class 10 maths Arithmetic Progressions Hots Questions

Q 1 :

In an A.P., if the first term a = 7, nth term an 84 and the sum of first n terms $S_{n}$ = 2093/2, then n is equal to:

22
24
23
26

1

2

3

4

(3)

$S_{n} = \frac{n}{2} (a + a_{n}) \Rightarrow \frac{2093}{2} = \frac{n}{2} (7 + 84)$

$\Rightarrow 91 n = 2093 \Rightarrow n = 23$

Q 2 :

Which term of the A.P. –29, –26, –23, ..., 61 is 16?

11th
16th
10th
31st

1

2

3

4

(2)

Given, $a = - 29$

$d = - 26 - (- 29) = - 26 + 29 = 3 \Rightarrow d = 3$

$a_{n} = 16$

Here we know, $a_{n} = a + (n - 1) d$

$\Rightarrow 16 = - 29 + (n - 1) 3$

$\Rightarrow 16 + 29 = (n - 1) 3 = 15$

$\Rightarrow n - 1 = \frac{45}{3} = 15$

$\Rightarrow 15 = n - 1 \Rightarrow n = 16$

Q 3 :

The 7th term from the end of the A.P. : – 8, – 5, – 2, ..., 49 is :

67
13
31
10

1

2

3

4

(3) 31

Q 4 :

If k + 7, 2k – 2 and 2k + 6 are three consecutive terms of an A.P., then the value of k is :

15
17
5
1

1

2

3

4

(2) 17

Q 5 :

The common difference of an A.P. in which a15 – a11 = 48, is

12
16
–12
–16

1

2

3

4

(1)

$a_{15} - a_{11} = 48$

$\Rightarrow a + 14 d - a - 10 d = 48$

$\Rightarrow 4 d = 48 \Rightarrow d = 12$

Q 6 :

The next (4th) term of the AP $\sqrt 18, \sqrt 50, \sqrt 98, \dots i s :$

$\sqrt 128$
$\sqrt 140$
$\sqrt 162$
$\sqrt 200$

1

2

3

4

(3)

$W e h a v e, \sqrt 18 = \sqrt 9 \times 2 = 3 \sqrt 2 \sqrt 50 = \sqrt 25 \times 2 = 5 \sqrt 2 a n d \sqrt 98 = \sqrt 49 \times 2 = 7 \sqrt 2 \Rightarrow 3 \sqrt 2, 5 \sqrt 2, 7 \sqrt 2, \dots a r e i n A P w h e r e a = 3 \sqrt 2 a n d d = 7 \sqrt 2 - 5 \sqrt 2 = 2 \sqrt 2 T h u s, 4 ᵗ ʰ t e r m = 3 \sqrt 2 + (4 - 1) \times 2 \sqrt 2 = 3 \sqrt 2 + 6 \sqrt 2 = 9 \sqrt 2 = \sqrt (9) ² \times 2 = \sqrt 162 O p t i o n (c) i s c o r r e c t .$

Q 7 :

The nth term of an AP is $7 n + 4 .$ The common difference is:

7n
4
7
1

1

2

3

4

(3)

$W e h a v e, a_{n} = 7 n + 4 \Rightarrow a_{n - 1} = 7 (n - 1) + 4 = 7 n - 3 \Rightarrow C o m m o n d i f f e r e n c e, d = a_{n} - a_{n - 1} = (7 n + 4) - (7 n - 3) = 7$

Q 8 :

The common difference of the AP $\frac{1}{2 x}, \frac{1 - 4 x}{2 x}, \frac{1 - 8 x}{2 x}, is :$

−2x
−2
2
2x

1

2

3

4

(2)

$G i v e n A P i s \frac{1}{2 x}, \frac{1 - 4 x}{2 x}, \frac{1 - 8 x}{2 x}, \dots \Rightarrow C o m m o n d i f f e r e n c e d = \frac{1 - 4 x}{2 x} - \frac{1}{2 x} = \frac{1 - 4 x - 1}{2 x} = \frac{- 4 x}{2 x} = - 2$

Q 9 :

If a, b, c from an AP with common difference d, then the value of $a - 2 b - c$ is equal to

$2 a + 4 d$
0
$- 2 a - 4 d$
$- 2 a - 3 d$

1

2

3

4

(3)

$a, b, c a r e i n A P w i t h c o m m o n d i f f e r e n c e = d ∴ b - a = c - b = d \Rightarrow b - a = d \Rightarrow a = b - d \Rightarrow b = a + d a n d c - b = d \Rightarrow c = b + d \Rightarrow c = a + 2 d N o w, a - 2 b - c = a - 2 (a + d) - (a + 2 d) = a - 2 a - 2 d - a - 2 d = - 2 a - 4 d$

Q 10 :

The 11th term from the end of the AP: 10, 7, 4, …, –62 is

25
16
-32
0

1

2

3

4

(3)

Given AP is 10, 7, 4, …, –62.

$∴ n^{t h} t e r m f r o m e n d = l - (n - 1) d$

$H e r e, l = - 62, d = 7 - 10 = - 3$

$\Rightarrow 11^{t h} t e r m f r o m e n d = - 62 - (11 - 1) (- 3) = - 62 + 30 = - 32$

Topic Question Set

Q 1 :

In an A.P., if the first term a = 7, nth term an 84 and the sum of first n terms $S_{n}$ = 2093/2, then n is equal to:

Q 2 :

Which term of the A.P. –29, –26, –23, ..., 61 is 16?

Q 3 :

The 7th term from the end of the A.P. : – 8, – 5, – 2, ..., 49 is :

Q 4 :

If k + 7, 2k – 2 and 2k + 6 are three consecutive terms of an A.P., then the value of k is :

Q 5 :

The common difference of an A.P. in which a15 – a11 = 48, is

Q 6 :

The next (4th) term of the AP $\sqrt 18, \sqrt 50, \sqrt 98, \dots i s :$

Q 7 :

The nth term of an AP is $7 n + 4 .$ The common difference is:

Q 8 :

The common difference of the AP $\frac{1}{2 x}, \frac{1 - 4 x}{2 x}, \frac{1 - 8 x}{2 x}, is :$

Q 9 :

If a, b, c from an AP with common difference d, then the value of $a - 2 b - c$ is equal to

Q 10 :

The 11th term from the end of the AP: 10, 7, 4, …, –62 is

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Topic Question Set

Q 2 : Which term of the A.P. –29, –26, –23, ..., 61 is 16? .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

Q 3 : The 7th term from the end of the A.P. : – 8, – 5, – 2, ..., 49 is : .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

Q 5 : The common difference of an A.P. in which a15 – a11 = 48, is .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

Q 6 : The next (4th) term of the AP √18, √50, √98, … is: .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

Q 7 : The nth term of an AP is 7n+4. The common difference is: .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

Q 8 : The common difference of the AP 12x,1-4x2x,1-8x2x, is: .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

Q 10 : The 11th term from the end of the AP: 10, 7, 4, …, –62 is .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

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Q 2 :

Which term of the A.P. –29, –26, –23, ..., 61 is 16?

Q 3 :

The 7th term from the end of the A.P. : – 8, – 5, – 2, ..., 49 is :

Q 5 :

The common difference of an A.P. in which a15 – a11 = 48, is

Q 6 :

The next (4th) term of the AP $\sqrt 18, \sqrt 50, \sqrt 98, \dots i s :$

Q 7 :

The nth term of an AP is $7 n + 4 .$ The common difference is:

Q 8 :

The common difference of the AP $\frac{1}{2 x}, \frac{1 - 4 x}{2 x}, \frac{1 - 8 x}{2 x}, is :$

Q 10 :

The 11th term from the end of the AP: 10, 7, 4, …, –62 is