2. In an AP, if a = 3.5, d = 0, n = 101, then an will be
(A) 0
(B) 3.5
(C) 103.5
(D) 104.5
Answer: (B)
Explanation:
For an A.P
an = a + (n – 1)d
= 3.5 + (101 – 1) × 0
= 3.5
3.The first four terms of an AP, whose first term is –2 and the common difference is –2, are
(A) – 2, 0, 2, 4
(B) – 2, 4, – 8, 16
(C) – 2, – 4, – 6, – 8
(D) – 2, – 4, – 8, –16
Answer: (C)
Explanation:
Let the first four terms of an A.P are a, a+d, a+2d and a+3d
Given that the first termis −2 and difference is also −2, then the A.P would be:
– 2, (–2–2), [–2 + 2 (–2)], [–2 + 3(–2)]
= –2, –4, –6, –8
4.The famous mathematician associated with finding the sum of the first 100 natural numbers is
(A) Pythagoras
(B) Newton
(C) Gauss
(D) Euclid
Answer: (C)
Explanation:
Gauss is the famous mathematician associated with finding the sum of the first 100 natural Numbers.
(A) –20
(B) 20
(C) –30
(D) 30
Answer: (B)
Explanation:
6.The 21st term of the AP whose first two terms are –3 and 4 is
(A) 17
(B) 137
(C) 143
(D) –143
Answer: (B)
Explanation:
First two terms are –3 and 4
Therefore,
a = −3
a + d = 4
⇒ d = 4 − a
⇒ d = 4 + 3
⇒ d = 7
Thus,
a21 = a + (21 – 1)d
a21 = –3 + (20)7
a21 = 137
7. If the 2nd term of an AP is 13 and the 5th term is 25, what is its 7th term?
(A) 30
(B) 33
(C) 37
(D) 38
Answer: (B)
Explanation:
Since
a2 = 13
a5 = 25
⇒ a + d = 13 ….(i)
⇒ a + 4d = 25 ….(ii)
Solving equations (i) and (ii), we get:
a = 9; d = 4
Therefore,
a7 = 9 + 6 × 4
a7 = 9 + 24
a7 = 33
8. If the common difference of an AP is 5, then what is a18 – a13?
(A) 5
(B) 20
(C) 25
(D) 30
Answer: (C)
Explanation:
Since, d = 5
a18 – a13 = a + 17d – a – 12d
= 5d
= 5 × 5
= 25
9. The sum of first 16 terms of the AP: 10, 6, 2,… is
(A) –320
(B) 320
(C) –352
(D) –400
Answer: (A)
Given A.P. is 10, 6, 2,…
10.The sum of first five multiples of 3 is
(A) 45
(B) 55
(C) 65
(D) 75
Answer: (A)
Explanation:
The first five multiples of 3 are 3, 6, 9, 12 and 15
11. The middle most term (s) of the AP:–11, –7, –3, …, 49 is:
(A) 18, 20
(B) 19, 23
(C) 17, 21
(D) 23, 25
Answer: (C)
Explanation:
Here, a = −11
d = − 7 – (−11) = 4
And an = 49
We have,
an = a + (n – 1)d
⇒ 49 = −11 + (n – 1)4
⇒ 60 = (n – 1)4
⇒ n = 16
As n is an even number, there will be two middle terms which are16/2th and [(16/2)+1]th, i.e. the 8th term and the 9th term.
a8 = a + 7d = – 11 + 7 × 4 = 17
a9 = a + 8d = – 11 + 8 × 4 = 21
12.Two APs have the same common difference. The first term of one of these is –1 and that of the other is – 8. Then the difference between their 4th terms is
(A) –1
(B) – 8
(C) 7
(D) –9
Answer: (C)
Explanation:
The 4th term of first series is
a4 = a1 + 3d
The 4th term of another series is
a`4 = a2 + 3d
Now,
As, a1 = –1, a2 = –8
Therefore,
a4 – a`4 = (–1 + 3d) – (–8 + 3d)
a4 – a`4 = 7
13.If 7 times the 7th term of an AP is equal to 11 times its 11th term, then its 18th term will be
(A) 7
(B) 11
(C) 18
(D) 0
Answer: (D)
Explanation:
According to question
7(a + 6d) = 11(a + 10d)
⇒ 7a + 42d = 11a + 110d
⇒ 4a + 68d = 0
⇒ 4(a + 17d) = 0
⇒ a + 17d = 0
Therefore,
a18 = a + 17d
a18 = 0
14.In an AP if a = 1, an = 20 and Sn = 399, then n is
(A) 19
(B) 21
(C) 38
(D) 42
Answer: (C)
Explanation:
15. If the numbers n – 2, 4n – 1 and 5n +2 are in AP, then the value of n is:
The roots of quadratic equation 5x2 – 4x + 5 = 0 are
(A) Real & Equal
(B) Real & Unequal
(C) Not real
(D) Non-real and equal
Answer: (C)
Explanation: To find the nature, let us calculate b2 – 4ac
b2 – 4ac = 42 – 4 x 5 x 5
= 16 – 100
= -84 < 0
2. Equation (x+1)2 – x2 = 0 has _____ real root(s).
(A) 1
(B) 2
(C) 3
(D) 4
Answer: (A)
Explanation:
Since (x + 1)2 – x2 = 0
⟹ x2 + 1 + 2x – x2 = 0
⟹ 1 + 2x = 0
⟹ x= -1/2
This gives only 1 real value of x.
3. Which constant should be added and subtracted to solve the quadratic equation 4x2 − √3x + 5 = 0 by the method of completing the square?
(A) 9/16
(B) 3/16
(C) 3/4
(D) √3/4
Answer: (B)
Explanation:
This can be written as
Hence the given equation can be solved by adding and subtracting 3/16.
4. If 1/2 is a root of the equation x2 + kx – (5/4) = 0 then the value of k is
(A) 2
(B) – 2
(C) 3
(D) –3
Answer: (A)
Explanation:
As one root of the equation x2 + kx – (5/4) = 0 is 1/2
5. A natural number, when increased by 12, equals 160 times its reciprocal. Find the number.
(A) 3
(B) 8
(C) 4
(D) 7
Answer: (B)
Explanation:
Let the number be x
Then according question,
x + 12 = 160/x
x2 + 12x – 160 = 0
x2 + 20x – 8x – 160 = 0
(x + 20) (x – 8) = 0
x = -20, 8
Since the number is natural, so we consider only positive value.
6. The product of two successive integral multiples of 5 is 300. Then the numbers are:
(A) 25, 30
(B) 10, 15
(C) 30, 35
(D) 15, 20
Answer: (D)
Explanation:
Let the consecutive integral multiple be 5n and 5(n + 1) where n is a positive integer.
According to the question:
5n × 5(n + 1) = 300
⇒ n2 + n – 12 = 0
⇒ (n – 3) (n + 4) = 0
⇒ n = 3 and n = – 4.
As n is a positive natural number so n = – 4 will be discarded.
Therefore the numbers are 15 and 20.
(A) 3.5
(B) 4
(C) 3
(D) – 3
Answer: (C)
Explanation:
Since y cannot be negative as negative square root is not real so y = 3.
8. If p2x2 – q2 = 0, then x =?
(A) ± q/p
(B) ±p/q
(C) p
(D) q
Answer:(A)
Explanation:
p2x2 – q2 = 0
⇒p2x2 = q2
⇒x = ±p/q
(A) 3
(B)5
(C) 4
(D) 7
Answer:(B)
Explanation:
10. If x2 (a2 + b2) + 2x (ac + bd) + c2 +d2 = 0 has no real roots, then
(A) ad≠bc
(B) ad<bc
(C) ad>bc
(D) all of these
Answer: (D)
Explanation:
If equation has no real roots then discriminant of the equation must be less than zero.
11. If the one root of the equation 4x2 – 2x + p – 4 = 0 be the reciprocal of other. Then value of p is
(A) 8
(B) – 8
(C) – 4
(D) 4
Answer:A
Explanation:
If one root is reciprocal of other, then product of roots is:
12. Rohini had scored 10 more marks in her mathematics test out of 30 marks, 9 times these marks would have been the square of her actual marks. How many marks did she get in the test?
(A) 14
(B) 16
(C) 15
(D) 18
Answer: (C)
Explanation:
Let her actual marks be x
Therefore,
9 (x + 10) = x2
⇒x2 – 9x – 90 = 0
⇒x2 – 15x + 6x – 90 = 0
⇒x(x – 15) + 6 (x – 15) = 0
⇒(x + 6) (x – 15) = 0
Therefore x = – 6 or x =15
Since x is the marks obtained, x ≠ – 6. Therefore, x = 15.
13. A train travels at a certain average speed for a distance of 63 km and then travels a distance of 72 km at an average speed of 6 km/h more than its original speed. If it takes 3 hours to complete the total journey, what is its original average speed?
(A) 42 km/hr
(B) 44 km/hr
(C) 46 km/hr
(D) 48 km/hr
Answer: (A)
Explanation:
Let the original speed be x,
Then according to question
This gives x = -3 and x = 42
Since speed cannot be negative, so we ignore –3,
Therefore original average speed is 42 km/hr.
14. Satvik observed that in a clock, the time needed by the minute hand of a clock to show 3 PM was found to be 3 min less than t2/4 minutes at t minutes past 2 PM. Then t is equal to
(a) 14
(b) 15
(c) 16
(d) None of these
Answer: (A)
Explanation: We know that the time between 2 PM to 3 PM = 1 hr = 60 min
Given that at t minutes past 2 PM, the time needed by the minute’s hand of a clock to show 3 PM was found to be 3 minutes less than t2/4minutes
Therefore,
15. A takes 6 days less than B to finish a piece of work. If both A and B together can finish the work in 4 days, find the time taken by B to finish the work.
(A)12 days
(B) 12 ½ Days
(C) 13 days
(D) 15days
Answer: (A)
Explanation: Let B alone finish the work in x days.
Therefore, A alone can finish the work in (x – 6) days
A’s one day work = 1/x-6
B’s one day work = 1/x
Given that (A + B) can finish the work in 4 days.
Therefore, A’s one day work + B’s one day work = (A + B)’s one day work
As, x ≠ 2 , because if x = 2 , then A alone can finish work in (2 – 6) = – 4 days which is not possible.
Therefore we consider x = 12.
This implies B alone can finish work in 12 days and A alone will finish the work in 12 – 6 = 6 days.
2. The pair of equations x + 2y – 5 = 0 and −3x – 6y + 15 = 0 have:
(A) A unique solution
(B) Exactly two solutions
(C) Infinitely many solutions
(D) No solution
Answer: (C)
Explanation:
Here,
Therefore, the pair of equations has infinitely many solutions.
3. If a pair of linear equations is consistent, then the lines will be:
(A) Parallel
(B) Always coincident
(C) Intersecting or coincident
(D) Always intersecting
Answer: (C)
Explanation: If a pair of linear equations is consistent the two lines represented by these equations definitely have a solution, this implies that either lines are intersecting or coincident.
4.The pair of equations y = 0 and y = –7 has
(A) One solution
(B) Two solutions
(C) Infinitely many solutions
(D) No solution
Answer: (D)
Explanation: The graph of equations will be parallel lines. So the equations have no solution.
5.If the lines given by
3x + 2ky = 2
2x + 5y + 1 = 0
are parallel, then the value of k is
(A) 5/4
(B) 2/5
(C) 15/4
(D) 3/2
Answer: (C)
Explanation:
For parallel lines
6. The value of c for which the pair of equations cx – y = 2 and 6x – 2y = 3 will have infinitely many solutions is
(A) 3
(B) – 3
(C) –12
(D) no value
Answer: (A)
Explanation: For infinitely many solutions:
7. One equation of a pair of dependent linear equations is –5x + 7y – 2 = 0. The second equation can be
(A) 10x + 14y + 4 = 0
(B) –10x – 14y + 4 = 0
(C) –10x + 14y + 4 = 0
(D) 10x – 14y = –4
Answer: (D)
Explanation: For dependent pair, the two lines must have
8.Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the numbers, the ratio becomes 4 : 5. Then the numbers are:
(A) 40, 42
(B) 42, 48
(C) 40, 48
(D) 44, 50
Answer: (C)
Explanation:
According to given information
9. The solution of the equations x – y = 2 and x + y = 4 is:
(A) 3 and 5
(B) 5 and 3
(C) 3 and 1
(D) –1 and –3
Answer: (C)
Explanation: Adding both equations, we have:
10.For which values of a and b, will the following pair of linear equations have infinitely many solutions?
x + 2y = 1
(a – b)x + (a + b)y = a + b – 2
(A) a = 2 and b = 1
(B) a = 2 and b = 2
(C) a = ̶ 3 and b = 1
(D) a = 3 and b = 1
Answer: (D)
Explanation: For infinitely many solutions:
Solving equation (i) and (ii), we get a = 3 and b = 1.
11.The father’s age is six times his son’s age. Four years hence, the age of the father will be four times his son’s age. The present ages, in years, of the son and the father are, respectively
(A) 4 and 24
(B) 5 and 30
(C) 6 and 36
(D) 3 and 24
Answer: (C)
Explanation: Let the age of father be x and of son is y.
Then according to question,
x = 6y …..(i)
Four years hence age of son will be y + 4 and age of father will be x + 4
Then according to question,
x + 4 = 4 (y + 4)
x – 4y = 12 …..(ii)
Solving equations (i) and (ii) we get:
y = 6 and x = 36
12. Rakshita has only Rs. 1 and Rs. 2 coins with her. If the total number of coins that she has is 50 and the amount of money with her is Rs 75, then the number of Rs.1 andRs.2 coins is, respectively
(A) 35 and 15
(B) 35 and 20
(C) 15 and 35
(D) 25 and 25
Answer: (D)
Explanation:
Let her number of Rs.1 coins are x
Let the number of Rs.2 coins are y
Then
By the given conditions
x + y = 50 …..(i)
1 × x + 2 × y = 75
⇒ x + 2y = 75 …..(ii)
Solving equations (i) and (ii) we get:
(x + 2y) – (x + y) = 75 – 50
⇒ y = 25
Therefore, x = 50 – 25 = 25
So the number of coins are 25, 25 each.
13.In a competitive examination, one mark is awarded for each correct answer while 1/2 mark is deducted for every wrong answer. Jayanti answered 120 questions and got 90 marks. How many questions did she answer correctly?
(A) 100
(B) 95
(C) 90
(D) 60
Answer: (A)
Explanation: Let x be the number of correct answers of the questions in a competitive exam.
Then, 120 − x be the number of wrong answers
Then by given condition
14. The angles of a cyclic quadrilateral ABCD are:
Then value of x and y are:
(A) x = 20o and y = 30o
(B) x = 40o and y = 10o
(C) x = 44o and y = 15o
(D) x = 15o and y = 15o
Answer: (A)
Explanation: In cyclic quadrilateral, sum of opposite angles is 1800
Therefore
6x + 10 + x + y = 180
⇒ 7x + y = 170 …..(i)
5x + 3y – 10 = 180
⇒ 5x + 3y = 190 …..(ii)
Multiplying equations (i) and (ii), we get:
x = 20o and y = 30o
15.A shopkeeper gives books on rent for reading. She takes a fixed charge for the first two days, and an additional charge for each day thereafter. Reema paid Rs. 22 for a book kept for six days, while Ruchika paid Rs 16 for the book kept for four days, then the charge for each extra day is:
(A) Rs 5
(B) Rs 4
(C) Rs 3
(D) Rs.2
Answer: (C)
Explanation: Let Rs. x be the fixed charge and Rs. y be the charge for each extra day.
Then by the given conditions
x + 4y = 22 …..(i)
x + 2y = 16 …..(ii)
Subtracting equation (ii) from (i), we get:
y = Rs. 3
Important Link
Quick Revision Notes : Pair of Linear Equations in Two
1.The zeroes of the quadratic polynomial x2 + 99x + 127are
(A) both positive
(B) both negative
(C) one positive and one negative
(D) both equal
Answer: (B)
2.If the zeroes of the quadratic polynomial x2 + bx + c , c ≠ 0are equal, then
(A) c and a have opposite signs
(B) c and b have opposite signs
(C) c and a have the same sign
(D) c and b have the same sign
Answer: (C)
3.The number of polynomials having zeroes as –2 and 5 is
(A) 1
(B) 2
(C) 3
(D) more than 3
Answer: (D)
4. The degree of the polynomial (x + 1)(x2 – x – x4 +1) is:
(A)2
(B) 3
(C) 4
(D) 5
Answer: (D)
Explanation: Since the highest degree variable in first bracket is x and in second bracket is x4 on multiplying x with x4.the highest power we obtain is 5.
5.If the zeroes of the quadratic polynomial x2 + (a + 1)x + b are 2 and –3, then
(A) a = –7, b = –1
(B) a = 5, b = –1
(C) a = 2, b = – 6
(D) a = 0, b = – 6
Answer: (D)
6.Given that one of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is zero, the product of the other two zeroes is
(A) –c/a
(B) c/a
(C) 0
(D) 3
Answer: (B)
7.If one of the zeroes of the cubic polynomial x3 + ax2 + bx + c is –1, then the
Product of the other two zeroes is
(A) b – a + 1
(B) b – a – 1
(C) a – b + 1
(D) a – b –1
Answer: (A)
8. If one of the zeroes of the quadratic polynomial (k – 1)x2 + kx + 1 is –3, then the value of k is
(A) 4/3
(B) – 4/3
(C) 2/3
(D) – 2/3
Answer: (A)
10.The value of p for which the polynomial x3 + 4x2 –px + 8 is exactly divisible by (x – 2) is:
(A) 0
(B) 3
(C) 5
(D) 16
Answer: (D)
11.If sum of the squares of zeroes of the quadratic polynomial 6x2 + x + k is 25/36, the value of k is:
(A) 4
(B) – 4
(C) 2
(D) – 2
Answer: (D)
12.If α and β are zeroes of x2 – 4x + 1, then 1/α + 1/β – αβ is
(A) 3
(B) 5
(C) –5
(D) –3
Answer: (A)
13.If (x + 1) is a factor of x2− 3ax +3a − 7, then the value of a is:
(A) 1
(B) –1
(C) 0
(D) –2
Answer: (A)
14. If one of the zeroes of a quadratic polynomial of the form x2 + ax + b is the negative of the other, then it
(A) has no linear term and the constant term is negative.
(B) has no linear term and the constant term is positive.
(C) can have a linear term but the constant term is negative.
(D) can have a linear term but the constant term is positive
Answer: (A)
15.If α, β are zeroes of x2 –6x + k, what is the value of k if 3α + 2β = 20?
1.O is the point of intersection of two equal chords ABand CD such that OB = OD, then triangles OAC and ODB are
(A) Equilateral but not similar
(B) Isosceles but not similar
(C) Equilateral and similar
(D) Isosceles and similar
Answer: (D)
Explanation:
Since O is the point of intersection of two equal chords AB and CD such that OB = OD,
As chords are equal and OB = OD, so AO will also be equal to OC
Also ∠AOC = ∠DOB = 450
Now in triangles OAC and ODB
AO/OB = CO/OD
And ∠AOC = ∠DOB = 450
So triangles are isosceles and similar
2. D and E are respectively the midpoints on the sides AB and AC of a triangle ABC and BC = 6 cm. If DE || BC, then the length of DE (in cm) is
(A) 2.5
(B) 3
(C) 5
(D) 6
Answer: B
Explanation:
By midpoint theorem,
If D and E are respectively the midpoints on the sides AB and AC of a triangle ABC, DE||BC and BC = 6 cm
So, DE will be half of BC i.e. 3cm
3. In triangle PQR, if PQ = 6 cm, PR = 8 cm, QS = 3 cm, and PS is the bisector of angle QPR, what is the length of SR?
(A) 2
(B) 4
(C) 6
(D) 8
Answer: (B)
Explanation:
Since, PS is the angle bisector of angle QPR
So, by angle bisector theorem,
QS/SR = PQ/PR
⇒ 3/SR = 6/8
⇒ SR = (3 X 8)/6 cm = 4 cm
4.The lengths of the diagonals of a rhombus are 16 cm and 12cm. Then, thelength of the side of the rhombus is
(A) 9 cm
(B) 10 cm
(C) 8 cm
(D) 20 cm
Answer:(B)
Explanation:
The diagonals of rhombus bisect each other at right angle, so side of rhombus is the hypotenuse for the triangles formed.
Therefore,
By Pythagoras theorem
(16/2)2 + (12/2)2 = Side2
⇒ 82 + 62 = Side2
⇒ 64 + 36 = Side2
⇒ Side = 10 cm
5. A flag pole 18 m high casts a shadow 9.6 m long. Find the distance of the top of the pole from the far end of the shadow.
(A) 25.6
(B) 20.4
(C) 23.7
(D) 32.5
Answer:(B)
Explanation:
According to given question
The far end of shadow is represented by point A,
Therefore we need to Find AC
By Pythagoras theorem,
(18)2 + (9.6)2 = (AC)2
⇒ AC2 = 416.16
⇒ AC = 20.4 m (approx)
6. Diagonals of a trapezium PQRS intersect each other at the point O, PQ || RS and PQ = 3 RS, Then the ratio of areas of triangles POQ and ROS is:
(A) 1:9
(B) 9:1
(C) 3:1
(D) 1:3
Answer:(B)
Explanation:According to given Question
Since
SR || PQ,
So, ∠OSR= ∠OQP (alternate interior angles)
Also ∠SOR= ∠POQ (vertically opposite angles)
So triangles SOR and POQ are similar,
Therefore,
ar(POQ)/ar(SOR) = (PQ/SR)2
ar(POQ)/ar(SOR) = (3 SR/SR)2
ar(POQ)/ar(SOR) = 9/1
7.ABCD is a trapezium in which AB|| DC and P, Q are points on ADand BC respectively such that PQ || DC. IfPD = 18 cm, BQ = 35 cm andQC = 15 cm, find AD.
(A) 55cm
(B) 57cm
(C) 60cm
(D) 62cm
Answer:(C)
Explanation:
According to question
ABCD is a trapezium in which AB || DC and P and Q are points on AD and BC, respectively such that PQ || DC. If PD = 18 cm, BQ = 35 cm and QC = 15 cm,
In triangle ABD
DP/AP = OD/OB
In triangle BDC
BQ/QC = OB/OD
This implies
DP/AP = QC/BQ
18/AP = 15/35
AP = (18 x 35)/15
AP = 42
Therefore, AD = AP + DP = 42 + 18 = 60cm
8.Areas of two similar triangles are 36 cm2 and 100 cm2. If the length of a side of the larger triangle is 20 cm, then the length of the corresponding side of the smaller triangle is:
(A) 12cm
(B) 13cm
(C) 14cm
(D) 15cm
Answer:(A)
Explanation:
Let the side of smaller triangle be x cm.
ar(Larger Triangle)/ar(Smaller Triangle) = (side of larger triangle/side of smaller triangle)2
100/36 = (20/x)2
x = √144
X = 12 cm
9. In the figure if ∠ACB = ∠CDA, AC = 8 cm and AD = 3 cm, find BD.
(A) 53/3 cm
(B) 55/3 cm
(C) 64/3 cm
(D) 35/7 cm
Answer:(B)
Explanation:
In triangle ACB and ADC
∠A=∠A
∠ACB = ∠CDA
Therefore triangle ACB and ADC are similar,
Hence
AC/AD = AB/AC
AC2 = AD X AB
82 = 3 x AB
⇒ AB = 64/3
This implies,
BD = 64/3 – AD
⇒ BD = 55/3
10. If ABCD is parallelogram, P is a point on side BC and DP when produced meets AB produced at L, then select the correct option
(A) DP/BL = DC/PL
(B) DP/PL = DC/BL
(C) DP/PL = BL/DC
(D) DP/PL = AB/DC
Answer: (B)
Explanation:
In ΔALD, we have
BP || AD
∴ LB/BA = LP/PD
⇒ BL/AB = PL/DP
⇒ BL/DC = PL/DP [∵ AB = DC
⇒ DP/PL = DC/BL
11. In the figure given below DE || BC. If AD = x, DB = x – 2, AE = x + 2 and EC = x – 1, the value of x is:
(A) 4
(B) 8
(C) 16
(D) 32
Answer: (A)
Explanation:
In triangle ABC, we have DE || BC
∴ AD/DB = AE/EC (By Thale’s Theorem)
⇒ x/x – 2 = (x + 2)/(x – 1)
⇒ x (x – 1) = (x – 2)(x + 2)
⇒ x2 – x = x2 – 4
⇒ x = 4
12. The length of altitude of an equilateral triangle of side 8cm is
(A) √3 cm
(B) 2√3 cm
(C) 3√3 cm
(D) 4√3 cm
Answer:(D)
Explanation:
The altitude divides the opposite side into two equal parts,
Therefore, BD = DC = 4 cm
In triangle ABD
AB2 = AD2 + BD2
82 = AD2 + 42
AD2 = 64 – 16
AD2 = 48
AD = 4√3 cm
13. If ΔABC ~ ΔDEF, AB = 4 cm, DE = 6 cm, EF = 9 cm and FD = 12 cm, find the perimeter of ABC.
(A) 18 cm
(B) 20 cm
(C) 21 cm
(D) 22 cm
Answer:(A)
Explanation:
According to question,
ΔABC ~ ΔDEF,
AB = 4 cm, DE = 6 cm, EF = 9 cm and FD = 12 cm,
Therefore,
AB/DE = BC/EF = AC/DF
4/6 = BC/9 = AC/12
⇒ 4/6 = BC/9
⇒ BC = 6 cm
And
4/6 = AC/12
⇒ AC = 8 cm
Perimeter = AB + BC + CA
= 4 + 6 + 8
= 18 cm
14. A 5 m long ladder is placed leaning towards a vertical wall such that it reaches the wall at a point 4 m high. If the foot of the ladder is moved 1.6 m towards the wall, then the distance by which the top of the ladder would slide upwards on the wall is:
(A) 2 m
(B) 1.2 m
(C) 0.8 m
(D) 0.5 m
Answer:(C)
Explanation:
Let AC be the ladder of length 5m and BC = 4m be the height of the wall where ladder is placed. If the foot of the ladder is moved 1.6m towards the wall i.e. AD = 1.6 m, then the ladder is slided upward to position E i.e. CE = x m.
In right triangle ABC
AC2 = AB2 + BC2
⇒52 = AB2 + 42
⇒ AB = 3m
⇒ DB = AB – AD = 3 – 1.6 = 1.4m
In right angled ΔEBD
ED2 = EB2 + BD2
⇒ 52 = EB2 + (1.4)2
⇒ EB = 4.8m
EC = EB – BC = 4.8 – 4 = 0.8m
Hence the top of the ladder would slide upwards on the wall at distance 0.8 m.
15.Corresponding sides of two similar triangles are in the ratio of 2 : 3. If the area of the smaller triangle is 48 cm2, then the area of the larger triangle is:
(A) 108 m2
(B) 107 m2
(C) 106 m2
(D) 230 m2
Answer:(A)
Explanation:
According to given Question
ar(Larger Triangle)/ar(Smaller Triangle) = (side of larger triangle/side of larger triangle)2
6. AOBC is a rectangle whose three vertices are vertices A (0, 3), O (0, 0) and B (5, 0). The length of its diagonal is
(A) 5
(B) 3
(C) √34
(D) 4
Answer: (C)
Explanation:
The length of the diagonal is distance between the points AB.
The distance is calculated as,
7. If P (a/3, 4) is the mid-point of the line segment joining the points Q (– 6, 5) and R (– 2, 3), then the value of a is
(A) – 4
(B) – 12
(C) 12
(D) – 6
Answer: (B)
Explanation: As (a/3, 4) is the mid – point of the line segment joining the points Q (– 6, 5) and R (– 2, 3). Therefore
8. The coordinates of the point which is equidistant from the three vertices of the Δ AOB as shown in the figure is:
(A) (x, y)
(B) (y, x)
(C) (x/2, y/2)
(D) (y/2, x/2)
Answer: (A)
Explanation: As we have to find the coordinates which are equidistant from A and B, Let the points be (a, b).
Then (a, b) will be the midpoint of AB.
Therefore,
Hence the coordinates are (x, y)
9. A circle drawn with origin as the centre passes through The point which does not lie in the interior of the circle is
(C) (5, –1/2)
(D) (–6, 5/2)
Answer: (D)
Explanation:If the point lies in the interior of circle, the distance of the point from the centre should be less than radius of circle.
The radius of circle is the distance between origin and the point
Distance between origin and (-3/4, 1) is
Similarly the distance of points (2, 7/3) and (5, –1/2) is also less than 6.5
But the distance of (–6, 5/2) is equal to 6.5.
So the point (–6, 5/2) does not lie in the interior of circle.
10. If the distance between the points (4, p) and (1, 0) is 5, then the value of p is
(A) 4 only
(B) ± 4
(C) – 4 only
(D) 0
Answer:B
Explanation:According to question:
11. The area of a triangle with vertices A (3, 0), B (7, 0) and C (8, 4) is:
(A) 14
(B) 28
(C) 8
(D) 6
Answer: (C)
Explanation: Area of triangle is calculated as,
12. The point which divides the line segment joining the points (7, –6) and (3, 4) in ratio 1 : 2 internally lies in the
(A) I quadrant
(B) II quadrant
(C) III quadrant
(D) IV quadrant
Answer: (D)
Explanation: Let the point be (x, y)
Then, by using section formula
Therefore, the point is (17/3, -8/3) which lies in fourth quadrant.
13. One of the two points of trisection of the line segment joining the points A (7, – 2) and B (1, – 5) which divides the line in the ratio 1:2 are:
(A) (5, –3)
(B) (5, 3)
(C) (–5, –3)
(D) (13, 0)
Answer: (A)
Explanation: Required point of trisection that divides the given line in the ratio 1: 2 is
14. A line intersects the y-axis and x-axis at the points P and Q, respectively. If (2, –5) is the mid – point of PQ, then the coordinates of P and Q are, respectively.
(A) (0, – 5) and (2, 0)
(B) (0, 10) and (– 4, 0)
(C) (0, 4) and (– 10, 0)
(D) (0, – 10) and (4, 0)
Answer: (D)
Explanation: As the line intersects the y and x axis, let the coordinates be (0, b) and (a, 0) respectively. Since (2, –5) is the midpoint of the axis. Therefore,
Therefore, the coordinates are (0, –10) and (4, 0).
15. The ratio in which the point P (3/4, 5/12)divides the line segment joining the Points A (1/2, 3/2) and B (2, –5) is:
(A) 1:5
(B) 5:1
(C) 1:3
(D) 3:1
Answer: (A)
Explanation: Let the ratio be m : n then, according to the question:
1.How many periods and groups are present in the periodic table?
a) 7 periods and 18 groups
b) 8 periods and 7 groups
c) 7 periods and 7 groups
d) 8 periods and 8 groups
Answer: (a) 7 periods and 18 groups
Explanation: Modern periodic table consists of 7 horizontal rows known as periods and 18 vertical columns named as groups.
2.Which of the following forms the basis of the modern periodic table?
a) Atomic mass
b) Atomic number
c) Number of nucleons
d) All of these
Answer: (b) Atomic number
Explanation: Modern periodic table is based on the atomic numbers of elements as according to the modern periodic law the properties of elements are a periodic function of their atomic numbers.
3. What happens to the electropositive character of elements on moving from left to right in a periodic table?
a) Increase
b) Decreases
c) First increases than decreases
d) First decreases than increases
Answer: (b) Decreases
Explanation: Electropositive character of an element is its ability to lose electrons and form positive ions. Now, as on moving from left to right in a period of periodic table, the nuclear charge increases due to the gradual increase in number of protons, so the valence electrons are pulled more strongly by the nucleus. Thus, it becomes more and more difficult for the atoms to lose electrons causing a decrease in the electropositive character of elements on moving from left to right in a periodic table.
4. The electronic configuration of an element M is 2, 8, 4. In modern periodic table, the element M is placed in
a) 4th group
b) 2nd group
c) 14th group
d) 18th group
Answer: (c) 14th group
Explanation: In the periodic table, elements having 4 valence electrons are placed in group 14.
5. Which of the following is the correct order of the atomic radii of the elements oxygen, fluorine and nitrogen?
a) O < F < N
b) N < F < O
c) O < N < F
d) F < O < N
Answer: (d) F < O < N
Explanation: Oxygen (8), fluorine (9) and nitrogen (7) belong to the same period of the periodic table, in the order nitrogen, oxygen and fluorine. Now in a period, on moving from left to right the atomic radius of the elements decreases. Therefore, the atomic radius of nitrogen is the largest.
6.What is the other name for group 18th elements?
a) Noble gases
b) Alkali metals
c) Alkali earth metals
d) Halogens
Answer: (a) Noble gases
Explanation: Group 18th elements are named as noble gases as they are very stable due to having the maximum number of valence electrons their outermost shell can hold, hence they rarely react with other elements.
7. Which of the following is the most reactive element of the group 17?
a) Oxygen
b) Sodium
c) Fluorine
d) Magnesium
Answer: (c) Fluorine
Explanation: As we move down in a group, the size of the atoms of elements goes on increasing. So, fluorine being on the top position in the halogen’s group, is the smallest element and has the maximum tendency to gain an electron to complete its octet. Thus fluorine is the most reactive element of the group 17.
8. Element X forms a chloride with the formula XCl2, which is a solid with a high melting point. X would most likely be in the same group of the Periodic Table as
a) Na
b) Mg
c) Al
d) Si
Answer: (b) Mg
Explanation: Group 2 alkaline earth metal atoms have two valence electrons each. They can donate their two valence electrons to two other chlorine atoms to form the solid compounds of the form XCl2.
This XCl2 compound being ionic in nature, has a very strong electrostatic forces of attraction between 2 chloride atoms and 1 metal atom. Thus a large amount of heat is required to break these strong bonds, causing the compound to have very high melting and boiling points.
9. Which group elements are called transition metals?
a) Group number 1 to 2
b) Group number 13 to 18
c) Group number 3 to 12
d) Group number 1 to 8
Answer: (c) Group number 3 to 12
Explanation: The elements occurring in the group 3 to 12 are named as transition metals because they are metallic elements that form a transition between the main group elements, which occur in groups 1 and 2 on the left side, and groups 13–18 on the right side of the periodic table.
10. Which of the following elements has 2 shells and both are completely filled?
a) Helium
b) Neon
c) Calcium
d) Boron
Answer: (b) Neon
Explanation: Neon with the atomic number 10, has the electronic configuration as:
Hence, both its K and L shells are completely filled.
11. Which of the following is the atomic number of an element that forms basic oxide?
a) 18
b) 17
c) 19
d) 15
Answer: (c) 19
Explanation: The elements which can donate their valence electrons to other atoms are the metallic elements which form basic oxides as they give hydroxides in their aqueous solutions.
12. The elements A, B and C belong to group 2, 14 and 16 respectively, of the periodic table. Which of the two elements will form covalent bonds?
a) A and B
b) B and C
c) C and A
d) None of these
Answer: (b) B and C
Explanation: The covalent bond is formed by the sharing of electrons between two atoms. As the element B (which belongs to group 14) has 4 valence electrons which it can share with two elements of C type (from group 16) electrons to complete the octet of each included atom:
13. Which of the following does not decrease while moving down the group of the periodic table?
a) Atomic radius
b) Metallic character
c) Number of shells in the atom
d) Valence electrons
Answer: (d) Valence electrons
Explanation: Number of valence electrons in a group remain the same.
14. An element X belongs to the 3rd period and 1st group of the periodic table. What is the number of valence electrons in its atom?
a) 1
b) 3
c) 6
d) 8
Answer: (a) 1
Explanation: As the element belongs to the 1st group of the periodic table, so the number of valence electrons in its atom is one.
15. An element M is in group 13th of the periodic table, the formula for its oxide is
a) MO
b) M2O3
c) M3O2
d) None of these
Answer: (b) M2O3
Explanation: As the element M belongs to group 13th of the periodic table so it has 3 valence electrons, i.e., it can have +3 oxidation state while oxygen atom (with 2 valency) has −2 oxidation state. So the formula for the corresponding oxide is M2O3.
Important Link
Quick Revision Notes :Periodic Classification of Elements
NCERT Solution :Periodic Classification of Elements
1.Which of the following structures correctly represents the electron dot structure of a chlorine molecule?
Answer: (a)
Explanation: In an electron dot structure of a molecule there must be shown eight electrons (in the form of dots or crosses) around each element of the molecule, to represent the complete octet of the element.
2. While cooking, if the bottom of the vessel is getting blackened on the outside, it means that:
a) The food is not cooked completely
b) The fuel is not burning completely
c) The fuel is wet
d) The fuel is burning completely
Answer: (b) The fuel is not burning completely
Explanation: In case the fuel doesn’t burn completely, i.e., there is not enough oxygen to react with the carbon to produce carbon dioxide, then the unburnt carbon particles are left behind in the form of black particles known as soot. These soot particles stick to the bottom of the vessel making it black.
3. Cation is formed when:
a) Atom gains electrons
b) Atom loses electrons
c) Proton is lost by the atom
d) Atom shares electrons
Answer: (b) Atom loses electrons
Explanation: A cation is formed by loss electrons from the atom of an element which acquires positive charge due to the presence of greater number of protons as compared to that of electrons.
4.The I.U.P.A.C name of CH3CH2CH=CH2 is?
a) 3-Butene
b) Prop-1-ene
c) But-1-ene
d) Butyne
Answer: (c) But-1-ene
Explanation: As the compound, CH3CH2CH=CH2 contains four carbon atoms and a double bond attached to the first carbon, so the I.U.P.A.C name of CH3CH2CH=CH2 is But-1-ene.
5.Which of the following compounds of carbon does not consist of ions?
a) CHCl3
b) CaCO3
c) NaHCO3
d) Ca2C
Answer: (a) CHCl3
Explanation: Carbon always forms covalent compounds by sharing its electrons with other atoms. Now, in covalent bonding, the two electrons shared by the atoms are attracted to the nucleus of both atoms and neither atom completely loses or gains electrons as in ionic bonding. So the compounds in which all the atoms are directly attached to C-atom, contain covalent bonding and no ionic bond.
In CHCl3, all the three chlorine atoms are bonded covalently to the carbon atom, not to the hydrogen atom. So CHCl3 is a covalent compound and does not consist of ions.
6.The property of self-linkage among identical atoms to form long chain compounds is known as:
a) Catenation
b) Isomerisation
c) Superposition
d) Halogenation
Answer: (a) Catenation
Explanation: Catenation is the property of self-linking of an element by which an atom combines with the other atoms of the same element to form long chains.
7. Which of the following is the molecular formula of cyclobutane?
a) C4H10
b) C4H6
c) C4H8
d) C4H4
Answer: (c) C4H8
Explanation: Cyclobutane is a cyclic hydrocarbon consisting of four carbon atoms where each carbon atom is attached to the two other carbon atoms and two hydrogen atoms, as shown below:
8.Which of the following statements about graphite and diamond is true?
a) They have the same crystal structure
b) They have the same degree of hardness
c) They have the same electrical conductivity
d) They can undergo the same chemical reactions
Answer: (d) They can undergo the same chemical reactions
Explanation: Both Graphite and diamond being the allotropes of the same element , carbon, have similar chemical properties. So they undergo the same chemical reactions.
9.How many number of carbon atoms are joined in a spherical molecule of buckminsterfullerene?
a) 30
b) 60
c) 90
d) 120
Answer: (b) 60
Explanation: Buckminsterfullerene is a molecule of carbon in the form of a hollow sphere consisting of 60 C-atoms and is having the formula C60.
10.Which of the followings is the major constituent of the liquefied petroleum gas?
a) Methane
b) Ethane
c) Propane
d) Butane
Answer: (d) Butane
Explanation: The major constituent of the liquefied petroleum gas is butane.
11.The organic compounds having functional group are known as:
a) Aldehyde
b) Ketone
c) Carboxylic acids
d) Alcohol
Answer: (c) Carboxylic acids
Explanation: Carboxylic acids are compounds which contain a group also known as carboxyl group.
12.From which of the following substance pencil lead is formed?
a) Charcoal
b) Wood
c) Lead
d) Graphite
Answer: (d) Graphite
Explanation: Pencil lead is formed of graphite. Graphite is an allotropic form of carbon in which each carbon atom is joined to three others, forming layers:
These layers are put together by weak van der Waals forces which enable the layers to slide over each other, making graphite soft and slippery. So graphite is used as pencil ‘lead’. As the pencil moves across the paper, layers of graphite rub off leaving the dark marks on paper.
13. Ester is formed by the reaction between:
a) An acid and an alcohol
b) An acid and a base
c) A base and an alcohol
d) An acid and an alkene
Answer: (a) An acid and an alcohol
Explanation: Reaction between an acid and an alcohol results in the formation of ester, and the reaction is named as estrification.
For example: Acetic acid reacts with ethyl alcohol in the presence of concentrated sulphuric acid to form Ethyl acetate:
14. What is denatured alcohol?
a) Ethyl alcohol which has been made unfit for drinking purpose by adding small amount of poisonous substance
b) Methyl alcohol which has been made unfit for drinking purpose by adding small amount of poisonous substance
c) Alcohol having properties of an acid
d) Ethyl alcohol containing 60% of water by weight
Answer: (a) Ethyl alcohol which has been made unfit for drinking purpose by adding small amount of poisonous substance
Explanation: Denatured alcohol is the ethyl alcohol which has been made unfit for drinking purpose by adding small amount of poisonous substance like methanol, pyridine, etc. This is mainly done to prevent the misuse of industrial alcohol for drinking purposes.
15.Which of the following substance produces brisk effervescence with baking soda solution?
a) Ethanoic acid
b) Table salt
c) Vinegar
d) Sunflower oil
Answer: (a) Ethanoic acid
Explanation: Ethanoic acid when treated with baking soda (Sodium hydrogencarbonate) gives brisk effervescence of Carbon dioxide gas.
9.An element reacts with oxygen to give a compound with a high melting point. This compound is also soluble in water. The element is likely to be
(a) Ca
(b) C
(c) Si
(d) Fe
Answer: (a) Ca
Explanation:
Calcium reacts with oxygen to give calcium oxide (CaO) which is having a high melting point and dissolves in water to form calcium hydroxide (Ca(OH)2)along with the release of large amount of thermal energy.
10.Which of the following pairs will give displacement reactions?
(a) NaCl solution and copper metal
(b) MgCl2 solution and aluminium metal
(c) FeSO4 solution and silver metal
(d) AgNO3 solution and copper metal
Answer: (d) AgNO3 solution and copper metal
Explanation: Copper (Cu) being more reactive than silver (Ag), displaces silver from silver nitrate (AgNO3) to form copper nitrate
2AgNO3 + Cu → Cu(NO3)2+ 2Ag
11.Which among the following is the most abundant metal found in the earth’s crust?
(a) Magnesium
(b) Aluminium
(c) Oxygen
(d) Iron
Answer: (b) Aluminium
Explanation: Aluminium is the most abundant metal found in the earth’s crust.
12.Which of the following pairs of reactants will go undergo a displacement reaction?
(a) CuSO4 + Fe
(b) ZnSO4 + Fe
(c) MgSO4 + Fe
(d) Ca(SO4)2 + Fe
Answer: (a) CuSO4 + Fe
Explanation: As per the reactivity series of metals, iron is more reactive than copper metal so it can displace copper from copper sulphate solution and form iron (II) sulphate and copper:
13.Galvanisation is a method of protecting steel and iron from rusting by coating them with a thin layer of
(a) Copper
(b) Aluminum
(c) Zinc
(d) Bauxite
Answer: (c) Zinc
Explanation: In this method a thin layer of zinc metal is deposited over the surface of steel or iron objects, which does not corrode on exposure to damp air and prevents the coated metals from rusting.
14.Which of the following alloys contains a non-metal as one of its constituents?
(a) Steel
(b) Brass
(c) Amalgam
(d) Bronze
Answer: (a) Steel
Explanation: Stainless steel is an alloy of iron (a metal) and carbon (a non metal).
15.An element X is soft and can becut with the help of a knife. It is very reactive to air and cannot be kept open in the air. It reacts vigorously with water. Identify the element from the following:
(a) Mg
(b) Na
(c) P
(d) Ca
Answer: (b) Na
Explanation: Na is a metal which is soft enough to be cut with a knife. It is so reactive that it reacts vigorously with air or moisture and catches fire when kept in open. So to prevent it from coming in contact with oxygen and moisture, it is kept in kerosene.
