Wednesday, 25 November 2015
Monday, 23 November 2015
EXTRA QUESTIONS (Quadratic Equations) SET-1
PODAR INTERNATIONAL SCHOOL (CBSE)
Topic : Quadratic Equations
1. The number of quadratic equations having real roots and which do not change by
squaring their root is :
(a) 4 (b) 3
(c) 2 (d) 1
2. For what value of k are the roots of the quadratic equation 3x+ 2 kx+ 27 = 0 are
real and equal?
3. An express train takes 1 hour less than a passenger train to travel 132 km between
Agra and Delhi( without taking into consideration the time they stop at intermediate
stations).If the average speed of the express train is 11km/hr more than that of the
passenger train, find the average speed of two trains
4. A takes 3 days longer than B to finish a work. But if they work together, then work
is completed in 2 days. How long would each take to do it separately? Can you say
cooperation helps to get more efficiency?
5. If x = 1 is a common root of ax2+ ax + 3 = 0 and x2+ x + b = 0, then ab = ?
a. 3 b. - 3
c. 4 d. 6
6. If x2+ 2 ( k + 2) x + 9 k = 0 has a repeated root, thenk = ?
(Repeated root means two roots are equal)
a. 1 or 4 b. 1 or - 4
c. - 1 or 4 d. - 1 or - 4
7. If x2- 4x + p = 0 has real roots, then
a. p ≥4 b. p ≤4
c. p ≥5 d. p ≤- 4
8. Which of the following is not a quadratic equation?
a. x2+ 2 x + 1 = 0 b. 2 x - x2= x2- 5
c. x2+ 9 = 3x2- 5x d. ( x2+ 1 )2= x2+ 3x + 9
9. If is a root of the equation x2+ kx - = 0, then the value of k is
a. 2 b. - 2
c. 1/4 d. 1/2
10 For what value of k ≠0, the polynomial kx2- 3 kx + 9 is a perfect square ?
a. k = 1 b. k = 2
c. k = 3 d. k = 4
11. If D is the discriminant of a quadratic polynomial,the false statement of the following is
a. D can hold negative value b. D can hold positive value
c. D can hold a zero value d. D = 0 always
12. The positive root of the quadratic equation x2+ ( x + 1)2 = 313 is
a. 12 b. 13
c. 12 and - 13 d. 12 and 13
13 The roots of the equation x2+ x - ( k + 1 ) ( k + 2) = 0 are
a. k + 1 b. - ( k + 2 )
c. k + 2 d. k + 1 and - ( k + 2)
14 .The equation 3x2 + 4√3x + 4 = 0 has
a. Two distinct real roots b. Two equal real roots
c. No real roots d. More than two real roots
15 Which of the following equations has the sum of itsroots as 3?
a. 2 x2- 3 x + 6 = 0 b. - x2+ 3 x - 3 = 0
c.√2x2- √x + 1 = 0 d. 3 x2 - 3x + 3 = 0
16.Which of the following equations has the product of its roots as
a. 2 x2+ 7 = 0 b. 2x2+ 4x + 7 = 0
c. 2x2- 4x + 7 = 0 d. 2 x2+ 4 x - 7 = 0
17 Which of the following has no real roots
a.x2- 4x + 3√2 = 0 b. x2+ 4 x - 3√2= 0
c. x2- 4x - 3√2 = 0 d. 3x2+ 4 √3 x + 4 = 0
xv If no roots of the equation x2- px + 1 = 0 is real, then
a. p >2 b. p < - 2
c. p = 2 d. - 2 <p < 2
18. Which constant must be added and subtracted to solve the quadratic equation
9x2+ 6 x - 5 = 0?
a. 1 b. 14
c. 18 d. 49
19. Two numbers whose sum is 27 and the product is 182,is
a. 8, 19 b. 11, 16
c. 13, 14 d. 15, 12
20. Two consecutive odd positive integers , sum of whose squares is 290 are
a. 9, 11 b. 11, 13
c. - 11, - 13 d. 12, 13
21. The hypotenuse of a right angled triangle is √52cm. If the smaller and the larger of the
remaining two sides are respectively tripled and doubled, then the new hypotenuse will be
√288cm. The original lengths of these two sides were ,respectively
a. 4cm and 6 cm b. 2 cm and 3 cm
c. 6cm and 8 cm c. 5 cm and 12 cm
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EXTRA QUESTIONS (Application of trigonometry) SET-1
PODAR INTERNATIONAL SCHOOL (CBSE)
Topic : Heights & Distances
1.The angle of elevation of the top of a tower is 30°. If the height of the tower is doubled ,
then the angle of elevation of its top will also be doubled. Is it true?
2. If the angle of depression of an object from a 75m high tower is 30°.Find the distance of the
object from the base of the tower.
3. A balloon is connected to a meteorological ground station by a cable of length 215 m
inclined at 60° to the horizontal. Determine the height of the balloon from the ground.
Assume that there is no slack in the cable.
4. A tree of 12 m height is broken by the wind in sucha way that its top touches the ground
and makes an angle 60° with the ground. At what height from the bottom, the tree is broken
by the wind?
5. The angle of elevation of the top of a hill from the foot of a tower is 60° and the angle of
elevation of top of the tower from the foot of the hill is 30°. If the tower is 50 m high, what
is the height of the hill?
6. Two men on opposite sides of the cliff 80 m high observe the angles of elevation of the top
of the cliff to be 30° and 60° respectively. Find the distance between the two men.
7. A ladder is placed against a wall such that it reaches the top of the wall. The foot of the
ladder is 1.5 m away from the wall and the ladder is inclinedat an angle of 60° with the ground. Find
the height of the wall.
8. Find the angle of elevation of the Sun (Sun's altitude) when the length of the shadow of a
vertical pole is equal to its height.
9. From a point 20 m away from the foot of a tower, the angle of elevation of the top of the
tower is 30°. Find the height of the tower.
10. The horizontal distance between two towers is 140 m. The angle of elevation of the top of
the first tower when seen from the top of the second tower is 30°.If the height of the second
tower is 60 m, find the height of the first tower.
11. Find the angular elevation of the Sun when the shadow of a 10 m long pole is 10 √3m.
12. The angles of elevation of the top of a hill, at the city centres of two towns on either side of
the hill are observed to be 30° and 60°.If the distance uphill from the first city centre is 9
km, find the distance uphill from the other city centre in kilometres up to two places of
decimals.
13. An aeroplane flying horizontally 1 km above the ground is observed at an elevation of 60°.
After 10 seconds its elevation is observed to be 30°. Find the speed of the aeroplane in km/
hr.
14. An aeroplane at an altitude of 200 m observes the angles of depression of opposite points
on the two banks of a river to be 45° and 60°. findin metres, the width of the river.
15. The angle of elevation of the top of the tower fromtwo points at a distance of 25m and
36m from the base of the tower and in the same straight line with it are complementary. Prove
that the height of the tower is 30m.
16. Monica is a 1.5 m tall girl. She is standing at a distance of 28.5 m from a multi-storeyed
building. The angle of elevation of the top of thebuilding from her eyes is 45°.Find the
height of the multi - storey building.
17. Two pillars of equal heights are on either side of a road, which is 100 m wide. At a point on
the road between the pillars, the angles of elevation of the top of the pillars are 60° and 30°
respectively. Find the position of the point between the pillars and the height of each pillar.
18. The angle of elevation of an aeroplane from a pointon the ground is 45°.After a flight of 15
seconds, the elevation changes to 30°.If the aeroplane is flying at a constant height of 3000
metres, find the speed of the aeroplane.
19. An aeroplane, when 3000 m high , passes vertically above another aeroplane at an instant
when the angles of elevation of the two aeroplanes from the same point on the ground are
60° and 45° respectively. Find the vertical distance between the two planes.
20. From the top and foot of a tower 40m high, the angles of elevation of the top of a lighthouse
are found to be 30° and 60° respectively. Find the height of the light house. Also find the
distance of the top of the light house from the foot of the tower.
21. A 1.6 m tall girl stands at a distance of 3.2 m from a lamp - post and casts a shadow of 4.8 m
on the ground. Find the height of the lamp - post by using
a. Trigonometric ratios b. property of similar triangles
22. A man on a cliff observes a boat at an angle of depression of 30° which is approaching the
shore to the point immediately beneath the observerwith a uniform speed. Six minutes later,
the angle of depression of the boat is found to be 60°.Find the time taken by the boat to
reach the shore.
23. A man standing on the deck of a ship, which is 10 mabove the water level, observes the
angle of elevation of the top of a hill as 60° and the angle of depression of the base of the
hill as 30°. Calculate the distance of the hill from the ship and the height of the hill.
24. A man standing on the deck of a ship, which is 10 mabove the water level, observes the
angle of elevation of the top of the hill as 60° and the angle of depression of the base of the
hill as 30°, calculate the distance of the hill from the ship and the height of the hill.
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EXTRA QUESTIONS (Application of trigonometry) SET-1
PODAR INTERNATIONAL SCHOOL (CBSE)
Topic :Heights & Distances
1. From a point A, in level with the foot of a vertical pole 25m from it, the angle of
elevation of the top of the pole from point A is 30°.Calculate the height of the pole.
2. A tree casts a shadow 4m long on the ground when the angle of elevation of the Sun
is 45°.Find the height of the tree.
3. An electrician has to repair an electric fault on apole of height 4m. He needs to
reach a point 1.3 m below the top of the pole to undertake the repair work. What
should be the length of the ladder that he should use which when inclined at an
angle of 60 to the horizontal would enable him to reach the required position?
4. While dashing to the destination point on the ground, the pilot of the aeroplane
declines his aeroplane by 30°and drives straight to the ground. The average speed
of the aeroplane is 200 km/hr. It takes 54 seconds to reach the ground. How high
was the aeroplane before it started its dash ?
5. The shadow of a flag staff is three times as long as its shadow, when the sunrays
meet the ground at an angle of 60°. Find the angle between the sunrays and the
ground at the time of the longer shadow.
6. A boy standing on a horizontal plane finds a bird flying at a distance of 100 m from
him at an elevation of 30°.A girl standing on the roof of a 20 m high building ,
finds the angle of elevation of the same bird to be45°.Both the boy and girl are
on the opposite sides of the bird. Find the distance of the bird from the girl.
7. A man on the top of a vertical observation tower observes a car moving at a uniform
speed coming directly towards it. If it takes 12 minutes for the angle of depression to
change from 30°to 45°, how soon after this will the car reach the tower ? Give your
answer correct to the nearest second.
8. The angle of elevation of the top Q of a vertical tower PQ from a point X on the
ground is 60°.At a point Y, 40 m vertically above X, the angle of elevation is 45°.
Find the height of the tower PQ and the distance XQ.
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Sunday, 22 November 2015
EXTRA QUESTIONS OF AP SET-3
PODAR INTERNATIONAL SCHOOL (CBSE)
Practice Sheet
Std : X Subject: Mathematics
Topic: Arithmetic Progression
1. In the following APs find the missing terms
(i) 2,____ , 26
(ii) 5,____ ,_____ , 9 (1/2)
2. Find a3, a5 and a8, if an = ( - 1)n × n + 1
3. Find the 21st and 42 nd terms of the sequence defined by
tn = 0, if n is odd.AND 1, if n is even
4. Find the sum of n terms of the sequence {an } where an = 5 - 6n , n Belongs to N
5. Find the 27 th and the nth terms of the sequence 5, 2, -1,- 4, -7, .....
6. A sequence { an } is given by the formula an = 10 - 3n. Prove that it is an A.P.
7 .If m times the m th term of an A. P. is equal to n times its n th term, prove that
( m + n ) th term of an A. P. is zero. [ Delhi 2004 ]
8. Find 10th term from each end of an A.P. 5, 7......, 159.
9. How many terms are there in an A.P. whose first and fifth terms are - 14 and 2 respectively
and the sum of the terms is 40?
10. The 4th term of an AP is equal to 3 times the first term and the 7th term exceeds twice the
third term by 1. Find the 1st term and the AP.
11. Find the sum of all three digit numbers which leave the remainder 3 when divided by 5.
4 marks questions:
12. Amrita buys a house for ` 22,000.She pays `4,000 cash and agrees to pay the balance in
annual installments of ` 1,000 plus 10 % interest on the unpaid amount. What will the house
cost for her?
13. How many terms of the A.P. - 6 , -11/2,- 5 , are needed to give the sum -25? Explain the
double answer.
14. If the p th, q th, r th terms of an A.P. be x, y, z respectively. Show that
x ( q - r ) + y ( r - p ) + z ( p - q) = 0
15. Find the sum of 32 terms of an A.P. whose third term is 1 and 6th term is -11.
16. Find the 17th term and the nth term of A.P
3, 3 + √2 , 3 + 2 √2 , 3 + 3 √2....
17. Supreet deposits a sum of ` 5000 in State Bank of India. Bank pays simple interest of 5 % per
annum on the money deposited. Calculate the interest at the end of 1, 2, 3,.... years .Verify that
the sequence of interest forms an AP. Also find the interest earned after 40 years by using the
idea of AP.
18. A contract on construction job specifies a penalty for delay of completion beyond certain date
as follows : `200 for the first day,`250 for the second day, `300 for the third day, etc. , the
penalty for each succeeding day being `50 more than for the preceding day. How much
money the contractor has to pay as penalty if he has delayed the work by 30 days?
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PODAR INTERNATIONAL SCHOOL (CBSE)
Std : X Subject : Mathematics
Topic : Arithmetic Progression Set : 9 (b
1. Find the 16 th term of the AP 3, 5, 7, 9, 11...
2. If the 2nd term of an AP is 13 and 5th term is 25, what is its 7th term?
3. The taxi fare after each km, when the fare is `15 for the first km and `8 for each
additional km, does not form an AP as the total fare ( in `) after each km is 15, 8, 8,
8,.....Is the statement true ? Give reasons.
4. Is 0 a term of the AP 31, 28, 25,...? Justify your answer.
5. Determine the AP whose 5th term is 19 and the difference of the eighth term from
the thirteenth term is 20.
6. How many numbers lie between 10 and 300, which divided by 4 leave a remainder
3?
7. Jaspal Singh repays his total loan of `1,18,000 by paying every month starting with
the first installment of `1000. If he increases the installment by `100 every month,
what amount will be paid by him in the 30 th installment? What amount of loan does
he still have to pay after the 30 th installment?
8. If the sum of first 4 terms of an AP is 40 and that of first 14 terms is 280, find the
sum of its first n terms.
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EXTRA QUESTIONS OF AP
PODAR INTERNATIONAL SCHOOL (CBSE)
MCQ
Std : X Subject: Mathematics
Topic : Arithmetic progression
Q1 Select the correct option.
i. Fourth term of the A.P. √2, √8, √18,……. is
a. √22 b. √32
c. 6 d. 5
ii. An A. P. whose second term and common difference are 7 and - 4 respectively is
a. 3, - 4, - 11,… b. -11, - 4, 3 ….
c. 11, 7, 3…. d. 1, 3, 7 …..
iii. In an A. P. if d = - 3 and a6 = 4, then a is
a. 19 b. 9
c. 7 d. 21
iv. If an = √7 n + √5 represents the nth term of an A.P., then the common difference is
equal to
a. 1 b. 2√7
c. √5 d. √7
v Which term of the A. P. 21, 42, 63, 84,….. is 231?
a. 9 th b. 10 th
c. 11 th d. 12 th
vi If x + 2, x2 – 2, 3x ,…. is an A.P. then the 5th term will be
a. - 7 or 13 b. - 1 or 3
c. 13 or 15 d. - 5 or - 7
vii In an A.P. , if d = - 4, n = 7, an = 4, then a is
a. 6 b. 7
c. 20 d. 28
viii In an A. P. , if a = 3.5, an = 3.5, n = 201, then d is
a. 0 b. 3.5
c. 203.5 d. 204.5
ix The 19 th term of an A. P. whose first two terms are - 3 and 4 is
a. 16 b. 23
c. 126 d. 123
x The 11th term of the A. P. - 7, -7/2 , 0,7/2 , …..is
a. - 28 b. 28
c. - 35 d. 35
xi. The 4th term from the end of the A. P.: - 11, - 8, - 5,…., 49 is
a. 37 b. 40
c. 43 d. 58
xii. If the common difference of an A.P. is 5, then a18 – a13 is
a. 5 b. 20
c. 25 d. 30
xiii The two A.P's have the same common difference. The first term of one AP is - 1 and
that of the other is - 8. Then the difference between their 4th terms is
a. -1 b. - 8
c. 7 d. -9
xiv. The sum of first seven multiples of 5 is
a. 130 b. 140
c. 160 d. 150
xv If 7 times the 7th term of an AP is equal to 11 times its 11th term, then its 18th term
will be
a. 7 b. 11
c. 18 d. 0
xvi The famous Mathematician associated with finding the sum of first 100 natural
numbers is
a. Pythagoras b. Newton
c. Gauss d. Euclid
xvii In an AP , if a = - 5, l = 2 l and S = 200, then n is equal to
a. 50 b. 40
c. 32 d. 25
xviii The number of two digit numbers which are divisible by 3 is
a. 33 b. 31
c. 30 d. 29
xix In an AP , if a = 3 and S8 = 192, then d is
a. 8 b. 7
c. 6 d. 4
xx If the 2nd term of an AP is 13 and the 5th term is 25, then its 7th term is
a. 30 b. 33
c. 37 d. 38
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