Maths : Algebra: Book Back, Exercise, Example Numerical Question with Answers, Solution: Exercise 3.15: One Mark Choose the correct Answers

**Exercise 3.15**

**Multiple choice questions**

1. If
*x*^{3} + 6*x*^{2} + *kx* + 6 is exactly divisible
by (*x* + 2), then *k*= ?

(1) –6

(2) –7

(3) –8

**(4) 11**

**Solution:**

P(-2) = (-2)^{3} + 6(-2)^{2} + *k*(-2) + 6 = 0

-8 + 24 – 2*k* +6
= 0

22 = 2*k*

*k* = 11

**[Answer: ****(4) 11 ]**

2. The
root of the polynomial equation 2*x* + 3 = 0 is

(1)
1/3

(2)
– 1/3

**(3) – 3/2 **

(4)
– 2/3

**[Answer: ****(3) – 3/2 ]**

3. The
type of the polynomial 4–3*x*^{3} is

(1)
constant polynomial

(2)
linear polynomial

(3)
quadratic polynomial

**(4) cubic polynomial.**

**[Answer: ****(4) cubic polynomial ]**

4. If
*x*^{51} + 51 is divided by *x* + 1, then the remainder is

(1)
0

(2)
1

(3)
49

**(4) 50**

**Solution: **P (−1) = (−1)^{51} + 51 = −1 +51 =50

**[Answer: ****(4) 50 ]**

5. The
zero of the polynomial 2*x*+5 is

(1)
5/2

**(2) – 5/2 **

(3)
2/5

(4)
– 2/5

**[Answer: ****(2) – 5/2 ]**

6. The
sum of the polynomials *p*(*x*) = *x*^{3} – *x*^{2}
– 2, *q*(*x*) = *x*^{2}–3*x*+ 1

**(1) x^{3} – 3x
– 1 **

(2)
*x*^{3} + 2*x*^{2} – 1

(3)
*x*^{3} – 2*x*^{2} – 3*x*

(4)
*x*^{3} – 2*x*^{2} + 3*x* –1

**Solution: **

**[Answer: ****(1) x^{3}
– 3x – 1 ]**

7. Degree
of the polynomial (*y*^{3}–2)(*y*^{3} + 1) is

(1)
9

(2)
2

(3)
3

**(4) 6**

**Solution: **(*y*^{3}–2)(*y*^{3}+1)
= (*y*^{3}–2)(*y*^{3}–2) × 1 = *y*^{6} –2*y*^{3}–2
= *y*^{6}–*y*^{3}–2

**[Answer: ****(4) 6 ]**

8. Let
the polynomials be

(A)
–13q^{5} + 4q^{2} + 12q

(B)
(*x*^{2} +4 )(*x*^{2} + 9)

(C)
4q^{8} – q^{6} + q^{2}

(D)
– 5/7* y*^{12}* *+*
y*^{3}* *+* y*^{5}

Then
ascending order of their degree is

(1)
A,B,D,C

(2)
A,B,C,D

(3)
B,C,D,A

**(4) B,A,C,D**

**Solution: **Degree of (A), (B) (C) & (D) are
respectively be 5,4,8,12

**[Answer: ****(4) B,A,C,D ]**

9. If
*p*(*a* ) = 0 then (*x* -*a*) is a ___________ of *p*(*x*)

(1)
divisor

(2)
quotient

(3)
remainder

**(4) factor**

**[Answer: ****(4) factor ]**

10.
Zeros of (2 − 3*x*) is ___________

(1)
3

(2)
2

**(3) 2/3 **

(4)
3/2

**Solution: **

2−3*x*
= 0

−3x = −2

*x*=2/3

**[Answer: ****(3) 2/3 ]**

11.
Which of the following has *x* -1 as a factor?

(1) 2*x* -1

**(2) 3 x − 3**

(3) 4*x* − 3

(4) 3*x* − 4

**Solution: **

*P(x) *= 3*x*−3

*P*(1)* *= 3(1)−3=0

So (*x*−1) is a factor
of *P(x)*

**[Answer: ****(2) 3 x − 3 ]**

12.
If *x* − 3 is a factor of *p*(*x*), then the remainder is

(1) 3

(2) –3

**(3) p(3)**

(4) *p*(–3)

**[Answer: ****(3) p(3) ]**

13.
(*x* + *y* )(*x*^{2} −*xy* +*y*^{2} ) is equal to

(1)
(*x* + *y*)^{3}

(2)
(*x* -*y*)^{3}

**(3) x ^{3} **

(4)
*x* ^{3} -*y*^{3}

**[Answer: ****(3) x ^{3}
**

14.
(*a* + *b* −*c*)^{2} is equal to __________

(1) (*a* −
*b* +*c*)^{2}

**(2) (****−***a*** ****−***b*** ****+***c***) ^{2} **

(3) (*a* + *b* +*c*)^{2}

(4) (*a* − *b* -*c*)^{2}

**Solution: **

**( a+b**−

**[Answer: ****(2) (****−***a*** ****−***b*** ****+***c***) ^{2} ]**

15.
If (*x* + 5) and (*x* − 3) are the factors of *ax* ^{2}
+
*bx* +*c*, then values of a,* b *and* c *are

(1) 1,2,3

(2) 1,2,15

**(3) 1,2, −15**** **

(4) 1, −2,15

**Solution: **

*P(**−5) = a*(−5^{2})+*b*(−5)+c = 25a−5b+c = 0 ….(1)

*P(**3) = a*(3^{2})+*bc*
+ 3 + c = 9 + 3*b* + *c* = 0 ….(2)

25*a*−5*b* = 9*a*−3*b*

25*a*−9*a* = 3*b*+5*b*

16*a*=8*b*

*a*/*b* = 8/16 = 1/2

Substitute *a*=1, *b*=2 in (1)

25(1) – 5(2) = − *c*

25 – 10 = 15 = −c

C = −15

** [Answer: ****(3) 1,2, −15 ]**

16.
Cubic polynomial may have maximum of ___________ linear factors

(1) 1

(2) 2

**(3) 3****
**

(4) 4

**[Answer: ****(3) 3 ]**

17.
Degree of the constant polynomial is __________

(1) 3

(2) 2

(3) 1

**(4) 0**

**[Answer: ****(4) 0 ]**

18.
In an expression *ax*^{2 }+ *bx *+ *c*
the sum and product of factors respectively,

(1)
*a,bc*

**(2) b,ac**

(3)
ac*,b*

(4)
*bc,a*

**[Answer: ****(2) b,ac ]**

19.
Find the value of *m* from the equation 2*x* +
3*y* = *m* . If its one solution is *x* =
2 and *y* = −2.

(1)
2

**(2) −2 **

(3)
10

(4)
0

**Solution:**

*x*=2, *y*=−2

2*x*+3*y*=*m,*

*m*=2(2)+3(−2)

=4−6= −2

**[Answer: ****(2) −2 ]**

20.
Which of the following is a linear equation

(1)* x *+ 1/*x* = 2

(2)* x *(*
x *− 1) = 2

**(3) 3 x + 5 = 2/3 **

(4)* x*^{3} −* x *= 5

**Solution:**

*x* + [1/*x*] = 2

*x*^{2}−2*x*+1=0

*x*(*x*−1) = 2

*x*^{2}−*x*−2=0

**[Answer: ****(3) 3 x + 5 = 2/3 ]**

21.
Which of the following is a solution of the equation 2*x* −
*y* = 6

(1)
(2,4)

**(2) (4,2)**** **

(3)
(3, −1)

(4)
(0,6)

**Solution:**

2*x*−*y*=6

2(4) – 2 =
8−2=6=RHS

**[Answer: ****(2) (4,2) ]**

22.
If (2,3) is a solution of linear equation 2*x* +
3*y* = *k* then, the value of *k* is

(1)
12

(2)
6

(3)
0

**(4) 13**

**Solution:**

2*x*+3*y*=*k*

2(2)+3(3)=4+9=13

**[Answer: ****(4) 13 ]**

23.
Which condition does not satisfy the linear equation *ax* +
*by* + *c* = 0

(1)
a ≠ 0 ,* b *= 0

(2)
a = 0 ,* b *≠ 0

**(3) a = 0 , b = 0 , c ≠ 0 **

(4)
a ≠ 0 ,* b *≠ 0

**Solution:**

*a*=0, *b*=0, c≠0

(0)*x* + (0)*y*+*c*=0 False

**[Answer: ****(3) a = 0 , b = 0 , c ≠ 0 ]**

24.
Which of the following is not a linear equation in two variable

(1) *ax *+* by *+*
c *=*
*0* *

**(2) 0x **

(3) 0*x*
+
*by* + *c* = 0

(4)
*ax* + 0 *y* + *c* =
0

**Solution:**

*a* and *b* both can not be zero

**[Answer: ****(2) 0x **

25.
The value of *k* for which the pair of linear equations
4*x* + 6 *y* −1 = 0 and 2*x* +
*ky* − 7 = 0 represents parallel lines is

**(1) k **

(2)
*k* = 2

(3)
*k* = 4

(4)
*k* = −3

**Solution:**

4x+6y = 1

6y = −4x +
1

*y* = −4/6 *x* + 1/6
………. (1)

2*x*+*ky*−7=0

*ky*=−2*x*+7

* y* = −2/*k* *x* +
7/*k*
………..(2)

Since the lines (1) and (2) parallel

*m*_{1} = *m*_{2}

−4/6 = −2/*k*

*k=3*

**[Answer: ****(1) k **

26.
A pair of linear equations has no solution then the graphical representation is

**Solution:**

Parallel lines have no solution

**[Answer: (2) ]**

27.
If *a*_{1}/*a*_{2} ≠ *b*_{1}/*b*_{2} where *a*_{1}*x *+ *b*_{2}*y *+* c*_{1}* *= 0 and *a*_{2}*x *+* b*_{2}*y *+* c*_{2} = 0 then
the given pair of linear equation has __________ solution(s)

(1)
no solution

(2)
two solutions

**(3) unique**

(4)
infinite

**Solution:**

*a*_{1}/*a*_{2} ≠ b_{1}/b_{2} ; unique solution

**[Answer: ****(3) unique ]**

28.
If *a*_{1}/*a*_{2} ≠ *b*_{1}/*b*_{2 }≠ *c*_{1}/*c*_{2}
where *a*_{1}*x *+ *b*_{1}*y *+*
c*_{1}* *= 0 and *a*_{2}*x *+ *b*_{2}*y *+*
c*_{2} = 0 then the given pair of linear equation has __________ solution(s)

**(1) no solution **

(2)
two solutions

(3)
infinite

(4)
unique

**Solution:**

*a*_{1}**/ a_{2} = b_{1}/b_{2} **≠

**[Answer: ****(1) no solution ]**

29.
GCD of any two prime numbers is __________

(1)
−1

(2)
0

**(3) 1**

(4)
2

**[Answer: ****(3) 1 ]**

30.
The GCD of *x* ^{4} -*y*^{4} and *x* ^{2} -*y*^{2} is

(1)
*x* ^{4} − *y*^{4}

**(2) x ^{2} **−

(3)
(*x* + *y*)^{2}

(4)
(*x* + *y*)^{4}

**Solution:**

*x*^{4}−*y*^{4} =
(*x*^{2})^{2} – (*y*^{2})^{2} = (*x*^{2}+*y*^{2}) (*x*^{2}−*y*^{2})

*x*^{2}−*y*^{2}=*x*^{2}−*y*^{2}

G.C.D is = *x*^{2} – *y*^{2}

**[Answer: ****(2) x ^{2}
**

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