Obtain the discriminant of the equation 2x³ - 23x²+82x-78=0. Hence provide the nature of its roots.
The discriminant (\(\Delta\)) of a cubic equation of the form: \[ ax^3 + bx^2 + cx + d = 0 \] is given by the formula: \[ \Delta = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2 \] ### Step 1: Identify the coefficients For the given equation: \[ 2x^3 - 23x^2 + 82x ______ ____ _________ __________ ______ _________ ______ _____ _________ ___.
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Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so. For instance, to show that '{1, padma, blue} is a set' is true, you need to say that this is true because it is a well-defined collection of 3 objects.)
i) The collection of Venn diagrams is a set.
ii) (A\B) UC=A\(B∩C) for any three sets A, B and C.
iii) If ze R, then |z|=z.
iv) xm + a1 xm -1+ ....am-1 x + am = 0, ai ∈ R∀ i = 1, ..., m, has a root in R only if m is an odd number.
v) x > 0 is necessary for x+3>1.
vi) Any system of three linear equations in two variables has no solution.
vii) If a matrix has n² entries, where n∈ N, then it is a square matrix.
viii) For n > 4, the AM of the first n natural numbers is greater than n +1.
Find the roots of the equation 2x³-x2-22x-24=0, if two of them are in the ratio 3:4.
Prove that
Draw a Venn diagram of sets A, B and C where A⊆B, A∩C, BOC≠ φ.B, ∩ C = φ. What is the universal set you have chosen? Justify your choice of sets in the diagram.
z1, z2 और z1,z2 के ध्रुवीय और घातांकीय निरूपण प्राप्त कीजिए, जहां और z2 = 3+ i हैं।
पुष्टि के साथ, निम्नलिखित के उदाहरण दीजिए :
i) दो शून्येतर 3 x 3 आव्यूह A और B, जिनके लिए | A | = 0, | B |
ii) दो ऐसे व्युत्क्रमणीय 2 x 2 आव्यूह C और D जिनके लिए, | C | = √2 | D |.
Express in standard (algebraic) form. Further, give an Argand diagram
in which and
are plotted