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Prism & PyramidSolved Questions· Unit 7
Mathematics · Chapter 07
32 solved questions

Chapter 7: Quadratic Equation

Every question in this chapter, answered and explained — step-by-step solutions drawn live from the chapter library across 3 question sections.

Section A — Exercise 7.1 (Solving Quadratic Equations)3Question 4 (Textbook Exercise 7.1) — Solve by Using the Formula3Section B — Exercise 7.2 (Word Problems)26
01

Section A — Exercise 7.1 (Solving Quadratic Equations)

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Question 1 (Textbook Exercise 7.1) — Identify Quadratic Equations

Ans.Which of the following are quadratic equations? Write with reason: (a) \((x-2)^2+1=2x-3\) (b) \(x(x+1)+8=(x+2)(x-2)\) (c) \(x(2x+3)=x^2+1\) (d) \((x+2)^3=x^3-4\) (e) \(x^2+3x+1=(x-2)^2\) (f) \((x+2)^3=2x(x^2-1)\)

Quadratic: (a), (c), (d). Not quadratic: (b), (e), (f).

2SEE Boardqb-q1

Question 2 (Textbook Exercise 7.1) — Solve by Factorization

Ans.Therefore, the roots of \(x^2 - 3x - 10=0\) are \(x = -2\) and \(x = 5\).

Therefore, the roots of \(2x^2 + x - 6=0\) are \(x = -2\) and \(x = \frac{3}{2}\).

3SEE Boardqb-q2

Question 3 (Textbook Exercise 7.1) — Solve by Completing the Square

Ans.Therefore, the roots of \(x^2 - 6x + 9=0\) are \(x = 3\) and \(x = 3\).

Therefore, the roots of \(9x^2 - 15x + 6=0\) are \(x = 1\) and \(x = \frac{2}{3}\).

02

Question 4 (Textbook Exercise 7.1) — Solve by Using the Formula

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Question 4 (Textbook Exercise 7.1) — Solve by Using the Formula

Ans.Therefore, the roots of \(x^2 - 9x + 20=0\) are \(x = 5\) and \(x = 4\).

Therefore, the roots of \(x^2 + 2x - 143=0\) are \(x = 11\) and \(x = -13\).

2SEE Boardqb-q4

Question 5 (Textbook Exercise 7.1) — Word Problem — Marks

Ans.Ramnaresh Mahato scored a total of 30 marks in two subjects, VR English and Mathematics, in the first terminal examination of grade 10. If he scored 2 more marks in Mathematics and 3 fewer marks in English, the product of his marks would be 210. Find his scores in both subjects.

His marks were 13 in Mathematics and 17 in English, or 12 in Mathematics and 18 in English.

3SEE Boardqb-q5

Question 6 (Textbook Exercise 7.1) — Word Problem — Playground

Ans.A rectangular playground's longer side is 30 m more than its shorter side, and its diagonal is 60 m more than its shorter side. (a) Find the length and breadth. (b) If 12 m × 3 m artificial-grass turfs are placed on the ground, how many turfs are needed? (c) If the ground is fenced 4 times with barbed wire costing Rs. 15 per metre, find the total cost.

(a) Length 120 m, breadth 90 m. (b) 300 turfs. (c) Rs. 25,200.

03

Section B — Exercise 7.2 (Word Problems)

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Question 1 (Textbook Exercise 7.2) — Number Problem

Ans.If 11 is added to the square of a natural number, the sum is 36. Find the number.

The number is 5.

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Question 2 (Textbook Exercise 7.2) — Number Problem

Ans.If 11 is subtracted from the square of a number, the remainder is 25. Find the number.

The number is \(\pm6\).

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Question 3 (Textbook Exercise 7.2) — Number Problem

Ans.If 7 is subtracted from double the square of a positive number, the remainder is 91. Find the number.

The number is 7.

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Question 4 (Textbook Exercise 7.2) — Number Problem

Ans.If 2 is subtracted from the square of a natural number, the remainder is 7. Find the number.

The number is 3.

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Question 5 (Textbook Exercise 7.2) — Number Problem

Ans.If 11 is subtracted from the square of a number and the remainder is 89, find that number.

The number is \(\pm10\).

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Question 6 (Textbook Exercise 7.2) — Number Problem

Ans.If 17 is subtracted from the square of a number, the remainder is 55. Find the number.

The number is \(\pm6\sqrt2\).

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Question 7 (Textbook Exercise 7.2) — Number Problem

Ans.If 3 is subtracted from double the square of a positive number, the remainder is 285. Find the number.

The number is 12.

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Question 8 (Textbook Exercise 7.2) — Number Problem

Ans.If the sum of a number and its square is 72, find the number.

The number is 8 (rejecting \(-9\) if a positive number is intended; both 8 and \(-9\) satisfy the equation algebraically).

9SEE Boardqb-q14

Question 9 (Textbook Exercise 7.2) — Consecutive Even Numbers

Ans.If the product of two consecutive even numbers is 80, find the numbers.

The numbers are 8 and 10 (or \(-10\) and \(-8\)).

10SEE Boardqb-q15

Question 10 (Textbook Exercise 7.2) — Consecutive Odd Numbers

Ans.If the product of two consecutive odd numbers is 225, find the numbers.

The numbers are 3 and 5 (or \(-5\) and \(-3\)).

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Question 11 (Textbook Exercise 7.2) — Reciprocal Problem

Ans.If the sum of a number and its reciprocal is \(\frac{10}{3}\), find the number.

Therefore, the roots of \(3x^2 - 10x + 3=0\) are \(x = \frac{1}{3}\) and \(x = 3\).

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Question 12 (Textbook Exercise 7.2) — Two Natural Numbers

Ans.If the sum of two natural numbers is 21 and the sum of their squares is 261, find the numbers.

The numbers are 6 and 15.

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Question 13 (Textbook Exercise 7.2) — Ages of Two Brothers

Ans.If the age difference between two brothers is 4 years and the product of their ages is 221, find their ages.

The ages are 17 years and 13 years.

14SEE Boardqb-q19

Question 14 (Textbook Exercise 7.2) — Present Ages of Two Brothers

Ans.The sum of the present ages of two brothers is 22 and the product of their ages is 120. Find their present ages.

Their present ages are 12 years and 10 years.

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Question 15 (Textbook Exercise 7.2) — Ages of Two Sisters

Ans.The age difference between two sisters is 3 years and the product of their ages is 180. Find their present ages.

Their present ages are 15 years and 12 years.

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Question 16 (Textbook Exercise 7.2) — Age-Product Problems (4 parts)

Ans.
  1. (a) Father 40, son 13: how many years ago was the product of their ages 198?
  2. (b) Mother 34, daughter 4: how many years later will the product be 400?
  3. (c) Father 35, son 1: how many years later will the product be 240?
  4. (d) Husband 35, wife 27: how many years ago was the product 425?

(a) 7 years ago.

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Question 17 (Textbook Exercise 7.2) — Geometry Problems (5 parts)

Ans.
  1. (a) Hypotenuse of a right triangle is 25 m; difference of the other two sides is 17 m. Find the remaining sides.
  2. (b) The hypotenuse is double and 6 m more than the shortest side; the remaining side is 2 m less than the hypotenuse. Find all sides.
  3. (c) Rectangular land: area \(150\text{ m}^2\), perimeter 50 m — find length and breadth.
  4. (d) Rectangular land: area \(54\text{ m}^2\), perimeter 30 m — find length and breadth.
  5. (e) Rectangle length 24 m, diagonal 16 m more than the breadth — find the area.

(a) The remaining sides are 24 m and 7 m.

18SEE Boardqb-q23

Question 18 (Textbook Exercise 7.2) — Two-Digit Number

Ans.A two-digit number is equal to four times the sum of its digits and three times the product of its digits. Find the number.

The number is 24.

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Question 19 (Textbook Exercise 7.2) — Pencil Distribution

Ans.An institute planned to distribute 180 pencils equally among the students enrolled in grade one. On the distribution day, 5 students were absent, so each student got 3 more pencils. (a) How many students were enrolled in grade one? (b) How many pencils did each student receive in total?

(a) 20 students were enrolled. (b) Each student received 12 pencils.

20SEE Boardqb-q25

Section C — Additional SEE-Style Practice

Ans.These additional questions are not from the textbook. They follow the same methods and difficulty level as Chapter 7, for extra exam-oriented practice.
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SEE Practice 1 — Direct Factorization

Ans.Solve \(x^2-8x+15=0\) by factorization.

Therefore, the roots of \(x^2 - 8x + 15=0\) are \(x = 3\) and \(x = 5\).

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SEE Practice 2 — Direct Formula

Ans.Solve \(4x^2-4x-3=0\) by using the formula.

Therefore, the roots of \(4x^2 - 4x - 3=0\) are \(x = \frac{3}{2}\) and \(x = - \frac{1}{2}\).

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SEE Practice 3 — Completing the Square

Ans.Solve \(x^2-4x+1=0\) by completing the square.

Therefore, the roots of \(x^2 - 4x + 1=0\) are \(x = \sqrt{3} + 2\) and \(x = 2 - \sqrt{3}\).

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SEE Practice 4 — Word Problem (Numbers)

Ans.The sum of a number and its square is 132. Find the number.

The number is 11 (rejecting \(-12\) if a positive number is required).

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SEE Practice 5 — Word Problem (Ages)

Ans.The product of a mother's and daughter's present ages is 175. If the mother is 18 years older than the daughter, find their present ages.

Daughter's age is 7 years, mother's age is 25 years.

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SEE Practice 6 — Word Problem (Right Triangle)

Ans.The hypotenuse of a right triangle is 17 m. If one side is 7 m longer than the other, find both sides.

The two sides are 8 m and 15 m.

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