Extra questions on Determinants notes for Class 12
Multiple Choice Questions
Question 1
If x,y∈R, then the determinant Δ=|cosx−sinx1sinxcosx1cos(x+y)−sin(x+y)0| lies in
(a) [−√2,√2]
(b) [−1,1]
(c) [−√2,1]
(d) [−1,−√2]
Answer
The correct choice is A. Indeed applying R3−>R3−cosyR1+sinyR2, we get Δ=|cosx−sinx1sinxcosx1cos(x+y)−sin(x+y)0| Δ=|cosx−sinx1sinxcosx100siny−cosy|
Now expanding Δ=siny−cosy=√2sin(y−π4)
Hence Δ lies in [−√2,√2]
Question 2
If A,B,C are the angles of the triangle, then the determinant
|−1cosCcosBcosC−1CosAcosBcosA−1| is equal to
(a) 1
(b) -1
(c) 0
(d) Nones of these
Answer
Doing C1−>aC1+bC2+cC3 |−a+bcosC+ccosBcosCcosBacosC−b+ccosA−1CosAacosB+bcosA−ccosA−1|
Now we know that in a triangle
a=bcos C + c cos A
b= acos C + c cos A
c= aCos B + b cos A
hence |−a+acosCcosBb−b−1CosAc−ccosA−1| |0cosCcosB0−1CosA0cosA−1|=0
Question 3
if f(x)=|(1+x)17(1+x)19(1+x)23(1+x)23(1+x)29(1+x)34(1+x)41(1+x)43(1+x)47|=A+Bx+Cx2+...
(a)0
(b)1
(c)-1
(d)Nones of these