Q. Is it possible to design a rectangular park of perimeter $80$ m and area $400m^2$ If so, find its length and breadth.

Q. The man insisted on buying the doves because he was fond of birds. Do you agree?

Q. Sum of the areas of two squares is 468$m^2$. If the difference of their perimeters is 24 m, find the sides of the two squares.

Q. Find two consecutive positive integers, sum of whose squares is $365$.

Q. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
(i) $2x^2-3x+5=0$
(ii) $3x^2-4\sqrt{3}x+4=0$
(iii) $2x^2-6x+3=0$

Q. 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 article (in rupees) was $3$ more than twice the number of articles produced on that day. If the total cost of production on that day was Rs $90,$ find the number of articles produced and the cost of each article.

Q. Find two numbers whose sum is 27 and product is 182.

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

Q. 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 of their ages in years was 48.

Q. In a class test, the sum of Shefali's marks in Mathematics and English is $30$. Had she got $2$ marks more in Mathematics and $3$ marks less in English, the product of their marks would have been $210$. Find her marks in the two subjects.

Q. Find the roots of the following quadratic equations, if they exist, by the method of completing the square:
(i) $2x^2-7x+3=0$
(ii) $2x^2+x-4=0$
(iii) $4x^2+4\sqrt{3x}+3=0$
(iv) $2x^2+x+4=0$

Q. 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.

Q. Find the roots of the following equations:
(i) $x-\frac{1}{x}=3,x\ne 0$
(ii) $\frac{1}{x+4}-\frac{1}{x-7}=\frac{11}{30},x\ne -4,7$

Q.

The sum of the reciprocals of Rehman's age, $3$ years ago and $5$ years from now is $\frac{1}{3}$. Find his present age.

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

Q.

Check whether the following are quadratic equations :
(i) ${(x+1)}^2=2(x-3)$
(ii) $x^2-2x=(-2)(3-x)$
(iii) $(x-2)(x+1)=(x-1)(x+3)$
(iv) $(x-3)(2x+1)=x(x+5)$
(v) $(2x-1)(x-3)=(x+5)(x-1)$
(vi) $x^2+3x+1={(x-2)}^2$
(vii) ${(x+2)}^3=2x(x^2-1)$
(viii) $x^3-4x^2-x+1={(x-2)}^3$

Q. Find the roots of the following quadratic equations by factorisation:
(i) $x^2-3x-10=0$
(ii) $2x^2+x-6=0$
(iii) $\sqrt{2}{x}^2+7x+5\sqrt{2}=0$
(iv) $2x^2-x+\frac{1}{8}=0$
(v) $100x^2-20x+1=0$

Q. Represent the following situations in the form of quadratic equations :
(i) The area of a rectangular plot is $528m^2$. 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.

Q.

The diagonal of a rectangular field is $60metres$ more than the shorter side. If the longer side is $30metres$ more than the shorter side, find the sides of the field.

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