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Questions
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Answers
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Section A
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1) If A= {0, π/6, π/4,π/3,π/2) and f: A→B is a surjection defined by f(x)=cosx, then find B
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B={1,3/2,1/2,1/2,0}
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2) Find the domain of real valued function f(x)= √x2-25
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(−∞,−5]∪[5,∞)
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3) Find the additive inverse of A, where A=[i 0 1
o -i 2
-1 1 5]
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(-i 0 -1
0 i -2
1 -1 -5)
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4) Find the unit vector in the direction of vector a=2i+3j+k
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(2/√14)i + (3/√14)j + (1/√14)k
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5) Find the vector equation of the line passing through the point 2i+3i and parallel to the vector 4i-2j+3k
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6) If |p|=2,|q|=3 and(p,q)=n/6, then find |pxq|2
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7) Find the value of sin 330. cos 120+cos 210. sin 300
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8) Find the range of 7 cosx-24 sinx+5
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9) For any x€R, prove that cosh4x-sinh4x=cosh(2x)
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cosh⁴x − sinh⁴x = cosh(2x)
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10) If A (o 2 1
-2 0 -1
-1 x 0), find x
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Section B
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11) If [x-1 2 5-y = [1 2 3
0 z-1 z-1 0 4 7
1 0 0] 1 0 0], find values of x,y,z, and a
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12) If A,B,C, and D be four points with position vectors a+2b, 2a-b, a, and 3a+b, respectively, express the vectors AC, DA, BA, and BC in terms of a and b
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AC = −2b
DA = −2a + b
BA = −a + 3b
BC = −a + b
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13) If a+b+c=0, then prove that axb=bxc=cxa
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a+b=−c. Take cross product with a: a×a+a×b=−a×c⇒a×b=c×a. Repeat for b. Hence Proved.
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14) If A+b=π4, then prove that (1+tan) (1+tanB)=2
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tan(A+B)=tan(π/4)=1. Use formula 1−tanAtanBtanA+tanB=1. Cross multiply and add 1 to both sides. Hence proved.
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15) Find the general solution of the equation sin2θ-cosθ=1/4
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θ ≈ 0.73 + 2nπ
θ ≈ 2.41 + 2nπ
(n ∈ ℤ)
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16) Prove that tan-1 3/4+tan-1 3/5-tan-1 8/19= π/4
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Use tan−1x+tan−1y formula twice. First: tan−11−1/101/2+1/5=tan−197. Then: tan−11−7/727/9+1/8=tan−1(1). Hence, proved.
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17) In ΔABC, if a=13, b=14,c=15, then find r1
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