# WBJEE: Important Questions and Preparation Tips – Theory of equations

In this article, Subject Experts of Mathematics bring to you chapter notes of chapter Theory of equations including important topics like roots of quadratic equation, relation between roots and coefficient of quadratic equation, discriminant of quadratic equation, conditions for nature of roots using discriminant, conditions for minimum and maximum value of a parabola etc.

*WBJEE 2018: Theory of equations*

Find chapter notes of chapter Theory of equations including important concepts, formulae and some previous year solved questions for WBJEE entrance examination 2018. Aspirants always get **1-2 questions** from this topic in the examination.

These chapter notes include important topics like **general form of quadratic equation, roots of quadratic equation, relation between roots and coefficient of quadratic equation, discriminant of quadratic equation, conditions for nature of roots using discriminant, conditions for minimum and maximum value of parabola, conditions for minimum and maximum value of a parabola** etc.

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**Important Concepts:**

**WBJEE: Important Questions and Preparation Tips – Application of Derivatives**

**Some previous years solved questions are given below:**

** **

**Question 1:**

If α, β are the roots of the quadratic equation x^{2} + px + q = 0, then the values of α^{3} + β^{3} and α^{4}+ α^{2}β^{2} + β^{4} are respectively

(A) 3pq – p^{3} and p^{4} – 3p^{2}q + 3q ^{2}

(B) –p(3q – p^{2}) and (p^{2} – q)(p^{2} + 3q)

(C) pq – 4 and p^{4} – q^{4}

(D) 3pq – p^{3} and (p^{2} – q) (p^{2} – 3q)

**Solution 1: **

** **

We have α^{3} + β^{3} = (α+β)^{3} – 3αβ (α + β)

= – p^{3} + 3pq

Again, α^{4} + β^{4} + α^{2}β^{2} = (α^{2} + β^{2})^{2} – α^{2}β^{2}

= (p^{2} – 2q)^{2} – q^{2}

= (p^{2} – 3q)(p^{2} – q)

Hence, the correct option is (D).

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**Question 2:**

Let p, q be real numbers. If α is the root of *x*^{2}+ 3*p*^{2 }*x* + 5*q*^{2} = 0, β is a root of x^{2 }+ 9p^{2}x + 15q^{2} = 0 and 0 < α < β, then the equation x^{2 }+ 6p^{2}x + 10q^{2} = 0 has a root γ that always satisfies

(A) γ = α/4 + β

(B) β < γ

(C) γ = α/2 + β

(D) α < γ < β

**Solution 2:**

Let us assume that,

f(x) = x^{2 }+ 6p^{2}x + 10q^{2}

As α is the root of *x*^{2}+ 3*p*^{2 }*x* + 5*q*^{2} = 0

So, α^{2 }+ 3p^{2}α + 5q^{2} = 0

Also, β is a root of x^{2 }+ 9p^{2}x + 15q^{2} = 0

So, β^{2}+9p^{2}β+15q^{2 }= 0

Now,

f(α) = α^{2 }+ 6p^{2}α + 10q^{2} = (α^{2}+3p^{2}α+5q^{2}) + (3p^{2}α+5q^{2}) = 0 + 3p^{2}α+5q^{2} > 0

Again, f(β) = β^{2 }+ 6p^{2}β + 10q^{2} = (β^{2}+9p^{2}β+15q^{2}) – (3p^{2}β+5q^{2}) = 0 – (3p^{2}β + 5q^{2}) < 0

So, there is one root γ such that, α < γ < β.

Hence, the correct option is (D).

**WBJEE: Important Questions and Preparation Tips – Area under Curve**

**Question 3:**

If α, β are the roots of ax^{2 }+ bx + c = 0 (a≠0) and α + h, β + h are the roots of px^{2} +qx+r = 0 (p≠0) then the ratio of the squares of their discriminants is

(A) a^{2}:p^{2}

(B) a:p^{2}

(C) a^{4}:p^{4}

(D) a:p^{2}

**Solution 3:**

We have the quadratic equation,

ax^{2 }+ bx + c = 0

of which α and β are the roots

So,

Hence, the correct option is (C).

**WBJEE: Important Questions and Preparation Tips – Limits**

**Question 4:**

Let f(x) = 2x^{2}+5x+1. If we write f(x) as f(x) = a(x + 1)(x – 2) +b(x – 2)(x – 1)+c(x – 1)(x + 1) for real numbers a,b,c then

(A) there are infinite number of choices for a,b,c

(B) only one choice for a but infinite number of choices for b and c

(C) exactly one choice for each of a,b,c

(D) more than one but finite number of choices for a,b,c

**Solution 4:**

We have,

f(x) = 2x^{2}+5x+1 …(1)

Given, f(x) can be written as:

f(x) = a(x + 1)(x – 2) +b(x – 2)(x – 1)+c(x – 1)(x + 1)

f(x) = a(x^{2} –x - 2) +b(x^{2} - 3x + 2) + c(x^{2} - 1)

f(x) = (a + b + c)x^{2} - (a + 3b)x -2a + 2b – c …(2)

From (1) and (2)

a + b + c = 2 …(3)

a + 3b = - 5 …(4)

-2a + 2b – c = 1 …(5)

Adding (3), (4), (5)

a + b + c + a + 3b -2a + 2b – c = 2 – 5 + 1

6b = -2

Putting value of b in (4) we get the value of a

Again putting the value of a and b in equation (3) we get the value of c.

So, we have exactly one choice for each of a,b,c.

Hence, the correct option is (C).

**WBJEE: Important Questions and Preparation Tips – Parabola**

**Question 5:**

Suppose that the equation f(x) = x^{2}+bx+c = 0 has two distinct real roots α and β. The angle between the tangent to the curve y = f(x) at the point , ((α + β)/2, f((α + β)/2))and the positive direction of the x-axis is

(A) 0°

(B) 30°

(C) 60°

(D) 90°

**Solution 5:**

As α and β are the roots of the equation f(x) = x^{2}+bx+c = 0. So, it represents upward parabola which cuts x-axis at α and β. As the graph is symmetric, so, tangent at ((α + β)/2, f((α + β)/2))** **parallel to x-axis.

Hence the angle between the tangent to the curve at the given point and positive direction of x axis is 0^{o}.

Hence, the correct option is (A).

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