Chapter 3: Real Numbers, Complex Numbers and Quadratic Expressions (Set-1)
Which set contains 0 and negative numbers?
A Natural numbers
B Whole numbers
C Integers
D Irrationals
Integers include positive numbers, negative numbers, and zero. Natural numbers start from 1, whole numbers include 0 but not negatives, and irrationals are non-rational numbers.
Which number is irrational?
A 0.25
B 3/7
C -5
D √2
√2 cannot be written as a fraction p/q with integers p and q. In contrast, 0.25 = 1/4, 3/7 is rational, and -5 is an integer.
Absolute value represents
A Distance from zero
B Number’s sign
C Square of number
D Reciprocal value
|x| gives the distance of x from 0 on the number line, so it is always non-negative. It ignores whether x is positive or negative.
For any real x, |x| is
A Always negative
B Always zero
C Always nonnegative
D Always integer
Absolute value measures distance, so it cannot be negative. It becomes zero only when x = 0, and it is not necessarily an integer (like |1/2|).
Which is true for all real a, b?
A |a+b| = |a|+|b|
B |a+b| ≤ |a|+|b|
C |a−b| = |a|−|b|
D |ab| = |a|+|b|
This is the triangle inequality. Equality happens only in special cases (same direction on number line). The other statements are not true in general.
Simplify √50
A 10√5
B 25√2
C 2√25
D 5√2
√50 = √(25×2) = √25·√2 = 5√2. Taking out the perfect square factor makes the surd simplest.
Rationalize 1/√3
A 3/√3
B √3
C √3/3
D 1/3
Multiply numerator and denominator by √3: 1/√3 = (1·√3)/(√3·√3) = √3/3. This removes the surd from the denominator.
Law of exponents: a^m · a^n equals
A a^(m+n)
B a^(m−n)
C a^(mn)
D (a^m)^n
When multiplying powers with the same base, add exponents: a^m·a^n = a^(m+n). This rule holds for a ≠ 0 in general exponent operations.
(a^m)^n equals
A a^(m+n)
B a^(m−n)
C a^(m/n)
D a^(mn)
Power of a power multiplies exponents: (a^m)^n = a^(mn). This is used to simplify expressions like (x^2)^3 = x^6.
log_a(a^x) equals
A ax
B a
C x
D 1/x
Logarithm base a and exponent base a undo each other: log_a(a^x) = x (for a>0, a≠1). It’s the inverse relationship of exponentials.
log_a(1) equals
A 1
B 0
C a
D Undefined
Since a^0 = 1 for a>0, a≠1, we get log_a(1)=0. This is a standard basic logarithm property.
Imaginary unit i satisfies
A i = 1
B i² = 1
C i² = −1
D i³ = 1
The imaginary unit i is defined by i² = −1. This allows square roots of negative numbers to be expressed using i, like √(-9)=3i.
Standard form of a complex number is
A a−b
B a/b
C ab+i
D a+bi
Any complex number is written as a+bi where a is real part and b is imaginary part coefficient. This form makes addition and multiplication rules clear.
If a+bi = c+di, then
A a=c and b=d
B a=c only
C b=d only
D a=d and b=c
Two complex numbers are equal only when both real parts are equal and imaginary parts are equal. So a=c and b=d must hold simultaneously.
Conjugate of a+bi is
A −a+bi
B −a−bi
C a−bi
D b+ai
Conjugate changes the sign of the imaginary part only. It is useful because (a+bi)(a−bi)=a²+b², which becomes a real number.
Product (a+bi)(a−bi) equals
A a²−b²
B a²+b²
C 2abi
D a²+b²i
Using (x+y)(x−y)=x²−y² with y=bi gives a²−(bi)² = a²−(−b²)=a²+b². Imaginary terms cancel out.
Modulus of a+bi is
A a+b
B a²+b²
C √(a²−b²)
D √(a²+b²)
|a+bi| represents distance from origin in Argand plane. By Pythagoras, distance = √(a²+b²). Modulus is always non-negative.
Argument of a complex number is
A Its modulus
B Its conjugate
C Angle from x-axis
D Its real part
The argument (arg) is the angle the line from origin to the point makes with positive real axis. It describes direction, while modulus describes distance.
Value of i^4 is
A 1
B i
C −1
D −i
Powers of i repeat every 4: i¹=i, i²=−1, i³=−i, i⁴=1. This cyclic nature helps simplify expressions quickly.
Value of i^7 is
A i
B 1
C −1
D −i
i^7 = i^(4+3) = i^4 · i^3 = 1 · (−i) = −i. Reduce power by multiples of 4 to simplify.
Reciprocal of (a+bi) equals
A (a+bi)/(a²+b²)
B (a−bi)/(a²−b²)
C (a−bi)/(a²+b²)
D (a+bi)/(a²−b²)
1/(a+bi) is found by multiplying by conjugate: (a−bi)/((a+bi)(a−bi)) = (a−bi)/(a²+b²). Denominator becomes real.
Polar form uses
A a+bi only
B r(cosθ+i sinθ)
C r(a+bi)
D θ(a+bi)
Polar form expresses z using modulus r and argument θ: z = r(cosθ+i sinθ). It simplifies multiplication, division, and powers using angle addition.
In De Moivre’s theorem, (cosθ+i sinθ)^n equals
A cos(nθ)+i sin(nθ)
B cosθ+i sin(nθ)
C cos(nθ)+i sinθ
D cosθ+i sinθ
De Moivre’s theorem states (cosθ+i sinθ)^n = cos(nθ)+i sin(nθ) for integer n. It’s key for finding powers and roots in polar form.
Argand plane represents
A Only real numbers
B Only integers
C Rational numbers only
D Complex numbers as points
In Argand plane, x-axis is real part and y-axis is imaginary part. A complex number a+bi is plotted as point (a, b), making geometry useful.
Locus |z| = r is a
A Straight line
B Parabola
C Circle at origin
D Hyperbola
|z| is distance from origin. All points at constant distance r form a circle centered at origin with radius r in the complex plane.
Region |z| < r represents
A Interior of circle
B Exterior region
C Circle boundary only
D A straight strip
|z| < r means distance from origin is less than r, so all points inside the circle are included, excluding the boundary circle.
Region |z−a| = r is a
A Line through origin
B Parabola focus a
C Segment length r
D Circle centered at a
|z−a| measures distance between point z and fixed point a. Constant distance r gives a circle with center at a and radius r.
Locus Re(z)=k is a
A Horizontal line
B Circle
C Vertical line
D Diagonal line
Re(z)=k means x-coordinate is fixed at k. That gives a vertical line parallel to imaginary axis, containing all points (k, y).
Locus Im(z)=k is a
A Vertical line
B Horizontal line
C Circle
D Ellipse
Im(z)=k means y-coordinate is fixed at k. That forms a horizontal line parallel to real axis, containing all points (x, k).
Region Re(z) > 0 is
A Left half-plane
B Upper half-plane
C Lower half-plane
D Right half-plane
Re(z)>0 means x-coordinate is positive. That is the right half of the Argand plane, separated by the imaginary axis.
Region |z−1| < 2 is a
A Outside circle
B Strip region
C Disc center 1
D Ray from origin
|z−1|<2 means points within distance 2 of the point 1+0i. That forms the interior (disc) of a circle centered at 1 on real axis.
Set of points equidistant from A and B is
A Perpendicular bisector
B Circle through AB
C Parallel to AB
D Angle bisector only
All points with equal distance from two fixed points A and B lie on the perpendicular bisector of segment AB. This is a basic locus result.
Annulus means
A Filled circle
B Straight strip
C Single point
D Ring between circles
An annulus is the region between two concentric circles, like r1 < |z| < r2. It includes points whose distance from origin lies between two radii.
Standard quadratic equation form is
A ax+b=0
B ax³+bx+c=0
C ax²+bx+c=0
D x²+1=0 only
A quadratic equation has degree 2 and is written as ax²+bx+c=0 with a≠0. Coefficients a, b, c are real or complex numbers.
Discriminant of ax²+bx+c is
A b²+4ac
B b²−4ac
C 4ac−b²
D (a+b+c)²
Discriminant Δ=b²−4ac determines nature of roots. It comes from quadratic formula and helps decide whether roots are real, equal, or complex.
If Δ > 0, roots are
A Equal real
B Pure imaginary
C No roots
D Distinct real
When discriminant is positive, square root of Δ is real and nonzero, so quadratic formula gives two different real roots.
If Δ = 0, roots are
A Distinct real
B Non-real
C Equal real
D Undefined
Δ=0 makes √Δ=0, so both roots become −b/(2a). This means the parabola touches the x-axis at one point (repeated root).
If Δ < 0, roots are
A Complex conjugates
B Real unequal
C Real equal
D Both integers
For real coefficients, Δ<0 gives √Δ as imaginary, producing non-real roots that come in conjugate pairs, like p±qi.
Quadratic formula gives roots
A (b±√Δ)/a
B (−b±√Δ)/(2a)
C (−a±√Δ)/(2b)
D (c±√Δ)/(2a)
The standard solution of ax²+bx+c=0 is x = (−b±√(b²−4ac))/(2a). It works for all quadratics with a≠0.
Sum of roots of ax²+bx+c is
A c/a
B −c/b
C −b/a
D b/a
If roots are α and β, then α+β = −b/a and αβ = c/a. These relations come from comparing coefficients after factoring a(x−α)(x−β).
Product of roots of ax²+bx+c is
A −b/a
B −c/a
C b/c
D c/a
For roots α, β of ax²+bx+c=0, product αβ equals c/a. This holds even when roots are irrational or complex.
Quadratic x²−5x+6=0 has roots
A 1 and 6
B −2 and −3
C 2 and 3
D 3 and 6
Factor x²−5x+6 = (x−2)(x−3). Setting each factor to zero gives x=2 and x=3. Quick factorization checks accuracy.
Vertex x-coordinate of y=ax²+bx+c is
A −b/2a
B b/2a
C −a/2b
D c/2a
The parabola’s vertex lies at x = −b/(2a). This comes from completing the square or symmetry of parabola around the vertical line x = −b/(2a).
If you multiply inequality by −1, sign
A Stays same
B Becomes equality
C Becomes absolute
D Reverses
Multiplying or dividing an inequality by a negative number reverses the inequality sign. Example: 2<5 implies −2>−5 after multiplying by −1.
Solution of x+3 < 7 is
A x > 4
B x < 4
C x ≤ 4
D x ≥ 4
Subtract 3 from both sides: x+3<7 ⇒ x<4. Adding or subtracting the same number does not change the inequality direction.
Solution of 2x ≥ 10 is
A x ≤ 5
B x > 5
C x ≥ 5
D x < 5
Divide both sides by 2 (positive), so inequality sign stays same: 2x≥10 ⇒ x≥5. Always check sign when dividing.
Solution of −3x < 6 is
A x < −2
B x ≤ −2
C x ≥ −2
D x > −2
Divide by −3 (negative), so inequality reverses: −3x<6 ⇒ x>−2. Reversal is the key step many learners miss.
|x| < 5 means
A −5 < x < 5
B x > 5
C x < −5
D x ≤ −5 or ≥5
|x|<5 means distance from zero is less than 5, so x lies between −5 and 5. This forms an open interval (−5, 5).
|x| ≥ 3 means
A −3 < x < 3
B x ≤ −3 or ≥3
C 0 ≤ x ≤ 3
D x = 3 only
|x|≥3 means distance from zero is at least 3, so x is outside the interval (−3, 3). It includes both sides: x≤−3 or x≥3.
Interval notation for x ≥ 2 is
A (2,∞)
B (−∞,2]
C [2,∞)
D (−∞,2)
x≥2 includes 2, so we use a closed bracket at 2. Infinity is never included, so it always has an open parenthesis: [2,∞).