(a) The perimeter of a right triangle is 60 cm and its hypotenuse is 25 cm. Find the lengths of other two sides of the triangle.
OR
(b) A train travels a distance of 480 km at a uniform speed. If the speed had been $8\text{ km/h}$ less, then it would have taken 3 hours more to cover the same distance. Find the speed of the train.
[5]
If the roots of equation $ax^{2}+bx+c=0$, $a \ne 0$ are real and equal, then which of the following relation is true ?
[1 mark]
A rectangular floor area can be completely tiled with 200 square tiles. If the side length of each tile is increased by 1 unit, it would take only 128 tiles to cover the floor.
(i) Assuming the original length of each side of a tile be x units, make a quadratic equation from the above information.
(ii) Write the corresponding quadratic equation in standard form.
(iii) (a) Find the value of x, the length of side of a tile by factorisation.
OR
(b) Solve the quadratic equation for x, using quadratic formula.
IMAGE
[4 marks]
The least positive value of k, for which the quadratic equation \( 2x^{2}+kx-4=0 \) has rational roots, is
[1 mark]
Case Study
While designing the school year book, a teacher asked the student that the length and width of a particular photo is increased by x units each to double the area of the photo. The original photo is 18 cm long and 12 cm wide.
Based on the above information, answer the following questions:
- (I) Write an algebraic equation depicting the above information.
- (II) Write the corresponding quadratic equation in standard form.(III) What should be the new dimensions of the enlarged photo?
OR - (III) Can any rational value of x make the new area equal to \( 220 \text{ cm}^{2} \)?
[4 marks]
(a) Find the value of m for which the quadratic equation \((m-1)x^{2}+2(m-1)x+1=0\) has two real and equal roots.
OR
(b) Solve the following quadratic equation for x: \(\sqrt{3}x^{2}+10x+7\sqrt{3}=0\)
[2 marks]
The product of Rehan's age (in years) 5 years ago and his age 7 years from now, is one more than twice his present age. Find his present age.
[2 marks]