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12th Maths Monthly Test - 3 ( Theory of Equations)

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 3 ( Theory of Equations)-Aug 2020

Time : 2.30 Hours

Part - A

Generic 2 - Answer All Question

Find all real numbers satisfying 4^x-3(2^x+2)+2⁵=0

Find value of a for which the sum of the squares of the equation x² - (a- 2) x - a-1=0 assumes the least value.
If x²+2(k+2)x+9k=0 has equal roots, find k.

Find all zeros of the polynomial x⁶-3x⁵-5x⁴+22x³-39x²-39x+135, if it is known that 1+2i and \(\sqrt{3}\) are two of its zeros.

Construct a cubic equation with roots 1,2, and 3

If the sides of a cubic box are increased by 1, 2, 3 units respectively to form a cuboid, then the volume is increased by 52 cubic units. Find the volume of the cuboid.

Find th Int rval for a for which 3x²+2(a²+1) x+(a²-3n+2) possesses roots of opposite sign.

If α, β and γ are the roots of the cubic equation x³+2x²+3x+4=0, form a cubic equation whose roots are,
2α, 2β, 2γ

Solve the cubic equations:
2x³-9x²-10x=3

Determine the number of positive and negative roots of the equation x⁹-5x⁸-14x²=0.

Find a polynomial equation of minimum degree with rational coefficients, having \(\sqrt{5}\)−\(\sqrt{3}\) as a root.

Solve: (2x-1)(x+3)(x-2)(2x+3)+20=0

Find the exact number of real roots and imaginary of the equation x⁹+9x⁷+7x⁵+5x³+3x.

Solve: \(8x^{ \frac { 3 }{ 2n } }-8x^{ \frac { -3 }{ 2n } }\)=63
Formalate into a mathematical problem to find a number such that when its cube root is added to it, the result is 6.

Find the monic polynomial equation of minimum degree with real coefficients having 2-\(\sqrt{3}\)i as a root.

Solve the following equations,
sin²x-5sinx+4=0

Part - B

Generic 3 - Answer All Question

If α and β are the roots of the quadratic equation 2x²−7x+13 = 0 , construct a quadratic equation whose roots are α² and β².

Solve the equation x³-5x²-4x+20=0

Form a polynomial equation with integer coefficients with \(\sqrt { \frac { \sqrt { 2 } }{ \sqrt { 3 } } } \) as a root.

Find the condition that the roots of x³+ax²+bx+c = 0 are in the ratio p:q:r.

Solve the equation 2x³+11x²−9x−18=0.
Solve: \({ (5+2\sqrt { 6 } ) }^{ { x }^{ 2 }-3 }+{ (5-2\sqrt { 6 } ) }^{ { x }^{ 2 }-3 }=10\)

Find the number .of real solu,tlons of sin (e^x) -5^x + 5^-x

Find the number of positive integral solutions of (pairs of positive integers satisfying) x² - y² =353702.

If α and β are the roots of the quadratic equation 17x²+43x−73 = 0 , construct a quadratic equation whose roots are α + 2 and β + 2.

Find the condition that the roots of ax³+bx²+cx+d=0 are in geometric progression. Assume a,b,c,d ≠0.

Find solution, if any, of the equation 2cos²x-9cosx+4=0

Part - C

Generic 5 - Answer All Question

If a, b, c, d and p are distinct non-zero real numbers such that (a²+b²+c²) p²-2 (ab+bc+cd) p+(b²+c²+d²)≤ 0 the n. Prove that a,b,c,d are in G.P and ad=bc

Form the equation whose roots are the squares of the roots of the cubic equation x³+ax²+bx+c = 0.

If the sum of the roots of the quadratic equation ax²+ bx + c = 0 (abe≠ 0) is equal to the sum of the squares of their reciprocals, then \(\frac { a }{ c } ,\frac { b }{ a } ,\frac { c }{ b } \) are H.P.

Solve the equation (x-2)(x-7)(x-3)(x+2)+19=0

If 2+i and 3-\(\sqrt{2}\) are roots of the equation x⁶-13x⁵+62x⁴-126x³+65x²+127x-140=0, find all roots.
Solve the equation (2x-3)(6x-1)(3x-2)(x-12)-7=0

If the equation x² + bx + ca = 0 and x² + cx + ab = 0 have a comnion root and b≠c, then prove that their roots will satisfy the equation x² + ax + bc =0.

Solve: (2x² - 3x + 1) (2x² + 5x + 1) = 9x².

Solve the following equation: x⁴-10x³+26x²-10x+1=0

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