Perfect solution to all problems Tips, Tricks, General Knowledge, Current Affairs, Latest Sample, Previous Year, Practice Papers with solutions. CBSE 12th Mathematics 2016 Solved Paper Outside Delhi Downloaded From www.4ono.com DEMO VERSION Note This pdf file is downloaded from www.4ono.com. Editing the content or publicizing this on any blog or website without the written permission of Rewire Media is punishable, the suffering will be decided under DMC
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Perfect solution to all problems
Tips, Tricks, General Knowledge, Current Affairs, Latest Sample, Previous Year, Practice Papers with solutions.
CBSE 12th Mathematics 2016 Solved Paper
Outside Delhi
Downloaded From www.4ono.com
DEMO VERSION
Note This pdf file is downloaded from www.4ono.com. Editing the content or publicizing this on any blog or
website without the written permission of Rewire Media is punishable, the suffering will be decided under DMC
Q.5. Write the number of vectors of unit length perpendicular to both the vectors of unit
length perpendicular to both the vectors �⃗⃗� = 𝟐�̂� + 𝒋̂ + 𝟐�̂� and �⃗⃗� = 𝒋̂ + �̂�. 1 mark
Ans. 1.
Q.6. Find the vector equation of the plane with intercepts 𝟑,−𝟒 and 𝟐 on 𝒙, 𝒚 and 𝒛 − 𝐚𝐱𝐢𝐬 respectively. 1 mark
Ans.
𝑥
3+
𝑦
−4+
𝑧
2= 1
⇒ 𝑥
3+
𝑦
4+
𝑧
2= 1 be the eq. of plane.
SECTION – B
Question numbers 7 to 19 carry 4 marks each.
Q.7. Find the coordinates of the point where the line through the points A (3, 4, 1) and B (5, 1, 6) crosses the XZ plane. Also find the angle which this line makes with the XZ plane. 4 marks
Ans. Eq. of line through A (3, 4, 1) and B (5, 1, 6) be:
𝑥 − 3
5 − 3=
𝑦 − 4
1 − 4=
𝑧 − 1
6 − 1
⇒𝑥 − 3
2=
𝑦 − 4
−3=
𝑧 − 1
5
Let the point of intersection of line and 𝑥𝑧 plane be (𝑥0, 𝑦0, 𝑧0) 𝑖. 𝑒. it lie online
Direct of line AB is (2, −3, 5) and Direction of plane 𝑥𝑧 is (1, 0, 1)
Let angle between line and plane is 𝜃 𝑖. 𝑒,. angle is sin 𝜃
= (2(1) + (−3)0 + 5(1)
√22 + (−3)2 + (5)2. √12 + 02 + 12)
= (2 + 5
√38. √2) = (
7
2√19)
𝜃 = sin−1 (7
2√19) .
Q.8. The two adjacent sides of parallelogram are 𝟐�̂� − 𝟒𝒋̂ − 𝟓�̂� and 𝟐�̂� + 𝟐𝒋̂ + 𝟑�̂�. Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram. 4 marks
Ans. Let OABC be a parallelogram with side 𝑂𝐴⃗⃗⃗⃗ ⃗ = 𝑎 = 2𝑖̂ − 4𝑗̂ − 5�̂� and
𝑖. 𝑒., unit vector along diagonal be 𝑂�̂� and 𝐶�̂�.
Now area of parallelogram be
=1
2|𝑂𝐵⃗⃗ ⃗⃗ ⃗×𝐶𝐴⃗⃗⃗⃗ ⃗|
𝑂𝐵⃗⃗ ⃗⃗ ⃗×𝐶𝐴⃗⃗⃗⃗ ⃗ = |𝑖̂ 𝑗̂ �̂�4 −2 −20 −6 −8
|
= 𝑖̂(16 − 12) − 𝑗̂(−32) + �̂�(−24)
= 4𝑖̂ + 32𝑗̂ − 24�̂�
|𝑂𝐵⃗⃗ ⃗⃗ ⃗×𝐶𝐴⃗⃗⃗⃗ ⃗| = √42 + (32)2 + (24)2
= √16 + 1024 + 576
= √1616 = 4√101
Area of parallelogram be = 1
2 (4√101)
= 2√101 𝑠𝑞. 𝑢𝑛𝑖𝑡.
Q.9. In a game, a man wins Rs 5 for getting a number greater than 4 and loses Rs 1 otherwise, when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a number greater than 4. Find the expected value of the amount he win/lose. 4 marks
Ans. Let 𝑥 denote the amount he win/loss 𝑖. 𝑒. , 𝑥 = 5, 4, 3, −3 win in first thrown
𝑃(𝑥 = 5) = win in first thrown 2/6 = 1/3 = 9/27 𝑃(𝑥 = 4) = win in second thrown
A bag contains 4 balls. Two balls are drawn at random (without replacement) and are found to be white. What is the probability that all balls in the bag are white?
Ans. Let 𝐸1 be event the bag has 4 white balls. 𝐸2 be event the bag has no white balls 𝐸3 be event the bag has 3 white balls 𝐸4 be event to draw 2 balls from balls
A be event to draw 2 balls from balls and are white
𝑃(𝐸1) = 1/4 𝑃(𝐸2) = 1/4 𝑃(𝐸3) = 1/4 𝑃(𝐸4) = 1/4
𝑃(𝐴 𝐸1⁄ ) = 1
𝑃(𝐴 𝐸2⁄ ) = 0
𝑃(𝐴 𝐸3⁄ ) =3
4×
1
3=
1
6
=
14×1
(14×1) (
14×0) (
14×
12) (
14×
16)
=1
1 +12 +
16
=1
6 + 3 + 16
=6
10=
3
5.
Q.10. Differentiate 𝒙𝐬𝐢𝐧𝒙 + (𝐬𝐢𝐧𝒙)𝐜𝐨𝐬 𝒙 with respect to 𝒙. 4 marks
