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WBJEE: Important Questions and Preparation Tips – Theory of equations

Jan 2, 2018 15:02 IST
    WBJEE 2018: Theory of equations
    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.

    WBJEE: Important Questions and Preparation Tips – Integrals

    Important Concepts:

    WBJEE Theory of equations

    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+ 3px + 5q2 = 0, β is a root of x+ 9p2x + 15q2 = 0 and 0 < α < β, then the equation x+ 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) = x+ 6p2x + 10q2

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

    So, α+ 3p2α + 5q2 = 0

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

    So, β2+9p2β+15q= 0

    Now,

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

    Again, f(β) = β+ 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 ax+ 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,

    ax+ bx + c = 0

    of which α and β are the roots

    So,

    WBJEE Theory f equations solution 3

    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.

    WBJEE Theory f equations solution 5

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

    Hence, the correct option is (A).

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