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If $$a$$, $$b$$ and $$c$$ are real numbers such that the polynomial $$x^3 + 6x^2 + ax + b$$ is the cube of $$x + c$$, then
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If $$a$$, $$b$$ and $$c$$ are real numbers such that the polynomial $$x^3 + 6x^2 + ax + b$$ is the cube of $$x + c$$, then
Expanding gives $$(x+c)^3 = x^3 + 3cx^2 + 3c^2x + c^3$$. Comparing coefficients gives $$3c = 6$$, so $$c = 2$$, $$a = 3c^2 = 12$$ and $$b = c^3 = 8$$. Hence $$a+b+c = 22$$, which is divisible by $$11$$.
In the adjoining figure, $$AB = 9\text{ cm}$$, $$AC = 7\text{ cm}$$, $$BC = 8\text{ cm}$$, $$AD$$ is the median and $$\angle C = 40^\circ$$. Then the measure of $$\angle ADB$$ in degrees is
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If $$x^2 + 6x + 1 = 0$$ and $$\frac{x^4+kx^2+1}{3x^3+kx^2+3x} = 2$$, then the value of $$k$$ is
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If $$x = \sqrt[3]{49}+\sqrt[3]{42}+\sqrt[3]{36}$$, then the value of $$x-\frac{1}{x^2}$$ is
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In the adjoining figure, $$AB = BC = CD$$. Point $$P$$ is the midpoint of $$AQ$$. If $$CR = 4$$ and $$QC = 12$$, then $$PQ$$ is equal to
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In the adjoining figure, $$A$$ is the midpoint of the arc $$BAC$$. Given that $$AB = 15$$ and $$AD = 10$$, the value of $$AE$$ is
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The number of real numbers $$x$$ which satisfy $$\frac{8^x+27^x}{12^x+18^x}=\frac{7}{6}$$ is
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$$a$$ , $$b$$ are real numbers such that $$2a^2+5b^2=20$$, then the maximum value of $$a^4b^6$$ is
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The number of ordered pairs $$(x,y)$$ of integers such that $$x-y^2=4$$ and $$x^2+y^4=26$$ is
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In the adjoining figure, three equal squares are placed. The squares are unit squares The area of the shaded region in $$(incm^2)$$ is
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In the adjoining figure, $$AB$$ is a diameter of the circle. Given $$\angle BAC=20^\circ$$ and $$\angle AEB=56^\circ$$, the measure of $$\angle BCD$$ in degrees is
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The number of ordered pairs $$(m,n)$$ of integers such that $$1\le m,n\le100$$ and $$m^n\cdot n^m$$ leaves a remainder of $$1$$ when divided by $$4$$ is
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The number of ordered pairs of positive integers $$(x,y)$$ satisfying the equation $$x^2+4y=3x+16$$ is
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The algebraic expression $$(a+b+ab+2)^2+(a-ab+2-b)^2-2b^2(1+a^2)$$ reduces to
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The sum of $$(1\times4)+(2\times7)+(3\times10)+(4\times13)+\cdots$$ for $$49$$ terms is equal to
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If the equations $$x^3+ax+1=0$$ and $$x^4-ax^2+1=0$$ have a common root, then the value of $$a^2$$ is
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If $$a$$, $$b$$, $$c$$ and $$d$$ are positive reals such that $$abcd=1$$, then the maximum value of $$a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd$$ is
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The sum of all natural numbers $$n$$ for which $$n(n+1)$$ is a perfect square is
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Point $$P$$ is inside the square $$ABCD$$ such that $$PA=PB$$ = Distance of $$P$$ from $$CD$$. The ratio of the area of triangle $$PAB$$ to the area of square $$ABCD$$ is $$\frac{m}{n}$$, where $$m$$ and $$n$$ are relatively prime integers. Then the value of $$m+n$$ is
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The sum of the roots of the simultaneous equations $$\sqrt[y]{4^x}=32\sqrt[x]{8^y}$$ and $$\sqrt[y]{3^x}=3\sqrt[y]{9^{1-y}}$$ is
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If $$2\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}=\sqrt{a}+\sqrt{b}$$, where $$a$$ and $$b$$ are natural numbers, then the value of $$a+b$$ is
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In the adjoining figure, $$\angle ACD=38^\circ$$. Then the measure of angle $$x$$ in degrees is
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If $$\frac{a}{b+c}+\frac{c}{a+b}=\frac{2b}{c+a}$$, where $$a+b$$, $$b+c$$, $$c+a$$ and $$a+b+c$$ are all nonzero, then the numerical value of $$\frac{a^2+c^2}{b^2}$$ is
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The geometric and arithmetic means of two positive numbers are respectively $$8$$ and $$17$$. The larger of the two numbers is
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The number of two-digit numbers in which the tens digit and the units digit are different and both odd is
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The value of $$(5\sqrt[3]{4}-3\sqrt[3]{\frac{1}{2}})(12\sqrt[3]{2}+\sqrt[3]{16}-2\sqrt[3]{2})$$ is
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If $$\frac{xy}{x+y}=1$$, $$\frac{yz}{y+z}=2$$ and $$\frac{zx}{z+x}=3$$, then the numerical value of $$15x-7y-z$$ is
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The sum of all natural numbers which satisfy the simultaneous inequations $$x+3<4+2x$$ and $$5x-3<4x-1$$ is
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In an increasing geometric progression with $$1st$$ term $$a$$ and $$n$$th term $$t_n$$, the difference between the fourth and first terms is $$52$$ and the sum of the first three terms is $$26$$. Then the numerical value of $$\frac{t_{2024}}{t_{2023}}+\frac{a^{2024}}{a^{2023}}$$ is
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The base of a triangle is $$4$$ units less than the altitude drawn to it. The area of the triangle is $$96\text{ units}^2$$. The ratio of the base to the height is $$\frac{p}{q}$$, where $$p$$ and $$q$$ are relatively prime. Then the value of $$p+q$$ is
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