The altitude of a right triangle is 7 cm less than its base. If the Hypotenuse is 13 cm, find the other two sides. [ncert 10th]

Let the base of the given rt|_d triangle=x cm.

Then its altitude = x-7 cm.

[Here Altitude=Height=perpendicular length] 

Hypotenuse = 13 cm.

Now in right-angled triangle, 

(Hypotenuse)²=(Based)²+(Altitude)²

=> (13)²=x²+(x-7)²

=> 169=x²+x2+49-14x

=> 2x²+49-14x-169=0

=> 2x²-14x-120=0

Dividing both sides by 2, we get,

x²-7x-60=0

=> x²-12x+5x-60=0

=> (x²-12x)+(5x-60)=0

=> x(x-12)+5(x-12)=0

=> (x-12)(x+5)=0

=> x=12, -5

But length of a side of a triangle cannot be negative. So x=12

=> Base=12 cm. Ans.

Altitude=x-7=12-7=5cm. Ans.

Rohan's mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan's present age. [ncert 10th]

Let the present age of Rohan= x years

Then present age of Rohan's mother =x+26 years.

After 3 years, age of Rohan = x+3 years.

After 3 years, Rohan's mother age = x+26+3 = x+29 years.

Now according to the given condition, 

(x+3)(x+29)=360

 => x²+32x+87=360

=> x²+32x+87-360=0

=> x²+32x-273=0

=> x²-7x+39x-273=0

=> (x²-7x)+(39x-273)=0

=> x(x-7)+39(x-7)=0

=> (x-7)(x+39)=0

=> x=7 or x= -39

Ignore x= -39 [∵ age cannot be negative]

=> Present age of Rohan = x = 7 years.

Find the two Numbers, whose sum is 27 and product is 182. [ncert 10th]

Find two consecutive positive integers, sum of whose squares is 365. [ncert 10th.]

Let the two consecutive positive integers

are  x and x+1.

Given that x²+(x+1)² = 365

Using (a+b)=a²+b²+2ab, we get,

=> x²+x²+1+2x=365.  

 => 2x²+2x+1-365=0

=> 2x²+2x-364=0

Dividing both sides by 2 (non zero), we get,

=> x²+x-182=0

=> x²+14x-13x-182=0

=> (x²+14x)-(13x+182)=0

=> x(x+14)-13(x+14)=0

=> (x+14)(x-13)=0

=> Either x+14=0 or x-13=0

=> Either x= -14 , 13

But we have to find positive integer, so -14 must be rejected.

So x= 13 is the required positive integer.

Now two consecutive positive integers shall be x, x+1 

=> 13 and 13+1=14

So 13 and. 14 are the required positive consecutive integers. Ans.

If A=[(1,3),(3,4)], B=[(-2,1),(-3,2)] and A²-5B²=5C. Find matrix C where C is a 2 by 2 matrix. [icse2017].

What must be subtracted from 16x ^ 3 - 8x ^ 2 + 4x + 7 that the resulting expression has (2x + 1) as a factor? [icse2017]

If (x+2) and (x+3) are factors of x^3+ax+b, find the values of 'a' and 'b'. [icse 2018]

 

If A=[(3,x),(0,1)] and B=[(9,16), (0,-y)], find x and y, when A²=B. [icse 2015]