Question 1

Q1: Check whether the following are quadratic equations: (i) $(x+1)^{2}=2(x-3)$ (ii) $x^{2}-2 x=(-2)(3-x)$ (iii) $(x-2)(x+1)=(x-1)(x+3)$ (iv) $(x-3)(2 x+1...

Accepted Answer

A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and $a 
eq 0$. (i) $(x+1)^{2}=2(x-3)$ Solution: We simplify the given equation: $$LHS = (x+1)^2 = x^2 + 2(x)(1) + 1^2 = x^2 + 2x + 1$$ $$RHS = 2(x-3) = 2x - 6$$ Equating LHS and RHS: $$x^2 + 2x + 1...

Question 2

Q1: Check whether the following are quadratic equations: (i) $(x+1)^{2}=2(x-3)$ (ii) $x^{2}-2 x=(-2)(3-x)$ (iii) $(x-2)(x+1)=(x-1)(x+3)$ (iv) $(x-3)(2 x+1...

Accepted Answer

A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and $a 
eq 0$. (i) $(x+1)^{2}=2(x-3)$ Solution: Expanding the terms: LHS: $(x+1)^2 = x^2 + 2x + 1$ RHS: $2(x-3) = 2x - 6$ Equating LHS and RHS: $$x^2 + 2x + 1 = 2x - 6$$ $$x^2 + 2x - 2x + 1 + 6 = 0...

Question 3

Q2: Represent the following situations in the form of quadratic equations : (i) The area of a rectangular plot is $528 	ext{ m}^{2}$. The length of the p...

Accepted Answer

(i) The area of a rectangular plot is $528 	ext{ m}^{2}$. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot. Solution: Let the breadth of the rectangular plot be $x$ metres. According to the problem, the length of the plot is o...

Question 4

Q2: Represent the following situations in the form of quadratic equations : (i) The area of a rectangular plot is $528 	ext{ m}^{2}$. The length of the p...

Accepted Answer

(i) The area of a rectangular plot is $528 	ext{ m}^{2}$. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot. Solution: Let the breadth of the rectangular plot be $x$ metres. According to the problem, the length of the plot is o...

Question 5

Q1: Find the roots of the following quadratic equations by factorisation: (i) $x^{2}-3 x-10=0$ (ii) $2 x^{2}+x-6=0$ (iii) $\sqrt{2} x^{2}+7 x+5 \sqrt{2}=0...

Accepted Answer

(i) $x^{2}-3 x-10=0$ Solution: To factorise the equation, we need to find two numbers whose product is $1 	imes (-10) = -10$ and whose sum is $-3$. These numbers are $-5$ and $2$. Splitting the middle term: $$x^2 - 5x + 2x - 10 = 0$$ Factoring by grouping: $$x(x - 5) + 2(x - 5) = 0$$ $$(x - 5)(x +...

Question 6

Q1: Find the roots of the following quadratic equations by factorisation: (i) $x^{2}-3 x-10=0$ (ii) $2 x^{2}+x-6=0$ (iii) $\sqrt{2} x^{2}+7 x+5 \sqrt{2}=0...

Accepted Answer

(i) $x^{2}-3 x-10=0$ Solution: We need to find two numbers whose product is -10 and sum is -3. These numbers are -5 and 2. Splitting the middle term: $$x^2 - 5x + 2x - 10 = 0$$ $$x(x - 5) + 2(x - 5) = 0$$ $$(x - 5)(x + 2) = 0$$ So, either $x - 5 = 0$ or $x + 2 = 0$. If $x - 5 = 0$, then $x = 5$. If...

Question 7

Q2: Solve the problems given in Example 1.

Accepted Answer

The problems in Example 1 require finding the roots of the quadratic equations derived. (i) From Example 1(i), the equation is $x^{2}-45 x+324=0$. This problem was about John and Jivanti's marbles, where $x$ is the number of marbles John had. Solution: We need to solve the equation $x^2 - 45x + 324...

Question 8

Q2: Solve the problems given in Example 1.

Accepted Answer

(i) John and Jivanti's marbles problem Given: From Example 1(i), the quadratic equation representing the situation is: $$x^{2}-45 x+324=0$$ where $x$ is the number of marbles John had. To Find: The number of marbles they had to start with. Solution: We need to solve the equation $x^2 - 45x + 324 = 0...

Question 9

Q3: Find two numbers whose sum is 27 and product is 182.

Accepted Answer

Given: Sum of two numbers = 27 Product of two numbers = 182 To Find: The two numbers. Let: One number be $x$. Then the other number is $(27 - x)$. Equation: According to the problem, the product of the two numbers is 182. $$x(27 - x) = 182$$ $$27x - x^2 = 182$$ $$x^2 - 27x + 182 = 0$$ Solution: We s...

Question 10

Q3: Find two numbers whose sum is 27 and product is 182.

Accepted Answer

Given: Sum of two numbers = 27 Product of two numbers = 182 To Find: The two numbers. Let: Let one number be $x$. Since the sum of the two numbers is 27, the other number will be $(27 - x)$. Equation: According to the problem, the product of the two numbers is 182. $$x(27 - x) = 182$$ $$27x - x^2 =...

Question 11

Q4: Find two consecutive positive integers, sum of whose squares is 365.

Accepted Answer

Given: The sum of the squares of two consecutive positive integers is 365. To Find: The two integers. Let: The two consecutive positive integers be $x$ and $(x+1)$. Equation: According to the problem, the sum of their squares is 365. $$x^2 + (x+1)^2 = 365$$ Solution: Expanding and simplifying the eq...

Question 12

Q5: The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

Accepted Answer

Given: Hypotenuse of a right triangle = 13 cm. The altitude is 7 cm less than its base. To Find: The lengths of the base and altitude. Let: The base of the right triangle be $x$ cm. Then, the altitude of the triangle is $(x - 7)$ cm. Equation: By the Pythagorean theorem, for a right triangle: $$(	e...

Question 13

Q6: A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each art...

Accepted Answer

Given: Total cost of production = ₹ 90. Cost of each article = 3 more than twice the number of articles produced. To Find: The number of articles produced and the cost of each article. Let: The number of articles produced on that day be $x$. Then, the cost of production of each article is $(2x + 3)$...

Question 14

Q1: Find the nature of the roots of the following quadratic equations. If the real roots exist, find them: (i) $2 x^{2}-3 x+5=0$ (ii) $3 x^{2}-4 \sqrt{3}...

Accepted Answer

The nature of the roots of a quadratic equation $ax^2 + bx + c = 0$ is determined by its discriminant, $D = b^2 - 4ac$. - If $D > 0$, the equation has two distinct real roots. - If $D = 0$, the equation has two equal real roots. - If $D  0$, the equation has two distinct real roots. The roots are gi...

Question 15

Q2: Find the values of $k$ for each of the following quadratic equations, so that they have two equal roots. (i) $2 x^{2}+k x+3=0$ (ii) $k x(x-2)+6=0$

Accepted Answer

For a quadratic equation to have two equal roots, its discriminant ($D = b^2 - 4ac$) must be equal to zero. (i) $2 x^{2}+k x+3=0$ Solution: Comparing with $ax^2 + bx + c = 0$, we have $a=2, b=k, c=3$. For equal roots, $D = 0$. $$b^2 - 4ac = 0$$ $$k^2 - 4(2)(3) = 0$$ $$k^2 - 24 = 0$$ $$k^2 = 24$$ $$k...

Question 16

Q3: Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is $800 	ext{ m}^{2}$? If so, find its length and...

Accepted Answer

Given: Area of rectangular mango grove = $800 	ext{ m}^2$. Length is twice its breadth. To Find: Whether the situation is possible, and if so, the length and breadth. Let: The breadth of the grove be $x$ metres. Then, the length of the grove is $2x$ metres. Equation: Area = length $	imes$ breadth...

Question 17

Q4: Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product...

Accepted Answer

Given: Sum of the present ages of two friends = 20 years. Four years ago, the product of their ages was 48. To Find: Whether the situation is possible, and if so, their present ages. Let: The present age of one friend be $x$ years. Then, the present age of the other friend is $(20 - x)$ years. Four...

Question 18

Q5: Is it possible to design a rectangular park of perimeter 80 m and area $400 	ext{ m}^{2}$? If so, find its length and breadth.

Accepted Answer

Given: Perimeter of a rectangular park = 80 m. Area of the rectangular park = $400 	ext{ m}^2$. To Find: Whether the situation is possible, and if so, its length and breadth. Let: The length of the park be $l$ metres and the breadth be $b$ metres. Equations: Perimeter = $2(l + b) = 80$ $$l + b = 40...

NCERT Solutions