Maharashtra Education Society’s

Jr. Colleges / Higher Secondary Schools

Practice Paper 1
Subject –Mathematics and Statistics
Std. -XII Date - /01/2022
Marks - 80 Time – 𝟑 𝐡𝐨𝐮𝐫𝐬

𝑮𝒆𝒏𝒆𝒓𝒂𝒍 𝒊𝒏𝒔𝒕𝒓𝒖𝒄𝒕𝒊𝒐𝒏𝒔:
𝑇ℎ𝑒 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛 𝑝𝑎𝑝𝑒𝑟 𝑖𝑠 𝑑𝑖𝑣𝑖𝑑𝑒𝑑 𝑖𝑛𝑡𝑜 𝐹𝑂𝑈𝑅 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑠.
1) 𝑺𝒆𝒄𝒕𝒊𝒐𝒏 𝑨: 𝑄: 1 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 𝑬𝒊𝒈𝒉𝒕 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑐ℎ𝑜𝑖𝑐𝑒 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛𝑠 𝑐𝑎𝑟𝑟𝑦𝑖𝑛𝑔 𝑻𝒘𝒐 𝑚𝑎𝑟𝑘𝑠 𝑒𝑎𝑐ℎ.
𝑄: 2 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 𝑭𝒐𝒖𝒓 𝑣𝑒𝑟𝑦 𝑠ℎ𝑜𝑟𝑡 𝑎𝑛𝑠𝑤𝑒𝑟 𝑡𝑦𝑝𝑒 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛𝑠 𝑐𝑎𝑟𝑟𝑦𝑖𝑛𝑔 𝑶𝒏𝒆 𝑚𝑎𝑟𝑘 𝑒𝑎𝑐ℎ.
2) 𝑺𝒆𝒄𝒕𝒊𝒐𝒏 𝑩: 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 𝑻𝒘𝒆𝒍𝒗𝒆 𝑠ℎ𝑜𝑟𝑡 𝑎𝑛𝑠𝑤𝑒𝑟 𝑡𝑦𝑝𝑒 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛𝑠 𝑐𝑎𝑟𝑟𝑦𝑖𝑛𝑔 𝑻𝒘𝒐 𝑚𝑎𝑟𝑘 𝑒𝑎𝑐ℎ.
(𝐴𝑡𝑡𝑒𝑚𝑝𝑡 𝑎𝑛𝑦 𝑬𝒊𝒈𝒉𝒕)
3) 𝑺𝒆𝒄𝒕𝒊𝒐𝒏 𝑪: 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 𝑻𝒘𝒆𝒍𝒗𝒆 𝑠ℎ𝑜𝑟𝑡 𝑎𝑛𝑠𝑤𝑒𝑟 𝑡𝑦𝑝𝑒 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛𝑠 𝑐𝑎𝑟𝑟𝑦𝑖𝑛𝑔 𝑻𝒉𝒓𝒆𝒆 𝑚𝑎𝑟𝑘 𝑒𝑎𝑐ℎ.
(𝐴𝑡𝑡𝑒𝑚𝑝𝑡 𝑎𝑛𝑦 𝑬𝒊𝒈𝒉𝒕)
4) 𝑺𝒆𝒄𝒕𝒊𝒐𝒏 𝑫: 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 𝑬𝒊𝒈𝒉𝒕 𝑙𝑜𝑛𝑔 𝑎𝑛𝑠𝑤𝑒𝑟 𝑡𝑦𝑝𝑒 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛𝑠 𝑐𝑎𝑟𝑟𝑦𝑖𝑛𝑔 𝑭𝒐𝒖𝒓 𝑚𝑎𝑟𝑘 𝑒𝑎𝑐ℎ.
(𝐴𝑡𝑡𝑒𝑚𝑝𝑡 𝑎𝑛𝑦 𝑭𝒊𝒗𝒆)
5) 𝑈𝑠𝑒 𝑜𝑓 𝑙𝑜𝑔𝑎𝑟𝑖𝑡ℎ𝑚𝑖𝑐 𝑡𝑎𝑏𝑙𝑒 𝑖𝑠 𝑎𝑙𝑙𝑜𝑤𝑒𝑑. 𝑈𝑠𝑒 𝑜𝑓 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑜𝑟 𝑖𝑠 𝒏𝒐𝒕 𝑎𝑙𝑙𝑜𝑤𝑒𝑑.
6) 𝐹𝑖𝑔𝑢𝑟𝑒𝑠 𝑡𝑜 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ𝑡 𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑒 𝑓𝑢𝑙𝑙 𝑚𝑎𝑟𝑘𝑠.
7) 𝑈𝑠𝑒 𝑜𝑓 𝑔𝑟𝑎𝑝ℎ 𝑝𝑎𝑝𝑒𝑟 𝑖𝑠 𝑛𝑜𝑡 𝑛𝑒𝑐𝑒𝑠𝑠𝑎𝑟𝑦. 𝑂𝑛𝑙𝑦 𝑟𝑜𝑢𝑔ℎ 𝑠𝑘𝑒𝑡𝑐ℎ 𝑜𝑓 𝑔𝑟𝑎𝑝ℎ 𝑖𝑠 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑.
8) 𝐹𝑜𝑟 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑐ℎ𝑜𝑖𝑐𝑒 𝑡𝑦𝑝𝑒 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛, 𝑜𝑛𝑙𝑦 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑎𝑡𝑡𝑒𝑚𝑝𝑡 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑐𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑒𝑑 𝑓𝑜𝑟 𝑒𝑣𝑎𝑙𝑢𝑎𝑡𝑖𝑜𝑛.
9) 𝑆𝑡𝑎𝑟𝑡 𝑎𝑛𝑠𝑤𝑒𝑟 𝑡𝑜 𝑒𝑎𝑐ℎ 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑛 𝑎 𝑛𝑒𝑤 𝑝𝑎𝑔𝑒.

𝐒𝐄𝐂𝐓𝐈𝐎𝐍 − 𝐀
𝐐. 𝐍𝐨. 𝟏: 𝐒𝐞𝐥𝐞𝐜𝐭 𝐚𝐧𝐝 𝐰𝐫𝐢𝐭𝐞 𝐭𝐡𝐞 𝐦𝐨𝐬𝐭 𝐚𝐩𝐩𝐫𝐨𝐩𝐫𝐢𝐚𝐭𝐞 𝐚𝐧𝐬𝐰𝐞𝐫 𝐟𝐫𝐨𝐦 𝐭𝐡𝐞 𝐠𝐢𝐯𝐞𝐧
𝐚𝐥𝐭𝐞𝐫𝐧𝐚𝐭𝐢𝐯𝐞𝐬 𝐟𝐨𝐫 𝐞𝐚𝐜𝐡 𝐬𝐮𝐛-𝐪𝐮𝐞𝐬𝐭𝐢𝐨𝐧: (𝟐 𝐦𝐚𝐫𝐤𝐬 𝐞𝐚𝐜𝐡)[𝟏𝟔]
(𝐢) The compound statement~pΛq is equivalent to … … ….

a)p → q b)q → p c) ~(p → q ) d) ~(q → p )

(𝐢𝐢)The principal solutions of the equation sinθ = √3cosθ are … …
𝜋 7π π 5π π 5π π 4π
a) , b), c) , d) ,
6 6 3 3 6 6 3 3
𝑥+1 y−m z−4
(𝐢𝐢𝐢) If the line = = lies in the plane
2 3 6
3𝑥 − 14y + 6z + 49 = 0 then value of m is …

a) − 5 b)5 c) 2 d) 3
2 −3
(𝐢𝐯) If A = [ ] then determinant of adj A = ⋯
3 4
a) − 1 b)1 c) − 17 d)1

𝑷𝒂𝒈𝒆 𝑵𝒐. 1 |𝟒
 dy
(𝐯) If y = sin−1 (√𝑥) + cos −1 (√𝑥) , then = − − −.
d𝑥
π
(a) (b) 0 (c) 1 (d) − 1
2
(𝐯𝐢) The equation of tangent to the curve y 2 = 2𝑥 & parallel to the line
𝑥 + y + 1 = 0 is − − −
(a) 2𝑥 + 2y = 1 (b) 2𝑥 − 2y = −1
(c) 2𝑥 + 2y = −1 (d) 2𝑥 − 2y = 1
(𝐯𝐢𝐢) The area between the parabola y 2 = 4𝑥 and the line y = 2𝑥
is − − − − sq. units.
2 1 1 3
(a) (b) (c) (d)
3 3 4 4
5
2
dy d2 y
(𝐯𝐢𝐢𝐢) Order & degree of a differential equation 𝑥 ( ) − 𝑥 ( 2 ) = 0
d𝑥 d𝑥
are respectively − − − −.
(a) 2,5 (b) 2,1 (c) 5,2 (d) 5,1
𝐐. 𝐍𝐨. 𝟐: 𝐀𝐧𝐬𝐰𝐞𝐫 𝐭𝐡𝐞 𝐟𝐨𝐥𝐥𝐨𝐰𝐢𝐧𝐠 𝐪𝐮𝐞𝐬𝐭𝐢𝐨𝐧: (𝟏 𝐦𝐚𝐫𝐤𝐬 𝐞𝐚𝐜𝐡)[𝟎𝟒]
(𝐢) Find vector equation of line through (−1, −1,2)and parallel to vector 2î + ĵ − 3k̂.
(𝐢𝐢) If equation a𝑥 2 + 2h𝑥y + by 2 = 0 represent pair of lines passing through origin,

then write condition for coincident lines.

(𝐢𝐢𝐢) Differentiate y = log (𝑥 5 + 4) w. r. t. 𝑥.
100

(𝐢𝐯) Evaluate ∫ 𝑥 101 d𝑥.

−100
𝐒𝐄𝐂𝐓𝐈𝐎𝐍 − 𝐁
𝐀𝐭𝐭𝐞𝐦𝐩𝐭 𝐚𝐧𝐲 𝐄𝐢𝐠𝐡𝐭 𝐨𝐟 𝐭𝐡𝐞 𝐟𝐨𝐥𝐥𝐨𝐰𝐢𝐧𝐠 𝐪𝐮𝐞𝐬𝐭𝐢𝐨𝐧: (𝟐 𝐦𝐚𝐫𝐤𝐬 𝐞𝐚𝐜𝐡)[𝟏𝟔]
𝐐. 𝐍𝐨. 𝟑: Show that( pvq)Λ(~p → q) is tautology.
𝐐. 𝐍𝐨. 𝟒: Find k, if slopes of the two lines given by 3𝑥 2 + k𝑥y − y 2 = 0 differ by 4.
𝐐. 𝐍𝐨. 𝟓: In ∆ABC, if a = 7, b = 5, c = 8, then find cosA.
𝐐. 𝐍𝐨. 𝟔: Show that the three points A(1, −2,3), B(2,3, −4) and C(0, −7,10) are
collinear.
𝐐. 𝐍𝐨. 𝟕: Find cartesian equation of line passing through A(1,2,3) and B(2,3,4).
𝐐. 𝐍𝐨. 𝟖: Find vector equation of plane passing through the point having position

vector î + ĵ + k̂ and perpendicular to the vector 4î + 5ĵ + 6k̂.

dy −1
𝐐. 𝐍𝐨. 𝟗: Find , if y = (𝑥)tan 𝑥 .
d𝑥
π
𝐐. 𝐍𝐨. 𝟏𝟎: Find the area of the region bounded by y = sin𝑥, 𝑥 = 0, 𝑥 = and 𝑋 axis.
2
𝐐. 𝐍𝐨. 𝟏𝟏: Form the differential equation of family of lines having intercepts a and b
on the coordinate axes respectively.

𝑷𝒂𝒈𝒆 𝑵𝒐. 2 |𝟒
 𝐐. 𝐍𝐨. 𝟏𝟐: The probability distribution of 𝑋 is as follows,
𝑋 0 1 2 3 4
P(𝑋) 0.1 k 2k 2k k
Find (i) k (ii) P(𝑋 ≥ 3).
b c b

𝐐. 𝐍𝐨. 𝟏𝟑: Prove that ∫ f(𝑥) d𝑥 = ∫ f(𝑥) d𝑥 + ∫ f(𝑥) d𝑥, a < c < b.
a a c
𝐐. 𝐍𝐨. 𝟏𝟒: let 𝑋~B(n, p). Given that E(𝑋) = 6, Var(𝑋) = 4.2, then find n and p.

𝐒𝐄𝐂𝐓𝐈𝐎𝐍 − 𝐂
𝐀𝐭𝐭𝐞𝐦𝐩𝐭 𝐚𝐧𝐲 𝐅𝐨𝐮𝐫 𝐨𝐟 𝐭𝐡𝐞 𝐟𝐨𝐥𝐥𝐨𝐰𝐢𝐧𝐠 𝐪𝐮𝐞𝐬𝐭𝐢𝐨𝐧: (𝟑 𝐦𝐚𝐫𝐤𝐬 𝐞𝐚𝐜𝐡)[𝟐𝟒]
π
𝐐. 𝐍𝐨. 𝟏𝟓: If acute angle between lines given by a𝑥 2 + 2h𝑥y + by 2 = 0 is ,
3
then show that (a + 3b)(3a + b) = 4h2 .
𝑥+y
𝐐. 𝐍𝐨. 𝟏𝟔: Prove that tan−1 𝑥 + tan−1 y = tan−1 ( ) , if 𝑥y < 1, 𝑥 > 0, y > 0.
1 − 𝑥y
𝐐. 𝐍𝐨. 𝟏𝟕: A(a̅)and B(b̅)be any two points in the space, R(r̅) is a point on segment AB,

mb̅ + na̅
dividing it internally in the ratio m: m, then show that r̅ =
.
m+n
𝑥−1 y−2 z−3
𝐐. 𝐍𝐨. 𝟏𝟖: Find shortest distance between the lines = = and
2 3 4
𝑥−2 y−4 z−5
= = .
3 4 5
𝐐. 𝐍𝐨. 𝟏𝟗: Direction ratios of two lines satisfy the relation 2a − b + 2c = 0 and

ab + bc + ca = 0, then show that lines are perpendicular to each other.

𝐐. 𝐍𝐨. 𝟐𝟎: Prove cosine rule using projection rule.
𝑥
𝐐. 𝐍𝐨. 𝟐𝟏: Find the values of 𝑥 for which the function f(𝑥) = 2 is
𝑥 +1
(i) strictly increasing (ii)strictly decreasing.
1
𝐐. 𝐍𝐨. 𝟐𝟐: Evaluate ∫ d𝑥.
3 − 2sin𝑥 + 5cos𝑥
du
𝐐. 𝐍𝐨. 𝟐𝟑: Prove that ∫(u. v) d𝑥 = u ∫ v d𝑥 − ∫ . (∫ v d𝑥) d𝑥.
d𝑥
Hence find ∫ log𝑥 d𝑥.
3π
10
sin𝑥
𝐐. 𝐍𝐨. 𝟐𝟒: Evaluate ∫ d𝑥.
sin𝑥 + cos𝑥
π
5

𝑷𝒂𝒈𝒆 𝑵𝒐. 3 |𝟒
 𝐐. 𝐍𝐨. 𝟐𝟓: Find expected value and variance of 𝑋, where 𝑋 is number obtained on
uppermost face when a fair die is thrown.
𝐐. 𝐍𝐨. 𝟐𝟔: The probability that a bulb produced by a factory will fuse after
150 days of use is 0.05. Find the probability that out of 5 such bulbs
(i) none (ii) more than one will fuse after 150 days of use.

𝐒𝐄𝐂𝐓𝐈𝐎𝐍 − 𝐃
𝐀𝐭𝐭𝐞𝐦𝐩𝐭 𝐚𝐧𝐲 𝐄𝐢𝐠𝐡𝐭 𝐨𝐟 𝐭𝐡𝐞 𝐟𝐨𝐥𝐥𝐨𝐰𝐢𝐧𝐠 𝐪𝐮𝐞𝐬𝐭𝐢𝐨𝐧: (𝟒 𝐦𝐚𝐫𝐤𝐬 𝐞𝐚𝐜𝐡)[𝟐𝟎]
𝐐. 𝐍𝐨. 𝟐𝟕: a̅ and b̅ are two non − collinear vectors. A vectorr̅ is coplaner with a̅ and b̅

if and only if there exist unique scalars t1 and t 2 such that r̅ = t1 a̅ + t 2 b̅.
𝐐. 𝐍𝐨. 𝟐𝟖: Show that ~(p ↔ q) ≡ (pΛ~q) ∨ (qΛ~p), using truth table.
2 −1 1
−1
𝐐. 𝐍𝐨. 𝟐𝟗: Find A , using adjoint method, where A = [−1 2 −1].
1 −1 2
𝐐. 𝐍𝐨. 𝟑𝟎: Minimise Z = 6𝑥 + 21y, subject to 𝑥 + 2y ≥ 3, 𝑥 + 4y ≥ 4, 3𝑥 + y ≥ 3,

𝑥 ≥ 0, y ≥ 0.
𝐐. 𝐍𝐨. 𝟑𝟏: If d𝑥 = f(t) and y = g(t) are differentiable functions of t,
dy
dy d𝑥
then y is a differentiable function of d𝑥 & = dt , ≠ 0.
d𝑥 d𝑥 dt
dt
dy
Hence find , if 𝑥 = cos(logt) , y = log(cost) .
d𝑥

𝐐. 𝐍𝐨. 𝟑𝟐: A box with a square base is to be have an open top. The surface area of the
box is 192 sq. cm. What should be its dimensions in order that the volume
is largest.
𝐐. 𝐍𝐨. 𝟑𝟑: Evaluate
2𝑥
(i) ∫ d𝑥
4 − 3𝑥 − 𝑥 2
𝑥 sec 2 (𝑥 2 )
(ii) ∫ d𝑥
√tan3 (𝑥 2 )
𝐐. 𝐍𝐨. 𝟑𝟒: Solve 𝑥 2 yd𝑥 − (𝑥 3 + y 3 )dy = 0.

**************

𝑷𝒂𝒈𝒆 𝑵𝒐. 4 |𝟒