16 Questions
Which of the following logical statements is equivalent to ~p ∨ ~q?
~p ∧ ~q
What is the solution of the inequation 4 - x + 0.05 - 7.2 - x < 4, x ∈ R?
(-2, ∞)
If cos(3x)sin(2x) = Σam sin(mx) is an identity in x, then what is the value of m?
3
If Re(z + 2) = z - 2, then the locus of z is
circle
If X2 – abx – a2 – 0 has
one positive root and one negative root
What is the total number of solutions of the equation cot(x) = cot(x) + 1, x ∈ [0, 3π/5]?
0
What is the minimum value of (sin^(-1) (x)) + (cos^(-1) (x))?
2π/3
If a + 2b + 3c = 12, (a, b, ∈ R+), then the maximum value of ab2c3 is
25
If the origin is shifted to (1, 2), then the equation y^2 - 8x - 4y + 12 = 0 changes to y^2 = 4ax, where a is equal to?
1
The sum of n terms of the infinite series 1.32 + 2.52 + 3.72 + ....∞ is
$\frac{n(n+1)}{6n^2 + 14n + 7}$
If log75 = α, log5 3 = b and log32 = c, then the logarithm of the number 70 to the base 225 is
$\frac{1+a+abc}{2a(1+b)}$
What are the equations of the bisectors of the angles between the straight lines 3x + 4y + 7 = 0 and 12x + 5y - 8 = 0?
7x - 9y - 17 = 0, 99x + 77y + 51 = 0
What is the equation of the circle which passes through the points (1, -2) and (3, -4) and touches the X-axis?
x^2 + y^2 + 6x + 2y + 9 = 0
The maximum number of points of intersection of 10 circles is
85
If $p \neq q \neq r$ and $x^2 - px - q + x - r = 0$, then the value of $x$ which satisfies the equation is
x = 0
If $A = \begin{bmatrix} x^3 & 2 \ 1 & y \ 4 & z \end{bmatrix}$, then $A(adj A)$ is equal to
$\begin{bmatrix} 64 & 0 & 0 \ 0 & 64 & 0 \ 0 & 0 & 64 \end{bmatrix}$
Study Notes
Algebra and Equations
- If Re(z+2) = z-2, then the locus of z is a circle.
- If X^2 – abx – a^2 – 0 has two distinct real roots with opposite signs, then the equation has one positive root and one negative root.
- If a + 2b + 3c = 12, (a, b, c ∈ R+), then the maximum value of ab^2c^3 is 24.
Series and Progressions
- The sum of n terms of the infinite series 1.32 + 2.52 + 3.72 + …∞ is n(n+1)(2n+1)/6.
Logarithms and Exponents
- If log_7 5 = α, log_5 3 = β, and log_3 2 = γ, then the logarithm of the number 70 to the base 225 is (1 - α + αγ)/(2α(1 + β)).
Geometry and Graphs
- The maximum number of points of intersection of 10 circles is 90.
- The equation of the circle which passes through the points (1, –2) and (3, –4) and touches the X-axis is x^2 + y^2 + 6x + 2y + 9 = 0.
Matrices and Determinants
- If A is a matrix, then A(adj A) is equal to [[64, 0, 0], [0, 64, 0], [0, 0, 64]].
Trigonometry
- If cos^3(x) sin(2x) = ∑am sin mx is an identity in x, then a_3 = 1/8, a_2 = 0, and a_1 = 1/4.
- The total number of solutions of cot(x) = cot(x) + (sin(x))/(sin(x) + 1) is 3 in the interval [0, 3π].
Inequalities and Functions
- The minimum value of (sin^(-1)x) + (cos^(-1)x) is π/3.
- The solution of the inequality 4^(x-1) + 0.05^(x-7.2) < 4 is (-2, ∞).
Coordinate Geometry
- The equation of the circle which passes through the points (1, –2) and (3, –4) and touches the X-axis is x^2 + y^2 + 6x + 2y + 9 = 0.
- The equation of the hyperbola x^2 - y^2 = 9 is 9x^2 - 8y^2 + 18x - 9 = 0.
This quiz contains algebra problems on equations, roots, and infinite series. Solve these problems to test your understanding of algebraic concepts.
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