Unit Test 2: Algebra
Class 10 Mathematics (Standard)
General Instructions:
- The question paper consists of 19 questions divided into 5 sections: A, B, C, D, and E.
- Section A: 8 MCQs of 1 mark each.
- Section B: 4 Short Answer questions of 2 marks each.
- Section C: 4 Short Answer questions of 3 marks each.
- Section D: 1 Case Study based question of 4 marks.
- Section E: 2 Long Answer questions of 4 marks each.
- Use of calculators is not permitted.
SECTION A (1 Mark Each)
[1]
1.
If one zero of the polynomial 2x² - 3x + k is reciprocal to the other, then the value of k is:
[1]
2.
The pair of equations x + 2y = 5 and 3x + 12y = 10 has:
[1]
3.
The roots of the quadratic equation x² - 0.04 = 0 are:
[1]
4.
The 30th term of the AP: 10, 7, 4, ... is:
[1]
5.
The degree of the zero polynomial is:
[1]
6.
For what value of k, do the equations 3x - y + 8 = 0 and 6x - ky = -16 represent coincident lines?
[1]
7.
The equation (x + 1)² - x² = 0 has number of real roots equal to:
[1]
8.
The sum of first n natural numbers is given by:
SECTION B (2 Marks Each)
[2]
9.
Find a quadratic polynomial whose zeroes are -3 and 4.
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[2]
10.
Solve for x and y: 2x + 3y = 11 and 2x - 4y = -24.
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[2]
11.
Find the roots of the quadratic equation 2x² - x + 1/8 = 0.
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[2]
12.
Which term of the AP: 3, 8, 13, 18, ... is 78?
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SECTION C (3 Marks Each)
[3]
13.
If α and β are the zeroes of the polynomial f(x) = x² - 6x + k, find the value of k such that 3α + 2β = 20.
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[3]
14.
The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.
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[3]
15.
Solve for x: 1/(x+4) - 1/(x-7) = 11/30, x ≠ -4, 7.
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[3]
16.
Find the sum of the first 24 terms of the list of numbers whose nth term is given by aₙ = 3 + 2n.
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SECTION D (Case Study - 4 Marks)
[4]
17.
Case Study: Classroom Arrangement
In a classroom, the seats are arranged in rows. The first row has 20 seats, the second row has 22 seats, the third row has 24 seats, and so on.
(i) If there are 15 rows in total, how many seats are there in the last row? (1 Mark)
(ii) How many total seats are there in the classroom? (2 Marks)
(iii) If 10 more seats are added to the last row, what will be the new number of seats in that row? (1 Mark)
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In a classroom, the seats are arranged in rows. The first row has 20 seats, the second row has 22 seats, the third row has 24 seats, and so on.
(i) If there are 15 rows in total, how many seats are there in the last row? (1 Mark)
(ii) How many total seats are there in the classroom? (2 Marks)
(iii) If 10 more seats are added to the last row, what will be the new number of seats in that row? (1 Mark)
SECTION E (Long Answer - 4 Marks Each)
[4]
18.
A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.
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[4]
19.
If the sum of the first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of the first n terms.
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TOTAL SCORE
0 / 8 (MCQ)
(Subjective answers submitted for review)
Solution Key (MCQs)
Q1. (a) 2
Product of roots = c/a = k/2. Reciprocal roots => product = 1. So k/2 = 1 => k=2.
Q2. (a) Unique solution
a1/a2 = 1/3, b1/b2 = 2/12 = 1/6. Since a1/a2 ≠ b1/b2, unique solution.
Q3. (a) ± 0.2
x² = 0.04 => x = ±√0.04 = ±0.2.
Q4. (c) -77
a=10, d=-3. a30 = 10 + 29(-3) = 10 - 87 = -77.
Q5. (d) Not defined
The degree of the zero polynomial is not defined.
Q6. (c) 2
Coincident lines: a1/a2 = b1/b2 = c1/c2. 3/6 = -1/-k => 1/2 = 1/k => k=2.
Q7. (a) 1
(x+1)² - x² = 0 => 2x+1=0. Linear equation has 1 root.
Q8. (c) n(n+1)/2
Standard formula for sum of first n natural numbers.