If \(\rm \vec{a},\vec{b},\vec{c},\vec{d}\) are four vectors such that \(\rm \vec{a}+\vec{b}+\vec{c}\) is collinear with \(\rm \vec d\) and \(\rm \vec{b}+\vec{c}+\vec{d}\) is collinear with \(\rm \vec{a}\), then \(\rm \vec{a}+\vec{b}+\vec{c}+\vec{d}\) is

collinear with \(\rm \vec{b}-\vec{c}\)

collinear with \(\rm \vec{a}+\vec{d}\)

collinear with \(\rm \vec{a}-\vec{d}\)

collinear with \(\rm \vec{b}-\vec{c}\)

A unit vector perpendicular to each of the vectors 2î - ĵ + k̂ and 3î - 4ĵ - k̂ is

\(\frac{1}{{\sqrt 2 }}\hat i + \frac{1}{2}\hat j + \frac{1}{2}\hat k\)

\(\frac{1}{{\sqrt 3 }}\hat i + \frac{1}{{\sqrt 3 }}\hat j - \frac{1}{{\sqrt 3 }}\hat k\)

\(\frac{1}{{\sqrt 2 }}\hat i + \frac{1}{2}\hat j + \frac{1}{2}\hat k\)

\(\frac{1}{{\sqrt 3 }}\hat i - \frac{1}{{\sqrt 3 }}\hat j - \frac{1}{{\sqrt 3 }}\hat k\)

\(\frac{1}{{\sqrt 3 }}\hat i - \frac{1}{{\sqrt 3 }}\hat j + \frac{1}{{\sqrt 3 }}\hat k\)

Consider a 2 × 2 matrix \(M = \left[ {\begin{array}{*{20}{c}}{{v_1}}&{{v_2}}\end{array}} \right]\), where, v1 and v2 are the column vectors. Suppose \({M^{ - 1}} = \left[ {\begin{array}{*{20}{c}}{u_1^T}\\{u_2^T}\end{array}} \right]\), where uT1 and uT2 are the row vectors. Consider the following statements.Statement: uT1v1 = 1 and uT2v2 = 1Statement: uT1v2 = 0 and uT2v1 = 0Which of the following options is correct?

Statement 1 is true and statement 2 is false

Statement 2 is true and statement 1 is false

In the given question, two equations numbered l and II are given. Solve both the equations and mark the appropriate answer.I. x2 – 8x + 16 = 0II. y2 – 7y + 12 = 0

x = y or no relationship could be established

Abdul reads 25 pages of a book containing 100 pages. Akbar read 1/2 of the same book. Which one of the following statements is true?

Akbar read ten pages less than twice that of Abdul

Abdul read five pages more than half that of Akbar

Consider two series \(\mathop \sum \limits_{n = 1}^\infty {\left( { - 1} \right)^{n - 1\;}}{a_n}\) and \(\mathop \sum \limits_{n = 2}^\infty {\left( { - 1} \right)^{n - 1\;}}{b_n}\), where \({a_n} = \frac{1}{{\sqrt n }},\;{b_n} = \frac{{{x^n}}}{{n\left( {n - 1} \right)}}\) 0 < x < 1. Then:

\(\mathop \sum \limits_{n = 1}^\infty {\left( { - 1} \right)^{n - 1\;}}{a_n}\) is divergent but \(\mathop \sum \limits_{n = 2}^\infty {\left( { - 1} \right)^{n - 1\;}}{b_n}\) is convergent

\(\mathop \sum \limits_{n = 1}^\infty {\left( { - 1} \right)^{n - 1\;}}{a_n}\) is convergent but \(\mathop \sum \limits_{n = 2}^\infty {\left( { - 1} \right)^{n - 1\;}}{b_n}\) is divergent.

\(\mathop \sum \limits_{n = 1}^\infty {\left( { - 1} \right)^{n - 1\;}}{a_n}\) is divergent but \(\mathop \sum \limits_{n = 2}^\infty {\left( { - 1} \right)^{n - 1\;}}{b_n}\) is convergent

Find the cofactor of b3 in the following matrix Δ: \(= \left[ {\begin{array}{*{20}{c}} a_1&{ b_1 }&{c_1}\\ a_2&b_2&{ c_2}\\ a_3&b_3&c_3 \end{array}} \right]\)

\(\left| {\begin{array}{*{20}{c}} a_1 &c_1\\ a_2&c_2 \end{array}} \right| \)

\(-\left| {\begin{array}{*{20}{c}} a_1 &c_1\\ a_2&c_2 \end{array}} \right| \)

1894 + Mcqs in Algebra in General Aptitude Page 3 McqOptions

101.	\(2x^2+7xy+6y^2= ?\)
A.	\((2x+3y)(x+2y)\)
B.	\((3x+2y)(x+2y)\)
C.	\((2xy+3y)(2x+y)\)
D.	\((3x+2y)(2x+y)\)
Answer» B. \((3x+2y)(x+2y)\)

102.	If \(a = \frac{2 +\sqrt{3}}{2 - \sqrt{3}}\) and b = \(\frac{2-\sqrt{3}}{2+\sqrt{3}}\), then the value of a2 + b2 + ab is:
A.	195
B.	185
C.	196
D.	186
Answer» B. 185

103.	If a + b = 5 and ab = 3, then (a3 + b3) is equal to:
A.	70
B.	65
C.	75
D.	80
Answer» E.

104.	If 4x + 5y = 14 and x – 5y = 16 then the value of x and y areA. 10 and –6/5B. 6 and 2C. 10 and 6/5D. 6 and – 2
A.	B
B.	D
C.	C
D.	A
Answer» C. C

105.	If \(\rm \vec{a},\vec{b},\vec{c},\vec{d}\) are four vectors such that \(\rm \vec{a}+\vec{b}+\vec{c}\) is collinear with \(\rm \vec d\) and \(\rm \vec{b}+\vec{c}+\vec{d}\) is collinear with \(\rm \vec{a}\), then \(\rm \vec{a}+\vec{b}+\vec{c}+\vec{d}\) is
A.	\(\rm \vec{0}\)
B.	collinear with \(\rm \vec{a}+\vec{d}\)
C.	collinear with \(\rm \vec{a}-\vec{d}\)
D.	collinear with \(\rm \vec{b}-\vec{c}\)
Answer» D. collinear with \(\rm \vec{b}-\vec{c}\)

106.	If the sum of two numbers is 11 and the sum of their squares is 65, then the sum of their cubes will be:
A.	407
B.	576
C.	615
D.	355
Answer» B. 576

107.	If x = 2 + √3 then the value of x3 + x-3 is:
A.	52√3
B.	52
C.	-52√3
D.	-52
Answer» C. -52√3

108.	If x = √3 - √2, then the value of x3 - x-3 is:
A.	-22√3
B.	22√3
C.	22√2
D.	-22√2
Answer» E.

109.	If x = 11, then the value of \(x^5 - 12x^4 + 12x^3 - 12x^2 + 12x - 1\) is
A.	11
B.	10
C.	12
D.	-10
Answer» C. 12

110.	If 15x2 – 26x + 8 = (Ax + B)(Cx + D), where A and C are positive integers, then what is the value of (2A + B – C – 2D)?
A.	2
B.	3
C.	0
D.	1
Answer» E.

111.	If α & β are the roots of the equation 3x2 – 7x + 5, then the sum of their reciprocals is:
A.	5/2
B.	7/5
C.	3/5
D.	7/2
Answer» C. 3/5

115.	If \({x^4} + \frac{1}{{{x^4}}} = 62\) , then what is the value of \({x^6} + \frac{1}{{{x^6}}}\)?
A.	144
B.	288
C.	396
D.	488
Answer» E.

116.	If a2 + b2 = 88 and ab = 6, (a > 0, b > 0) then what is the value of (a3 + b3)?
A.	980
B.	1180
C.	820
D.	1000
Answer» D. 1000

118.	If 7x4 - 6x3 + 8x2 - 20x - 300 is divided by x + 2, then the remainder is:
A.	68
B.	-68
C.	-244
D.	-212
Answer» C. -244

120.	If a + b + c = 0 then the value of \(\frac{1}{{\left( {a + b} \right)\left( {b + c} \right)}} + \frac{1}{{\left( {b + c} \right)\left( {c + a} \right)}} + \frac{1}{{\left( {c + a} \right)\left( {a + b} \right)}}\) is
A.	0
B.	1
C.	3
D.	2
Answer» B. 1

112.	If a – b = 5 and ab = 6, then (a3 – b3) is equal to:
A.	155
B.	225
C.	90
D.	215
Answer» E.

113.	A unit vector perpendicular to each of the vectors 2î - ĵ + k̂ and 3î - 4ĵ - k̂ is
A.	\(\frac{1}{{\sqrt 3 }}\hat i + \frac{1}{{\sqrt 3 }}\hat j - \frac{1}{{\sqrt 3 }}\hat k\)
B.	\(\frac{1}{{\sqrt 2 }}\hat i + \frac{1}{2}\hat j + \frac{1}{2}\hat k\)
C.	\(\frac{1}{{\sqrt 3 }}\hat i - \frac{1}{{\sqrt 3 }}\hat j - \frac{1}{{\sqrt 3 }}\hat k\)
D.	\(\frac{1}{{\sqrt 3 }}\hat i - \frac{1}{{\sqrt 3 }}\hat j + \frac{1}{{\sqrt 3 }}\hat k\)
Answer» B. \(\frac{1}{{\sqrt 2 }}\hat i + \frac{1}{2}\hat j + \frac{1}{2}\hat k\)

114.	Find the discriminant of the quadratic equation 2x2 - 4x + 3 = 0 and hence the nature of the roots
A.	6, two real roots
B.	- 8, no real roots
C.	0, two equal roots
D.	- 6, no real roots
Answer» C. 0, two equal roots

117.	If α and β satisfy the equations 3x + 7y = 5 and 11x + 5y = 7 then 31 (α + β) =
A.	12
B.	29
C.	31
D.	1
Answer» C. 31

119.	Consider a 2 × 2 matrix \(M = \left[ {\begin{array}{{20}{c}}{{v_1}}&{{v_2}}\end{array}} \right]\), where, v1 and v2 are the column vectors. Suppose \({M^{ - 1}} = \left[ {\begin{array}{{20}{c}}{u_1^T}\\{u_2^T}\end{array}} \right]\), where uT1 and uT2 are the row vectors. Consider the following statements.Statement: uT1v1 = 1 and uT2v2 = 1Statement: uT1v2 = 0 and uT2v1 = 0Which of the following options is correct?
A.	Statement 1 is true and statement 2 is false
B.	Statement 2 is true and statement 1 is false
C.	Both the statements are true
D.	Both the statements are false
Answer» D. Both the statements are false

121.	In the given question, two equations numbered l and II are given. Solve both the equations and mark the appropriate answer.I. x2 – 8x + 16 = 0II. y2 – 7y + 12 = 0
A.	x > y
B.	x < y
C.	x ≥ y
D.	x ≤ y
E.	x = y or no relationship could be established
Answer» D. x ≤ y

122.	Consider an n × n matrix A and a non-zero n × 1 vector p. Their product Ap = α2p, where α ∈ ℜ and α ∉ {-1, 0, 1}. Based on the given information, the eigen value of A2 is:
A.	α2
B.	√α
C.	α
D.	α4
Answer» E.

123.	Abdul reads 25 pages of a book containing 100 pages. Akbar read 1/2 of the same book. Which one of the following statements is true?
A.	Abdul read half that of Akbar
B.	Abdul read more than Akbar
C.	Akbar read ten pages less than twice that of Abdul
D.	Abdul read five pages more than half that of Akbar
Answer» B. Abdul read more than Akbar

124.	Find the value of ‘a’ for which \(x + \sqrt {3x} + \frac{{{a^2}}}{4}\) is a perfect square.
A.	√3
B.	2√3
C.	3√3
D.	4√3
Answer» B. 2√3

125.	If x4 + x2y2 + y4 = 91, and x2 - xy + y2 = 13, then what is the value of \|x - y\|?
A.	8
B.	6
C.	2
D.	4
Answer» E.

Explore topic-wise MCQs in General Aptitude.

\(2x^2+7xy+6y^2= ?\)

If \(a = \frac{2 +\sqrt{3}}{2 - \sqrt{3}}\) and b = \(\frac{2-\sqrt{3}}{2+\sqrt{3}}\), then the value of a2 + b2 + ab is:

If a + b = 5 and ab = 3, then (a3 + b3) is equal to:

If 4x + 5y = 14 and x – 5y = 16 then the value of x and y areA. 10 and –6/5B. 6 and 2C. 10 and 6/5D. 6 and – 2

If \(\rm \vec{a},\vec{b},\vec{c},\vec{d}\) are four vectors such that \(\rm \vec{a}+\vec{b}+\vec{c}\) is collinear with \(\rm \vec d\) and \(\rm \vec{b}+\vec{c}+\vec{d}\) is collinear with \(\rm \vec{a}\), then \(\rm \vec{a}+\vec{b}+\vec{c}+\vec{d}\) is

If the sum of two numbers is 11 and the sum of their squares is 65, then the sum of their cubes will be:

If x = 2 + √3 then the value of x3 + x-3 is:

If x = √3 - √2, then the value of x3 - x-3 is:

If x = 11, then the value of \(x^5 - 12x^4 + 12x^3 - 12x^2 + 12x - 1\) is

If 15x2 – 26x + 8 = (Ax + B)(Cx + D), where A and C are positive integers, then what is the value of (2A + B – C – 2D)?

If α & β are the roots of the equation 3x2 – 7x + 5, then the sum of their reciprocals is:

If a – b = 5 and ab = 6, then (a3 – b3) is equal to:

A unit vector perpendicular to each of the vectors 2î - ĵ + k̂ and 3î - 4ĵ - k̂ is

Find the discriminant of the quadratic equation 2x2 - 4x + 3 = 0 and hence the nature of the roots

If \({x^4} + \frac{1}{{{x^4}}} = 62\) , then what is the value of \({x^6} + \frac{1}{{{x^6}}}\)?

If a2 + b2 = 88 and ab = 6, (a > 0, b > 0) then what is the value of (a3 + b3)?

If α and β satisfy the equations 3x + 7y = 5 and 11x + 5y = 7 then 31 (α + β) =

If 7x4 - 6x3 + 8x2 - 20x - 300 is divided by x + 2, then the remainder is:

If a + b + c = 0 then the value of \(\frac{1}{{\left( {a + b} \right)\left( {b + c} \right)}} + \frac{1}{{\left( {b + c} \right)\left( {c + a} \right)}} + \frac{1}{{\left( {c + a} \right)\left( {a + b} \right)}}\) is

In the given question, two equations numbered l and II are given. Solve both the equations and mark the appropriate answer.I. x2 – 8x + 16 = 0II. y2 – 7y + 12 = 0

Consider an n × n matrix A and a non-zero n × 1 vector p. Their product Ap = α2p, where α ∈ ℜ and α ∉ {-1, 0, 1}. Based on the given information, the eigen value of A2 is:

Abdul reads 25 pages of a book containing 100 pages. Akbar read 1/2 of the same book. Which one of the following statements is true?

Find the value of ‘a’ for which \(x + \sqrt {3x} + \frac{{{a^2}}}{4}\) is a perfect square.

If x4 + x2y2 + y4 = 91, and x2 - xy + y2 = 13, then what is the value of |x - y|?

If x = (1/8), which of the following has the largest values?

If a × b = a2 + b2 – ab for all the natural numbers a and b, then the value of 9 × 10 is

How much does a watch lose per day, if the hands coincide every 64 minutes?

If (10.24 × 10.24) – (10.24 × A) + (0.24 × 0.24) is a perfect square, then find the value of 'A'.

If a is positive and \({a^2} + \frac{1}{{{a^2}}} = 7,\;{\rm{then\;find}}\:{a^3} + \frac{1}{{{a^3}}}.\)

If 2x2 - 3x + 5 = 4, then the value of x is :

At which value of k, the linear equations 3x - 2y = 13 and kx - 8y = 40 have no solutions?

A person carries Rs. 500 and wants to buy apples and oranges out of it. If the cost of one apple is Rs. 5 and the cost of one orange is Rs. 7 then what is the number of ways in which a person can buy both apples and oranges using total amount?

If \(x+\dfrac{1}{x}=6\), then find \(x^2+\dfrac{1}{x^2}\):

If x4 + x2y2 + y4 = 21/256 and x2 + xy + y2 = 3/16, then 2(x2 + y2) = ?

If α and β are the roots of the equation ax2 + bx + c = 0, then what is the value of the expression (α + 1) (β + 1)?

Find the scalar triple product of the vectors \(\vec a = \hat i + 3\hat j + 4\hat k\;,\;\vec b = 3\hat i + 4\hat j + 2\hat k\;and\;\vec c = 2\hat i + 4\hat j + 5\hat k\)

If (3x - 6)/(x - 6) (x + k ) = 2/[(x - 6) + 1/(x + k)] then what is the value of k?

If a + b – c = 7, ab – bc – ca = 21, then a3 + b3 – c3 + 3abc =

Find the value of 1006 × 994?

Determine the value of ‘x’, if \(x = \frac{{{{\left( {943 + 864} \right)}^2} - {{\left( {943 - 864} \right)}^2}}}{{\left( {1886 \times 1728} \right)}}\).

If x + y + z = 19, xy + yz + zx = 114, then the value of \(\sqrt {{{\rm{x}}^3}{\rm{\;}} + {\rm{\;}}{{\rm{y}}^3}{\rm{\;}} + {\rm{\;}}{{\rm{z}}^3} - 3{\rm{xyz}}} \) is:

If x2 –4x + 1 = 0, then what is the value of (x6 + x-6)?

Consider the matrix\(P = \left[ {\begin{array}{*{20}{c}} 1&1&0\\ 0&1&1\\ 0&0&1 \end{array}} \right]\)The number of distinct eigen value of P is

Eight years ago, the ratio of ages of A and B was 9 : 10. The ratio of their ages 4 years from now will be 12 : 13. What in the age (in years) of C now, if his age is 6 years more than that of A?

Consider two series \(\mathop \sum \limits_{n = 1}^\infty {\left( { - 1} \right)^{n - 1\;}}{a_n}\) and \(\mathop \sum \limits_{n = 2}^\infty {\left( { - 1} \right)^{n - 1\;}}{b_n}\), where \({a_n} = \frac{1}{{\sqrt n }},\;{b_n} = \frac{{{x^n}}}{{n\left( {n - 1} \right)}}\) 0 < x < 1. Then:

Find the cofactor of b3 in the following matrix Δ: \(= \left[ {\begin{array}{*{20}{c}} a_1&{ b_1 }&{c_1}\\ a_2&b_2&{ c_2}\\ a_3&b_3&c_3 \end{array}} \right]\)

A box contains 38 coins each one of which is either a two rupee coin or a five rupee coin. If the total value of these count is Rs. 157, then the number of five rupee coins exceeds the number of two rupee coins by:

If (-5x)/3 + 2 = x - 6 then find the value of ‘x’

Multiplication of real valued square matrices of same dimension is

126.	If x = (1/8), which of the following has the largest values?
A.	x/2
B.	x2
C.	√x
D.	1/x
Answer» E.

127.	If a × b = a2 + b2 – ab for all the natural numbers a and b, then the value of 9 × 10 is
A.	90
B.	91
C.	181
D.	182
Answer» C. 181

128.	How much does a watch lose per day, if the hands coincide every 64 minutes?
A.	\(32\frac{8}{{11}}\;minutes\)
B.	32/11 minutes
C.	\(17\frac{5}{{11}}\;minutes\)
D.	16/11 minutes
Answer» B. 32/11 minutes

129.	If (10.24 × 10.24) – (10.24 × A) + (0.24 × 0.24) is a perfect square, then find the value of 'A'.
A.	0.24
B.	0.36
C.	0.48
D.	0.12
Answer» D. 0.12

130.	If a is positive and \({a^2} + \frac{1}{{{a^2}}} = 7,\;{\rm{then\;find}}\:{a^3} + \frac{1}{{{a^3}}}.\)
A.	21
B.	3√7
C.	18
D.	7√7
Answer» D. 7√7

131.	If 2x2 - 3x + 5 = 4, then the value of x is :
A.	-1
B.	2
C.	1/2
D.	1/4
Answer» D. 1/4

132.	At which value of k, the linear equations 3x - 2y = 13 and kx - 8y = 40 have no solutions?
A.	17
B.	15
C.	8
D.	12
Answer» E.

133.	A person carries Rs. 500 and wants to buy apples and oranges out of it. If the cost of one apple is Rs. 5 and the cost of one orange is Rs. 7 then what is the number of ways in which a person can buy both apples and oranges using total amount?
A.	10
B.	14
C.	15
D.	17
Answer» C. 15

134.	If \(x+\dfrac{1}{x}=6\), then find \(x^2+\dfrac{1}{x^2}\):
A.	36
B.	30
C.	34
D.	32
Answer» D. 32

135.	If x4 + x2y2 + y4 = 21/256 and x2 + xy + y2 = 3/16, then 2(x2 + y2) = ?
A.	4
B.	5/8
C.	3/4
D.	5/16
Answer» C. 3/4