Board Boosters
The questions your board exam loves to ask
800 most-asked Class 11 & 12 (+1 / +2) questions across Physics, Chemistry, Maths and Biology — each with a model answer and the exact marking-scheme points examiners reward. Revise smart, walk in calm.
MathsClass 112 marksmedium
Trigonometric Functions
Find the value of sin 15 degrees.
Reveal model answer + marking pointsHide answer▾
Write 15 = 45 - 30. Using sin(A - B) = sin A cos B - cos A sin B: sin 15 = sin 45 cos 30 - cos 45 sin 30 = (1/sqrt2)(sqrt3/2) - (1/sqrt2)(1/2) = (sqrt3 - 1)/(2 sqrt2) = (sqrt6 - sqrt2)/4.
sin(A - B) = sin A cos B - cos A sin B
Marking-scheme points
- ✓15 = 45 - 30
- ✓sin(A-B) = sinA cosB - cosA sinB
- ✓sin 15 = (sqrt6 - sqrt2)/4
MathsClass 113 marksmedium
Trigonometric Functions
Prove that sin 3x = 3 sin x - 4 sin^3 x.
Reveal model answer + marking pointsHide answer▾
Write sin 3x = sin(2x + x) = sin 2x cos x + cos 2x sin x. Substitute sin 2x = 2 sin x cos x and cos 2x = 1 - 2 sin^2 x: = (2 sin x cos x) cos x + (1 - 2 sin^2 x) sin x = 2 sin x cos^2 x + sin x - 2 sin^3 x. Replace cos^2 x = 1 - sin^2 x: = 2 sin x (1 - sin^2 x) + sin x - 2 sin^3 x = 2 sin x - 2 sin^3 x + sin x - 2 sin^3 x = 3 sin x - 4 sin^3 x. Hence proved.
sin 3x = 3 sin x - 4 sin^3 x
Marking-scheme points
- ✓sin 3x = sin(2x + x), expand
- ✓Use sin 2x = 2 sinx cosx, cos 2x = 1 - 2 sin^2 x
- ✓Replace cos^2 x = 1 - sin^2 x to get 3 sinx - 4 sin^3 x
MathsClass 115 markshard
Trigonometric Functions
Prove that cos A/(1 - tan A) + sin A/(1 - cot A) = sin A + cos A.
Reveal model answer + marking pointsHide answer▾
Write tan A = sin A/cos A and cot A = cos A/sin A. First term: cos A/(1 - sin A/cos A) = cos A/((cos A - sin A)/cos A) = cos^2 A/(cos A - sin A). Second term: sin A/(1 - cos A/sin A) = sin A/((sin A - cos A)/sin A) = sin^2 A/(sin A - cos A) = -sin^2 A/(cos A - sin A). Adding: [cos^2 A - sin^2 A]/(cos A - sin A) = [(cos A - sin A)(cos A + sin A)]/(cos A - sin A) = cos A + sin A. Hence proved.
cos^2 A - sin^2 A = (cosA - sinA)(cosA + sinA)
Marking-scheme points
- ✓Convert tan A and cot A into sin/cos
- ✓First term = cos^2 A/(cosA - sinA); second = -sin^2 A/(cosA - sinA)
- ✓Add: (cos^2 A - sin^2 A)/(cosA - sinA) = cosA + sinA
MathsClass 112 markseasy
Complex Numbers and Quadratic Equations
Express (1 + i)(2 - i) in the form a + i b.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 112 marksmedium
Complex Numbers and Quadratic Equations
Find the multiplicative inverse of the complex number 4 - 3i.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 113 marksmedium
Complex Numbers and Quadratic Equations
Find the modulus and argument of the complex number z = 1 + i sqrt(3), and write it in polar form.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 113 marksmedium
Complex Numbers and Quadratic Equations
Express (3 + 4i)/(1 - 2i) in the form a + i b.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 112 markseasy
Complex Numbers and Quadratic Equations
Solve the quadratic equation x^2 + x + 1 = 0.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 113 marksmedium
Complex Numbers and Quadratic Equations
Solve the equation x^2 - 2x + 5 = 0.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 112 markseasy
Complex Numbers and Quadratic Equations
Find the value of i^9 + i^19.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 113 markshard
Complex Numbers and Quadratic Equations
Find the square root of the complex number -5 + 12i.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 112 markseasy
Linear Inequalities
Solve the inequality 3x - 5 < 7 for real x and represent the solution on a number line.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 113 marksmedium
Linear Inequalities
Solve 5x - 3 >= 3x - 5 and show the solution set on a number line.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 113 marksmedium
Linear Inequalities
Solve the system of inequalities 2x - 1 < 5 and 3x + 2 > -4, and find the common solution.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 113 marksmedium
Linear Inequalities
The longest side of a triangle is three times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum length of the shortest side.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 112 markseasy
Permutations and Combinations
Evaluate 8! / (6! x 2!).
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 112 markseasy
Permutations and Combinations
How many 3-digit numbers can be formed using the digits 1 to 9 if no digit is repeated?
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 113 marksmedium
Permutations and Combinations
How many distinct arrangements can be made using all the letters of the word 'MATHEMATICS'?
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 113 marksmedium
Permutations and Combinations
Find the value of n if nC2 = 15.
🔒 Model answer + marking pointsUnlock with Premium →MathsClass 112 markseasy
Permutations and Combinations
In how many ways can a committee of 3 persons be chosen from 10 persons?
🔒 Model answer + marking pointsUnlock with Premium →You are more ready than you feel.
One question at a time is how every topper started. Bookmark this, revise a few each day, and watch the fear shrink. And if a friend is stressing about boards — send this their way. You both win.