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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.

800 board questions•Model answers•Marking-scheme points•Every chapter•CBSE · ISC · State boards

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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 points▾

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
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MathsClass 115 markshard

Sequences and Series

Find the sum of the series 1 x 2 + 2 x 3 + 3 x 4 + ... up to n terms.

Reveal model answer + marking points▾

The rth term is a_r = r(r + 1) = r^2 + r. So the sum S = sum over r of (r^2 + r) = sum of r^2 + sum of r. Using sum of r^2 = n(n + 1)(2n + 1)/6 and sum of r = n(n + 1)/2: S = n(n + 1)(2n + 1)/6 + n(n + 1)/2. Take n(n + 1)/6 common: S = [n(n + 1)/6][(2n + 1) + 3] = [n(n + 1)/6](2n + 4) = [n(n + 1) x 2(n + 2)]/6 = n(n + 1)(n + 2)/3.

S = n(n + 1)(n + 2)/3

Marking-scheme points

  • ✓rth term = r(r + 1) = r^2 + r
  • ✓Sum = sum of r^2 + sum of r
  • ✓= n(n + 1)(2n + 1)/6 + n(n + 1)/2 = n(n + 1)(n + 2)/3
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MathsClass 115 markshard

Straight Lines

Find the image (reflection) of the point (3, 8) in the line x + 3y - 7 = 0.

Reveal model answer + marking points▾

For the image (h, k) of point (x1, y1) in line Ax + By + C = 0, we use (h - x1)/A = (k - y1)/B = -2(A x1 + B y1 + C)/(A^2 + B^2). Here A = 1, B = 3, C = -7, (x1, y1) = (3, 8). A x1 + B y1 + C = 3 + 24 - 7 = 20, and A^2 + B^2 = 1 + 9 = 10, so the common ratio = -2(20)/10 = -4. Then h = 3 + 1(-4) = -1 and k = 8 + 3(-4) = 8 - 12 = -4. Hence the image is (-1, -4).

(h - x1)/A = (k - y1)/B = -2(A x1 + B y1 + C)/(A^2 + B^2)

Marking-scheme points

  • ✓(h - x1)/A = (k - y1)/B = -2(A x1 + B y1 + C)/(A^2 + B^2)
  • ✓Ratio = -2(20)/10 = -4
  • ✓Image = (-1, -4)
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