2)y = 4(p+q)+1
Q. For what value of k will the following pair of linear equations have infinitely many solutions
2x-3y=7
(k+1)x +(1-2k) y = 5k -4
Q. For what value of K will the following pair of linear equation have no solution?
3x+y=1
(2k-1) x+ (k-1)y = 2k+1
Q. For what value of k will the pair of equation kx+3y=k-3, 12x+ky=k have no solutions
Q. Find the value(s) of k so that the pair of equations x + 2y = 5 and 3x + ky + 15 = 0 has a unique solution.
Q. Find c, if the system: cx +3y (3-c)=0 and 12x+ cy-c=0,has infinitely many solutions
Q. Draw the graph of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Using this graph, find the values of x
and y which satisfy both the equations.

5. Q. Students are made to stand in rows. If one student is extra in a row there would be 2 rows less. If one
student is less in a row there would be 3 rows more. Find the number of students in the class.
Q. A part of monthly hostel charges in a college hostel are fixed and the remaining depends on the number
of days one has taken food in the mess. When a student A takes food for 25 days, he has to pay Rs4,500,
whereas a student B who takes food for 30 days, has to pay Rs 5,200. Find the fixed charges per month and
the cost of food per day.
Q. The larger of two supplementary angles exceeds the smaller by 180, find them. (Ans:990,810)
Q. A train covered a certain distance at a uniform speed. If the train would have been 6km/hr faster, it would
have taken 4hours less than the scheduled time. And if the train were slower by 6km/hr, it would have taken
6 hours more than the scheduled time. Find the distance of the journey.
Q. A chemist has one solution which is 50% acid and a second which is 25% acid. How much of each
should be mixed to make 10 litres of 40% acid solution. (Ans:6L,4L)
Q.In an election contested between A and B, A obtained votes equal to twice the no. of persons on the
electoral roll who did not cast their votes & the number of person who did not caste vote was equal to twice
the majority of A over B. If there were 18000 persons on the electoral roll.
Q. The population of the village is 5000. If in a year, the number of males were to increase by 5% and that
of a female by 3% annually, the population would grow to 5202 at the end of the year. Find the number of
males and females in the village.
Q. The income of a and b are in the ratio of 3:2 and their expenditures are in the ratio of 5:3 . If each saves rs
1000 the income of B is?
Q. If the ordered pair (sin𝜽, cos𝜽) satisfies the system of equations ax+by+c=0 and bx+ay+c=0, then find
the value of𝜽.
2 3 2 3 2
Q. The length of the sides of a triangle are 2x + 𝑦
, 5𝑥
+y +1
and 2
x +2y +5
. If the triangle is equilateral. Find
its perimeter.
Q. When 6 boys were admitted & 6 girls left the percentage of boys increased from 60% to 75%. Find the
original no. of boys and girls in the class.
Q. When the son will be as old as the father today their ages will add up to 126 years. When the father was
old as the son is today, their ages add upto 38 years. Find their present ages.
Q. A cyclist, after riding a certain distance, stopped for half an hour to repair his bicycle, after which he
completes the whole journey of 30km at half speed in 5 hours. If the breakdown had occurred 10km farther
off, he would have done the whole journey in 4 hours. Find where the breakdown occurred and his original
speed. (Ans: 10km, 10km/hr)
Q. A takes 3 hours more than B to walk 30km but if A doubles his pace he is ahead of B by 1 and half hours
find the speed of walking.(10/3 and 5)
Q. Ankita travels 14 km to her home partly by rickshaw and partly by bus. She takes half an hour if she
travels 2 km by rickshaw, and the remaining distance by bus. On the other hand, if she travels 4 km by

6. rickshaw and the remaining distance by bus, she takes 9 minutes longer. Find the speed of the rickshaw
and of the bus. (10 and 40)
Q. A train covers a certain distance at a uniform speed. if the train had been 10 km/hr faster, it would have
taken 2 hour less than the scheduled time. if the train was slower by 10 km/hr, it would have taken 3 hours
more than the scheduled time. Write the two equation(Ans: 50 and 12)
Q. A motor boat can travel 30 km upstream and 28 km downstream in 7 hours. It can travel 21 km
upstream and return in 5 hours. Find the speed of the boat in still water and the speed of the stream.(10 and
4)
Q. On selling a tea set at 5% loss and a lemon at 15% gain, a shopkeeper gains rs70. If he sells the tea set at
5% gain and the lemon set at 10% gain, he gains rs130. Find the cost price of the lemon set.(Ans: 1000 and
800)
Q. The sum of a two digit number and the number obtained by reversing the digit is 66 if the digits of the
number differ by 2 find the number how many such numbers are there.
Q. If three times the larger of two number is divided by the smaller one we get 4 as quotient and 3 as
remainder also if 7 time the smaller number is divided by the larger no. and we get 5 as a quotient and 1 as
reminders find the no.
Q. Find the four angles of a cyclic quadrilateral ABCD in which angle A=(2x-1) , angle B=(y+5) , angle
C=(2y+15) and angle D=(4x-7).
Q. Draw the graph of the following equations on same graph paper ,
2x+y=2, 2x + y=6
Find the coordinates of the vertices of the trapezium formed by these lines. Also find the area of the
trapezium so formed.
Q. A says to B "my present age is five times your that age when I was as old as you are now." If the sum of
their present ages is 48 years. Find their present ages. (Ans: 30 and 18) (HOTS)
Q. Father's age is three times the sum of ages of his two children. After 5 years, his age will be twice the
sum of ages of two children. Find the age of father.
QUADRATIC EQUATION:
Q. Write whether the following statements are true or false. Justify your answer.
(i) Every quadratic equation has exactly one root.
(ii) Every quadratic equation has at least one real root.
(iii) Every quadratic equation has at least two roots.
(iv) Every quadratic equation has at most two roots.
(v)If the coefficient of x2 and the constant term of a quadratic equation have
opposite signs, then the quadratic equation has real roots.
Q. A quadratic equation with integral coefficient has integral roots. Justify your answer.

7. Q. Does there exist a quadratic equation whose coefficient are all distinct irrationals but both the roots are
rationals? Why?(Ans: yes) (hint: Take example: 3√3
x2 + 5
x - 2√2
=0)
2 √6 3
Q. Is 0.2 a root of equation x2 – 0.4 = 0? Justify your answer.
Q. Write all the values of p for which the quadratic equation x2 + px + 16 = 0 has equal roots. Find the roots
of the equation so obtained.
Q. If b = 0, c < 0, is it true that the roots of x2 + bx + c = 0 are numerically equal and opposite in sign?
Justify your answer.
Solve:
i) 4x2 – 4a2x + (a4 – b4) = 0
ii) 1
= 1
+1
+ 1
𝑎+𝑏+𝑥 𝑎 𝑏 𝑥
iii)
X=
1
2−
1
2− 1
2−𝑥
Ans: 1, 1
iv)
X=√6 + √6 + √6 + ⋯
v) √𝑎+𝑥−√𝑎+𝑥 = 𝑎
√𝑎+𝑥−√𝑎−𝑥 𝑥
vi)
𝑥2 𝑥
(x2 + 1
) – (x- 1
) =8 (x≠ 0)
vii)
√𝑥2 − 6𝑥 + 19 + √𝑥2 − 6𝑥 + 16 =1
viii)
a2x2 +b2x –a2x -1 =0
ix) 𝑥−1
+ 𝑥−3
=31
𝑥−2 𝑥−4 3
x)
x2 + 5x – (a2 + a – 6) = 0
xi)
12abx2 –(9a2 -8b2)x - 6ab=0
xii)
2 1
𝑥3 + 𝑥3 -2 =0
xiii)
9x2 -9(a+b)x +(2a2 +5ab + 2b2)=0
xiv)
(a+b)2x2 -4abx –(a-b)2=0
xv)
−1
3√𝑥 + 5(𝑥) 2 =√2
Q. By the method of completion of squares shows that the equation 4x2 +3x +5 = 0 has no real roots.
Q. Find the quadratic equation whose roots are 2+√3 and 2-√3

8. Q. The sum of areas of two squares is 468m2 If the difference of their perimeters is 24cm, find the sides of
the two squares.
Q. A dealer sells a toy for Rs.24 and gains as much percent as the cost price of the toy. Find the cost price of
the toy.
Q. A fox and an eagle lived at the top of a cliff of height 6m, whose base was at a distance of 10m from a
point A on the ground. The fox descends the cliff and went straight to the point A. The eagle flew vertically
up to a height x meters and then flew in a straight line to a point A, the distance traveled by each being the
same. Find the value of x.
Q. A lotus is 2m above the water in a pond. Due to wind the lotus slides on the side and only the stem
completely submerges in the water at a distance of 10m from the original position. Find the depth of water in
the pond.
Q. The areas of two similar triangles are 49 cm sq 64 cm sq respectively. If the difference of the
corresponding altitudes is 10 cm, then find the length of their altitudes.
Q. The hypotenuse of a right triangle is 20m. If the difference between the length of the
other sides is 4m. Find the sides.
Q. The positive value of k for which x2 +Kx +64 = 0 & x2 - 8x + k = 0 will have real roots.
Q. A pole has to be erected at a point on the boundary of a circular park of diameter 13m in such a way that
the differences of its distances from two diametrically opposite fixed gates A & B on the boundary in 7m. Is
it possible to do so? If answer is yes at what distances from the two gates should the pole be erected.
Q. If the roots of the equation (a-b)x2 + (b-c) x+ (c - a)= 0 are equal. Prove that 2a=b+c.
Q. If the equation (1 +m2)x2 +2mcx +(c2 –a2) has equal roots then prove that c2=a2(1 +m2)
Q. If p , q , r and s are real numbers such that pr = 2 (q + s) , then show that at least one of the equations x² +
px + q = 0 and x² + rx + s = 0 , has real roots
Q. If the roots of the equation x2 + 2cx + ab are real unequal, prove that the equation x2 -2(a+b ) x +
a2+b2+2c2 no real roots.
Q. If ad≠ bc, then prove that the equation (a2 + b2) x2 + 2 (ac + bd) x + (c2 + d2) = 0 has no real roots.
Q. Find he value of k for which the quadratic equation is (k+4)x2 +(k+1)x +1=0 has equal roots.
Q. X and Y are centers of circles of radius 9cm and 2cm and XY = 17cm. Z is the centre of a circle of radius
4 cm, which touches the above circles externally. Given that <XZY=900, write an equation in r and solve it
for r.
Q. Find a natural number whose square diminished by 84, is equal to thrice of 8 more than the given
number.
Q. A natural number, when increased by 12, equals 160 times its reciprocal. Find the number.
Q. A train, travelling at a uniform speed for 360 km, would have taken 48 minutes less to travel the same
distance if its speed were 5 km/h more. Find the original speed of the train.

9. Q. If Zeba were younger by 5 years than what she really is, then the square of her age (in years) would have
been 11 more than five times her actual age. What is her age now?
Q. At present Asha’s age (in years) is 2 more than the square of her daughter Nisha’s age. When Nisha
grows to her mother’s present age, Asha’s age would be one year less than 10 times the present age of
Nisha. Find the present ages of both Asha and Nisha. (Ans: 27 ,5)
Q. In the centre of a rectangular lawn of dimensions 50 m × 40 m, a rectangular pond has to be constructed
so that the area of the grass surrounding the pond would be 1184 m2 [see Fig]. Find the length and breadth of
the pond.
Q.if sin𝛼 and sin𝛽 are the roots of the equation ax2 –bx -1=0 then prove a2 –b2=2a
Q. If one root of the quadratic equation ax2 +bx+c=0 is triple the other show that 3b2 = 16ac.
Q. One fourth of a herd of camels was seen int the forest.twice the square root of herd had gone to mountain
and 15camels were seen on the river.find total no of camels
Q. If one zero of the polynomial 3x2 - 8x +2k +1 is seven times of other then find the value of k.
Q. Find the condition that one root of the quadratic equation ax2+bx+c=0 shall be n times the other,where n
is a positive integer.
Q.A two digit number is such that the product of its digits is 18.when 63 is subtracted from the number,the
digits interchange their places. Find the number.
Q. The denominator of a fraction is one more than twice the numerator. If the sum of fraction and its
reciprocal is 58/21. Find the numbers.
Q. Three consecutive positive integer are such that the sum of square of the first and product of other two is
46 find the integer
ARITHMATIC PROGRESSION:
Q. find the common difference in the AP
1
,3−𝑎
,3−2𝑎
𝑎 3𝑎 3𝑎
Q. The fourth term of an AP is 0. Prove that its 25th term is triple of its 11th term.
Q. Determine the A.P. whose third term is 16 and 7th term exceeds the 5th term by 12.
Q. How many terms of the Arithmetic Progression 45, 39, 33, ... must be taken so that their sum is 180 ?
Explain the double answer.
Q. Which term of the A.P. 3, 15, 27, 39,... will be 120 more than its 21st term?
Q. If the pth, qth & rth term of an AP is x, y and z respectively, show that x(q-r) + y(r-p) + z(p-q) = 0

10. Q. Find the sum of first 40 positive integers divisible by 6 also find the sum of first 20 positive integers
divisible by 5 or 6.
Q. Find the sum of all 3 digit numbers which leave remainder 3 when divided by 5.
Q. How many two digits numbers are divisible by 3.
Q. Find the value of x if 2x + 1, x2 + x +1, 3x2 - 3x +3 are consecutive terms of anAP.
Q. Raghav buys a shop for Rs.1,20,000.He pays half the balance of the amount in cash and agrees to pay the
balance in 12 annual installments of Rs.5000 each. If the rate of interest is 12% and he pays with the
installments the interest due for the unpaid amount. Find the total cost of the shop.
Q. Prove that am+n + am-n =2am
𝑎 𝑛+𝑏 𝑛
𝑛+1 𝑛+1
Q. If 𝑎 +𝑏
is the A.M between a and b find the value of n.
Q. If m times the mth term of an Arithmetic Progression is equal to n times its nth term and m ≠ n, show that
the (m + n)th term of the A.P. is zero.
Q.The sum of the first three numbers in an Arithmetic Progression is 18. If the product of the first and the
third term is 5 times the common difference, find the three numbers.
Q. If the roots of the equation (b-c)x2+(c-a)x +(a-b) = 0 are equal show that a, b, c are in AP.
Q. Balls are arranged in rows to form an equilateral triangle .The first row consists of one ball, the second
two balls and so on. If 669 more balls are added, then all the balls can be arranged in the shape of a square
and each of its sides then contains 8 balls less than each side of the triangle. find the initial number of balls.
𝑛 𝑛 𝑛
Q. Find the sum of (1- 1
) +(1- 2
) +(1- 3
) +……….
Q. If the following terms form a AP. Find the common difference & write the next 3 terms3, 3+ √2,3+2√2,
3+3√2……….
1 1 1
Q. if , , , are in AP find x.𝑥+2 𝑥+3 𝑥+5
Q. Find the middle term of the AP 1, 8, 15….505.
Q. Find the common difference of an AP whose first term is 100 and sum of whose first 6 terms is 5 times
the sum of next 6 terms.
Q. Find the sum of all natural no. between 101 & 304 which are divisible by 3 or 5. Find their sum.
Q. Find the sum of all 3 digits numbers which are multiple of 11.
Q. The ratio of the sum of first n terms of two AP’s is 7n+1:4n+27. Find the ratio of their 11th terms .
Q. If there are (2n+1)terms in an AP ,prove that the ratio of the sum of odd terms and the sum of even terms
is (n+1):n
Q. Find the sum of first 25th term if an =7 -3n
Q. If Sn' the sum of first n terms of an A.P is given by Sn = 5n2 + 3n then find its nthterm.
n n
𝟐𝟑𝒏 𝟏𝟑
𝒏 𝟐 𝟐
Q. If S ' the sum of first n terms of an A.P is given by S = + then find its 25th term.
Q. In an AP first term is a second term is b ,last term is c and the number of term is n, then show that
n= 𝒃+𝒄−𝟐𝒂
prove also the sum of AP=
(𝒃+𝒄−𝟐𝒂)(𝒂+𝒄)
𝒃−𝒂 𝟐(𝒃−𝒂)
Q. If a,b,c are in A.P show a(b+c)/bc,b(c+a)/ca,c(a+b)/ab are in a.p.
Q. If a,b,c are in A.P show that (a-c)2 =4(b2 –ac)
Q.solve:

11. 1 +6 +11+16+…………………..+x =148(Ans: 36)
Q. Show that if the length of the sides of a right angled triangle are in AP then they are in the ratio of 3:4:5
(Hints: take the sides are a-d,a, a+d)
COORDINATE GEOMETRY:
Q. Find the perimeter of a triangle with vertices (0, 4), (0, 0), and (3, 0)
Q. Find the centroid of triangle whose vertices are(3, -7),(-8,6)and (5, 10).
Q. Find the area of a triangle whose vertices are given as (1, – 1) (– 4, 6) and (– 3, – 5).
Q. Find the point which lies on the perpendicular bisector of the line segment joining the points A(–2, –5)
and B(2, 5)
Q. Show that the three points (a, a), (-a, -a) & (-a√3, a√3) are the vertices of an equilateral triangle.
Q. Find the fourth vertex D of a parallelogram ABCD whose three vertices are A(–2, 3), B(6, 7) and C(8, 3).
Q. Find the point that cut the perpendicular bisector of the line segment joining the points A(1, 5) and B(4,6)
at y axis.
Q. The coordinates of the point which is equidistant from the three vertices of the ΔAOB as shown in the
figure is
Q. A line intersects the y–axis and x axis at points P and Q respectively. If (2, –5) is the mid–Point of PQ,
and then find co–ordinates of P and Q.
Q. Check whether ΔABC with vertices A (–2, 0), B (2, 0) and C (0, 2) and ΔDEF with vertices D(–4, 0),
E (4, 0) and F (0, 4) are similar or not.
Q.Show that the point P(–4, 2) lies on the line segment joining the points A(–4, 6) and B(–4, –6).
Q. Show that the points A(3, 1), B(12, –2) and C(0, 2) cannot be the vertices of a triangle.
Q. Points A(–6, 10), B(–4, 6) and C(3, –8) are collinear such that AB =2
AC
9
Q. Check whether point P(–2, 4) lies on a circle of radius 6 and centre (3, 5).
Q. The mid-point of the line segment joining A(2a, 4) and B(–2, 3b) is (1, 2a + 1). Find the values of a and
b.

12. Q. If (–4, 3) and (4, 3) are two vertices of an equilateral triangle, find the coordinates of the third vertex,
given that the origin lies in the interior of the triangle.
Q. A(6, 1), B(8, 2) and C(9, 4) are three vertices of a parallelogram ABCD. If E is the midpoint of DC, then
find the area of ΔADE.
Q. Find the area of the triangle formed by joining the mid-points of the sides of the triangle ABC, whose
vertices are A(0, – 1), B(2, 1) and C(0, 3).
Q. If the points A(1, –2), B(2, 3), C(a, 2) and D(–4, –3) from a parallelogram, find the value of a and height
13
of the parallelogram taking AB a base.(Ans: a=-3 and height=12
√26)
Q. Prove that the points A(3, 0), B(4, 5), C(-1, 4) and D (-2, -1) taken in order, are the vertices of a rhombus.
Show that it is not a square, Also, find the area of the rhombus.
Q. If two vertices of an equilateral triangle are (3,0) and (6,0),find the third vertex.
Q. If the points (-1, 3), (1,-1) and (5, 1) are the vertices of a triangle. Find the length of the median through
the first vertex.
Q. Find a relation between x and y if the points A(x, y), B(– 4, 6) and C(– 2, 3) are collinear.
Q. For what value of p, are the points (-3, 9),(2,p) and(4,-5) collinear?
Q. If the points A (-2, 1),B (a,b) and C(4,-1) are collinear and a-b=1.Find the value of a and b.
Q. Find the centre of a circle passing through the points (3,-7),(3,3),(6,-6).
Q. Find the ratio in which the line segment joining the points (1,-3,) and (4,5) is divided by x axis.
Q. The line segment joining the points A (2,1) and B (5,–8) is trisected at the points P and Q such that P is
nearer to A. If P also lies on line give by 2x - y + k = 0, find the value of k.
INTRODUCTION TO TRIGONOMETRY:
5
Q. If cos A =4
then the value of all trigonometric ratio.
Q.if SinA= 𝑎
find the value of all trigonometric ratio.
𝑏
Q. Find the value of (tan 1° tan2° tan 3° … tan 89°)
Q. Find the value of the expression (cos2 23° – sin2 67°)
Q. If cos 9α = sinα and 9α < 90°, then find the value of tan 5α
Q. If ΔABC is right angled at C, then find the value of cos (A + B)
Q. If sin A + sin2A= 1, then find the value of the expression (cos2A + cos4A)
Q. If cos A + cos2 A = 1, then show sin2 A + Sin4A = 1
Q. If sin θ – cos θ = 0, then find the value of (sin2 θ + cos4 θ)
Q. If √3tan θ = 1, then find the value of sin2 θ – cos2θ.
Q. If 2 sin2 θ – cos2 θ = 2, then find the value ofθ.
Q. If xcos𝜃 – ysin𝜃 = a, xsin𝜃 + ycos 𝜃 = b, prove that x2+y2=a2+b2.

13. 2
Q. If cosec θ + cot θ = p, then prove that cos θ = 𝑝 +1
𝑝2−1
2
Q. If sec θ + tan θ = p, then prove that sinθ = 𝑝 −1
𝑝2+1
Q. if x=rsinAcosC, y=rsinAsinC, z=rcosA prove that r2 =x2 +y2 +z2
Q. If 1 + sin2 θ = 3 sin θ cos θ, then prove that tan θ = 1, or 1⁄2
Q. If sin θ + 2 cos θ = 1, then prove that 2 sin θ – cos θ = 2.
2
Q. If tan θ + sec θ = m, then prove that sec θ= 𝑚 +1
2 𝑚
Q. If a sin θ + b cos θ = c, then prove that a cos θ – b sin θ =√(𝑎2 + 𝑏2 − 𝑐2)
Q. If cosϕ+sinϕ= 2cosϕ,prove that cosϕ- sinϕ= sin ϕ
Q. If tanA+sinA=m and tanA-sinA=n, show that m²-n² = 4√mn Q.
if sin𝜃 + 𝑠𝑖 𝑛2 𝜃+sin3 𝜃=1 prove that cos6 𝜃 -4cos4 𝜃 +8cos2 𝜃 = 4
𝑥2 𝑦2
2 2
Q.If x = a sin𝜃 and If y= b tan𝜃 prove that 𝑎
- 𝑏
=1
Q.if cosA - sinA= √2 sinA then prove cosA+ sinA= √2 cosA
Q.if 𝑐𝑜𝑠𝐴
=m and 𝑐𝑜𝑠𝐴
=n the show that(m2+n2)cos2B=n2
𝑐𝑜𝑠𝐵 𝑠𝑖𝑛𝐵
𝑥
Q. The value of 2 sinA can be (x +1
), where x is a positive number, and x ≠ 1.
Q. if SecA=x + 1
then show that SecA +tana= 2x or 1
4𝑥 2𝑥
𝒙 𝟐
Q. If secA = 2x and tanA = 2/x .find the value of 2(x2 - 𝟏
)
2
Q. Find A and B if sin (A + 2B) =√3
and Cos (A + 4B) = 0, where A and B are acute angles.
𝐶𝑜𝑠𝐴+𝑠𝑖𝑛𝐴 1+ √3
Q. Solve for A if 𝐶𝑜𝑠𝐴−𝑠𝑖𝑛𝐴
= 1−√3
(Hint: use componendo and dividend)
2
Q. If A +B=90° then prove that √𝑡𝑎𝑛𝐴𝑡𝑎𝑛𝐵+𝑡𝑎𝑛𝐴.𝑐𝑜𝑡𝐵
− 𝑠𝑖 𝑛 𝐵
= tanA(entire LHS is over square root)
𝑠𝑖𝑛𝐴𝑠𝑒𝑐𝐵 𝑐𝑜𝑠2 𝐴
Q. if sinA + 𝑠𝑖𝑛2 𝐴 = 1 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒value of 𝑐𝑜𝑠12 𝐴 + 3𝑐𝑜𝑠10 𝐴 + 3𝑐𝑜𝑠8 𝐴 + 𝑐𝑜𝑠6 𝐴 + 2𝑐𝑜𝑠4 𝐴 + 2𝑐𝑜𝑠2 𝐴 -2
(Hint: Arrange in the form of (𝑐𝑜𝑠4 𝐴 + 𝑐𝑜𝑠2 𝐴)3
𝑎 𝑏 𝑎 𝑏 𝑎2 𝑏2
2 2
Q. if 𝑥
cosA + 𝑦
sinA=1 and 𝑥
sinA - 𝑦
cosA=1 prove that 𝑥
+ 𝑦
=2(HOTs)
3 3 𝑥
Q.If x= a𝑐𝑜𝑠 𝜃 and If y= b𝑠𝑖 𝑛 𝜃 prove that ( )
𝑦
𝑎 𝑏
2 2
3 3+( ) =1
Q. if sin(A +B-C)=1/2 cos(B+C-A)=1/2 and tan(C+A-B)=1 find A,B,C
Q.if 𝑠𝑖𝑛2 𝐴 + 2𝑠𝑖𝑛2 𝐴 +3𝑠𝑖𝑛2 𝐴 + 4𝑠𝑖𝑛2 𝐴 + ⋯………….upto40 terms=405
Find 𝑠𝑖𝑛2 𝐴 + 𝑐𝑜𝑠2 𝐴
Q.
Prove that:
Q. (sin α + cos α) (tan α + cot α) = sec α + cosec α
Q. ( √3+1) (3 – cot 30°) = tan3 60° – 2 tan 60°

14. 2 2
Q. cos (45°+θ )+ cos (45°−θ )
=1
tan(60°+θ ) tan(30°+θ )
Q. Prove that √𝑠𝑒𝑐2 𝜃 + 𝑐𝑜𝑠𝑒𝑐2 𝜃 = tan θ + cot θ.
Q.1+𝑠𝑒𝑐𝜃−𝑡𝑎𝑛𝜃
=1−𝑠𝑖𝑛𝜃
1+𝑠𝑒𝑐𝜃+𝑡𝑎𝑛𝜃 𝑐𝑜𝑠𝜃
2
Q.tanA –cotA=2𝑠𝑖𝑛 𝐴−1
𝑠𝑖 𝑛𝐴𝑐𝑜𝑠 𝐴
Q.
1 1
- = -
1 1
𝑐𝑜𝑠𝑒𝑐𝜃−𝑐𝑜𝑡𝜃 𝑠𝑖𝑛𝜃 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝑒𝑐𝜃+ 𝑐𝑜𝑡𝜃
Q.
1 1
- = -
1 1
𝑠𝑒𝑐𝜃−𝑡𝑎𝑛𝜃 𝑐𝑜𝑠𝜃 𝑐𝑜𝑠𝜃 𝑠𝑒𝑐𝜃+ 𝑡𝑎𝑛𝜃
𝐶𝑜𝑠𝐴+𝑆𝑖𝑛𝐴−1
Q 𝐶𝑜𝑠𝐴−𝑆𝑖𝑛𝐴+1
= CosecA + CotA (using identity cosec2 A=1+cot2A)
Q.
𝐶𝑜𝑡𝐴+𝑐𝑜𝑠𝑒𝑐𝐴−1
= 1+𝑐𝑜𝑠𝐴
𝐶𝑜𝑡𝐴−𝑐𝑜𝑠𝑒 𝑐 𝐴 +1 𝑠𝑖 𝑛𝐴
Q. Sinθ (1+tanθ)+cosθ (1+cotθ)=secθ+cosecθ
Q. 2(sin6 𝐴+cos6A) – 3(sin4A+cos4A)+1 = 0
Q. 5(sin 8A- cos 8A) = (2sin 2A – 1) (1- 2sin 2A cos 2 A)
Q. = 2 +
𝑠𝑖 𝑛𝜃 𝑠𝑖 𝑛𝜃
𝑐𝑜𝑡𝜃+𝑐𝑜𝑠𝑒𝑐𝜃 𝑐𝑜𝑡𝜃− 𝑐𝑜𝑠𝑒𝑐𝜃
Q.(1+ cotA tanA ) (sinA – cosA )
= Sin2A Cos2A
𝑠𝑒𝑐3 𝐴−𝐶𝑜𝑠𝑒𝑐3 𝐴
Q. (sin 𝐴 + 1 + cos A) (sinA – 1 + cos A) . sec A cosec A = 2
Q. (1 +cotθ - cosec θ) (1+tanθ+secθ) = 2
Q.(sinA –cosecA)(cosA – secA)=
1
𝑡𝑎 𝑛 𝐴 +𝐶𝑜𝑡𝐴
Q.(𝑠𝑖𝑛8 𝐴 − 𝑐𝑜𝑠8 𝐴) = (𝑠𝑖𝑛2 𝐴 − 𝑐𝑜𝑠2 𝐴)( 1 − 2 𝑠𝑖𝑛2 𝐴𝑐𝑜𝑠2 𝐴)
2 2 2 2
Q. (𝑡𝑎𝑛2 𝐴 − 𝑡𝑎𝑛2 𝐵)= 𝑐𝑜𝑠 𝐵−𝑐𝑜𝑠 𝐴
= 𝑠𝑖 𝑛 𝐴−𝑠𝑖𝑛 𝐵
𝑐𝑜𝑠2 𝐵𝑐𝑜𝑠2 𝐴 𝑐𝑜𝑠2 𝐵𝑐𝑜𝑠2 𝐴
EVALUATE:
Q. (1 + tan2 𝜃) (1 – sin θ) (1 + sin θ)
Q. (3𝑠𝑖𝑛43°
)2 –
𝑐𝑜𝑠47° 𝑡𝑎𝑛5°𝑡𝑎𝑛25°𝑡𝑎𝑛45°𝑡𝑎𝑛65°𝑡𝑎𝑛85°
𝑐𝑜𝑠37°𝑐𝑜𝑠𝑒𝑐53°
Q.
𝑠𝑖𝑛2 𝐴+ 𝑠𝑖𝑛2(90°−𝐴)
- 3𝑐𝑜𝑡230°𝑠𝑖𝑛254°𝑠𝑒𝑐236°
3(𝑠𝑒𝑐261°−𝑐𝑜𝑡229°) 2(𝑐𝑜𝑠𝑒𝑐265°−𝑡𝑎𝑛225°)
Q. 𝑠𝑖𝑛18°
+ √3(𝑡𝑎𝑛10°𝑡𝑎𝑛30°𝑡𝑎𝑛40°𝑡𝑎𝑛50°𝑡𝑎𝑛80°)
𝑐𝑜𝑠72°
Q. cosec (65° +A) –sec(25° − 𝐴) − tan(55° − 𝐴) + cot (35° + 𝐴)
2 2
Q 𝑠𝑒𝑐𝐴𝑐𝑜𝑠𝑒𝑐(90−𝐴)−𝑡𝑎𝑛𝐴𝑐𝑜𝑡(90−𝐴)+ 𝑠𝑖 𝑛 55+ 𝑠𝑖 𝑛 35
𝑡𝑎𝑛10𝑡𝑎𝑛20𝑡𝑎𝑛60𝑡𝑎𝑛70𝑡𝑎𝑛80
𝑠𝑒𝑐257− 𝑡𝑎𝑛233
2 2
Q. 𝑐𝑜𝑠 20+ 𝑐𝑜𝑠 70
+2𝑐𝑜𝑠𝑒𝑐258 − 2𝑐𝑜𝑡58𝑡𝑎𝑛32 −4𝑡𝑎𝑛13𝑡𝑎𝑛37𝑡𝑎𝑛45𝑡𝑎𝑛53𝑡𝑎𝑛77
Q. an equilateral triangle is inscribed in a circle of radius 6 cm. Find its side.

15. Q. In the fig. AD=4cm BD = 3cm and BC=12cm, find cot𝜽.
SOME APPLICATION OF TRIGONOMETRY:
Q. A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height ‘h’. At a
point on the plane, the angles of elevation on the bottom and top of the flag staff are α and β, respectively.
Prove that the height of the tower is
ℎ 𝑡𝑎𝑛𝛼
𝑡𝑎𝑛𝛽−𝑡𝑎𝑛𝛼
Q. The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is
60°, and the angle of elevation of the top of second tower from the foot of first tower is 30°. Find the
distance between the two towers and also the height of the other tower.( distance between the two towers is
17.32 m and the height of the second tower is 10 m.)

16. Q. From the top of a tower h m high, the angles of depression of two objects, which are in line with the foot
of the tower are α and β, (β > α). Find the distance between the two objects.( distance between the two
objects is h(cot α – cot β) m.)
Q. From a window, h metres high above the ground, of a house in a street, the angles of elevation and
depression of the top and the foot of another house on the opposite side of the street are α and β respectively.
Show that the height of the opposite house is h(1+tanα.cotβ) metres
Q. Two ships are sailing in the sea on the either side of the lighthouse, the angles of depression of two ships
as observed from the top of the top of the lighthouse are 60 degree and 45 degree respectively. if the
√𝟑
distance between the ships is( 𝟐𝟎𝟎(√𝟑+𝟏
) metres, find the height of the lighthouse.( ans: 200m)
Q. A ladder rests against a vertical wall at an inclination x to the horizontal . Its foot is pulled away from the
wall through a distance p, so that its upper end slides a distance q down the wall and then the ladder makes
an angle y to the horizontal. Show that 𝒑
= 𝒄𝒐𝒔𝒚−𝒄𝒐𝒔𝒙
𝒒 𝒔𝒊𝒏𝒙−𝒔𝒊𝒏𝒚
Q. Amit, standing on a horizontal plane, finds a bird flying at a distance of 200 m from him at an elevation
of 30°. Deepak standing on the roof of a 50 m high building, finds the angle of elevation of the same bird to
be 45°.Amit and Deepak are on opposite sides of the bird. Find the distance of the bird from Deepak.
Q. The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun’s altitude is
30° than when it was 60°. Find the height of the tower. (Given √3= 1·732)
Q. A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted
by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm3 of iron has
approximately 8 gm mass. (Use 𝜋 = 3·14)
Q. The angle of elevation of the top of a vertical tower from a point on the ground is 60°. From another point
10 m vertically above the first, its angle of elevation is 45°. Find the height of the tower.( height of the tower
= 23.66 m.)
Q. A man in a boat rowing away from a lighthouse 100m high takes 2 minutes to change angle of elevation
of the top of a lighthouse from 60 degree to 30 degree find the speed of boat in metres per minute.
Q. A window of a house is h m above the ground. From the window, the angles of elevation and depression
of the top and the bottom of another house situated on the opposite side of the lane are found to be α and β
respectively. Prove that the height of the other house is h(1 + tan α cot β) m.
Q. The lower window of a house is at a height of 2 m above the ground and its upper window is 4 m
vertically above the lower window.At certain instant the angles of elevation of a balloon from these
windows are observed to be 60° and 30°, respectively. Find the height of the balloon above the ground.
Q. A plane left 30 minutes later than the scheduled time and in order to reach its destination 1500 km away
on time, it has to increase its speed by 250 km/hr from its usual speed. Find the usual speed of the plane.
Q. A moving boat is observed from the top of a 150 m high cliff moving awayfrom the cliff. The angle of
depression of the boat changes from 60° to 45° in 2 minutes. Find the speed of the boat in m/min.
TRIANGLE:

17. Q. Is the triangle with sides 25 cm, 5 cm, and 24 cm a right triangle? Give reasons for your answer.
Q. Find the altitude of an equilateral triangle of side 8 cm.
Q. It is given that ΔDEF ~ ΔRPQ. Is it true to say that ∠D = ∠R and ∠F = ∠P? Why?
Q. Is the following statement true? Why?
“Why quadrilaterals are similar if their corresponding angles are equal”.
Q. D is the point on side QR of ΔPQR such that PD ⏊ QR. Will it be correct to say that ΔPQD ~ ΔRPD?
Why?
Q. A and B are respectively the points on the sides PQ and PR of a ΔPQR such that PQ= 12.5 cm, PA = 5
cm, BR = 6 cm and PB = 4 cm. Is AB || QR? Give reasons for your answer.
Q. The lengths of the diagonals of a rhombus are 16 cm and 12 cm. Then, Find the length of the side of the
rhombus.
Q. If ΔA B C ~ ΔEDF and ΔABC is not similar to ΔDEF, then which of the following is not true?
(A) BC ∙ EF = A C ∙ FD (B) AB ∙ EF = AC ∙ DE (C) BC ∙ DE = AB ∙ EF (D) BC ∙ DE = AB ∙ FD
Q. In ΔABC and ΔDEF, ∠B = ∠E, ∠F = ∠C and AB = 3DE. Then, the two triangle are
(a) congruent but not similar
(b) similar but not congruent
(c) neither congruent nor similar
(d) congruent as well as similar
Q. It is given that ΔABC ~ ΔPQR, with 𝐵𝐶
= 1
then find 𝑎𝑟𝑒𝑎(∆𝑃𝑅𝑄)
𝑄 𝑅 3 𝑎𝑟𝑒𝑎(𝐵𝐶𝐴)
Q. In ∆𝑨𝑩𝑪 DE ∥BC ,if AD=1cm, BD=2cm ,then find 𝑨𝒓𝒆𝒂(∆𝑨𝑩𝑪)
𝑨𝒓𝒆𝒂(∆𝑨 𝑫
𝑬 )
Q. If S is a point on side PQ, of a ΔPQR such that PS = SQ = RS, then
(a) PR ∙ QR = RS2
(b) QS2 + RS2 =QR2
(c) PR2 + QR2 =PQ2
(d) PS2+ RS2 = PR2
Q. In the given figure, BD and CE intersect each other at P. Is ΔPBC ~ ΔPDE? Why?
Q. Find the value of x for which DE || AB in the given figure

18. Q. In the given figure, if ∠1 = ∠2 and ΔNSQ ≅ ΔMTR. Prove that ΔPTS ~ ΔPRQ.
Q. Diagonals of a trapezium PQRS intersect each other at the point O, PQ || RS and PQ= 3 RS. Find the ratio
of the areas of triangles ΔPOQ and ΔROS.
Q. Corresponding sides of two similar triangles are in the ratio of 2 : 3. If the area of the smaller triangle is
48 cm2, find the area of the larger triangle.
Q. Areas of two similar triangles are 36 cm2 and 100 cm2. If the length of a side of the larger triangle is 20
cm, find the length of the corresponding side of the smaller triangle.
Q. Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides, then the two
sides are divided in the same ratio.
Q. In ΔPQR, PD ⏊ QR such that D lies on QR. If PQ = a, PR = b, QD = c and DR = d, prove that (a + b) (a
– b) = (c + d) (c – d).
Q. In a quadrilateral ABCD, ∠A+ ∠D = 90°. Prove that AC2 + BD2 = AD2 + BC2 [Hint: Produce AB and
DC to meet at E.]
Q. The perpendicular from A on side BC of a ∆ABC meets BC at D such that DB = 3CD. Prove that 2AB2
= 2AC2 + BC2.
Q. Prove that the area of the semi–circle drawn on the hypotenuse of a right angled triangle is equal to the
sum of the areas of the semicircles drawn on the other two sides of the triangle.
Q. ABC is a right triangle, right angled at C. let BC=a,CA=b,AB=c and let p be the length of the
perpendicular frm C on AB. Prove that 1) cp=ab 2) 𝟏⁄ 𝒑 𝟐= 𝟏⁄ 𝒂 𝟐 + 𝟏⁄ 𝒃 𝟐
Q. in trapezium ABCD, AB is parallel to DC and DC = 2AB. EF drawn parallel to AB cuts AD in F and BC
in E such that BE/EC = 3/4. Diagonal DB intersects EF at G. prove that 7FE = 10AB
Q. In a trapezium ABCD, AB || DC and DC = 2AB. EF || A B, where E and F lie on BC and AD respectively
BE/EC = 4/3. Diagonal DB intersects EF at G. Prove that, 7EF = 11AB
Q.Vertical angles of two isosceles triangles are equal. If their areas are in the ratio 16 : 25, then find the
ratio of their altitudes.
Q. If ABC is a right angled triangle, right angled at A. A circle is inscribed in it. The lengths of two sides
containing the right angle are 6 cm and 8 cm. then what is the radius of the circle?
Q. Prove that three times the sum of the squares of the sides of a triangle is equal to four times the sum of
the squares of the medians of the triangle.

19. Q. Two right triangles ABC and DBC are drawn on the same hypotenuse BC and on the same side of BC. If
AC and BD intersect at P, prove that AP × PC = BP × DP.
Q. Diagonals of a trapezium PQRS intersect each other at the point O, PQ parallel to RS and PQ = 3RS.
Find the ratio of the areas of triangles POQ and ROS.
Q. In the given figure, PA, QB and RC are perpendicular to AC. prove that 1/x + 1/y = 1/z
Q. In triangle ABC, AD is perpendicular to BC prove that
i) AB2 + CD2= BD2 + AC2 ii) AB2 - BD2= AC2 - CD2
Q. In a figure D,E trisects the side BCof a triangle ABC prove that 8AE2=3AC2+5AD2
Q. P and Q are two points on the sides CA and CB respectively of ABC right angled at C. Prove that
AQ²+BP²=AB²+PQ².
Q. In ∆ABC, AD perpendiculars to BC. Prove that AC2 = AB2 + BC2 – 2BC ×BD
Q. In ∆ABC, <B = 90° and D is the mid-point of BC. Prove that AC2 = AD2 + 3CD2
Q. ABCD is a parallelogram. AB is divided at P and CD at Q such that AP: PB is 3 : 2 and CQ : QD is 4 : 1.
if PQ meets AC at R, prove that AR= 3/7 AC .
Q. BL and CM are medians of a ∆ABC right-angled at A. Prove that 4 (BL2 + CM2) = 5 BC2.
Q. In a quadrilateral ABCD, ∠ B = 90°, AD² = AB² + BC² + CD². Prove that ∠ACD = 90°.
Q. If a be the area of right triangle and b one of its sides containing the right angle. Prove that the length of
altitude on the hypotenuse is
𝟐𝒂
𝒃√ 𝒃 𝟐+𝟒𝒂 𝟐
Q. ABC is a right triangle right angled at A . Circle is inscribed in it . The lengths of two sides containing
the right angles are 6cm and 8cm . Find the radius of the incircle.
Q. Triangle ABC is right-angled at B and D is the mid-point of BC. Prove that AC2 = 4AD2 - 3AB2

20. Q. In given figure, the bisectors of the angles B and C of a triangle ABC, meet the opposite sides in D and E
respectively. If DE||BC, prove that the triangle is isosceles.
Q. In triangle ABC < ABC=135° prove that AC2 =AB2 +BC2 4Ar(∆𝑨𝑩𝑪)
Q. In triangle ABC, ray AD bisects angle A and intersects BC in D. If BC = a, AC = b and AB =c, prove
that BD = ac/(b+c) and DC=ab/b+c
Q. D and E are the point on side AB and AC respectively of a triangle ABC. Such that DE parallel to BC
and divides Triangle ABC into two parts, equal in area, find BD/AB
Q.If two sides and median of a triangle is proportional to the corresponding two sides and median of another
triangle then show that both the triangle are similar.
Q. ABC is a triangle right angled at C and AC= √3 BC. Prove that angle ABC = 60°
Q. A point O is in the interior of a rectangle ABCD, is joined with each of the vertices A, B, C & D. Prove
that OA2+OC2 = OB2+OD2
Circle:
Q. If the radii of two concentric circles are 4 cm and 5 cm, then find the length of each chord of one circle
which is the tangent to the other circle.
Q.Two circles touch externally the sum of the areas is 130 cm2 and distance between there centre is 14 cm.
Find the radius of circle.
Q. Two circle touch internally. The sum of their areas is 116 cm2 and the distance between their centres is 6
cm. Find the radii of circles
Q. In the given figure, if ∠AOB = 125°, then ∠COD is equal to
Q. In the given figure, AB is a chord of the circle and AOC is its diameter, such that
∠ACB = 50°. If AT is the tangent to the circle at the point A, then find ∠BAT.

21. Q. in the figure OP is equal to the diameter of the circle. Prove that ∆𝐴𝐵𝑃 𝑖𝑠 𝑒𝑞𝑢𝑖𝑙 𝑎𝑡𝑒𝑟𝑎𝑙.
Q. At one end A of diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle. Find the
length of the chord CD parallel to XY and at a distance 8 cm from A.
Q. In the given figure, ‘O’ is the centre of circle, PQ is a chord and the tangent PR at Pmakes an angle of 50°
with PQ, then ∠POQ is equal to
Q. In the given figure, if PA and PB are tangents to the circle with centre O such that ∠APB = 50°, then
∠OAB is equal to
Q. In the given figure, if PQR is the tangent to a circle at Q, whose centre is O, AB is a chord parallel to PR
and ∠BQR = 70°, then ∠AQB is equal to
𝑄. 𝑖 𝑛 𝑓𝑖𝑔𝑢𝑟𝑒 𝐵𝑂𝐴 𝑖𝑠 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑎𝑡 𝑝𝑜𝑖 𝑛𝑡 𝑃 𝑚𝑒𝑒𝑡 𝐵𝐴 𝑒𝑥𝑡𝑒𝑛𝑑𝑒𝑑 𝑎𝑡 𝑇. 𝑖𝑓
< 𝑃𝐵𝑂 = 30° find <PTA

22. Q.PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at point T. Find
the length of TP.
Q. In Figure PQ and RS are two parallel tangents to a circle with centre O and another tangent AB with
point of contact C intersecting PQ at A and RS at B. Prove that <AOB = 90 .
Q.. If a,b,c, are the sides of a right angled triangle where c is hypotenuse, prove that the radius of the circle
which touches the sides of triangle is give by r= 𝑎+𝑏−𝑐
2
Q. Prove that the intercept of a tangent between the pair of parallel tangents to a circle subtend a right angle
at the centre of the circle.
AREA RELATED TO CIRCLE:
Q. If the sum of the areas of two circles with radii R1 and R2 in equal to the area of a
circle of radius R, then which one of the followings is true

23. 2> 2 2 2 2
(a) R1 + R2 = R (b) R1
2 +R2 R (c) R1 + R2 < R (b) R1 +R2 <R
Q. If the sum of circumferences of two circles with radii R1 and R2 is equal to the circumference of a circle
of radius R, then
(a) R1 + R2 = R (b) R1 + R2 < R (c) R1 + R2 > R (d) Nothing definite can be said about the relation
among R1, R2 and R.
Q. Find the radius of a circle whose circumference is equal to the sum of the circumferences of the two
circles of diameters 36 cm and 20 cm .
Q.Find the diameter of a circle whose area is equal to the sum of the areas of the two circles of radii 24 cm
and 7 cm.
Q. Is the area of circle inscribed in a square of side a cm, πr2 cm2? Give reasons for your answer.
Q. The area of two sectors of two different circles with equal corresponding arc length are equal. Is this
statement true? Why?
Q. A circular pond is 17.5 m in diameter. It is surrounded by a 2m wide path. Find the cost of constructing
the path at the rat of Rs25 per m2.
Q. Four circular cardboard pieces of radii 7 cm are placed on a paper in such a way that each piece touches
other two pieces. Find the area of the portion enclosed between these pieces.
Q. On a square cardboard sheet of area 784 cm2, four congruent circular plates of maximum size are places
such that each circular plate touches the other two plates and each side of square sheet is tangent to two
circular plates. Find the area of the square sheet not covered by the circular plates.
Q. Floor of a room is of dimensions 5m × 4 m and it is covered with circular tiles of diameters 50 cm each
as shown in figure. Find the area of the floor that remains uncovered with tiles. (Use π = 3.14)
Q. All the vertices of a rhombus lie on a circle. Find the area of the rhombus if area of the circle is 1256 cm2.
(Use π = 3.14)
Q. The central angles of two sectors of circles of radii 7 cm and 21 cm are respectively 120° and 40°. Find
the areas of the two sectors as well as the lengths of the corresponding arcs. What do you observe?
Q. Find the area of the shaded region given in figure here.
Q. In Figure 4, a square OABC is inscribed in a quadrant OPBQ. IfOA = 15 cm, find the area of the shaded
region. (Use 𝜋 = 3·14)

24. Q. In Figure 5, ABCD is a square with side 2 2 cm and inscribed in a circle.Find the area of the shaded
region. (Use 𝜋 = 3·14)
Q. In Figure 2, three sectors of a circle of radius 7 cm, making angles of 60°, 80° and 40° at the centre are
shaded. Find the the area of the shaded region.
CONSTRUCTION
Q. By geometrical construction, it is possible to divide a line segment in ratio of √3 : 1
√3
Q. Draw a line AB = 12 cm and divide it in the ratio 3 : 5. Measure the two parts.
Q. Construct a triangle similar to a given triangle ABC with its sides 4/5 th of the corresponding sides of
ΔABC. Also, write the steps of construction.
Q.Draw a circle of radius 3 cm. Take a point at a distance of 5 cm from the centre of the circle. Measure the
length of each tangent.
Q. Divide a line segment of length 6 cm internally in the ratio 3 : 2.
Q. Construct a triangle similar to a given ΔABC with sides equal to 5/3 of the corresponding sides of
ΔABC.
Q. Draw a right triangle in which the sides (other than hypotenuse) are of length 4 cm and 3 cm. Then
construct a triangle similar to it and of scale factor 5/3
Q. Draw a circle of radius 1.5 cm. Take a point P outside it. Without using the centre, draw two tangents to
the circle from the point P?
Q. Two line-segments AB and AC include an angle of 60°, where AB = 5 cm and AC = 7 cm. Locate points
P and Q on AB and AC respectively such that AP =3/4AB and AQ=1/4AC Join P and Q and measure the
length PQ.

25. Q. Draw two concentric circles of radii 3 cm and 5 cm. Taking a point on outer circle construct the pair of
tangents to the other. Measure the length of a tangent and verify it by actual calculation
Q. Draw an isosceles ΔABC in which AB = AC = 6 cm and BC = 5 cm. Construct a triangle PQR similar to
ΔABC in which PQ = 8 cm. Also justify the construction.
Q. Draw a circle of radius 4 cm. Construct a pair of tangents to it, the angle between which is 60°. Also,
justify the construction. Measure the distance between the centre of the circle and the point of intersection of
tangents.
Q. Draw a circle of radius 4 cm. Construct a pair of tangents to it, the angle between which is 120°. Also,
justify the construction. Measure the distance between the centre of the circle and the point of intersection of
tangents
Q. Draw a ΔABC in which AB = 4 cm, BC = 6 cm, and AC = 6 cm. Construct a triangle similar to ΔABC
with scale factor 3/2 .Justify the construction.Are the two triangle congruence? Note that, all the three
angles and two sides of the two triangles are equal.
SURFACE AREAAND VOLUME:
Q. A cylindrical pencil sharpened at one edge is the combinations of ………….. and …………
Q. A surahi is the combination of………………….And ………………
Q. plumbline (Sahul) is the combination of……………… and ……………..
Q. The shape of a gilli, in the gilli–danda game (see in figure) is the combination of…………. And ………..
Q.A shuttle cock used for playing badminton has the shape of the combination of
Q. The shape of a glass (Tumbler) (see figure) is usually in the form of a………………
Q. A cone is cut through a plane parallel to its base and then the cone is left over on the other side of the
plane is called…………………..
Q. During conversion of a solid from one shape to another, the volume of new sphere Will
remain………………

26. Q. Two identical solid hemispheres of equal base radius r cm are stuck together along their bases. Find the
total surface area of the combination.
Q. A solid cylinder of radius r and height h is placed over other cylinder of same height and radius. Find the
total surface area of the shape so formed.
Q. A solid cone of radius r and height h is placed over a solid cylinder having same base radius and height as
that of a cone. The total surface area of the combination.
Q. A solid ball is exactly fitted inside the cubical box of side a. Find the volume of the ball.
Q. A solid piece of iron in the form of a cuboid of dimensions 49 cm × 33 cm × 24 cm, is moulded to form a
solid sphere. Find the radius of the sphere.
Q. Volume of two spheres are in the ratio 64 : 27. Find the ratio of their surface areas.
Q. Two cones have their heights in the ratio 1:3 and radii 3:1. What is the ratio of their volumes?
Q. How many cubic centimeters of iron is required to construct an open box whose external dimensions are
36 cm, 25 cm and 16.5 cm provided the thickness of the iron is 1.5 cm. If one cubic centimeter of iron
weight 7.5 g, then find the weight of the box.
Q. Water flows at the rate of 10 m per min. through a cylindrical pipe 5 mm in diameter. How long would it
take to fill a conical vessel whose diameter at the base is 40 cm and depth 24 cm?
Q. A help of rice is in the form of a cone of diameter 9 m and height 3.5 m. Find the volume of rice. How
much canvas cloth is required to just cover the heap?
Q. Water is flowing at the rate of 15 km/hr through a pipe of diameter 14 cm into a cuboidal pond which is
50 m long and 44 m wide. In what time will the level of water in the pond rise by 21 cm?(Ans: 2 hours)
Q. 500 persons are taking a dip into a cuboidal pond which is 80 m long and 50 m broad. What is the rise of
water level in the pond, if the average displacement of the water by a person is 0.04 m3?(Ans: 0.5cm)
Q. A cylindrical bucket of height 32 cm and base radius 18 cm is filled with sand.This bucket is emptied on
the ground, and a conical heap of sand is formed. If the height of conical heap is 24 cm, find the radius and
slant height of the heap.( radius and slant height are 36 cm and 43.2666 cm respectively.)
Q. A rocket is in the form of a right circular cylinder closed at the lower end and surmounted by a cone with
the same radius as that of cylinder. The diameter and height of cylinder are 6 cm and 12 cm, respectively. If

27. the slant height of the portion is 5 cm, then find the total surface are and volume of rocket. (Useπ = 3.14)(
surface area of rocket is 301.44 cm2 ,Volume of Rocket = 376.8 cm3)
Q. A hemispherical bowl of internal radius 9 cm is full of liquid. The liquid is to be filled into cylindrical
shaped bottles each of radius 1.5 cm and height 4 cm. How many bottles are needed to empty the bowl?( 54
bottles)
Q. Water flows through a cylindrical pipe, whose inner radius is 1 cm at the rate of 80 cm per second in an
empty cylindrical tank, the radius of whose base is 40 cm. What is the rise of water level in tank in half an
hour? (90 cm)
Q. A solid is in the form of a right circular cone mounted on a hemisphere. The radius of the hemisphere is
3.5 cm and the height of the cone is 4 cm. The solid is placed in a cylindrical tub, full of water, in such a
way that the whole solid is submerged in water. If the radius of the cylindrical tub is 5 cm and its height is
10.5 cm, find the volume of water left in the cylindrical tub (use p =22⁄7) (Ans= 683.83 cm3)
Q. A bucket of height 8 cm and made up of copper sheet is in the form of frustum of right circular cone with
radii of its lower and upper ends as 3 cm and 9 cm respectively. Calculate
i) the height of the cone of which the bucket is a part
ii) the volume of water which can be filled in the bucket
iii)the area of copper sheet required to make the bucket (Leave the answer in
terms of p (Ans: 129 pcm2)
Q. A right triangle whose sides are 15 cm and 20 cm is made to revolve about its hypotenuse. Find the
volume and surface area of the double cone so formed. (Ans : 3768cu.cm,1318.8 Sq.cm)
Q. If the areas of the circular bases of a frustum of a cone are 4cm2 and 9cm2 respectively and the height of
the frustum is 12cm. What is the volume of the frustum. (Ans:44cm2).
Q. Water in a canal 30 dm wide and 12 dm deep is flowing with a velocity of 10km/h. How much area will
it irrigate in 30 minutes if 8 cm of standing water is required for irrigation?
(Ans:225000 cu. m)
Q. The height of the Cone is 30 cm A small cone is cut of f at the top by a plane parallel to its base if its
volume be1⁄27 of the volume of the given cone at what height above the base is the section cut.(Ans; 20cm)
Q. Two right circular cones X and Y are made X having 3 times the radius of Y and Y having half the
Volume of X. Calculate the ratio of heights of X and Y. (Ans: 9 : 2)
A shuttlecock used for playing badminton has the shape of a frustum of a Cone mounted on a hemisphere.
The external diameters of the frustum are 5 cm and 2cm, and the height of the entire shuttlecock is 7cm.
Find the external surface area. (Ans: 74.26cm2)
Q. A conical vessel of radius 6cm and height 8cm is completely filled with water. A sphere is lowered into
the water and its size is such that when it touches the sides, it is just immersed as shown in the figure. What
fraction of water flows out.(ans; 3/8)

28. Q. A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 20 cm and the
diameter of the cylinder is 7 cm. Find the total volume of the solid. (Use 𝜋 = 22⁄7)
Q. A pole has to be erected at a point on the boundary of a circular park of diameter 13 m in such a way that
the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 m.
Is it possible to do so ? If yes, at what distances from the two gates should the pole be erected ?
Q. In Figure , a decorative block is shown which is made of two solids, a cube and a hemisphere. The base
of the block is a cube with edge 6 cm and the hemisphere fixed on the top has a diameter of 4·2 cm. Find
(a) the total surface area of the block.
(b) the volume of the block formed. (Take 𝜋 =22/7)
Q. A container opened at the top and made up of a metal sheet, is in the form of a frustum of a cone of
height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of milk
which can completely fill the container, at the rate of Rs 50 per litre. Also find the cost of metal sheet used
to make the container, if it costs rs10 per 100 cm2. (Take 𝜋 = 3·14)
Q. A juice seller was serving his customers using glasses as shown in Figure . The inner diameter of the
cylindrical glass was 5 cm but bottom of the glass had a hemispherical raised portion which reduced the
capacity of the glass. If the height of a glass was 10 cm, find the apparent and actual capacity of the glass.
(Use 𝜋 = 3·14)
Q. An open metallic bucket is in the shape of a frustum of a cone. If the diameters of the two circular ends of
the bucket are 45 cm and 25 cm and the vertical height of the bucket is 24 cm, find the area of the metallic
sheet used to make the bucket. Also find the volume of the water it can
hold. (Use 𝜋 = 22/7)

29. STATISTICS:
Q. If xi’s are the mid points of the class intervals of grouped data fi’s are the corresponding frequencies and
x is the mean, then ∑(𝑓𝑖 𝑥𝑖− 𝑥) is equal to…………………
Q. The median of an ungrouped data and the median calculated when the same data is grouped are always
the same. Do you think that this is a correct statement? Give reason
Q. Is it true to say that the mean, mode and median of grouped data will always be different? Justify your
answer
Q. The sum of deviations of a set of values x1, x2, x3,…………xn, measured from 50 is -10 and the sum of
deviations of the values from 46 is 70. Find the value of n and the mean. (Ans:20,.49.5)
Q. Prove that ∑(xi – xbar ) = 0
Q. In a frequency distribution mode is 7.88, mean is 8.32 find the median. (Ans: 8.17)
Q. The mean of 30 numbers is 18, what will be the new mean, if each observation is increased by 2?
(Ans:20)
Q. The median of the following frequency distribution is 35. Find the value of x.
C.I. f
0-10 2
10-20 3
20-30 X
30-40 6
40-50 5
50-60 3
60-70 2
Q. The mean of the following distribution is 18. Find f
Class
interval
11-13 13-15 15-17 17-19 19-21 21-23 23-25
Frequencies 3 6 9 13 f 5 4
Q. Find the missing frequencies when the mean of dats is 53.
Age (in
years)
0-20 20-40 40-60 60-80 80-100 Total
Number of
people
15 F1 21 F2 17 100
Q. The median of the following data is 52.5. Find value of x and y if the total frequency is 100.
Class Frequency
0-10 5
10-20 2
20-30 X
30-40 12
40-50 20
50-60 17
60-70 Y

30. 70-80 7
80-90 9
90-100 4
Q. If the median of the following data is 32.5, find the value of f1 and f2 frequencies.
Q. Change the following data into ‘less than type’ distribution and draw its ogive:
Class Interval : 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80 80 – 90 90 – 100
Frequency : 7 5 8 10 6 6 8
Q. Find the value of p, if the mean of the following distribution is 18.
Variate
(xi)
13 15 17 19 20+p 23
Frequen
cy (fi)
8 2 3 4 5p 6
PROBABILITY
Fill in the blanks:
Q. If an even that cannot occur, then its probability is……………….
Q. If P(E) = 0.05, what is the probability of 'not E
Q. If the probability of an event is p, then the probability of its complementary event
will be……………………
Q. If a card is selected from a deck of 52 cards, then the probability of its being a red face card is
……………….
Q. The probability that a non-leap year selected at random will contains 53 Sundays is…………..
Q. When a die is thrown, the probability of getting an odd number less than 3 is ……………..
Q.Someone is asked to take a number from 1 to 100. The probability that it is a prime is……………..
Q. Which of the following cannot be the probability of an event?
(a)1/3 (b) 0.1 (c) 3% (d)17/16
Q. The probability expressed as a percentage of a particular occurrence can never be
(a) less than 100 (b) less than 0 (c) greater than 1 (d) anything but a whole number
.QIf P(E) denotes the probability of an event , then
(a) P(E) < 0 (b) P(E) > 1 (c) 0 ≤ P(E) ≤ 1 (d) -1 ≤ P(E) ≤ 1
Q. A card is drawn from a deck of 52 cards. The event E is that card is not an ace of hearts. The number of
outcomes favorable to E is

31. (a) 4 (b) 13 (c) 48 (d) 51
Q. In a family having three children, there may be no girl, one girl, two girls or three girls. So, the
probability of each is 1/4. Is this correct? Justify your answer.
Q. Cards marked with numbers 5 to 50 (one number on one card) are placed in a box and mixed thoroughly.
One card is drawn at random from the box. Find the probability that the number on the card taken out is (i) a
prime number less than 10, (ii) a number which is a perfect square.
Q. Apoorva throws two dice once and computes the product of the numbers appearing on the dice. Peehu
throws one die and squares the number that appears on it. Who has better chance of getting the number 36?
Why?
Q. The probability of selecting a blue marble at random from a jar that contains only blue, black and green
marbles is 1/5 The probability of selecting a black marble at random from the same jar is ¼ .If the jar
contains 11 green marbles, find the total number of marbles in the jar.
Q. A card is drawn at random from a well shuffled deck of 52 playing cards. Find the probability that the
card drawn is a) a black king b) an ace .
Q. What is probability of having 53 Sundays in a leap year and in a non leap Year?
2 2Q.Let A, B, C are three mutually exclusive and exhaustive events such that P(B)=3
P(A), P(C)=1⁄ P(B) Find
P(A).
Q. A letter is chosen at random from the word “MATHEMATICS” find the probability that the letter chosen
is a vowel and a consonant.
Q. A letter is chosen at random from the word “ENTERTAINMENT” find the probability that the letter
chosen is a vowel or a letter “t”
Q. Find the probability that
i)a leap year chosen at random will have 53 Fridays.
ii) a leap year chosen at random will have 52 Fridays
iii) a non-leap year chosen at random will have 53 Fridays.
Q. A bag contains 5 red balls and some blue balls if the probability of drawing blue ball is double that of red
ball then find the number of blue balls in the bag
Q. Two unbiased dice are rolled once. what is the probability to get
i) a doublet
ii) a sum equal to 7
iii) two identical prime number
iv) two doublet prime number.
Q. Tossing a fair coin twice. what is the probability of getting
i) two heads ii) at least two head iii) at most two heads iv) exactly one tail.
Q. If we have numbers ranging from 1-100, what is the probability of a single number chosen to not be
divisible by a 2 or 5?

32. Q. An integer is chosen at random from 1 to 50. Find the probability that the number is: (i) divisible by 5
(ii) a perfect cube
(iii) a prime number
iii) divisible by 3 and 5.(i.e. divisible by 15)
Q. A die has its six faces marked 0,1,1,1,6,6.Two such dice are thrown together and the total score is
recorded. a)_ How many different scores are possible ? b) What is the probabilty of getting a total of 7 ?
Q. A game consists of tossing a coin 3 times and noting its outcomes each time. If getting the same result in
all the tosses a success, find the probability of losing the game.