CBSE Class 10th Mathematics Question Paper Solution Set 30/3
|10th class Solution of cbse board paper 2015 (30_3)Maths set 3|
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CBSE Class 10th Mathematics Question Paper Solution Set 30/3
GEOMETRICAL REPRESENTATION OF QUADRATIC EXPRESSION Consider the quadratic expression, y = ax2 + bx + c, a ≠0 & a,b,c ∈R then : (i) The graph between x,y is always a parabola. If a > 0, then the shape of the parabola is concave upwards & if a < 0 then the shape of the parabola is concave downwards. (ii) The graph of y = ax2
(Let the roots of the equation ax + bx + c can be divided into 6 categories which are as follows : 2α + bx + c = 0 be and β)
Definition :- An Arithmetic Progression is a sequence in which the difference between a term and it’s preceding term is a constant. The constant is the common difference (c.d) and is denoted by ‘d’. (The Arithmetic Progression is abbreviated as A.P).
The general form and term of an A.P :
T1, T2, T3 ………….. Tn .......... be an A.P
T2 –T1 = T3 –T2 = T4 –T3 = ………….. = Tn – Tn-1 = d
Let T1 = a = a + (1 – 1) d
∴ T2 = T1 + d = a + d = a + (2 – 1)d
T3 = T2 + d = a + 2d = a + (3 – 1)d
T4 = T3 + d = a + 3d = a + (4 – 1)d
Tn = Tn-1 + d = a + (n – 1)d
∴a, a + d, a + 2d, a + 3d, ...................., a + (n – 1)d is the general form of an
arithmetic progression with ‘a’ as the first term and having common difference ‘d’. This is the Standard form of an A.P. The last term a + (n – 1) d is denoted by Tn or l
Note : 1. In an A.P succeeding term of a given term is obtained by adding ‘d’to it. [Tn+1 = Tn+ d].
2. In an A.P preceding term of a given term is obtained by subtracting ‘d’ from it [Tn-1 = Tn – d].
3. Tn = a + (n – 1)d is the general term of an A.P
Find the nth and the 20th terms of the A.P. 3, 7, 11, 15, ...............
Solution : a = 3, d = T2 – T1 = 7 – 3 = 4, Tn = ?, T20 = ?
Tn = a + (n – 1)d.
Tn = 3 + (n – 1)4
Tn = 4n – 1
T20 = 4(20) – 1
T20 = 79
(OR) T20 = 30 + (20 – 1)4
= 80 – 1 = 3 + 19 x 4
T20 = 3+76=79
How many terms are there in the A.P. 4, –1, –6, ............., (–106)
Solution : Given : a = 4, d = –5, Tn = –106, n = ?
Tn = a + (n – 1)d.
∴ a + (n – 1)d = Tn
4 + (n – 1) (–5) = –106
4 – 5n + 5 = –106
–5n + 9 = –106
∴ –5n = –106 – 9
∴ –5n = –115
n = 23
There are 23 terms in the given A.P.
Q . The fourth and eighth terms of an A.P. are in the ratio 1 : 2 and tenth term is 30, Find the Common difference and the first term.[ d = 3; a = 3]
Q. The angles of a triangle are in A.P. The smallest angle is 300. Show that the triangle is a right angled triangle.[ In triangle ABC, let us say ∠A = 30 0 then <∠B = 300+d and <∠C = 300 +2d ]
Q. Find the three numbers of an A.P. whose sum is 12 and their product is 48.
[2, 4 and 6 (or) 6, 4 and 2]
Q. Find the sum of all positive multiples of 3 less than 50 
Sum of First n Terms of an AP
Let us denote the first term of an AP by a1, second term by a2 , . . ., nth term by an and the common difference by d. Then the AP becomes a1, a2, a3. . . a + [n-1]d
Let S denote the sum of the first n terms of the AP. We have
S = a + (a + d ) + (a + 2d ) + . . . + [a + (n – 1) d ] ---------------------(i)
Rewriting the terms in reverse order, we have
S = [a + (n – 1) d ] + [a + (n – 2) d ] + . . . + (a + d ) + a ----------------(ii)
On adding (i) and (ii), term-wise. we get
2S = [2a + (n – 1) d ] + [2a + (n – 1) d ] + . . . + [2a + (n – 1) d ] ----- n times
S = [2a + (n – 1) d ]
OR, S = [a + a + (n – 1) d ] = [a + an ]
OR, if there are only n terms in an AP, then an = l, the last term.
S = (a + l )
This form of the result is useful when the first and the last terms of an AP are given and the common difference is not given.
Find the sum of all natural numbers between 91 and 170 which are divisible by 5.
Solution : The numbers between 91 and 170 which are divisible by 5 are 95, 100, 105, ...... 165
a = 95 ; d = 5
First find the number of terms in the A.P.
Tn = 165
∴ a + (n – 1)d = 165
95 + (n – 1)5 = 165
∴ 95 + 5n – 5 = 165
∴ 5n + 90 = 165
5n = 165 – 90
n = 15
Sn = n/2 (a + l) =15/2 (95 + 165) =1950
The sum of n terms of an arithmetic series is Sn = 2n2 + 6n. Find the first term and the common difference.
Solution : Given : Sn = 2n2 + 6n and S1 = T1
S1 = 2(1)2 + 6(1) = 2+6 = 8
T1 = 8
∴ a = 8
To find 'd'
S2 = 2(2)2 + 6(2) = 2(4) + 12 = 8 + 12
∴ S2 = 20
S2 = T1 + T2 = 20
20 = 8 + T2
T2 = 12
d = T2 – T1 = 12 – 8
Q. How many terms of the series 10 + 8 + 6 + ............ should be added to get the sum –126. [n = 18 ]
Q. Sanganbasava rides a bicycle from his home to the ashram. He covers 125 meters at the end of first minute, 135 meters at the end of second minute and so on. If he reaches the Ashram at the end of 10 minutes. Find the distance between hishome and ashram. [1.7 Kms]
Q. Find the A.P in which a)Tn = 2n + 1 b) Tn = 4 – 5n c) Sn = 5n2 + 3n
[I. a) 3, 5, 7, ............. b) –1, –6, –11, .............. c) 8, 18, 28 ............]
Q. Find the sum of the Arithmetic series which contains 25 terms and whose middle term is 20. 
Q. The angles of a quadrilateral are in A.P. If the smallest angle is 150, find the angles of the quadrilateral.[ 650, 115 0 and 1650]
Q. Veershree climbed 23 steps of Golgumbaz in the first minute. After that she climbed two steps less than what she had climbed in the previous minute. If she reached the whispering gallery of Golgumbaz after 7 minutes how many steps did she climb to reach the whispering gallery? 
For more Test paper/Solved paper/E-Book Visit Arithmetic progressions
10th Maths - Quadratic Equation
1. Standard form of a quad. Equations: ax2 + bx + c = 0 (a â 0)
2. Solution of the quadratic equations conly real roots by
(b) Completing the square
(c) Quadratic formula
3. Relationship between discriminant and nature of roots.
4. Problems related to day to day activities.
Learning Objective: To learn the following facts
1. A second degree eqns of the form ax2 + bx + c = 0 where a, b, c are real numbers and a â 0 are called quadratic eqn.
2. If ax2 + bx + c = 0 equivalent to (x â a) (x â b) = 0 then x = a and x = b is a solution.
3. If ax2 + bx + c = 0 has a & b as roots of eqn then sum of roots = â b/a. Product of roots = c/a.
4. (b2 â 4ac) is called the discriminant of the eqn and
if D > 0 => Real & Distinct Roots
if D = 0 => Real & Equal Roots
if D < 0 => No Real Roots
1. Roots of an eqn.
3. Real & Non Real Roots
4. Quadratic Formula =>
X Ch : Quadratic Equation Solved Question With Self Evaluation Question
Visit [study guide pdf ] Download
If the polynomial x4 – 6x3 + 16x2 – 26x + 10 – a is divided by another polynomial x2 – 2x + k, the remainder comes out to be x + a. Find k and a.
For full study material visit : Polynomial : Algebra
1. A solid metallic hemisphere of radius 8 cm is melted and recasted into a right circular cone of base radius 6 cm. Determine the height of the cone.
2. A rectangular water tank of base 11 m × 6 m contains water upto a height of 5 m. If the water in the tank is transferred to a cylindrical tank of radius 3.5 m, find the height of the water level in the tank.
3. How many cubic centimetres 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 cm of iron weighs 7.5 g, find the weight of the box.
4. The barrel of a fountain pen, cylindrical in shape, is 7 cm long and 5 mm in diameter. A full barrel of ink in the pen is used up on writing 3300 words on an average. How many words can be written in a bottle of ink containing one fifth of a litre?
5. Water flows at the rate of 10m/minute 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?
6. A heap of rice is in the form of a cone of diameter 9 m and height 3.5 m. Find thevolume of the rice. How much canvas cloth is required to just cover the heap?
7. A factory manufactures 120000 pencils daily. The pencils are cylindrical in shape each of length 25 cm and circumference of base as 1.5 cm. Determine the cost of colouring the curved surfaces of the pencils manufactured in one day at Rs 0.05 per dm2.
8. Water is flowing at the rate of 15 km/h 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 pond rise by 21 cm?
9. A solid iron cuboidal block of dimensions 4.4 m × 2.6 m × 1m is recast into a hollow cylindrical pipe of internal radius 30 cm and thickness 5 cm. Find the length of the pipe.
10. 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 0.04m3?
1. 28.44 cm 2. 8.6 m 3. 3960 cm3, 29.7 kg 4. 480000 words
5. 51 minutes 12 sec 6. 4.25m3, 80.61 m2 7. Rs 2250 8. 2 hours
9. 112 m 10. 0.5 cm
Mathematics X : Sample Question Papers (March 2013)
sample -Paper - 02
sample- paper - 04
SAMPLE PAPERS Mathematics (Solved) (KV)
Sample papers -II- 5
CBSE BOARD New X : Sample Question Papers for Term II (March 2013) By J.Sunil
Class: X Subject: Mathematics Assignment: 2013 Chapter: Probability.
1. 17 cards numbered 1, 2, 3 …, 16, 17 are put in a box and mixed thoroughly. One person draws a card from the box. Find the probability that the number on the card is:
(i) Odd (ii) Even (iii) a Prime (iv) Divisible by 3 (v) Divisible by 3 and 2 both.
2. Two customers are visiting a particular shop in the same week (Monday to Saturday). Each is equally likely to visit the shop on any one day as on another. What is probability that both will visit the shop on (i) same day (ii) different days?
3. Three coins are tossed once. Find the probability of: (i) 3 heads (ii) Exactly 2 heads (iii) At least 2 heads
4. Cards marked with numbers 3 to 152 are thoroughly mixed. If one card is drawn at random, find the probability that the number on the card is: - (i) An odd number (ii) A number less than 25 (iii) A number greater than 140 (iv) A number which is a perfect number (v) A prime number between 10 and 40.
5. In a class, there are 18 girls & 16 boys. The class teacher wants to choose 1 pupil for class monitor. She writes the name of each pupil on a card & puts them into a basket & mixes thoroughly. A child is asked to pick up one card from the basket. What is the probability that name written on the card is:- (i) the name of a girl (ii) the name of a boy.
6. 1500 families with 2 children were selected randomly, and the following data were recorded:
Compute the probability of a family chosen at random, having: - (i) 2 girls (ii)1girl (iii) no girl Also check whether the sum of these probabilities is 1.
7. Find the Probability of getting 53 Sundays in a leap year.
8. Find the probability of getting 53 Sundays in a non-leap year.
9. A dice is thrown twice. What is the Probability of (i) 5 will not come up either time?
(ii) 5 will come up at least once? [Hint: throwing a dice twice and throwing two dice simultaneously are treated as the same experiment.]
10.Two dice are thrown. Determine Probability of getting a multiple of 2 on the first dice and a multiple of 3 on the other.
11.Two dice are thrown simultaneously. Find the Probability of getting: -
(i) A total of 2 (ii) A sum of 7 (iii) A sum of 6 (iv) A sum of 8 (v) Doublets of even numbers.
Number of girls in family: 2 1 0
Number of families: 475 814 211
12.There are 840 tickets sold in a raffle. Bhawna bought 5 tickets and Shilpy bought 4 tickets.
What is Probability that
(a) Bhawna has a winning ticket? (b) Shilpy has a winning ticket?
13.A card is drawn from an ordinary pack and gambler bets that it is a spade or ace. What are odds against his winning the bet?
14.Two customers Shyam and Ekta are visiting a particular shop in the same week (Tuesday to Saturday) each is equally to visit the shop on any day as on another day. What is the Probability that both will visit the shop on :-
(i) Same day (ii) Consecutive days (iii) Different days.
15. Two dice are thrown simultaneously. List the sample space for this experiment.
16.In a single throw of 2 dices find the probability of getting:-
(a) A total of 7 (ii) A total of 11 (iii) Doublets (iv) Six as a product.
17.A box contains 19 balls bearing numbers 1, 2… 19. A ball is drawn at random from the box. What is Probability that number of the ball is:-(i) a prime number (ii) divisible by 3 or 5 (iii) neither divisible by 5 nor 10 (iv)an even number
18.From a pack of 52 playing cards, jacks, queens, kings and aces of red colour are removed.
From the remaining cards one card is drawn. Find the Probability that the card drawn is:
(a) A black queen (b) A red card (c) A jack (d) A diamond (e) A black card.
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