Explanation

For the parabola y^2 = −4x, three normals can be drawn from a point (h, 2) to the parabola only if h < −2.

This condition ensures that the point lies in the correct region where three normals are possible.

Explanation

A group generated by one of its elements is called a cyclic group.

This means that the group can be generated by repeatedly applying the group operation to a single element, called the generator.

In other words, all elements of the group can be expressed as powers of the generator.

Explanation

A monoid is a set with an associative binary operation and an identity element.

A group is a monoid where every element has an inverse.

Explanation

C.P. (Cost Price) of each share = Rs. 25 + (2% of 25) = Rs. 25 + 0.50 = Rs. 25.50

And therefore, the Number of shares = Total amount / C.P. of each share = Rs. 12750 / Rs. 25.50 = 500

Explanation

3x² - 7x - 6 = 0

We can factor the equation:

(3x + 2)(x - 3) = 0

This gives us two possible solutions:

3x + 2 = 0 --> 3x = -2 --> x = -2/3

x - 3 = 0 --> x = 3

So, the roots of the equation are x = -2/3 and x = 3.

The correct answer is:

3, -2/3

Explanation

The quadratic equation whose roots are reciprocal of:

2x2 + 5x + 3 = 0 can be obtained by replacing x by 1/x.

Hence, 2(1/x)2 + 5(1/x) + 3 = 0

=> 3x^2 + 5x + 2 = 0

Explanation

6(A's Capital) = 8(B's Capital) = 10(C's Capital)

We can write this as:

A's Capital = x

B's Capital = x × (6/8) = 0.75x

C's Capital = x × (6/10) = 0.6x

Now, we can write the ratio of their capitals as:

A's Capital : B's Capital : C's Capital = x : 0.75x : 0.6x

Divide each term by x:

A's Capital : B's Capital : C's Capital = 1 : 0.75 : 0.6

Multiply each term by 20 (the least common multiple of 1, 0.75, and 0.6) to eliminate decimals:

A's Capital : B's Capital : C's Capital = 20 : 15 : 12

So, the ratio of their capitals is:

20 : 15 : 12

Explanation

Let the two consecutive positive integers be x and x + 1

x2 + (x + 1)2 - x(x + 1) = 91

x2 + x - 90 = 0

(x + 10)(x - 9) = 0 => x = -10 or 9.

As x is positive x = 9

Hence the two consecutive positive integers are 9 and 10.

Explanation

Convert the race distance from 1 km to meters: 1 km = 1000 m

Find the distance covered by X in the race: 48 m (given)

Find the distance covered by Y in the race: 42 m (given)

Since X beats Y, X covers the entire race distance (1000 m), while Y covers a shorter distance.

To find the distance by which X beats Y, subtract the distance covered by Y from the race distance: 1000 m - 42 m x (1000/48) = 1000 - 875 = 125 m

So, X beats Y by 125 meters.

Explanation

Track length = 1 km

Formula used:

Distance Speed x Time

Calculation:

"A" beats "B" by 50 m or 10 sec, means "B" covers 50 m in 10 sec

Speed of "B" = 50/10 = 5 m/s

Time to cover 1000 m = 1000/5 = 200 sec

"A" beats "B" by 10 sec means

"A" reach 10 sec early

Time taken by "A" to complete the race = 200 - 10 = 190 sec

∴ Time taken by "A" to complete the race is 190 sec.