Introduction to Linear Polynomials

Which of the following is a linear polynomial?

1. a\(y^{3} + 2y - 1\)
2. b\(2x^{2} - 5x + 3\)
3. c\(4z - 3\)
4. d\(-9\)

What is the coefficient of \(x^{3}\) in the polynomial \(x^{4} - 3x^{3} + 6x^{2} - 2x + 7\)?

1. a\(-3\)
2. b6
3. c3
4. d4

Match List I with List II:

1. a(i)-R, (ii)-P, (iii)-Q, (iv)-S
2. b(i)-R, (ii)-S, (iii)-P, (iv)-Q
3. c(i)-Q, (ii)-S, (iii)-R, (iv)-P
4. d(i)-R, (ii)-S, (iii)-Q, (iv)-P

1. I. In the expression \(4x + 5y + 3\), the numbers 4 and 5 are the coefficients of \(x\) and \(y\), and 3 is a constant.
2. II. The expression \(4x + 5y + 3\) is a univariate polynomial.

Which of the statements above about the expression \(4x + 5y + 3\) is/are correct?

1. aOnly II
2. bOnly I
3. cBoth I and II
4. dNeither I nor II

What is the value of the polynomial \(7s^{2} - 4s + 6\) when \(s = -3\)?

1. a57
2. b\(-45\)
3. c81
4. d69

A chess club charges a joining fee of ₹200 plus ₹50 for every match played. A player paid ₹750 in all. How many matches did he play?

1. a15
2. b11
3. c19
4. d10

Assertion (A): The expression \(10x - x^{2}\), which gives the area of a rectangle made by bending a 20 cm wire, is a linear polynomial.

Reason (R): A polynomial of degree 1 is called a linear polynomial.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

In a growing pattern of square tiles, Stage \(n\) has \(2n - 1\) tiles. Which stage contains 47 tiles?

1. aStage 24
2. bStage 23
3. cStage 25
4. dStage 93

1. I. \(h(t) = 3 - 0.5t\), the height (in m) of water in a tank at the end of \(t\) months, represents linear decay.
2. II. In \(C(d) = 100 + 60d\), the cost rises by ₹60 for every extra km travelled.
3. III. Linear decay is represented by a straight line with positive slope.

Which of the statements above about linear growth and linear decay is/are correct?

1. aOnly I
2. bOnly III
3. cOnly I and II
4. dI, II and III

The height of water in a tank at the end of \(t\) months is \(h(t) = 3 - 0.5t\) metres. At the end of how many months will the tank become empty?

1. a5 months
2. b1.5 months
3. c3 months
4. d6 months

A school library has 1200 books. Every month it buys 35 new books, and no book is removed. How many books will the library have at the end of one year (12 months)?

1. a1585
2. b1620
3. c1655
4. d14435

An auto-rickshaw fare is ₹25 for the first 2 km, and then it increases by ₹15 for every further km. For how many km will the fare be ₹130?

1. a7 km
2. b10 km
3. c9 km
4. d11 km

Assertion (A): The lines \(y = 2x - 1\) and \(y = 2x + 5\) are parallel to each other.

Reason (R): Linear growth is represented by a straight line with positive slope.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

A water-tanker service charges a fixed fee plus a fixed amount for each kilolitre (kL) supplied. A family's bill was ₹380 for 4 kL and ₹530 for 7 kL. If the bill \(y\) (in ₹) for \(x\) kL is \(y = ax + b\), which relation is correct?

1. a\(y = 50x + 180\)
2. b\(y = 150x - 220\)
3. c\(y = 50x + 330\)
4. d\(y = 180x + 50\)

1. I. The line \(y = 3x\) is steeper than the line \(y = x\).
2. II. The line \(y = \frac{1}{2}x\) is steeper than the line \(y = x\).
3. III. The line \(y = x\) is equally inclined to both axes.

Which of the statements above about the steepness of lines is/are correct?

1. aOnly I
2. bOnly II and III
3. cI, II and III
4. dOnly I and III

Assertion (A): The line \(y = 5x\) passes through the origin.

Reason (R): Any straight line written in the form \(y = ax + b\) cuts the \(y\)-axis at the point \((0, b)\).

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

What are the slope and the \(y\)-intercept of the line \(2y = 4x + 7\)?

1. aSlope 4, \(y\)-intercept 7
2. bSlope 2, \(y\)-intercept 7
3. cSlope 4, \(y\)-intercept \(\frac{7}{2}\)
4. dSlope 2, \(y\)-intercept \(\frac{7}{2}\)

Match each line in List I with the point where it cuts the \(y\)-axis in List II:

1. a(i)-S, (ii)-R, (iii)-P, (iv)-Q
2. b(i)-R, (ii)-S, (iii)-P, (iv)-Q
3. c(i)-R, (ii)-Q, (iii)-P, (iv)-S
4. d(i)-P, (ii)-S, (iii)-R, (iv)-Q

The graph of a linear polynomial \(p(x)\) passes through the points \((1, 5)\) and \((3, 11)\). At which point does this graph cut the \(x\)-axis?

1. a\((\frac{2}{3}, 0)\)
2. b\((0, 2)\)
3. c\((-\frac{2}{3}, 0)\)
4. d\((-\frac{3}{2}, 0)\)

Riya makes a row of hexagons with matchsticks. Stage 1 is one hexagon made of 6 matchsticks. At every stage she adds a new hexagon that shares one side with the last hexagon. She has 200 matchsticks. What is the largest complete stage she can make, and how many matchsticks are left over?

1. aStage 39, with 4 matchsticks left
2. bStage 40, with no matchsticks left
3. cStage 38, with 9 matchsticks left
4. dStage 33, with 2 matchsticks left