CBSE Class 10 Mathematics Arithmetic Progression MCQs Set 09

Question. If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are
(a) 5, 10, 15, 20
(b) 4, 10, 16, 22
(c) 3, 7, 11, 15
(d) None of these
Answer: A
Let the four numbers in A.P. be \( a - 3d, a - d, a + d, a + 3d \).
Given, \( (a - 3d) + (a - d) + (a + d) + (a + 3d) = 50 \)
\( 4a = 50 \Rightarrow a = \frac{25}{2} \)
Greatest number \( = 4 \times \) Least number
\( (a + 3d) = 4(a - 3d) \)
\( (\frac{25}{2} + 3d) = 4(\frac{25}{2} - 3d) \)
\( \frac{25}{2} + 3d = 50 - 12d \)
\( 15d = 50 - \frac{25}{2} = \frac{75}{2} \)
\( d = \frac{5}{2} \)
\( a - 3d = \frac{25}{2} - 3 \times \frac{5}{2} = \frac{25}{2} - \frac{15}{2} = \frac{10}{2} = 5 \)
\( a - d = \frac{25}{2} - \frac{5}{2} = \frac{20}{2} = 10 \)
\( a + d = \frac{25}{2} + \frac{5}{2} = \frac{30}{2} = 15 \)
\( a + 3d = \frac{25}{2} + 3 \times \frac{5}{2} = \frac{25}{2} + \frac{15}{2} = \frac{40}{2} = 20 \)

Question. The first four terms of an AP whose first term is -2 and the common difference is -2 are
(a) -2, 0, 2, 4
(b) -2, 4, -8, 16
(c) -2, -4, -6, -8
(d) -2, -4, -8, -16
Answer: C
Let the first four terms of an AP are \( a, a + d, a + 2d \) and \( a + 3d \).
Given, that first term, \( a = -2 \) and common difference, \( d = -2 \), then we have an AP as follows
\( -2, -2 + (-2), -2 + 2(-2), -2 + 3(-2) \)
\( = -2, -4, -6, -8 \)

Question. If the first, second and the last terms of an A.P. are \( a, b, c \) respectively, then the sum is:
(a) \( \frac{(a + b)(a + c - 2b)}{2(b - a)} \)
(b) \( \frac{(b + c)(a + b - 2c)}{2(b - a)} \)
(c) \( \frac{(a + c)(b + c - 2a)}{2(b - a)} \)
(d) None of these
Answer: C
First term \( = a \)
Common differences \( = b - a \)
\( a_n = c \)
\( c = a + (n - 1)(b - a) \)
\( \frac{c - a}{b - a} = n - 1 \)
\( \frac{c - a}{b - a} + 1 = n \)
\( \frac{c - a + b - a}{b - a} = \frac{c + b - 2a}{b - a} = n \)
\( S_n = \frac{n}{2} [a + a_n] = \frac{n}{2} [a + c] \)
\( = \frac{b + c - 2a}{2 \times (b - a)} (a + c) \)
\( = \frac{(b + c - 2a)(a + c)}{2(b - a)} \)

Question. The \( 21^{th} \) term of an AP whose first two terms are -3 and 4, is
(a) 17
(b) 137
(c) 143
(d) -143
Answer: B
Given, first two terms of an AP are \( a = -3 \)
and \( a + d = 4 \)
\( -3 + d = 4 \)
Common difference, \( d = 7 \)
\( a_{21} = a + (21 - 1)d \)
[Since, \( a_n = a + (n - 1)d \)]
\( = -3 + (20)7 \)
\( = -3 + 140 = 137 \)

Question. The number of terms of the series 5, 7, 9, \dots that must be taken in order to have the sum 1020 is
(a) 20
(b) 30
(c) 40
(d) 50
Answer: B
Let \( n \) number be taken in the series.
Given first term, \( a = 5 \)
Common difference, \( d = 7 - 5 = 2 \)
and Sum of \( n \) terms, \( S_n = 1020 \)
We know that, \( S_n = \frac{n}{2} [2a + (n - 1)d] \dots (1) \)
Putting above values in eq (1)
\( 1020 = \frac{n}{2} [2 \times 5 + (n - 1)2] \)
\( 2040 = n [10 + 2n - 2] \)
\( 2040 = 10n + 2n^2 - 2n \)
\( 2040 = 2n^2 + 8n \)
\( 1020 = n^2 + 4n \)
\( n^2 + 4n - 1020 = 0 \)
\( n^2 + 34n - 30n - 1020 = 0 \)
\( n(n + 34) - 30(n + 34) = 0 \)
\( (n - 30)(n + 34) = 0 \)
\( n = 30, -34 \)
Negative term is neglected Hence, \( n = 30 \) number be taken in the series.

Question. A circle with area \( A_1 \) is contained in the interior of a larger circle with area \( A_1 + A_2 \). If the radius of the larger circle is 3 and \( A_1, A_2 \) and \( A_1 + A_2 \) are in AP, then the radius of the smaller circle is
(a) 3
(b) \( \sqrt{3} \)
(c) 2
(d) \( \sqrt{2} \)
Answer: B
Let \( R \) and \( r \) be the radius of larger and smaller circle respectively.
We know that, Area of circle \( = \pi \times (\text{Radius})^2 \)
Area of the larger circle be,
\( A_1 + A_2 = \pi R^2 = \frac{22}{7} \times 3^2 \)
\( = \frac{198}{7} \) [ \( R = 3 \) ] \dots (1)
Since, \( A_1, A_2, A_1 + A_2 \) are in AP
\( 2A_2 = A_1 + (A_1 + A_2) \Rightarrow A_2 = 2A_1 \)
\( A_1 + A_2 = A_1 + 2A_1 \) [adding both sides \( A_1 \)]
\( A_1 + A_2 = 3A_1 \)
\( \frac{198}{7} = 3A_1 \Rightarrow A_1 = \frac{66}{7} \) [From Eq. (1)]
\( \pi r^2 = \frac{66}{7} \Rightarrow \frac{22}{7} \times r^2 = \frac{66}{7} \)
\( r^2 = 3 \Rightarrow r = \sqrt{3} \)
[Since, Radius cannot be negative]

Question. The number of two digit numbers which are divisible by 3 is
(a) 33
(b) 31
(c) 30
(d) 29
Answer: C
Two digit numbers which are divisible by 3 are 12, 15, 18, \dots, 99;
So, \( 99 = 12 + (n - 1) \times 3 \)
\( 99 - 12 = 3n - 3 \)
\( 99 - 12 + 3 = 3n \)
\( 90 = 3n \)
\( n = 30 \)

Question. The sum of the series \( 45^2 - 43^2 + 44^2 - 42^2 + 43^2 - 41^2 + 42^2 - 40^2 + \dots \) upto 30 terms.
(a) 1110
(b) 2220
(c) 3330
(d) 4440
Answer: B
Let \( S = (45^2 - 43^2) + (44^2 - 42^2) + (43^2 - 41^2) + (42^2 - 40^2) + \dots \) upto 15 terms
\( = (45 + 43)(45 - 43) + (44 + 42)(44 - 42) + (43 + 41)(43 - 41) + (42 + 40)(42 - 40) + \dots \) upto 15 terms
[Since, \( a^2 - b^2 = (a - b)(a + b) \)]
\( = (45 + 43)2 + (44 + 42)2 + (43 + 41)2 + (42 + 40)2 + \dots \) upto 15 terms
\( = 2 [ \{45 + 44 + 43 + \dots \text{ upto 15 terms} \} + \{43 + 42 + 41 + \dots \text{ upto 15 terms} \} ] \)
\( = 2 [ \frac{15}{2}\{2 \times 45 + (15 - 1)(-1)\} + \frac{15}{2}\{2 \times 43 + (15 - 1)(-1)\} ] \)
\( = 15 [ 90 - 14 + 86 - 14 ] = 15 [ 148 ] = 2220 \)

Question. Take a point \( A(3, 4) \) on the graph and draw two lines from it, one is parallel to X-axis and another parallel to Y-axis. Again, take four points on both lines on both sides of A, such that their x-coordinates and y-coordinates form an AP with common difference 2. Then, the area of circle, passing through these four points is
(a) 12 sq units
(b) 13 sq units
(c) 12.56 sq units
(d) 13.56 sq units
Answer: C

Question. The sum of n terms of sequence \( \frac{1}{1 \times 2}, \frac{1}{2 \times 3}, \frac{1}{3 \times 4}, \dots \) is
(a) \( \frac{1}{n+1} \)
(b) \( \frac{1}{n} \)
(c) \( \frac{n+1}{n} \)
(d) \( \frac{n}{n+1} \)
Answer: D

Question. Suppose \( b_1, b_2, \dots, b_{24} \) are in AP, such that \( b_1 + b_5 + b_{10} + b_{15} + b_{20} + b_{24} = 300 \). Then the sum of first 24 terms of the AP is
(a) 1200
(b) 900
(c) 600
(d) 1500
Answer: A

Question. If the nth term of an A.P. is \( 4n + 1 \), then the common difference is :
(a) 3
(b) 4
(c) 5
(d) 6
Answer: B

Question. If \( a, b, c, d, e, f \) are in A.P., then \( e - c \) is equal to :
(a) \( 2(c - a) \)
(b) \( 2(d - c) \)
(c) \( 2(f - d) \)
(d) \( (d - c) \)
Answer: B

Question. Along a road line, an odd number of stones placed at intervals of 10 m. These stones have to be assembled around the middle stone. A person can carry only one stone at a time. A man carried the job with one of the end stone by carrying them in succession. In carrying, all the stones he covered a distance of 3 km. Then, the total number of stones is
(a) 10
(b) 15
(c) 12
(d) 25
Answer: D

Question. If \( S_1 = 3, 7, 11, 15, \dots \) upto 125 terms and \( S_2 = 4, 7, 10, 13, 16, \dots \) upto 125 terms, then how many terms are there in \( S_1 \) that are in \( S_2 \)?
(a) 29
(b) 30
(c) 31
(d) 32
Answer: C

FILL IN THE BLANK

Question. An arithmetic progression is a list of numbers in which each term is obtained by .......... a fixed number to the preceding term except the first term.
Answer: adding

Question. Sum of \( 1 + 3 + 5 + \dots + 1999 \) is ..........
Answer: \( \frac{1000}{2}[2(1) + (1000 - 1)2] \)

Question. If \( S_n \) denotes the sum of n term of an AP, then \( S_{12} - S_{11} \) is the .......... term of the AP.
Answer: twelfth

Question. The nth term of an AP whose first term is a and common difference is d is ..........
Answer: \( a + (n - 1)d \)

Question. The nth term of an AP is always a .......... expression.
Answer: linear

Question. The difference of corresponding terms of two A.P.’s will be ..........
Answer: another A.P.

Question. The sum of 8 A.Ms between 3 and 15 is ..........
Answer: 72

Question. The sum of the AP, \( 1 + 2 + \dots + 10 \) is ..........
Answer: 55

Question. Sum of all the integers between 100 and 1000 which are divisible by 7 is ..........
Answer: 70336

TRUE/FALSE

Question. The balance money (in ₹) after paying 5% of the total loan of ₹1000 every month is 950, 900, 850, 800, .......... 50. represented A.P.
Answer: True

Question. In an AP the letter d is generally used to denote the first term.
Answer: False

Question. 2, 4, 8, 16, .......... is not an A.P.
Answer: True

Question. The amount of money in the account of a person at the end of every year when the interest is calculated at 5% compound interest forms an AP.
Answer: False

Question. 10th term of A.P. 2, 7, 12, .......... is 45.
Answer: False

Question. Common difference of an AP may be positive, negative or zero.
Answer: True

Question. 301 is a term of A.P. 5, 11, 17, 23, .......... .
Answer: False

Question. If \( a_{n+1} - a_n = \) same for all ‘n’, then the given numbers form an A.P.
Answer: True

Question. If \( S_n \) of A.P. is \( 3n^2 + 2n \), then the first term of A.P. is 3.
Answer: False

Question. If a, b, c are in AP then, \( 2a = b + c \).
Answer: False

ASSERTION AND REASON

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Question. Assertion : If nth term of an A.P. is \( 7 - 4n \), then its common differences is -4.
Reason : Common difference of an A.P. is given by \( d = a_{n+1} - a_n \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: A

Question. Assertion : The sum of the first n terms of an AP is given by \( S_n = 3n^2 - 4n \). Then its nth term \( a_n = 6n - 7 \).
Reason : nth term of an AP, whose sum to n terms is \( S_n \), is given by \( a_n = S_n - S_{n-1} \)
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: A

Question. Assertion : If \( S_n \) is the sum of the first n terms of an A.P., then its nth term \( a_n \) is given by \( a_n = S_n - S_{n-1} \).
Reason : The 10th term of the A.P. 5, 8, 11, 14, .......... is 35.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: C

Question. Assertion : Common difference of an AP in which \( a_{21} - a_7 = 84 \) is 14.
Reason : nth term of AP is given by \( a_n = a + (n - 1)d \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: D

Question. Assertion : Sum of first hundred even natural numbers divisible by 5 is 500.
Reason : Sum of first n-terms of an A.P. is given by \( S_n = \frac{n}{2}[a + l] \) where \( l = \text{last term} \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: D

Question. Assertion : Arithmetic between 8 and 12 is 10.
Reason : Arithmetic between two numbers ‘a’ and ‘b’ is given as \( \frac{a + b}{2} \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: A