12th Maths Monthly Test - 2 ( Theory of Equations)

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 2 ( Theory of Equations)-Aug 2020

Time : 2.30 Hours

Part - A

Generic 2 - Answer All Question

Solve:
(x-5)(x-7)(x+6)(x+4)=504

Discuss the nature of the roots of the following polynomials:
x⁵-19x⁴+2x³+5x²+11

Construct a cubic equation with roots 2,−2, and 4.

Solve the equation 6x⁴-5x³-38x²-5x+6=0 if it is known that \(\frac{1}{3}\) is a solution.

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

If sin ∝, cos ∝ are the roots of the equation ax²+ bx + c-0 (c ≠ 0), then prove that (n + c)²- b² + c²

Prove that a straight line and parabola cannot intersect at more than two points.

If α, β, and γ are the roots of the polynomial equation ax³+bx²+cx+d=0 , find the value of \(\Sigma \frac { \alpha }{ \beta \gamma } \) in terms of the coefficients.

If p and q are the roots of the equation lx²+nx+n = 0, show that \(\sqrt { \frac { p }{ q } } +\sqrt { \frac { q }{ p } } +\sqrt { \frac { n }{ l } } \)=0.

Find a polynomial equation of minimum degree with rational coefficients, having \(\sqrt{5}\)−\(\sqrt{3}\) as a root.
Solve the equation 3x³-16x²+23x-6=0 if the product of two roots is 1.

Find a polynomial equation of minimum degree with rational coefficients, having 2+\(\sqrt{3}\)i as a root.

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.
Find the sum of squares of roots of the equation 2x⁴-8x³+6x²-3=0.

Solve the cubic equation : 2x³−x²−18x+9=0 if sum of two of its roots vanishes.

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.

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

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

Obtain the condition that the roots of x³+px²+qx+r=0 are in A.P.

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.

If x²+2(k+2)x+9k=0 has equal roots, find k.

Formalate into a mathematical problem to find a number such that when its cube root is added to it, the result is 6.

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

Solve the equation 3x³-26x²+52x-24=0 if its roots form a geometric progression.

If α, β, and γ are the roots of the equatio x³+pz²+qx+r=0, find the value of \(\Sigma \frac { 1 }{ \beta \gamma } \) in terms of the coefficients.

Solve the equation 9x³-36x²+44x-16=0 if the roots form an arithmetic progression.

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

Part - B

Generic 5 - Answer All Question

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

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

Solve the equation (2x-3)(6x-1)(3x-2)(x-12)-7=0

If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p.

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 following equation: x⁴-10x³+26x²-10x+1=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: (2x² - 3x + 1) (2x² + 5x + 1) = 9x².

Comment

Add Comment

InstaQB

Institutions

12th Standard English Medium

12th Standard Tamil Medium

11th Standard English Medium

11th Standard Tamil Medium

10th Standard English Medium

9th Standard English Medium

8th Standard English Medium

7th Standard English Medium

Class XII

Class XI

12Th Standard

11Th Standard

10Th Standard

9Th Standard

12th Maths Monthly Test - 2 ( Theory of Equations)

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 2 ( Theory of Equations)-Aug 2020

Part - A

Generic 2 - Answer All Question

Solve: (x-5)(x-7)(x+6)(x+4)=504

Discuss the nature of the roots of the following polynomials: x5-19x4+2x3+5x2+11

Construct a cubic equation with roots 2,−2, and 4.

Solve the equation 6x4-5x3-38x2-5x+6=0 if it is known that \(\frac{1}{3}\) is a solution.

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

If sin ∝, cos ∝ are the roots of the equation ax2 + bx + c-0 (c ≠ 0), then prove that (n + c)2 - b2 + c2

Prove that a straight line and parabola cannot intersect at more than two points.

If α, β, and γ are the roots of the polynomial equation ax3+bx2+cx+d=0 , find the value of \(\Sigma \frac { \alpha }{ \beta \gamma } \) in terms of the coefficients.

If p and q are the roots of the equation lx2+nx+n = 0, show that \(\sqrt { \frac { p }{ q } } +\sqrt { \frac { q }{ p } } +\sqrt { \frac { n }{ l } } \)=0.

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

Solve the equation 3x3-16x2+23x-6=0 if the product of two roots is 1.

Find a polynomial equation of minimum degree with rational coefficients, having 2+\(\sqrt{3}\)i as a root.

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.

Find the sum of squares of roots of the equation 2x4-8x3+6x2-3=0.

Solve the cubic equation : 2x3−x2−18x+9=0 if sum of two of its roots vanishes.

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.

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

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

Obtain the condition that the roots of x3+px2+qx+r=0 are in A.P.

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.

If x2+2(k+2)x+9k=0 has equal roots, find k.

Formalate into a mathematical problem to find a number such that when its cube root is added to it, the result is 6.

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

Solve the equation 3x3-26x2+52x-24=0 if its roots form a geometric progression.

If α, β, and γ are the roots of the equatio x3+pz2+qx+r=0, find the value of \(\Sigma \frac { 1 }{ \beta \gamma } \) in terms of the coefficients.

Solve the equation 9x3-36x2+44x-16=0 if the roots form an arithmetic progression.

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

Part - B

Generic 5 - Answer All Question

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

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

Solve the equation (2x-3)(6x-1)(3x-2)(x-12)-7=0

If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p.

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 following equation: x4-10x3+26x2-10x+1=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: (2x2 - 3x + 1) (2x2 + 5x + 1) = 9x2.

Comment

Other Study material

12th Maths Monthly Test - 2 ( Discrete Mathematics

12th Maths Monthly Test - 3 ( Discrete Mathematics

12th Maths Monthly Test - 1 ( Applications of Vect

12th Maths Monthly Test - 2 ( Applications of Vect

12th Maths Monthly Test - 3 ( Applications of Vect

12th Maths Monthly Test - 1 ( Ordinary Differentia

12th Maths Monthly Test - 2 ( Ordinary Differentia

12th Maths Monthly Test - 3 (Ordinary Differential

12th Maths Monthly Test - 1 (Probability Distribut

12th Maths Monthly Test - 2 ( Probability Distribu

12th Maths Monthly Test - 3 (Probability Distribut

12th Maths Monthly Test - 1 ( Discrete Mathematics

12th Maths Weekly Test -1 (Ordinary Differential E

12th Maths Weekly Test -2 (Ordinary Differential E

12th Maths Weekly Test -3 (Ordinary Differential E

Tamilnadu Stateboardszs - Class

Solve:
(x-5)(x-7)(x+6)(x+4)=504

Discuss the nature of the roots of the following polynomials:
x⁵-19x⁴+2x³+5x²+11

Solve the equation 6x⁴-5x³-38x²-5x+6=0 if it is known that \(\frac{1}{3}\) is a solution.

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

If sin ∝, cos ∝ are the roots of the equation ax²+ bx + c-0 (c ≠ 0), then prove that (n + c)²- b² + c²

If α, β, and γ are the roots of the polynomial equation ax³+bx²+cx+d=0 , find the value of \(\Sigma \frac { \alpha }{ \beta \gamma } \) in terms of the coefficients.

If p and q are the roots of the equation lx²+nx+n = 0, show that \(\sqrt { \frac { p }{ q } } +\sqrt { \frac { q }{ p } } +\sqrt { \frac { n }{ l } } \)=0.

Solve the equation 3x³-16x²+23x-6=0 if the product of two roots is 1.

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.

Find the sum of squares of roots of the equation 2x⁴-8x³+6x²-3=0.

Solve the cubic equation : 2x³−x²−18x+9=0 if sum of two of its roots vanishes.

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

Obtain the condition that the roots of x³+px²+qx+r=0 are in A.P.

If x²+2(k+2)x+9k=0 has equal roots, find k.

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

Solve the equation 3x³-26x²+52x-24=0 if its roots form a geometric progression.

If α, β, and γ are the roots of the equatio x³+pz²+qx+r=0, find the value of \(\Sigma \frac { 1 }{ \beta \gamma } \) in terms of the coefficients.

Solve the equation 9x³-36x²+44x-16=0 if the roots form an arithmetic progression.

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

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.

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 following equation: x⁴-10x³+26x²-10x+1=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: (2x² - 3x + 1) (2x² + 5x + 1) = 9x².