Mathematics · Class 12 Science

Bihar Bseb Class 12 Mathematics — Real Previous-Year Papers

with complete answers

Real previous-year board papers, year by year — the official exam pattern, the full question paper, and every question solved the concept-first way. Distinct from the chapter-wise textbook bank.

2018–2026
Years of papers: 8
Total Papers: 8
Real Board Papers: 0
Sample papers: 987
Real-paper Q & A: 0
Sample-paper Q & A

Real board-paper questions available, by year

138 Q2026complete

138 Q2025complete

138 Q2024complete

138 Q2023complete

138 Q2022complete

138 Q2021complete

—2020Not available

83 Q2019complete

76 Q2018complete

Bihar Board Intermediate 2026 · Set A

Real board examination

About this paper

The real Class-12 board examination held in 2026. Every question below is solved the concept-first way. Sample papers are labelled honestly — never shown as a past exam.

Total marks: 100
Questions: 69
Duration: 195 min
Sections: 3

The marks / questions / duration above are the official exam pattern. We currently have 138 of this paper’s questions (200% of the full paper), with 138 fully solved. Questions we couldn’t yet extract or verify are held — never shown as complete.

Sections & marks

Section	Type	Questions	Marks each	Total
Section A (Objective, Q1–100)Section Section A (Objective, Q1–100)	Objective (MCQ) — answer any 50 of 100, on OMR	50	1	50
Section B — Short AnswerSection Section B — Short Answer	Short answer — answer any 15 of 30	15	2	30
Section B — Long AnswerSection Section B — Long Answer	Long answer (internal choice) — answer any 4 of 8	4	5	20
Total		69		100

The question paper

The questions we hold for this paper, laid out by section. Solutions are on the Answers tab.

Board Examination

Mathematics

Bihar Board Intermediate 2026 · Set A

Series/Set: ARoll No. ________

Time Allowed: 3 hr 15 minMaximum Marks: 100

General Instructions

This question paper contains 69 questions divided into 3 sections — Section A (Objective, Q1–100), Section B — Short Answer, Section B — Long Answer.
Section Section A (Objective, Q1–100) comprises 50 questions of 1 mark each (Objective (MCQ) — answer any 50 of 100, on OMR).
Section Section B — Short Answer comprises 15 questions of 2 marks each (Short answer — answer any 15 of 30).
Section Section B — Long Answer comprises 4 questions of 5 marks each (Long answer (internal choice) — answer any 4 of 8).

Above is the official exam pattern. The questions printed below are those we currently hold for this paper.

Section A

Q1.

\tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) =

(a) $\frac{\pi}{3}$
(b) $\frac{\pi}{6}$
(c) $-\frac{\pi}{3}$
(d) $-\frac{\pi}{6}$

[1]

Q2.

2\tan^{-1}\frac{1}{3} =

(a) $\tan^{-1}\frac{3}{2}$
(b) $\tan^{-1}\frac{3}{4}$
(c) $\tan^{-1}\frac{4}{3}$
(d) $\tan^{-1}\frac{2}{3}$

[1]

Q3.

x \in R,\ \cot(\tan^{-1}x + \cot^{-1}x) =

(a) $1$
(b) $\frac{1}{2}$
(c) $0$
(d) $\frac{1}{3}$

[1]

Q4.

\sin(\cos^{-1}3/5) =

(a) $\frac{3}{4}$
(b) $\frac{4}{5}$
(c) $\frac{3}{5}$
(d) $\frac{5}{4}$

[1]

Q5.

\tan^{-1}(1) + \cos^{-1}\left(-\frac{1}{2}\right) + \sin^{-1}\left(-\frac{1}{2}\right) =

(a) $\pi$
(b) $\frac{2\pi}{3}$
(c) $\frac{3\pi}{4}$
(d) $\frac{\pi}{2}$

[1]

Q6.

\cos^{-1}\left(\cos\frac{7\pi}{6}\right) =

(a) $\frac{\pi}{6}$
(b) $\frac{\pi}{3}$
(c) $\frac{5\pi}{6}$
(d) $\frac{7\pi}{6}$

[1]

Q7.

\tan^{-1}2 + \tan^{-1}3 =

(a) $-\frac{\pi}{4}$
(b) $\frac{\pi}{4}$
(c) $\frac{3\pi}{4}$
(d) $\pi$

[1]

Page 1 of 16

Q8.

If |x| ≤ 1, then tan(cos⁻¹x) = (a) frac√(1-x²)x (b) (x)/(1+x²) (c) frac√(1+x²)x (d) √(1-x²)

[1]

Q9.

tan⁻¹(x)/(y) - tan⁻¹(x-y)/(x+y) = (a) -(3π)/(4) (b) (π)/(2) (c) (π)/(4) (d) (π)/(3)

[1]

Q10.

|x| ≤ 1,cos⁻¹((1-x²)/(1+x²)) = (a) 2cos⁻¹x (b) 2sin⁻¹x (c) 2tan⁻¹x (d) tan⁻¹2x

[1]

Q11.

23 12 11 \36 10 26 \63 26 37 = (a) 1 (b) -1 (c) 0 (d) 2

[1]

Q12.

a-b b-c c-a b-c c-a a-b c-a a-b b-c = (a) 1 (b) 0 (c) -1 (d) a+b+c

[1]

Q13.

x 4 \2 2x = 0 ⇒ x = (a) ± 2 (b) ± 1 (c) ± 3 (d) 0

[1]

Q14.

cos 15^° sin 15^° ; sin 75^° cos 75^° = (a) 1 (b) 0 (c) -1 (d) (1)/(2)

[1]

Q15.

1 - P(A' ∩ B') = (a) P(A ∩ B) (b) P(A ∪ B) (c) P(A) (d) P(B)

[1]

Q16.

If A, B and C are three independent events then P(ABC) = (a) P(A) + P(B) + P(C) (b) P(A) - P(B) - P(C) (c) P(A) · P(B) · P(C) (d) None of these

[1]

Q17.

P(A) = (7)/(13),P(B) = (9)/(13),P(A ∩ B) = (4)/(13) ⇒ P(A/B) = (a) (4)/(9) (b) (4)/(7) (c) (12)/(13) (d) (1)/(6)

[1]

Q18.

If the odds against the event A is 3 : 7 then P(A) = (a) (3)/(10) (b) (7)/(10) (c) (3)/(7) (d) (7)/(3)

[1]

Page 2 of 16

Q19.

(d)/(dx)(log xⁿ) = (a) (1)/(xⁿ) (b) n (c) (1)/(x) (d) (n)/(x)

[1]

Q20.

(d²)/(dx²)(sin 2x) = (a) 4sin 2x (b) 4cos² 2x (c) -4sin 2x (d) 2sin 4x

[1]

Q21.

(d)/(dx)(ex-a) = (a) ex-a (b) (x-a)ex-a (c) ex (d) -ex-a

[1]

Q22.

(d)/(dx)√(x²+ax+1) = (a) fracx+a2√(x²+ax+1) (b) frac2x+a2√(x²+ax+1) (c) frac2x+a√(x²+ax+1) (d) frac12√(x²+ax+1)

[1]

Q23.

(d)/(dx)(sin x²) = (a) 2xcos x² (b) cos x² (c) x²cos x² (d) xcos x²

[1]

Q24.

(d)/(dx)√(cot x) = (a) frac12√(cot x) (b) √(csc² x) (c) frac-csc² x2√(cot x) (d) fraccsc² x2√(cot x)

[1]

Q25.

(d)/(dx)(tan⁻¹√(x) + cot⁻¹√(x)) = (a) (π)/(2) (b) 0 (c) 1 (d) π

[1]

Q26.

(d)/(dx)(2tan⁻¹x) = (a) (1)/(1+x²) (b) (2)/(1+x²) (c) (1)/(2(1+x²)) (d) (1)/(1-x²)

[1]

Q27.

(d)/(dx)(cos√(x)) = (a) sin√(x) (b) frac-sin√(x)2√(x) (c) fracsin√(x)2√(x) (d) frac12√(x)

[1]

Q28.

(d)/(dx)\limx→ 0(x⁵-a⁵)/(x-a)\ = (a) a (b) 0 (c) 5a⁴ (d) 5

[1]

Q29.

(d)/(dx)(cot⁻¹x) = (a) (1)/(1+x²) (b) (-1)/(1+x²) (c) (1)/(x) (d) (-1)/(x)

[1]

Page 3 of 16

Q30.

The equation of the tangent to the curve y = x² + 4x + 1 at the point x = 3 is (a) x + 10y = 8 (b) 10x + y = 8 (c) 10x - y = 8 (d) x - 10y = 8

[1]

Q31.

If y = √sin x + √sin x + √(sin x + …) then (dy)/(dx) = (a) (1)/(2y-1) (b) (cos x)/(2y-1) (c) (sin x)/(2y-1) (d) (2y-1)/(cos x)

[1]

Q32.

If xⁿ + yⁿ = aⁿ then (dy)/(dx) = (a) -fracxⁿ⁻¹yⁿ⁻¹ (b) fracxⁿ⁻¹yⁿ⁻¹ (c) -fracyⁿ⁻¹xⁿ⁻¹ (d) nxⁿ⁻¹

[1]

Q33.

If x = a(1 - cosθ),y = a(θ + sinθ), then (dy)/(dx) = (a) tan(θ)/(2) (b) -tan(θ)/(2) (c) cot(θ)/(2) (d) -cot(θ)/(2)

[1]

Q34.

If y = xx then (dy)/(dx) = (a) xx(log x + 1) (b) log x (c) (log x + 1) (d) nxⁿ⁻¹

[1]

Q35.

(d)/(dx)sin⁻¹(3x - 4x³) = (a) frac3√(1-x²) (b) frac-3√(1-x²) (c) frac1√(1-x²) (d) frac-1√(1-x²)

[1]

Q36.

(d)/(dx)(sin 3x · cos 5x) = (a) 4cos 8x (b) 4cos 8x - cos 2x (c) cos 2x (d) cos 2x - 4cos 8x

[1]

Q37.

(d)/(dx)(cos x³) = (a) -3x²sin x³ (b) sin x³ (c) 3x²sin x³ (d) 3x²

[1]

Q38.

If f : R → R, where f(x) = 3x - 5, then f⁻¹(x) = (a) (1)/(3)(x + 5) (b) (1)/(3)(x - 5) (c) 3x - 5 (d) none of these

[1]

Q39.

∫(x + 2)dx = (a) (x + 2)³ + k (b) (x²)/(2) + k (c) (x²)/(2) + 2x + k (d) log(x + 2) + k

[1]

Page 4 of 16

Q40.

∫(cos 2x)/((sin x + cos x)²)dx = (a) 2log(sin x + cos x) + k (b) log(sin x + cos x) + k (c) log(sin x - cos x) + k (d) -(1)/(sin x + cos x) + k

[1]

Q41.

∫(1 - cos 2x)/(1 + cos 2x)dx = (a) tan x + x + k (b) tan x - x + k (c) x - tan² x + k (d) tan(x)/(2) + k

[1]

Q42.

∫(dx)/(x - 1) = (a) log|x + 1| + k (b) -log|x + 1| + k (c) log|x - 1| + k (d) log x + k

[1]

Q43.

∫log xdx = (a) (1)/(x) + k (b) xlog x + k (c) xlog x - x + k (d) xlog x + x + k

[1]

Q44.

∫cos√(x)dx = (a) sin√(x) + cos√(x) + k (b) (1)/(2)(√(x)sin√(x) - cos√(x)) + k (c) 2(√(x)sin√(x) + cos√(x)) + k (d) sin√(x) + k

[1]

Q45.

∫ ex(tan⁻¹x + (1)/(1+x²))dx = (a) extan⁻¹x + k (b) ex · (1)/(1+x²) + k (c) ex + k (d) tan⁻¹x + k

[1]

Q46.

[1]

Q47.

∫sin²(x)/(2)dx = (a) (1)/(2)x - (1)/(2)sin x + k (b) (1)/(2)x - (1)/(2)cos x + k (c) (1)/(2)sin x + k (d) -(1)/(2)sin x + k

[1]

Q48.

[1]

Q49.

∫fracdxex + e-x = (a) cot⁻¹(ex) + k (b) tan⁻¹(ex) + k (c) log|ex + 1| + k (d) sin⁻¹(ex) + k

[1]

Page 5 of 16

Q50.

∫₀¹(dx)/(1 + x²) = (a) (π)/(2) (b) (π)/(3) (c) (π)/(4) (d) π

[1]

Q51.

∫₀¹(x³dx)/(1 + x⁸) = (a) (π)/(2) (b) (π)/(4) (c) (π)/(8) (d) (π)/(16)

[1]

Q52.

∫₁e((log x)²)/(x)dx = (a) (1)/(3) (b) (1)/(3)e³ (c) (1)/(3)(e³ - 1) (d) e³

[1]

Q53.

A = [423] ⇒ A' = (a) [423] (b) 3 \2 \4 (c) [324] (d) 4 \2 \3

[1]

Q54.

∫₀π/4fracetan xcos² xdx = (a) e - 1 (b) e + 1 (c) (1)/(e) + 1 (d) (1)/(e) - 1

[1]

Q55.

∫₀π/2cos² xdx = (a) (π)/(2) (b) π (c) (π)/(4) (d) 1

[1]

Q56.

∫₀π/3cos³ xdx = (a) frac3√(3)8 (b) frac√(3)8 (c) (3)/(8) (d) (1)/(8)

[1]

Q57.

∫₀2π|sin x|dx = (a) 2 (b) 4 (c) 1 (d) 3

[1]

Q58.

∫₀π/2(sin x)/(sin x + cos x)dx = (a) π (b) (π)/(2) (c) 0 (d) (π)/(4)

[1]

Q59.

∫-ππsin⁵ xdx = (a) (3π)/(4) (b) 2π (c) (5π)/(6) (d) 0

[1]

Q60.

∫₀afrac√(x)√(a-x) + √(x)dx = (a) a (b) (a)/(2) (c) 2a (d) 3a

[1]

Page 6 of 16

Q61.

∫₀¹frace√(x)√(x)dx = (a) 2(e - 1) (b) e - 1 (c) 2(e + 1) (d) e + 1

[1]

Q62.

∫₀¹ x exdx = (a) 1 (b) 0 (c) 2 (d) -1

[1]

Q63.

∫₁⁴fracdx√(x) = (a) 1 (b) -2 (c) 2 (d) -1

[1]

Q64.

∫₀a√(a² - x²)dx = (a) (π)/(4) (b) (a²)/(4) (c) (π a²)/(4) (d) π

[1]

Q65.

∫-2²|x|dx = (a) 4 (b) 3 (c) 2 (d) 0

[1]

Q66.

The order and degree of the differential equation ((d² s)/(dt²))² + ((ds)/(dt))³ + 4 = 0 is (a) order = 2, degree = 1 (b) order = 2, degree = 2 (c) order = 1, degree = 2 (d) order = 1, degree = 1

[1]

Q67.

The integrating factor of the differential equation (1 + x²)(dy)/(dx) + y = e^tan⁻¹x is (a) e^tan⁻¹x (b) e^sin⁻¹x (c) tan⁻¹x (d) sin⁻¹x

[1]

Q68.

The solution of differential equation (dy)/(dx) = ex+y is (a) ex + e-y = k (b) ex + ey = k (c) e-x + ey = k (d) e-x + e-y = k

[1]

Q69.

The solution of differential equation x(dy)/(dx) = cot y is (a) xcos y = k (b) xtan y = k (c) xsec y = k (d) xsin y = k

[1]

Q70.

If the operation * is defined as a * b = a + 2b, then (2 * 3) * 4 is (a) 30 (b) 20 (c) 16 (d) 15

[1]

Q71.

If A = 1 -1 \-1 1 , then A³ = (a) 3A (b) 4A (c) 2A (d) None of these

[1]

Page 7 of 16

Q72.

If A = cosα sinα \-sinα cosα and A + A' = I then α = (a) π (b) (π)/(3) (c) (3π)/(2) (d) (π)/(6)

[1]

Q73.

If A = 3 -5 \-1 2 then operatornameadj A = (a) 2 5 \1 3 (b) 2 3 \1 5 (c) 1 3 \2 5 (d) none of these

[1]

Q74.

If A = [1234] and B = 1 \2 \3 \4 then AB = (a) [30] (b) [10] (c) [20] (d) [40]

[1]

Q75.

A = [aᵢⱼ]m × n is a square matrix if (a) m = n (b) m < n (c) m > n (d) none of these

[1]

Q76.

A = 3 -4 \1 -1 ⇒ A + A' = (a) 6 -3 \-3 -2 (b) 6 3 \3 2 (c) 6 3 \-3 -2 (d) 6 3 \-3 2

[1]

Q77.

f : A → B will be an into function if (a) f(A) ⊂ B (b) f(A) = B (c) f(A) supset B (d) none of these

[1]

Q78.

What type of a relation is "less than" in the set of real numbers? (a) Only symmetric (b) Only transitive (c) Only reflexive (d) Equivalence

[1]

Q79.

|3veci + 4vecj + 7veck| = (a) √(14) (b) √(74) (c) √(61) (d) √(94)

[1]

Q80.

veci · (vecj × veck) = (a) 1 (b) 0 (c) -1 (d) veci

[1]

Q81.

veci × veck = (a) 1 (b) veck (c) vecj (d) -vecj

[1]

Page 8 of 16

Q82.

veca · (veca × veca) = (a) 1 (b) 0 (c) veca (d) -1

[1]

Q83.

If 3veci + vecj - 2veck and veci + λvecj - 3veck are perpendicular to each other then the value of λ = (a) -3 (b) -6 (c) -9 (d) -1

[1]

Q84.

The projection of the vector veci + 3vecj + 7veck on the vector 2veci - 3vecj + 6veck is (a) 5 (b) 25 (c) 6 (d) none of these

[1]

Q85.

[1]

Q86.

If veca = veci - vecj + 2veck and vecb = 2veci + 3vecj - 4veck then |veca × vecb| = (a) √(174) (b) √(87) (c) √(93) (d) none of these

[1]

Q87.

[1]

Q88.

2vecj · (-3)veck = (a) 6 (b) -6 (c) 0 (d) -6veci

[1]

Q89.

The direction ratios of a straight line are 2, 6, -3. Then its direction cosines are (a) (1)/(7), (2)/(7), (3)/(7) (b) (2)/(7), (-6)/(7), (3)/(7) (c) (2)/(7), (6)/(7), (-3)/(7) (d) none of these

[1]

Q90.

If a line makes angles α, β and γ with the positive directions of x, y and z axes respectively, then (a) cos²α + cos²β + cos²γ = 1 (b) sin²α + sin²β + sin²γ = 4 (c) cos²α + cos²β + cos²γ = 2 (d) sin²α + sin²β + sin²γ = 1

[1]

Q91.

The angle between the straight lines (x-2)/(2) = (y-1)/(7) = (z+3)/(-3) and (x+2)/(-1) = (y-4)/(2) = (z-5)/(4) is (a) (π)/(2) (b) 0 (c) (π)/(6) (d) (π)/(4)

[1]

Page 9 of 16

Q92.

The distance of the plane x + 2y - 2z = 9 from the point (2, 3, -5) is (a) 1 (b) 2 (c) 3 (d) 4

[1]

Q93.

If two planes x - 4y + λ z + 3 = 0 and 2x + 2y + 3z = 5 are perpendicular to each other then λ = (a) 1 (b) 2 (c) 3 (d) 4

[1]

Q94.

If the line (x-x₁)/(a₁) = (y-y₁)/(b₁) = (z-z₁)/(c₁) is parallel to the plane a₂ x + b₂ y + c₂ z + d = 0 then (a) (a₁)/(a₂) = (b₁)/(b₂) = (c₁)/(c₂) (b) a₁ x + b₁ y + c₁ z + d = 0 (c) a₁ a₂ + b₁ b₂ + c₁ c₂ = 0 (d) none of these

[1]

Q95.

The equation of the plane whose intercepts on the x, y and z-axes are -2, 3 and 4 respectively, will be (a) 6x - 4y - 3z + 12 = 0 (b) 6x + 4y + 3z + 12 = 0 (c) 6x - 4y - 3z = 0 (d) none of these

[1]

Q96.

If the direction cosines of two straight lines are l₁, m₁, n₁ and l₂, m₂, n₂ then the cosine of the angle between the lines will be (a) (l₁ + m₁ + n₁)(l₂ + m₂ + n₂) (b) (l₁)/(l₂) + (m₁)/(m₂) + (n₁)/(n₂) (c) l₁ l₂ + m₁ m₂ + n₁ n₂ (d) none of these

[1]

Q97.

If operation * is defined as a * b = a³ + b³, then 4 * (1 * 2) = (a) 729 (b) 793 (c) 783 (d) 792

[1]

Q98.

If n(A) = 4 and n(B) = 2, then n(A × B) = (a) 6 (b) 8 (c) 16 (d) none of these

[1]

Q99.

a+ib c+id \-c+id a-ib = (a) a² + b² + c² + d² (b) a² - b² - c² - d² (c) a² - b² + c² + d² (d) a² + b² + c² - d²

[1]

Q100.

The maximum value of Z = 4x + y subject to the constraints x + y ≤ 50, x ≥ 0, y ≥ 0 is (a) 50 (b) 250 (c) 0 (d) none of these

[1]

Page 10 of 16

Section B

Q1.

Prove that tan⁻¹1 + tan⁻¹2 + tan⁻¹3 = π.

[2]

Q2.

Find the value of 2tan⁻¹(1)/(3) + tan⁻¹(1)/(7).

[2]

Q3.

Prove that sec²(tan⁻¹2) + csc²(cot⁻¹3) = 15.

[2]

Q4.

Prove that the value of the determinant 102 18 36 \1 3 4 \17 3 6 = 0.

[2]

Q5.

Prove that 0 a b c 0 b c a 0 = 2abc.

[2]

Q6.

Find the value of x, when 2x 8 \4 x = 0.

[2]

Q7.

Find the slope at the point (1, √(2)) of the curve x² + y² = 3.

[2]

Q8.

Find (dy)/(dx) when x = at²,y = 2at.

[2]

Page 11 of 16

Q9.

If y = sec\tan(√(x))\ then find (dy)/(dx).

[2]

Q10.

Find (dy)/(dx) when xsin y = ysin x.

[2]

Q11.

Find the inverse matrix of the following matrix: 4 1 \2 3 .

[2]

Q12.

If matrix A = 1 2 \-3 4 , B = 0 7 \2 -4 then find the value of (4A - B).

[2]

Q13.

Integrate: ∫(dx)/(1 + cos x).

[2]

Q14.

Integrate: ∫sec⁴ xdx.

[2]

Q15.

Integrate: ∫fracsin⁻¹x√(1 - x²)dx.

[2]

Q16.

Integrate: ∫(dx)/(x² + 6x + 13).

[2]

Q17.

Find the value of ∫₀afracxdx√(a² + x²).

[2]

Page 12 of 16

Q18.

Solve: loge((dy)/(dx)) = ax + by.

[2]

Q19.

Solve: ∫₁²(log x)/(x)dx.

[2]

Q20.

Find the value of ∫₀π/2frac√(cos x)√(cos x) + √(sin x)dx.

[2]

Q21.

Solve: (dy)/(dx) + 2ytan x = sin x.

[2]

Q22.

If veca + vecb + vecc = vec0, then prove that veca × vecb = vecb × vecc = vecc × veca.

[2]

Q23.

Find the area of the parallelogram whose adjacent sides are vectors veci + 2vecj + 3veck and -3veci - 2vecj + veck.

[2]

Q24.

Find the sine of the angle between the two vectors 3veci + vecj + 2veck and 2veci - 2vecj + 4veck.

[2]

Q25.

If A and B are two events such that P(A) = (1)/(4), P(B) = (1)/(3) and P(A ∪ B) = (1)/(2) then prove that A and B are independent events.

[2]

Q26.

Two dice are thrown. Find the probability that the numbers appeared have the sum 8 if the second die always exhibits 4.

[2]

Page 13 of 16

Q27.

Maximize Z = 3x + 2y subject to 3x + y ≤ 15, x ≥ 0, y ≥ 0.

[2]

Q28.

Find the distance between the planes 2x - y + 3z + 4 = 0 and 6x - 3y + 9z - 3 = 0.

[2]

Q29.

Find the value of p so that the straight lines (x-1)/(-1) = (y+3)/(p) = (z-7)/(2) and (x+1)/(p) = (y)/(2) = (z+6)/(-3) are perpendicular to each other.

[2]

Q30.

If f : R → R; f(x) = (3 - x³)1/3, then prove that (f ° f)(x) = x.

[2]

Section C

Q1.

Prove that x y z x² y² z² yz zx xy = (x - y)(y - z)(z - x)(xy + yz + zx).

[5]

Q2.

If sin⁻¹x + sin⁻¹y + sin⁻¹z = π then prove that x√(1 - x²) + y√(1 - y²) + z√(1 - z²) = 2xyz.

[5]

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Q3.

Solve: (x² - y²)(dy)/(dx) = 2xy.

[5]

Q4.

Prove that ∫₀π/2(√(tan x) + √(cot x))dx = π√(2).

[5]

Q5.

Find (dy)/(dx), when xy + yx = 1.

[5]

Q6.

Prove that |veca × vecb|² + (veca · vecb)² = a² b².

[5]

Q7.

Maximize Z = x + y subject to x - y ≤ -1, -x + y ≤ 0, x ≥ 0, y ≥ 0.

[5]

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Q8.

10 coins are tossed. What is the probability that exactly 5 heads appear?

[5]

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