Olympiad Mathematics - Algebraic Expressions
Exam Duration: 45 Mins Total Questions : 30
What is the difference between 3a + 2b and -2a - 5b?
- (a)
5a + 7b
- (b)
-5a - 7b
- (c)
5a - 7b
- (d)
a - 3b
(3a + 2b) - (-2a - 5b)
= 3a + 2b + 2a + 5b = 5a + 7b
A and B are polynomials and each is the additive inverse of the other. What does it mean?
- (a)
A = B
- (b)
A + B is a zero polynomial.
- (c)
A - B is a zero polynomial.
- (d)
A - B = B - A
If C=\(\frac { x-a }{ x-b } \) , find the value of x.
- (a)
\(\frac { bC-a }{ C-b } \).
- (b)
\(\frac { C-a }{ C-b } \).
- (c)
\(\frac { C+a }{ C+b } \).
- (d)
\(\frac { 1-C }{ a-bC } \).
C=\(\frac { x-a }{ x-b } \) ⇒ C(x-b)=x-a
⇒ x=\(\frac { bC-a }{ C-1 } \)..
Which of the following is true?
- (a)
The product of numbers p and q subtracted from 7 is 7 + pq.
- (b)
Y - y3 is a monomial.
- (c)
The coefficient of y2 in 2x2y + 7y2 is 7
- (d)
100z3 is a binomial
Simplify x2y3 - 1.5 x2y3 + 1.4x2y3.
- (a)
0.9x2y3
- (b)
-0.9x2y3
- (c)
0.9
- (d)
-0.9
The third term of the series 7n + 20 is 41. What is the 10th term?
- (a)
90
- (b)
56
- (c)
63
- (d)
87
What is the expression related to the pattern 5, 8, 11, .......?
- (a)
2n-1
- (b)
3n+2
- (c)
4n+1
- (d)
n2-1
Which expression gives the predecessor of a natural number 'n'?
- (a)
2n-1
- (b)
n+1
- (c)
n-1
- (d)
2n+1
Identify the like terms in
21p - 32 - 7p + 20p.
- (a)
21p, - 32 and 20p
- (b)
-32, -7p and 20p
- (c)
21p, -7p and 20p
- (d)
-7p, 21p, and 32
Like terms have the same literal coefficients.
What is the coefficient of 'y' in the expression 3xy - 13?
- (a)
3x
- (b)
3
- (c)
-13
- (d)
Either (A) or (B)
A rectangle is 3p cm long and 2p cm wide. Find the perimeter of the rectangle when p = 12.
- (a)
102 cm
- (b)
120 cm
- (c)
210 cm
- (d)
10p cm
Perimeter = 2 (l + b)
= 2 (3p + 2p)
= 2 (5p) = 10p cm
∴ Perimeter when p = 12 cm is 10(12)
= 120 cm.
The angles of a quadrilateral are (p+25)0, 2p0, (p -20)0 and (p + 20)0. What is the value of the smallest angle?
- (a)
1050
- (b)
650
- (c)
1150
- (d)
650
Sum of angles in a quadrilateral is 360°.
⇒ 6p + 30° = 360°
⇒ p = 60° + 5° = 55°
∴ (p + 25)° = 90°, 2po = 130°
(p + 20)° = 65° + 20° = 85°
∴ The smallest angle is 65°.
The sides of a right angled triangle are 2a cm. (2a + 2) cm and (4a - 2) cm long. What is the length of the shortest side of the triangle if its perimeter is 24 cm?
- (a)
8 cm
- (b)
6 cm
- (c)
10 cm
- (d)
3 cm
Perimeter of the triangle
= [2a + (2a + 2) + (4a - 2)] cm
⇒ 8a cm = 24 cm or a = 3 cm
∴ The length ofthe shortest side is 2a = 6 cm.
What is the value of the expression 2x2y + xy2 + xy for x = 1 and y = -2?
- (a)
-2
- (b)
-3
- (c)
-4
- (d)
-5
How is "4 is less than half of x" written in symbolic form?
- (a)
4>\(\frac { x }{ 2 } \) .
- (b)
\(\frac { x-4 }{ 2 } \).
- (c)
\(\frac { x }{ 2 } \)+4
- (d)
4<\(\frac { x }{ 2 } \) .
Simplify \(\frac { 4 }{ 11 } \)(66x+44)+\(\frac { 3 }{ 11 } \)(33x-3)
- (a)
33x + 7
- (b)
33x - 7
- (c)
33x - 7x
- (d)
33 + 7x
\(\frac { 4 }{ 11 } \)(66x+44)+\(\frac { 3 }{ 11 } \)(33x-33)
=33x+7
Addition of \(\frac{a^2}{2}+\frac{b^3}{3}-\frac{c^3}{4},\frac{2a^2}{3}+\frac{3b^3}{4}-\frac{4c^3}{5}\) and a2+b3+c3 is_______.
- (a)
\(\frac{13}{6}a+\frac{25}{12}b^3-\frac{1}{20}c^3\)
- (b)
\(\frac{13}{6}a^2-\frac{1}{20}b^3+\frac{25}{12}c^3\)
- (c)
\(\frac{13}{6}a^2+\frac{25}{12}b^3-\frac{1}{20}c^3\)
- (d)
\(\frac{13}{6}a^2-\frac{25}{12}b^3+\frac{1}{20}c^3\)
\((\frac{a^2}{2}+\frac{b^3}{3}-\frac{c^3}{4})+(\frac{2a^2}{3}+\frac{3b^3}{4}-\frac{4c^3}{5})\)
\(=\frac{13}{6}a^2+\frac{25}{12}b^3-\frac{1}{20}c^3\)
Simplify \((2x+\frac{1}{3y})^2-(2x-\frac{1}{3y})^2.\)
- (a)
\(\frac{4x}{3y}\)
- (b)
\(2(4x^2+\frac{1}{9y^2})\)
- (c)
\(\frac{8x}{3y}\)
- (d)
\(\frac{4y}{3x}\)
\((2x+\frac{1}{3y})^2-(2x-\frac{1}{3y})^2\)
\(=[2x+\frac{1}{3y}][2x+\frac{1}{3y}]-\left\{(2x-\frac{1}{3y})(2x-\frac{1}{3y})\right\}\)
\(=[4x^2+\frac{1}{9y^2}+\frac{4x}{3y}]-\left\{4x^2+\frac{1}{9y^2}-\frac{4x}{3y}\right\}=\frac{8x}{3y}\)
The product of \((\frac{4p}{5}-3)\ and(\frac{5p}{8}-6)\) is_______.
- (a)
\(\frac{p^2}{2}+\frac{267}{40}p-18\)
- (b)
\(\frac{p^2}{2}-\frac{267}{40}p-18\)
- (c)
\(\frac{p^2}{2}+\frac{267}{40}p+18\)
- (d)
\(\frac{p^2}{2}-\frac{267}{40}p+18\)
We have \((\frac{4p}{5}-3)\times(\frac{5p}{8}-6)\)
\(=\frac{p^2}{2}-\frac{24}{5}p-\frac{15}{8}p+18=\frac{p^2}{2}-\frac{267}{40}p+18\)
The value of 9x2 + 49y2 - 42xy when x = 15 and y = 3 is_______.
- (a)
636
- (b)
576
- (c)
456
- (d)
386
When x = 15 and y = 3, we have
9 x (15)2 + 49 x (3)2 - 42 x 15 x 3
= 2025 + 441 - 1890 = 576
Simplify:
(a3 - 2a2 + 4a - 5) - (- a3 - 8a + 2a2 + 5)
- (a)
2a3+7a2+6a-10
- (b)
2a3 + 7a2 + 12a - 10
- (c)
2a3 - 4a2 + 12a - 10
- (d)
2a3 - 4a2 + 6a - 10
(a3 - 2a2 + 4a - 5) - (-a3 - 8a + 2a2 + 5)
= a3 - 2a2 + 4a - 5 + a3 + 8a - 2a2 - 5
= 2a3 - 4a2 + 12a - 10
By how much is a4 + 4a2b2 + b4 more than a4 - 8a2b2 + b4?
- (a)
12a2b2
- (b)
- 12a2b2
- (c)
2a4 + 2b4
- (d)
10a2b2
Required expression
= (a4 + 4a2b2 + b4) - (a4 - 8a2b2 + b4)
= a4 + 4a2b2 + b4 - a4 + 8a2b2 - b4 = 12a2b2
If \(\frac{x}{y}=\frac{3}{4},\) then the value of \((\frac{6}{7}+\frac{y-x}{y+x})\) equals______.
- (a)
\(\frac{5}{7}\)
- (b)
\(1\frac{1}{7}\)
- (c)
1
- (d)
2
Subtract (2a - 3b + 4c) from the sum of (a + 3b - 4c), (4a - b + 9c) and (-2b + 3c - a).
- (a)
3a + 2b + 4c
- (b)
2a - 2b + 4c
- (c)
3a - 4b - 2c
- (d)
2a + 3b + 4c
Sum = (a + 3b - 4c) + (4a - b + 9c) + (- 2b + 3c - a)
= (a + 4a - a) + (3b - b - 2b) + (- 4c + 9c + 3c)
= 4a + 8c
\(\therefore\) Required difference
= (4a + 8c) - (2a - 3b + 4c)
= 4a + 8c - 2a + 3b - 4c = 2a + 3b + 4c
In a school, 8a2 + 4a + 9 students were enrolled. 2a2 - 9a + 2 students were boys. How many girls were enrolled?
- (a)
6a2-13a + 7
- (b)
4a2 + 13a + 7
- (c)
6a2 + 13a + 7
- (d)
4a2 - 13a + 7
Number of girls enrolled
= (8a2 + 4a + 9) - (2a2 - 9a + 2)
= 8a2 + 4a + 9 - 2a2 + 9a - 2 = 6a2 + 13a + 7
Mohit's monthly salary was Rs 5445q. He saved 30% of it and gave \(\frac{1}{2}\) of the remainder to his parents. If Mohit used \(\frac{3}{4}\) of the amount of money, he had left to buy a guitar, then how much money would he have left, if q = 8?
- (a)
Rs 1100.50
- (b)
Rs 11434.50
- (c)
Rs 11079.50
- (d)
Rs 3811.50
A vacuum cleaner set costs Rs 154.25 k. Additional pipe costs Rs 15.2 k. What is the total cost of 3 vacuum cleaner sets and 5 additional pipes?
- (a)
Rs 400 k
- (b)
Rs 530.75 k
- (c)
Rs 538.75 k
- (d)
Rs 600 k
Cost of 1 vacuum cleaner set = Rs 154.25 k
\(\therefore\) Cost of 3 vacuum cleaner sets
= Rs (3 x 154.25 k)
Cost of 1 additional pipe =Rs15.2 k
\(\therefore\) Cost of 5 additional pipes
= Rs (5 x 15.2 k) = Rs 76 k
So, total cost = Rs(462.75 k + 76 k) = Rs 538.75 k
From 2012-2016, the amount (in crores) spent on natural gas N and electricity E by Indian residents can be described by the following expressions, where t is the number of years since 2012.
Gas spending model,
N = 2.13t2 - 4.21t + 37.40
Electricity spending model,
E = - 0.209t2 + 5.393t + 307.735
What is the total amount A spent on natural gas and electricity by Indian residents from 2012 to 2016?
- (a)
1.467t2 + 7.423 + 121.721
- (b)
1.339t2 - 8.729t + 76.245
- (c)
1.01t2 + 7.083 + 97.83
- (d)
1.921t2 + 1.183t + 345.135
The value of \(\frac{1}{4}\left\{x-5(q-x)\right\}-\frac{3}{2}\left\{\frac{1}{3}(q-\frac{x}{3})-\frac{2}{9}[x-\frac{3}{4}(q-\frac{4x}{5})]\right\}\) is______.
- (a)
\(\frac{9x}{5}+\frac{q}{2}\)
- (b)
\(\frac{9x}{2}-\frac{q}{2}\)
- (c)
\(\frac{9x}{5}-\frac{q}{2}\)
- (d)
\(\frac{11x}{5}-2q\)
Fill in the blanks.
(i) Any expression with one or more terms is called a____P_______.
(ii) Terms which have the same algebraic factors are____Q_____terms.
(iii) The___R____is the numerical factor in the term.
(iv) Algebraic expressions are formed from____S_____and____T_____.
- (a)
P Q R S T Binomial unlike term factors constants - (b)
P Q R S T Polynomial like term factors constants - (c)
P Q R S T Trinomial unlike coefficient variables constant - (d)
P Q R S T Polynomial like coefficient variables constants