Exercises
Mathematics I
Week 4 Practice Assignment

Week 4 Practice Assignment


Mathematics I

Multiple Choice Questions (MCQ)

Question 1: Which of the following polynomial functions represents the profit from selling Tamil books?

  • ○ 2x³ + 4x² - 2x - 4
  • ○ x³ - 2x² - x + 2
  • ○ x³ + 2x² - x - 2
  • ○ 2x³ - 4x² - 2x + 4

Solution: The profit from selling Tamil books is $x^3 + 2x^2 - x - 2$.

Question 2: In which year was the profit from Hindi books zero?

  • ○ 2001
  • ○ 2002
  • ○ 2004
  • ○ 2010

Solution: The profit from Hindi books was zero in the year 2001.

Question 3: Find the quadratic polynomial which when divided by $x$, $x – 1$, and $x + 1$ gives the remainders 7, 14, and 8 respectively.

  • ○ 4x² - 3x +7
  • ○ x² + 7x +7
  • ○ 7x² + x +7
  • ○ 4x² + 3x + 7

Solution: The quadratic polynomial is $4x^2 + 3x + 7$.

Question 4: Box A has length $x$ unit, breadth $(x + 1)$ unit, and height $(x + 2)$ unit. Box B has length $(x + 1)$ unit, breadth $(x + 1)$ unit, and height $(x + 2)$ unit. There are two more boxes C and D of cubic shape with side $x$ unit. The total volume of A and B is $y$ cubic unit more than the total volume of C and D. Find $y$ in terms of $x$.

  • ○ $x^3 + 7x^2 + 7x + 2$
  • ○ $7x^2 + 7x + 2$
  • ○ $7x^2 + 7x - 2$
  • ○ $x^3 + 7x^2 + 7x - 2$

Solution: $y = 7x^2 + 7x + 2$.

Question 5: The population of a bacteria culture in laboratory conditions is known to be a function of time of the form $p(t) = at^5 + bt^2 + c$, where $p$ represents the population (in lakhs) and $t$ represents the time (in minutes). Suppose a student conducts an experiment to determine the coefficients $a, b,$ and $c$ in the formula and obtains the following data:

  • $p(0) = 3$
  • $p(1) = 5$
  • $p(2) = 39$

Which of the following options is correct?

  • ○ $p(t) = 3t^5 – t^2 + 3$
  • ○ $p(t) = 4t^5 - 2t^2 + 3$
  • ○ $p(t) = t^5 + t^2 + 3$
  • ○ $p(t) = 39t^5 + 5t^2 + 3$

Solution: The correct function is $p(t) = t^5 + t^2 + 3$.

Question 6: If the polynomials $x^3 + ax^2 + 5x + 7$ and $x^3 + 2x^2 + 3x + 2a$ leave the same remainder when divided by $(x - 2)$, then the value of $a$ is:

  • ○ $-3/2$
  • ○ $3/2$
  • ○ $-5/2$
  • ○ $5/2$

Solution: The value of $a$ is $-3/2$.

Question 7: Let $r(x)$ be a polynomial function which is obtained as the remainder after dividing the polynomial $2x^3 + x^2 – 5$ by the polynomial $2x – 3$. Choose the correct option which represents the polynomial $r(x)$ most appropriately.

(The question includes figures labeled Option A, B, C, D, but the exact images are not shown here.)

Solution: The remainder $r(x) = 4$, which is a constant polynomial. Hence, the correct option is the one representing a constant value.

Multiple Select Questions (MSQ)

Question 8: By dividing a polynomial $p(x)$ with another polynomial $q(x)$ we get $h(x)$ as the quotient and $r(x)$ as the remainder. The maximum degree of $r(x)$ can be:

  • ○ deg $p(x)$
  • ○ deg $(p(x)) - 1$
  • ○ deg $q(x)$
  • ○ deg $(q(x)) – 1$

Solution: The maximum degree of $r(x)$ is deg $(q(x)) – 1$.

Question 9: If deg $p(x) <$ deg $q(x)$, then choose the set of correct answers:

  • ☐ $h(x) = 0$
  • ☐ deg $h(x) =$ deg $q(x)$
  • ☐ deg $r(x) =$ deg $q(x)$
  • ☐ deg $r(x) =$ deg $p(x)$

Solution: The correct answers are:

  • $h(x) = 0$
  • deg $r(x) =$ deg $p(x)$

Numerical Answer Type (NAT)

Question 10: An open box can be made from a piece of cardboard of length $7x$ unit and breadth $5x$ unit, by cutting squares of side $x$ unit out of the corners of the rectangular cardboard, then folding up the sides as shown in the Figure P-6.1.

What will be the coefficient of $x^3$ in the polynomial representing the volume of the box?

Solution: The coefficient of $x^3$ is 15.

Question 11: What will be the coefficient of $x^2$ in the polynomial representing the volume of the box?

Solution: The coefficient of $x^2$ is 0.

Question 12: What should be subtracted from the polynomial $P(x) = 6x^4 + 5x^3 + 4x – 4$ to make it divisible by $2x^2 + x – 1$?

  • ○ 4x
  • ○ 4x – 3
  • ○ 6x – 3
  • ○ 2x – 3

Solution: The correct answer is 4x – 3.

Question 13: Table 1 provides the information regarding some polynomials. Which is the most suitable (not exact) representation of $h(x)$ where $h(x)$ is known to be a polynomial in $x$, and if $h(x) = (P(x)Q(x)-R(x)S(x)+S(x)P(x))/(P(x)+P(x)Q^2(x))$?

Solution: The most suitable representation is option D (a constant polynomial).

Question 14: A manufacturing company produces three types of products A, B, and C from one raw material in a single continuous process. This process generates total solid wastes (W) (in kg) as $W(r) = -0.0001r^3 + 0.1r^2 + r$, where $r$ is the amount of raw material used in kg. If instead, the company uses three different batch-processes (one batch process for one product) to produce the above products, then the amount of waste generated because of products A, B, and C are given as $W_A = -0.00001r^4 + 0.015r^3$, $W_B = -0.005r^3 + 0.05r^2$, and $W_C = 0.05r^2$ respectively.

What is the total amount of waste generated because of the three different batch-processes?

  • ○ $-0.00001r^4 + 0.01r^3 + 1.5r^2$
  • ○ $-0.00001r^4 + 0.015r^3 + 1.5r^2$
  • ○ $-0.00001r^4 + 0.01r^3 + 0.1r^2$
  • ○ $-0.00001r^4 + 0.01r^3 + 0.5r^2$
  • ○ $-0.00001r^4 + r^3 + 1.5r^2$
  • ○ $0.0001r^4 + 0.01r^3 + 1.5r^2$

Solution: The total amount of waste generated is $-0.00001r^4 + 0.01r^3 + 0.1r^2$.

Question 15: What is the ratio of the total waste generated by the three-batch-processes with respect to the single continuous process?

  • ○ $-0.001r^2$
  • ○ $-0.001r$
  • ○ $-0.01r$
  • ○ $-0.1r$
  • ○ $0.1r$
  • ○ $0.01r$

Solution: The ratio is 0.1r (Note: The solution in the PDF shows the ratio as $\frac{WA + WB + WC}{W(r)} = 0.1r$, but this is a simplification. The correct option is 0.1r as per the options given.)

Question 16: Let the company wastes Rs. 5,000 in waste treatment when it uses the single continuous process by consuming 100 kg of raw material. If instead of continuous process the company uses the three-batch-processes, then how much extra amount (in Rs.) will the company have to pay for waste treatment with respect to the continuous process?

  • ○ 50,000
  • ○ 500
  • ○ 45,000
  • ○ 5,000
  • ○ 4,000

Solution: The extra amount required is 45,000.

Multiple Select Questions (MSQ) (continued)

Question 17: Given $P(x)$ and $Q(x)$ be two non zero polynomials of degrees $m$ and $n$ respectively. If $f(x) = P(x) + Q(x)$, $g(x) = P(x)Q(x)$, and $h(x) = P(x){P(x)Q(x) + P(x)}$, If $h(x)$ is known to be a polynomial in $x$, then choose the set of correct options.

  • The degree of $f(x)$ is $m + n$.
  • The degree of $g(x)$ is $m + n$.
  • The degree of $f(x)$ is $\max{m, n}$ if $m \neq n$, where $\max{m, n}$ represents the maximum value from $m$ and $n$.
  • The degree of $h(x)$ is $m^3$.
  • The degree of $h(x)$ is $n^3$.
  • The degree of $h(x)$ is $2m + n$.

Solution: The correct options are:

  • The degree of $g(x)$ is $m + n$.
  • The degree of $f(x)$ is $\max{m, n}$ if $m \neq n$.
  • The degree of $h(x)$ is $2m + n$.

Question 18: Given a polynomial $P(x) = (2x+5)(1-3x)(x^2 + 3x + 1)$, then choose the set of correct options.

  • Coefficient of $x^5$ is 0.
  • Coefficient of $x^3$ is -18.
  • Degree of $P$ is 4.
  • Coefficient of $x^3$ is -31.
  • Degree of $P$ is 7.
  • All of the above.

Solution: The correct options are:

  • Coefficient of $x^5$ is 0.
  • Degree of $P$ is 4.
  • Coefficient of $x^3$ is -31.

Question 19: A sheet ABCD of dimensions 10 ft x 3 ft is shown in Figure 7. A box is made by removing two squares of equal dimensions AEFG and DHIJ and two rectangles of equal dimensions BKLM and CNOP respectively.

  • The volume of the box is $2x^2 – 23x + 30$.
  • The volume of the box is $2x^3 – 13x^2 + 15x$.
  • If $x = 0.5$, then the volume of the box is 5.625 cubic ft.
  • To create the box, value of $x$ should always be greater than 0 but less than 1.5.

Solution: The correct options are:

  • The volume of the box is $2x^3 – 13x^2 + 15x$.
  • To create the box, value of $x$ should always be greater than 0 but less than 1.5.

Numerical Answer Type (NAT) (continued)

Question 20: A curious student created a performance profile of his favourite cricketer as $R = -x^5 + 6x^4 - 30x^3 + 80x^2 + 70x + c$, where $R$ is the total cumulative runs scored by the cricketer in $x$ matches. He picked three starting values shown in Table 2 and tried to find the value of $c$. If he uses Sum Squared Error method, then what will be the value of $c$?

Solution: The value of $c$ is -2.

Question 21: What is the minimum value of $x$-coordinate for the points of intersection of functions $f(x) = -x^5 + 5x^4 - 7x - 2$ and $g(x) = -x^5 + 5x^4 – x^2 - 2$?

Solution: The minimum value of $x$-coordinate is 0.

All questions and solutions from the provided PDF have been extracted above1.

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  1. Week-4-Practice-Assignment-Solution.pdf ↩︎

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1. Multiple Choice Questions (MCQ) 1. What will be the equation of the tangent to the curve $ f(x) = 2x^2 + 9x + 20 $ at point $(-3, 11)$? Options: $ y = 3x $ $ y = -3x + 2 $ $ y = -3x + 20 $ $ y = -\frac{x}{3} + 2 $ $ y = \frac{x}{3} + 20 $ $ y = -\frac{x}{3} $ Solution: Slope at $ x = -3 $: $ m = 2 \times 2 \times (-3) + 9 = -3 $ Equation of tangent: $ y = -3x + c $ Passes through $(-3, 11)$: $ 11 = -3 \times (-3) + c \implies c = 2 $ Answer: $ y = -3x + 2 $123

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