CBSE Board 2020
Mathematics
Question Paper Solution
Section B
2 Marks Questions
Question 21: A teacher asked 10 of his students to write a polynomial in one variable on a paper and then to handover the paper. The following were the answers given by the students.
`2x+3`, `3x^2+7x+2`, `4x^3+3x^2+2`, `x^3sqrt(3x)+7`, `7x+sqrt7`, `5x^3-7x+2`, `2x^2+3-5/x`, `5x-1/2`, `ax^3+bax^2+cx+d`, `x+1/x`
Answer the following questions:
- How many of the above ten, are not polynomials?
- How many of the above ten are quadratic polynomials?
Answer: (i) 3, (ii) 1
Question 22: A child has a die whose six faces show the letters as shown here: A, B, C, D, E, A
The die is thrown once. What is the probability of getting (i) A, (ii) D?
Answer: (i) `1/3` (ii) `1/6`
Question 23: In following figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that
`text(arΔABC)/text(areΔDBC)-(AO)/(DO)`
Answer: Let us draw altitudes AM and DN on BC; respectively from A and D
`text(ar ABC)/text(ar DBC)=(1/2xxBCxxAM)/(1/2xxBCxxDN)`
`=(AM)/(DN)`
In ΔAMO and ΔDNO;
∠ AMO = ∠ DNO (Right angle)
∠ AOM = ∠ DON (Opposite angles)
Hence; ΔAMO ∼ ΔDNO
Hence;
`(AM)/(DN)=(AO)/(DO)`
Or, `text(ar ABC)/text(ar DBC)=(AO)/(DO)`
OR
In following figure, if AD ⊥ BC, then prove that AB2 + CD2 = BD2 + AC2
Answer: In Δ ADB
AD2 = AB2 - BD2
In Δ ADC
AD2 = AC2 - CD2
So, AB2 - BD2 = AC2 - CD2
Or, AB2 - BD2 + CD2 = AC2
Or, AB2 + CD2 = BD2 + AC2 proved
Question 24: Prove that `1+(text(cot)^2α)/(1+text(cosecα))=text(cosecα)`
Answer: LHS =`1+(text(cot)^2α)/(1+text(cosecα))`
`=1+(text(cosecα+1)text(cosecα-1))/(1+text(cosecα))`
`=1+text(cosecα)-1=text(cosecα)` proved
OR
Show that tan4θ + tan2θ = sec4θ - sec2θ
Answer: RHS = sec4θ - sec2θ
= (1 + tan2θ)2 - (1 + tan2θ)
= 1 + tan4θ + 2 tan2θ - 1 – tan2θ
= tan4θ + tan2θ = LHS proved
Question 25: Find the mode of the following frequency distribution:
Class | 15-20 | 20-25 | 25-30 | 30-35 | 35-40 | 40-45 |
---|---|---|---|---|---|---|
Frequency | 3 | 8 | 9 | 10 | 3 | 2 |
Answer: Here, modal class is 30 – 35, l = 30, h = 5, f1 = 10, f0 = 9, f2 = 3
Mode can be calculated as follows:
`=l+(f_1-f_0)/(2f_1-f_0-f_2)× x`
`=30+(10-9)/(2×10-9-3)× 5`
`=30+1/8×5`
`=(240+5)/8=(245)/8=30.625`
Question 26: From a solid right circular cylinder of height 14 cm and base radius 6 cm, a right circular cone of the same height and same base radius is removed. Find the volume of the remaining solid.
Answer: Volume of cylinder `=πr^2h`
`=(22)/7×6^2×14=1584` cm3
Volume of cone `=1/3×πr^2h`
`=(1584)/3=528` cm3
So, volume of remaining solid `=1584-528=1056` cm3