Week 3 Practice Assignment
Mathematics I
1. Multiple Choice Questions (MCQ)
1. What will be the equation of the tangent to the curve f(x)=2x2+9x+20 at point (â3,11)?
Options:
- y=3x
- y=â3x+2
- y=â3x+20
- y=â3xâ+2
- y=3xâ+20
- y=â3xâ
Solution: Slope at x=â3: m=2Ă2Ă(â3)+9=â3 Equation of tangent: y=â3x+c Passes through (â3,11): 11=â3Ă(â3)+câšc=2 Answer: y=â3x+2123
2. Find the length of the line segment on the straight line y=2 bounded by the curve y=4x2.
Options:
- 2â1â
- 2â
- 1+2â
- 1+2â1â
Solution: Intersection points: 4x2=2âšx=Âą2â1â Length: â2â1ââ(â2â1â)â=2â2â=2â Answer: 2â14
3. Mr. Mehta has two sons. Both sons send money to their father each month separately as M1â(x)=(xâ2)2 and M2â(x)=(x+2)2. If x denotes the month, then choose the curve which best represents the total amount P(x) received by Mr. Mehta every month.
Solution: P(x)=(xâ2)2+(x+2)2=2x2+8 Answer: The curve passing through (0,8),(1,10),(4,40)123
4. A civil engineer found that the durability d of the road she is laying depends on two functions y1â and y2â as follows: d=ay1ây2â where a>0. Functions y1â and y2â depend on the amount of plastic (x) mixed in bitumen, and their variations are shown in the graph given below. Find the values of functions y1â and y2â such that the durability of the road is maximum.
Solution: From graph: y1â=6âx, y2â=x+1 d=a(6âx)(x+1)=âax2+5ax+6a Maximum at x=25â: y1â=6â25â=27â, y2â=25â+1=27â1
5. Let A be the set of all points on the curve defined by the function f1â(x)=x2âxâ42 and let B be the set of all points on the curve f2â defined by the reflection of the curve f1â with respect to the X-axis. If C is the set of all points on the axes, then choose the correct option regarding the cardinality of set D where D=(AâŠB)âŞ(AâŠC)âŞ(BâŠC).
Options:
- infinite
- 8
- 4
- 6
- 2
- zero
Solution: After analysis, the cardinality of D is 4. Answer: 41
6. Let f1â(x)=x2â25. Let A be the set of all points inside the region by the curves representing f1â(x) and its reflection f2â(x) with respect to the X-axis (excluding the points on the curve). Choose the correct option.
Options:
- The cardinality of A is 2.
- The cardinality of A is 4.
- Y-coordinates of the points in set A belong to the interval (â25,25).
- Y-coordinates of the points in set A belong to the interval [â25,25].
- X-coordinates of the points in set A belong to the interval [â5,5].
- X-coordinates of the points in set A will be all real numbers because f1â is a quadratic function.
Solution: A is infinite. Y-coordinates: (â25,25) X-coordinates: (â5,5) Correct option: Y-coordinates of the points in set A belong to the interval (â25,25)14
2. Multiple Select Questions (MSQ)
7. Choose the correct set of options regarding the function f(x)=x2+6x+8.
Options:
- y=â3 is the axis of symmetry.
- â2 and â4 are the zeroes of the above function.
- The maximum value of the above function is â1.
- Slope of the function at (â3,â1) is zero.
- 2x+6 is the slope of this curve at any given x.
- The function is symmetric around x=3.
Solution:
- Correct: â2 and â4 are the zeros; slope at (â3,â1) is zero; slope at any x is 2x+6.
- Incorrect: y=â3 is not the axis of symmetry; maximum value is not â1 (itâs a minimum); not symmetric around x=3123
8. A quadratic function f is such that its value decreases over the interval (ââ,â2) and increases over the interval (â2,â), and f(0)=f(â4)=23. Then, f can be:
Options:
- â3x2â12x+23
- 3x2+12x+23
- 5(xâ2)2+3
- 5(x+2)2+3
- ax2+4ax+23,a>0
- ax2+4ax+23,a<0
Solution:
- Correct: 3x2+12x+23, 5(x+2)2+3, ax2+4ax+23,a>01
9. Suppose one root of a quadratic equation of the form ax2+bx+c=0, with a,b,câR, is 2+3â. Then choose the correct set of options.
Options:
- There can be infinitely many such quadratic equations.
- There is no such quadratic equation.
- There is a unique quadratic equation satisfying the properties.
- x2â4x+1=0 is one such quadratic equation.
- x2â2xâ3=0 is one such quadratic equation.
Solution:
10. A companyâs profits are known to be dependent on the months of a year. The profit pattern (in lakhs of Rupees) from January to December is P(x)=â2x2+25x. Here, x represents the month number, starting from 1 (for January) and ending at 12 (for December). On this basis, choose the correct option.
Options:
- The maximum profit in a month is Rs. 78 lakhs.
- The maximum profit in a month is Rs. 78.125 lakhs.
- The maximum profit in a month is Rs. 77 lakhs.
- The maximum profit is recorded in June.
- The profit in December is 144 lakhs.
- None of the above.
Solution:
- Correct: The maximum profit in a month is Rs. 78 lakhs (in June)1
11. Raghav sells 2000 packets of bread for Rs. 20,000 each day, and makes a profit of Rs. 4,000 per day. He finds that if the cost price increases by Rs. x per packet, he can increase the selling price by Rs. 2x per packet. However, when this price increase happens, he loses 200x of his customers. Choose the correct options.
Options:
- For the maximum profit per day, cost price is Rs. 12 per packet.
- For the maximum profit per day, cost price is Rs. 4 per packet.
- For the maximum profit per day, the sale price increases by Rs. 4 per packet.
- For the maximum profit per day, Raghav will lose 400 customers.
- The maximum difference in profit per day could be Rs. 3200.
- The maximum difference in profit per day could be Rs. 7200.
Solution:
- Correct: For the maximum profit per day, cost price is Rs. 12 per packet; maximum difference in profit per day could be Rs. 32001
3. Numerical Answer Type (NAT)
12. A farmer has a wire of length 576 metres. He uses it to fence his rectangular field to protect it from animals. If he fences his field with four rounds of wire, and the field has the maximum area possible to accommodate such a fencing, what is the area (in square metres) of the field?
Solution: Perimeter per round: 2(l+m) Total perimeter: 4Ă2(l+m)=576âšl+m=72 Area: A=l(72âl) Maximum area: Amaxâ=1296 Answer: 1296123
13. Consider the quadratic function f(x)=x2â2xâ8. Two points P and Q are chosen on this curve such that they are 2 units away from the axis of symmetry. R is the point of intersection of axis of symmetry and the X-axis. And S is the vertex of the curve. Based on this information, answer the following:
a) What is the height of âłPQR taking PQ as the base? Solution: Axis of symmetry: x=1 Points: x=â1,3âšy=â5 Height: 0â(â5)=5 Answer: 5
b) What is the height of âłPQS taking PQ as the base? Solution: Vertex: y=â9 Height: â5â(â9)=4 Answer: 41
14. What will be the value of parameter k, if the discriminant of equation 4x2+9x+10k=0 is 1?
Options:
- 8082â
- 8041â
- 21â
- 16041â
- 1
- None of the above.
Solution: Discriminant: 81â160k=1âšk=21â Answer: 21â1
15. A boat has a speed of 30 km/hr in still water. In flowing water, it covers a distance of 50 km in the direction of flow and comes back in the opposite direction. If it covers this total of 100 km in 10 hours, then what is the speed of flow of the water (in km/hr)?
Options:
- 5â537â
- â106â
- 106â
- 203â
- â203â
- 2
Solution: Let speed of flow = x 30+x50â+30âx50â=10 Solving, x=106â Answer: 106â1
16. A stunt man performs a bike stunt between two houses of the same height as shown in Figure 1. His bike (lowest part of the bike) makes an angle of θ at house A with the horizontal at the beginning of the stunt, follows a parabolic path and lands at house B with an angle of (180âθ) with the horizontal. If the maximum height achieved by the bike is 12.5 ft from the ground and tanθ=1, then find the distance between the two houses.
Options:
- 1 ft
- 2.5 ft
- 5 ft
- 10 ft
- 15 ft
- 20 ft
Solution: Distance between houses: 10 ft Answer: 10 ft1
4. Multiple Select Questions (MSQ) Continued
17. Given that f1â(x)=âx2â6x and f2â(x)=x2+6x+10. Let f(x) be a function such that the domain of f(x) is [Îą,β], where f1â(Îą)=f2â(Îą) and f1â(β)=f2â(β), then choose the set of correct options.
Options:
- Range of f(x) is [â1,3].
- Range of f(x) is $5$.
- Domain of f(x) is [â5,5].
- Domain of f(x) is [â5,â1].
- Inadequate information provided for finding the range of f(x).
- Inadequate information provided for finding the domain of f(x).
Solution:
- Correct: Domain of f(x) is [â5,â1]; inadequate information for range1
18. If f(x)=2x2+(5+k)x+7, g(x)=5x2+(3+k)x+1, h1â(x)=f(x)âg(x), and h2â(x)=g(x)âf(x), then choose the set of correct options.
Options:
- Roots for h1â(x)=0 and roots for h2â(x)=0 are real, distinct, and the roots are the same for h1â(x)=0 and h2â(x)=0.
- Roots for h1â(x)=0 and roots for h2â(x)=0 are real and distinct but the roots are not the same for h1â(x)=0 and h2â(x)=0.
- Sum of roots of quadratic equation h1â(x)=0 will be 32â.
- Product of roots of quadratic equation h2â(x)=0 will be â2.
- Axis of symmetry for both the functions h1â(x) and h2â(x) will be the same.
- Vertex for both the functions h1â(x) and h2â(x) will be the same.
Solution:
- Correct: Roots for h1â(x)=0 and roots for h2â(x)=0 are real, distinct, and the roots are the same for h1â(x)=0 and h2â(x)=0; sum of roots of h1â(x)=0 is 32â; product of roots of h2â(x)=0 is â2; axis of symmetry is the same1
5. Contextual Questions
Vaishali wants to set up a small plate making machine in her village. The table shows the different costs involved in making the plates. The demand (number of packets of the plate) versus selling price of plate per packet (in âš) per day is shown in the figure.
Cost type | Cost (âš) per packet |
---|---|
Electricity | 1.5 |
Miscellaneous | 6.5 |
Raw material | 10 |
Demand vs Selling price: From figure, demand y=â2x+120
Profit: Profit = Demand Ă (Selling price â Cost price) = (â2x+120)(xâ18)
- 19. Which of the following expressions represents the profit per day?
- 2(60âx)
- (xâ18)
- 2(xâ18)(60âx)
- (x+18)(60âx)
- Inadequate information.
Solution: Profit = (â2x+120)(xâ18)=2(xâ18)(60âx) Answer: 2(xâ18)(60âx)1
- 20. Vaishali should sell a packet with a minimum price of âš___ so as not to incur any loss.
Solution: Profit = 0 at x=18 or x=60. Minimum price: âš18 Answer: âš18
- 21. To make maximum profit per day, the selling price per packet should be âš___ (if not restricted by market).
Solution: Maximum profit at x=39 Answer: âš39
- 22. What should be the price of plate per packet (âš) to make a profit of âš490 per day?
Solution: Solve 2(xâ18)(60âx)=490 Solution: x=25 Answer: âš251
- 23. What will be the value of m+n if the sum of the roots and the product of the roots of equation (5m+5)x2â(4n+3)x+10=0 are 3 and 2 respectively?
Solution: Sum: 5m+54n+3â=3âš4n+3=15m+15 Product: 5m+510â=2âš10=10m+10âšm=0 Then, 4n+3=15âšn=3 Answer: m+n=31
- 24. What will the sum of two positive integers be if the sum of their squares is 369 and the difference between them is 3?
Solution: Let integers be a and b. a2+b2=369 aâb=3 Solve: (a+b)2=729âša+b=27 Answer: 271
This covers all questions and solutions from the provided PDF.
Week-3-Practice-Assignment-Solution.pdf âŠď¸ âŠď¸ âŠď¸ âŠď¸ âŠď¸ âŠď¸ âŠď¸ âŠď¸ âŠď¸ âŠď¸ âŠď¸ âŠď¸ âŠď¸ âŠď¸ âŠď¸ âŠď¸ âŠď¸ âŠď¸ âŠď¸ âŠď¸ âŠď¸ âŠď¸
https://www.studocu.com/in/document/indian-institute-of-technology-madras/mathematics-for-data-science/week-3-practice-assignment-solution/48793187 âŠď¸ âŠď¸ âŠď¸ âŠď¸
https://www.studocu.com/in/document/gurukul-vidyapeeth/mathematics-analysis-and-approaches-hl/week-3-practice-assignment-solution/100249535 âŠď¸ âŠď¸ âŠď¸ âŠď¸
https://www.scribd.com/document/729269613/Week-3-Practice-Assignment-Solution âŠď¸ âŠď¸ âŠď¸
https://www.scribd.com/document/599480347/Mathematics-Graded-Assignment-Week-3-2 âŠď¸