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Showing 15 questions in Mathematics for Class 10. Tap a card to reveal the answer.
MathematicsReal NumbersmediumFind the HCF of 96 and 404 using Euclid's division algorithm.
Reveal answer ↓
What it is
Euclid's algorithm finds the HCF by repeated division until the remainder becomes zero.
Answer
Apply repeated division: 404 = 96 x 4 + 20; 96 = 20 x 4 + 16; 20 = 16 x 1 + 4; 16 = 4 x 4 + 0. The last non-zero remainder is 4, so HCF(96, 404) = 4.
a = bq + r, then HCF(a,b) = HCF(b,r)
- •404 = 96 x 4 + 20
- •96 = 20 x 4 + 16
- •20 = 16 x 1 + 4
- •16 = 4 x 4 + 0 -> HCF = 4
Why learn this
HCF simplifies fractions, shares things equally and even underpins computer cryptography (RSA).
💡 Memory trick
Divide, take the remainder, divide again. Last non-zero remainder = HCF.
MathematicsPolynomialseasyFind the zeroes of the polynomial x^2 - 7x + 10 and verify the relationship between the zeroes and the coefficients.
Reveal answer ↓
What it is
The zeroes of a quadratic are where it equals zero; their sum is -b/a and product is c/a.
Answer
Factorise: x^2 - 7x + 10 = (x - 2)(x - 5), so the zeroes are 2 and 5. Sum of zeroes = 2 + 5 = 7 = -(-7)/1 = -b/a. Product of zeroes = 2 x 5 = 10 = 10/1 = c/a. Both relations hold.
Sum = -b/a ; Product = c/a
- •(x - 2)(x - 5) -> zeroes 2, 5
- •Sum = 7 = -b/a
- •Product = 10 = c/a
Why learn this
Finding roots is how we solve projectile paths, profit models and design problems in engineering.
💡 Memory trick
Sum = -b/a (has the minus), Product = c/a (plain).
MathematicsPair of Linear Equations in Two VariablesmediumSolve the pair of equations: 2x + 3y = 13 and x - y = -1.
Reveal answer ↓
What it is
Two linear equations meet at one point - the (x, y) that satisfies both at once.
Answer
From the second equation, x = y - 1. Substitute into the first: 2(y - 1) + 3y = 13, so 5y - 2 = 13, giving 5y = 15 and y = 3. Then x = y - 1 = 2. Solution: x = 2, y = 3.
Substitution: express one variable, substitute into the other
- •x = y - 1 from second equation
- •2(y-1) + 3y = 13 -> 5y = 15
- •y = 3, x = 2
Why learn this
It solves 'two unknowns' problems: prices, mixtures, speed-time and business break-even.
💡 Memory trick
Substitution: make one variable the subject, then plug it into the other equation.
MathematicsQuadratic EquationsmediumFind the roots of x^2 - 5x + 6 = 0 and state the nature of the roots using the discriminant.
Reveal answer ↓
What it is
The discriminant D = b^2 - 4ac reveals the nature of a quadratic's roots before you solve it.
Answer
Factorise: (x - 2)(x - 3) = 0, so the roots are x = 2 and x = 3. The discriminant D = b^2 - 4ac = (-5)^2 - 4(1)(6) = 25 - 24 = 1. Since D > 0 and is a perfect square, the roots are real, distinct and rational.
D = b^2 - 4ac ; x = (-b +/- sqrt(D)) / 2a
- •(x-2)(x-3) = 0 -> roots 2, 3
- •D = b^2 - 4ac = 1
- •D > 0 -> two distinct real roots
Why learn this
Engineers use it to know at a glance if a design has real solutions (bridges, projectiles).
💡 Memory trick
D > 0 real and distinct; D = 0 real and equal; D < 0 no real roots.
MathematicsArithmetic ProgressionseasyThe first term of an AP is 3 and the common difference is 5. Find its 10th term.
Reveal answer ↓
What it is
In an AP each term rises by a fixed step d; the nth term is a + (n - 1)d.
Answer
The nth term of an AP is a_n = a + (n - 1)d. Here a = 3, d = 5, n = 10, so a_10 = 3 + (10 - 1) x 5 = 3 + 45 = 48.
a_n = a + (n - 1)d
- •a_n = a + (n - 1)d
- •a = 3, d = 5, n = 10
- •a_10 = 48
Why learn this
APs model salaries with fixed raises, seating rows, EMIs and steady savings.
💡 Memory trick
nth term = start + (steps taken) x step = a + (n - 1)d.
MathematicsArithmetic ProgressionsmediumFind the sum of the first 20 terms of the AP 2, 5, 8, 11, ...
Reveal answer ↓
What it is
The sum of an AP pairs the first and last terms: S = n/2 [2a + (n - 1)d].
Answer
Here a = 2 and d = 3. The sum of n terms is S_n = n/2 [2a + (n - 1)d]. So S_20 = 20/2 [2(2) + 19(3)] = 10 [4 + 57] = 10 x 61 = 610.
S_n = n/2 [2a + (n - 1)d]
- •a = 2, d = 3, n = 20
- •S_n = n/2 [2a + (n-1)d]
- •S_20 = 10 x 61 = 610
Why learn this
It quickly totals things that grow steadily - savings, stacked logs, salaries over years.
💡 Memory trick
Sum = (number of terms / 2) x (2a + (n - 1)d): half the count times the bracket.
MathematicsTrianglesmediumState the Basic Proportionality Theorem (Thales' theorem).
Reveal answer ↓
What it is
A line parallel to one side of a triangle cuts the other two sides in the same ratio.
Answer
The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, then it divides those two sides in the same ratio. In triangle ABC, if DE is parallel to BC and meets AB at D and AC at E, then AD/DB = AE/EC.
If DE || BC then AD/DB = AE/EC
- •Line parallel to one side of a triangle
- •Divides the other two sides in the same ratio
- •AD/DB = AE/EC
- •Basis of similarity of triangles
Why learn this
It's the basis of similarity, scale drawings, maps and finding heights from shadows.
💡 Memory trick
Parallel line inside a triangle -> equal ratios: AD/DB = AE/EC.
MathematicsCoordinate GeometryeasyFind the distance between the points (2, 3) and (5, 7).
Reveal answer ↓
What it is
The distance between two points is the hypotenuse of the right triangle of their coordinate gaps.
Answer
The distance formula is d = sqrt[(x2 - x1)^2 + (y2 - y1)^2]. Here d = sqrt[(5 - 2)^2 + (7 - 3)^2] = sqrt[9 + 16] = sqrt[25] = 5 units.
d = sqrt[(x2 - x1)^2 + (y2 - y1)^2]
- •d = sqrt[(x2-x1)^2 + (y2-y1)^2]
- •= sqrt[9 + 16] = sqrt[25]
- •Distance = 5 units
Why learn this
GPS, video games, robotics and maps all compute 'how far' with this exact formula.
💡 Memory trick
It's just Pythagoras: sqrt of (x-gap)^2 + (y-gap)^2.
MathematicsIntroduction to TrigonometrymediumIf sin(theta) = 3/5, find cos(theta) and tan(theta), where theta is acute.
Reveal answer ↓
What it is
The identity sin^2 + cos^2 = 1 lets you find one trig ratio from another for the same angle.
Answer
Using the identity sin^2(theta) + cos^2(theta) = 1, cos^2(theta) = 1 - (3/5)^2 = 1 - 9/25 = 16/25, so cos(theta) = 4/5 (positive since theta is acute). Then tan(theta) = sin/cos = (3/5) / (4/5) = 3/4.
sin^2(theta) + cos^2(theta) = 1 ; tan = sin/cos
- •sin^2 + cos^2 = 1
- •cos(theta) = 4/5
- •tan(theta) = sin/cos = 3/4
Why learn this
Trig ratios drive navigation, physics waves, construction and computer graphics.
💡 Memory trick
Think of the 3-4-5 triangle: sin 3/5 -> cos 4/5 -> tan 3/4.
MathematicsSome Applications of TrigonometrymediumThe angle of elevation of the top of a tower from a point 30 m from its base is 30 degrees. Find the height of the tower.
Reveal answer ↓
What it is
An angle of elevation turns a triangle into a height problem using tan = opposite / adjacent.
Answer
Let the height be h. tan(30 degrees) = h / 30, and tan(30 degrees) = 1/sqrt(3). So h = 30 x (1/sqrt(3)) = 30/sqrt(3) = 10 sqrt(3) m, which is about 17.3 m.
tan(theta) = height / base distance
- •tan(angle) = opposite/adjacent = h/30
- •tan(30 deg) = 1/sqrt(3)
- •h = 10 sqrt(3) m (about 17.3 m)
Why learn this
Surveyors, pilots and builders measure unreachable heights (towers, hills) exactly this way.
💡 Memory trick
Draw the right triangle; tan(angle) = height / base distance.
MathematicsCirclesmediumA point P is 13 cm from the centre of a circle of radius 5 cm. Find the length of the tangent drawn from P to the circle.
Reveal answer ↓
What it is
A tangent touches a circle at one point and is perpendicular to the radius at that point.
Answer
The tangent is perpendicular to the radius at the point of contact, so the radius, tangent and line OP form a right-angled triangle with OP as the hypotenuse. Length of tangent = sqrt[OP^2 - r^2] = sqrt[13^2 - 5^2] = sqrt[169 - 25] = sqrt[144] = 12 cm.
tangent length = sqrt[OP^2 - r^2]
- •Tangent is perpendicular to radius at contact
- •Right triangle: OP hypotenuse
- •Length = sqrt[13^2 - 5^2] = 12 cm
Why learn this
It's used in gear design, road curves and belt-and-pulley engineering.
💡 Memory trick
Radius, tangent and line to the point form a right triangle: tangent = sqrt(OP^2 - r^2).
MathematicsSurface Areas and VolumeseasyFind the volume of a cone whose base radius is 3 cm and height is 4 cm. (Take pi = 22/7.)
Reveal answer ↓
What it is
A cone's volume is exactly one-third of a cylinder with the same base and height.
Answer
Volume of a cone = (1/3) x pi x r^2 x h = (1/3) x (22/7) x 3^2 x 4 = (1/3) x (22/7) x 36 = (22 x 12) / 7 = 264/7, which is about 37.7 cubic cm.
V(cone) = (1/3) pi r^2 h
- •V = (1/3) pi r^2 h
- •= (1/3)(22/7)(9)(4)
- •= 264/7 = about 37.7 cm^3
Why learn this
It's how we measure ice-cream cones, funnels, heaps of grain and tent capacity.
💡 Memory trick
Cone = 1/3 x pi r^2 h - a cone is a third of its cylinder.
MathematicsStatisticsmediumThe mean of a distribution is 27 and its median is 30. Estimate the mode using the empirical relationship.
Reveal answer ↓
What it is
For a moderately skewed data set, the three averages link as Mode = 3 Median - 2 Mean.
Answer
The empirical relationship between the three measures of central tendency is Mode = 3 Median - 2 Mean. Substituting, Mode = 3(30) - 2(27) = 90 - 54 = 36.
Mode = 3 Median - 2 Mean
- •Mode = 3 Median - 2 Mean
- •= 3(30) - 2(27)
- •= 90 - 54 = 36
Why learn this
It estimates the most common value from the other two - handy in surveys and economics.
💡 Memory trick
Mode = 3 Median - 2 Mean ('3M minus 2-Mean gives the Mode').
MathematicsProbabilityeasyOne card is drawn at random from a well-shuffled deck of 52 playing cards. What is the probability that it is a king?
Reveal answer ↓
What it is
Probability = favourable outcomes / total outcomes, always a number between 0 and 1.
Answer
There are 4 kings in a deck of 52 cards. Probability = number of favourable outcomes / total outcomes = 4/52 = 1/13.
P(event) = favourable outcomes / total outcomes
- •4 kings in 52 cards
- •P = favourable/total = 4/52
- •= 1/13
Why learn this
It powers weather forecasts, insurance, games, medical trials and AI.
💡 Memory trick
Favourable over Total: 4 kings out of 52 -> 4/52 = 1/13.
MathematicsIntroduction to TrigonometrymediumEvaluate 2 tan^2(45 degrees) + cos^2(30 degrees) - sin^2(60 degrees).
Reveal answer ↓
What it is
The standard angles 0, 30, 45, 60 and 90 degrees have fixed trig values worth memorising.
Answer
Use standard values: tan(45 deg) = 1, cos(30 deg) = sqrt(3)/2, sin(60 deg) = sqrt(3)/2. So the expression = 2(1)^2 + (sqrt(3)/2)^2 - (sqrt(3)/2)^2 = 2 + 3/4 - 3/4 = 2.
tan45 = 1 ; cos30 = sqrt(3)/2 ; sin60 = sqrt(3)/2
- •tan45 = 1, cos30 = sin60 = sqrt(3)/2
- •2(1) + 3/4 - 3/4
- •= 2
Why learn this
They appear in nearly every physics and maths problem - knowing them saves huge time.
💡 Memory trick
For sin, use sqrt(0..4)/2 across 0,30,45,60,90; cos is the same list reversed.