Chapter 2 practice
Practice: Introduction to Linear Polynomials
A question bank of 50 exam-style questions on this chapter only: 20 one-mark, 12 two-mark, 10 three-mark, 8 four-mark. Every question is followed by its answer and a full explanation. Use the buttons to list only the questions of one mark value.
Sources are shown on each question. "NCERT Exemplar" questions are adapted from NCERT's official Exemplar Problems. "Board pattern" questions follow standard CBSE exam questions on the same skills. "Written for this site" questions cover this chapter's own topics. Every answer has been worked out and checked.
The degree of the polynomial 5x³ − 2x + 7 is
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(c) 3, the highest power of x.
The degree of the constant polynomial −8 is
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(a) 0, because −8 = −8x⁰.
Which of these is a linear polynomial?
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(b) 3x − 7. Its highest power of x is 1. (a) is quadratic, (c) is cubic, (d) is a constant.
The coefficient of x in 2x³ − x + 4 is
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(b) −1. −x means −1 × x, and the minus sign belongs to the coefficient.
The coefficient of x² in 3x³ + 5x − 2 is
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(c) 0. There is no x² term at all.
The constant term of 4 − 7x + x² is
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(c) 4. The order of terms does not matter; 4 is the term without x.
The value of 5x − 4x² + 3 when x = −1 is
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(a) −6. 5 × (−1) − 4 × (−1)² + 3 = −5 − 4 + 3 = −6.
How many points (x, y) satisfy y = 2x + 3?
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(d) infinitely many. Every point on the straight line works, and a line goes on forever.
The graph of y = 5x passes through
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(a) (0, 0). Any line y = ax passes through the origin, since x = 0 gives y = 0.
The slope of the line y = −4x + 9 is
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(c) −4, the number multiplying x.
The y-intercept of y = 7x − 2 is
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(b) −2. The line crosses the y-axis at (0, −2).
Which point lies on the line y = 3x − 1?
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(b) (2, 5): 3 × 2 − 1 = 5. The others fail: (1, 3) gives 2, (0, 1) gives −1, (−1, 2) gives −4.
The graph of y = 6 is a line
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(a). y = 6 is y = 0x + 6: slope 0 (flat) and y-intercept 6.
Which pair of lines is parallel?
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(b). Both have slope 3, with different y-intercepts.
Which rule shows linear decay?
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(b). It drops by the same 15 for every step in x (negative slope). (d) is not linear at all.
The next number in the linear pattern 5, 9, 13, 17, … is
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(b) 21. The pattern jumps by 4 each time.
Is 4, 7, 11, 16 a linear pattern? Why?
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No. The jumps are 3, 4, 5. A linear pattern needs the same jump every time.
If p(x) = 2x + 7, find p(−3).
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2 × (−3) + 7 = −6 + 7 = 1.
Write a linear polynomial whose value at x = 0 is 5 and which grows by 3 for every increase of 1 in x.
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3x + 5. The growth per step (3) is the slope, and the value at 0 (5) is the constant.
True or false: the line y = −2x passes through the origin.
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True. It has the form y = ax (b = 0), so x = 0 gives y = 0.
Classify as linear, quadratic or cubic: (i) 2x − 3 (ii) x² + x³ (iii) 7 − y² (iv) √2 x
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(i) linear (ii) cubic, highest power 3 (iii) quadratic (iv) linear: √2 is just the coefficient; x still has power 1.
Find the value of 3t² − 5t + 2 at t = 2 and at t = −1.
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t = 2: 3 × 4 − 10 + 2 = 4.
t = −1: 3 × 1 − 5 × (−1) + 2 = 3 + 5 + 2 = 10.
The sum of three consecutive integers is 51. Find them.
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Let them be x, x + 1, x + 2. Then 3x + 3 = 51, so 3x = 48, so x = 16.
16, 17 and 18
One number is 12 more than another, and their sum is 90. Find the numbers.
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x + (x + 12) = 90, so 2x = 78, so x = 39.
39 and 51
A taxi charges ₹50 fixed plus ₹14 per km. Write the cost C(d) for d km, and find the distance if the fare was ₹246.
Answer
C(d) = 50 + 14d. Then 50 + 14d = 246, so 14d = 196, so d = 14 km.
Find the rule for the n-th term of 7, 12, 17, 22, … and use it to find the 20th term.
Answer
Jump d = 5, first term a = 7. Rule: 5n + (7 − 5) = 5n + 2. 20th term: 5 × 20 + 2 = 102.
A tank holds 1200 litres and leaks 40 litres every hour. Write W(t), the water left after t hours, and find when the tank is empty.
Answer
W(t) = 1200 − 40t. Empty when 1200 − 40t = 0, so t = 30 hours.
Check whether (2, 7) and (−1, 1) lie on the line y = 2x + 3.
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(2, 7): 2 × 2 + 3 = 7. Yes. (−1, 1): 2 × (−1) + 3 = 1. Yes. Both lie on the line.
Find k if the point (2, 3) lies on the line y = kx − 5.
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Substitute: 3 = 2k − 5, so 2k = 8, so k = 4.
Find the slope and y-intercept of 4y = 8x − 12.
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Divide by 4: y = 2x − 3. Slope 2, y-intercept −3.
Write the equation of the line parallel to y = −3x + 2 that passes through the origin.
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Parallel means the same slope, −3. Through the origin means b = 0. y = −3x.
The line y = ax + b passes through (0, 4) and (2, 10). Find a and b.
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(0, 4) gives b = 4. Then 10 = 2a + 4, so a = 3. y = 3x + 4.
A mother is 26 years older than her daughter. In 5 years, the mother will be 3 times as old as the daughter. Find their present ages.
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- Daughter now: d. Mother now: d + 26.
- In 5 years: daughter d + 5, mother d + 31.
- d + 31 = 3(d + 5) = 3d + 15, so 16 = 2d, so d = 8.
Daughter 8 years, mother 34 years
Check: in 5 years, 13 and 39, and 39 = 3 × 13.
A rectangle's perimeter is 64 cm. Its length is 2 cm more than twice its width. Find its length and width.
Answer
Width w, length 2w + 2. Perimeter = 2(w + 2w + 2) = 6w + 4 = 64, so 6w = 60, so w = 10.
Width 10 cm, length 22 cm
A bag has only ₹5 and ₹10 coins: 25 coins worth ₹175 in total. How many of each are there?
Answer
Let x be the number of ₹5 coins; then 25 − x are ₹10 coins. 5x + 10(25 − x) = 175, so 250 − 5x = 175, so x = 15.
15 five-rupee coins and 10 ten-rupee coins
Check: 75 + 100 = 175.
A mobile bill is y = ax + b for x GB of data. 5 GB cost ₹330 and 12 GB cost ₹540. Find a and b, and the bill for 20 GB.
Answer
7 more GB cost ₹210 more, so a = 210 ÷ 7 = 30. Then b = 330 − 30 × 5 = 180. So y = 30x + 180.
For 20 GB: 30 × 20 + 180 = ₹780.
A 30 cm candle burns down 2.5 cm every hour. Write its height h after t hours, find the height after 4 hours and when it burns out. Is this growth or decay?
Answer
h = 30 − 2.5t. After 4 hours: 30 − 10 = 20 cm. Burns out when 2.5t = 30, so t = 12 hours. It is linear decay: the height drops by the same 2.5 cm every hour.
Draw the graph of y = 2x − 4 using a table of values. Where does it cut each axis?
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| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | −4 | −2 | 0 | 2 |
It cuts the y-axis at (0, −4) and the x-axis at (2, 0) (where y = 0, so 2x = 4).
Compare y = −x + 3 and y = −x − 2: slopes, y-intercepts, whether they are parallel, and how far apart they cross the y-axis.
Answer
Both have slope −1. Their y-intercepts are 3 and −2. Same slope, different intercepts, so they are parallel. On the y-axis they cross at (0, 3) and (0, −2), which are 5 units apart.
Squares made of matchsticks are placed in a row, each sharing a side with the previous one: 4, 7, 10, … sticks. Find the rule, the sticks for 25 squares, and whether exactly 100 sticks make a complete row.
Answer
Jump 3, first term 4: rule 3n + 1. For 25 squares: 3 × 25 + 1 = 76.
3n + 1 = 100 gives 3n = 99, so n = 33, a whole number. Yes, 100 sticks make exactly 33 squares.
The rule F = (9/5)C + 32 changes Celsius to Fahrenheit. Convert 25 °C and −40 °C. What is special about the second answer?
Answer
25 °C: (9/5) × 25 + 32 = 45 + 32 = 77 °F.
−40 °C: (9/5) × (−40) + 32 = −72 + 32 = −40 °F. At −40 the two scales read the same number.
p(x) = ax + b with p(1) = 4 and p(3) = 10. Find p(x), p(0), and the value of x where p(x) = 0.
Answer
a + b = 4 and 3a + b = 10. Subtract: 2a = 6, so a = 3, b = 1. p(x) = 3x + 1.
p(0) = 1. p(x) = 0 when 3x + 1 = 0, so x = −1/3.
A school fair charges ₹30 entry plus ₹20 for each game. (i) What do 4 games cost in total? (ii) Write the cost C for n games. (iii) How many games for ₹190? (iv) Can someone spend exactly ₹175? Explain.
Answer
(i) 30 + 80 = ₹110. (ii) C = 30 + 20n. (iii) 20n = 160, so 8 games. (iv) 20n = 145 gives n = 7.25, not a whole number, so no.
A laptop costs ₹24,000 and loses ₹2,500 of value each year. (i) Value after 4 years? (ii) Write v in terms of t. (iii) When is it worth ₹9,000? (iv) Is this growth or decay, and what is the sign of the slope?
Answer
(i) 24000 − 10000 = ₹14,000. (ii) v = 24000 − 2500t. (iii) 2500t = 15000, so t = 6 years. (iv) Linear decay, with a negative slope (−2500).
Draw y = x + 1 and y = −x + 5 on the same axes. Where do they meet? Are they parallel?
Answer
Where they meet, both give the same y: x + 1 = −x + 5, so 2x = 4, so x = 2, and y = 3. They meet at (2, 3).
Their slopes are 1 and −1, which are different, so they are not parallel (that is why they meet).
p(x) and q(x) are linear, with p(x) + q(x) = 5x + 3 and p(x) − q(x) = x − 1. Find p(x) and q(x), and where q(x) cuts the x-axis.
Answer
Add the two facts: 2p(x) = 6x + 2, so p(x) = 3x + 1. Then q(x) = (5x + 3) − (3x + 1) = 2x + 2.
q(x) = 0 when 2x + 2 = 0, so x = −1: the point (−1, 0).
Check: p − q = x − 1. Correct.
The digits of a two-digit number add up to 9. Adding 27 to the number reverses its digits. Find the number.
Answer
- Tens digit a, units digit b: the number is 10a + b, its reverse is 10b + a.
- 10a + b + 27 = 10b + a, so 9b − 9a = 27, so b − a = 3.
- Also a + b = 9. Adding: 2b = 12, so b = 6 and a = 3.
The number is 36
Check: 36 + 27 = 63.
A line passes through (−2, −1) and (4, 11). Find its equation y = ax + b, both intercepts, and whether (1, 5) lies on it.
Answer
−2a + b = −1 and 4a + b = 11. Subtract: 6a = 12, so a = 2; then b = −1 + 4 = 3. y = 2x + 3.
y-axis: (0, 3). x-axis: 2x + 3 = 0, so (−1.5, 0).
(1, 5): 2 × 1 + 3 = 5. Yes, it lies on the line.
An auto-rickshaw charges ₹30 for the first 1.5 km, then ₹16 for every extra km. (i) Write the fare for n km (n at least 1.5). (ii) Fare for 6 km? (iii) Distance for ₹166? (iv) Why do we subtract 1.5 in the rule?
Answer
(i) Fare = 30 + 16(n − 1.5) = 30 + 16n − 24 = 16n + 6.
(ii) 16 × 6 + 6 = ₹102. (iii) 16n + 6 = 166, so 16n = 160, so 10 km.
(iv) The first 1.5 km are already paid for by the ₹30, so only the km after that are charged at ₹16.
Plan A: start with ₹500 and save ₹200 a month. Plan B: start with ₹1400 and save ₹120 a month. (i) Write each amount after m months. (ii) Which is ahead after 11 months? After 12? (iii) Which plan's line is steeper, and what does that mean?
Answer
(i) A = 500 + 200m, B = 1400 + 120m.
(ii) Month 11: A = 2700, B = 2720, so B is ahead. Month 12: A = 2900, B = 2840, so A is ahead. (They would be equal at m = 11.25.)
(iii) A has the bigger slope (200 against 120), so its line is steeper: Plan A grows faster, which is why it overtakes B even though it starts lower.