WBJEE: Important Questions and Preparation Tips – 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.

WBJEE: Important Questions and Preparation Tips – Integrals

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 x² + px + q = 0, then the values of α³ + β³ and α⁴+ α²β² + β⁴ are respectively

(A) 3pq – p³ and p⁴ – 3p²q + 3q ²

(B) –p(3q – p²) and (p² – q)(p² + 3q)

(C) pq – 4 and p⁴ – q⁴

(D) 3pq – p³ and (p² – q) (p² – 3q)

Solution 1:

We have α³ + β³ = (α+β)³ – 3αβ (α + β)

= – p³ + 3pq

Again, α⁴ + β⁴ + α²β² = (α² + β²)² – α²β² 

= (p² – 2q)² – q²

= (p² – 3q)(p² – q)

Hence, the correct option is (D).

WBJEE: Important Questions and Preparation Tips – Circles

Question 2:

Let p, q be real numbers. If α is the root of x²+ 3p² x + 5q² = 0, β is a root of x² + 9p²x + 15q² = 0 and 0 < α < β, then the equation x² + 6p²x + 10q² = 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) = x² + 6p²x + 10q²

As α is the root of x²+ 3p² x + 5q² = 0

So, α² + 3p²α + 5q² = 0

Also, β is a root of x² + 9p²x + 15q² = 0

So, β²+9p²β+15q² = 0

Now,

f(α) = α² + 6p²α + 10q² = (α²+3p²α+5q²) + (3p²α+5q²) = 0 + 3p²α+5q² > 0

Again, f(β) = β² + 6p²β + 10q² = (β²+9p²β+15q²) – (3p²β+5q²) = 0 – (3p²β + 5q²) < 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 ax² + bx + c = 0 (a≠0) and α + h, β + h are the roots of px² +qx+r = 0 (p≠0) then the ratio of the squares of their discriminants is

(A) a²:p²

 (B) a:p²

(C) a⁴:p⁴

(D) a:p²

Solution 3:

We have the quadratic equation,

ax² + 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) = 2x²+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) = 2x²+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(x² –x - 2) +b(x² - 3x + 2) + c(x² - 1)

f(x) = (a + b + c)x² - (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) = x²+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) = x²+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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