Q1
mcq
If $n(A\times B)=6$ and $A=\{1,3\}$ then $n(B)$ is:
- A. 1
- B. 2
- C. 3
- D. 6
Solution hidden
Log in to view solution →
Q2
mcq
The sum of the exponents of the prime factors in the prime factorization of 1729 is:
- A. 1
- B. 2
- C. 3
- D. 4
Solution hidden
Log in to view solution →
Q3
mcq
Given $F_1=1$, $F_2=3$ and $F_n=F_{n-1}+F_{n-2}$, then $F_5$ is:
- A. 3
- B. 5
- C. 8
- D. 11
Solution hidden
Log in to view solution →
Q4
mcq
The square root of $\dfrac{8^4 10}{6^6 256\,25\,x y z}$ is equal to:
- A. $\dfrac{2^4 z}{5x^2 y}$
- B. $\dfrac{2}{y^2x^4z}$
- C. $\dfrac{2}{xz^4y}$
- D. $\dfrac{2}{xz^5y}$
Solution hidden
Log in to view solution →
Q5
mcq
Graph of a linear equation is a __________.
- A. Straight line
- B. Circle
- C. Parabola
- D. Hyperbola
Solution hidden
Log in to view solution →
Q6
mcq
The G.C.D. of $a^m$, $a^{m+1}$, $a^{m+2}$ is:
- A. $a^m$
- B. $a^{m+1}$
- C. $a^{m+2}$
- D. 1
Solution hidden
Log in to view solution →
Q7
mcq
In $\triangle ABC$, $DE \parallel BC$, $AB=3.6$ cm, $AC=2.4$ cm and $AD=2.1$ cm then, the length of $AE$ is:
- A. 1.4 cm
- B. 1.8 cm
- C. 1.2 cm
- D. 1.05 cm
Solution hidden
Log in to view solution →
Q8
mcq
How many tangents can be drawn to the circle from an exterior point?
- A. one
- B. two
- C. infinite
- D. zero
Solution hidden
Log in to view solution →
Q9
mcq
The area of a triangle formed by the points $(-5,0)$, $(0,-5)$ and $(5,0)$ is:
- A. 0 sq. units
- B. 25 sq. units
- C. 5 sq. units
- D. none of these
Solution hidden
Log in to view solution →
Q10
mcq
If the ratio of the height of a tower and the length of its shadow is $3:1$ then, the angle of elevation of the sun has measure:
- A. 45$^\circ$
- B. 30$^\circ$
- C. 90$^\circ$
- D. 60$^\circ$
Solution hidden
Log in to view solution →
Q11
mcq
The height of a right circular cone whose radius is 5 cm and slant height is 13 cm will be:
- A. 12 cm
- B. 10 cm
- C. 13 cm
- D. 5 cm
Solution hidden
Log in to view solution →
Q12
mcq
The total surface area of a hemisphere is how many times the square of its radius?
- A. $\pi$
- B. $4\pi$
- C. $3\pi$
- D. $2\pi$
Solution hidden
Log in to view solution →
Q13
mcq
A page is selected at random from a book. The probability that the digit at units place of the page number chosen is less than 7 is:
- A. $\dfrac{3}{10}$
- B. $\dfrac{7}{10}$
- C. $\dfrac{3}{9}$
- D. $\dfrac{7}{9}$
Solution hidden
Log in to view solution →
Q14
short answer
If $A=\{1,3,5\}$ and $B=\{2,3\}$ then show that $n(A\times B)=n(A)\times n(B)$.
Solution hidden
Log in to view solution →
Q15
short answer
Let $A=\{1,2,3,4,\ldots,45\}$ and $R$ be the relation defined as “is square of a number” on $A$. Write $R$ as a subset of $A\times A$. Also, find the domain and range of $R$.
Solution hidden
Log in to view solution →
Q16
short answer
Find the number of terms in the A.P. $3, 6, 9, 12, \ldots, 111$.
Solution hidden
Log in to view solution →
Q17
short answer
If $31k$, $18-k$, $5k11$ are in A.P. then find $k$.
Solution hidden
Log in to view solution →
Q18
short answer
Determine the quadratic equations, whose sum and product of roots are $-9$ and 20.
Solution hidden
Log in to view solution →
Q19
short answer
Determine the nature of the roots for the quadratic equation $15x^2-11x+12=0$.
Solution hidden
Log in to view solution →
Q20
short answer
In $\triangle ABC$, $D$ and $E$ are points on the sides $AB$ and $AC$ respectively such that $DE \parallel BC$. If $\dfrac{AD}{DB}=\dfrac{3}{4}$ and $AC=15$ cm, find $AE$.
Solution hidden
Log in to view solution →
Q21
short answer
Show that the points $(-3,-4)$, $(7,2)$ and $(12,5)$ are collinear.
Solution hidden
Log in to view solution →
Q22
short answer
Calculate the slope and $y$ intercept of the straight line $8x-7y16=0$.
Solution hidden
Log in to view solution →
Q23
short answer
Find the intercepts made by the line $3x-2y-6=0$ on the coordinate axes.
Solution hidden
Log in to view solution →
Q24
short answer
Find the angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of a tower of height $10\sqrt{3}$ m.
Solution hidden
Log in to view solution →
Q25
short answer
Find the volume of a cylinder whose height is 2 m and base area is 250 m$^2$.
Solution hidden
Log in to view solution →
Q26
short answer
A die is rolled and a coin is tossed simultaneously. Find the probability that the die shows an odd number and the coin shows a head.
Solution hidden
Log in to view solution →
Q27
short answer
The heights of two right circular cones are in the ratio $1:2$ and the perimeters of their bases are in the ratio $3:4$. Find the ratio of their volumes.
Solution hidden
Log in to view solution →
Q28
short answer
Let $A=\{x\in W\mid x<2\}$, $B=\{x\in N\mid 1<x\le 4\}$ and $C=\{3,5\}$. Verify that $A\times(B\cap C)=(A\times B)\cap(A\times C)$.
Solution hidden
Log in to view solution →
Q29
short answer
The sum of three consecutive terms that are in A.P. is 27 and their product is 288. Find the three terms.
Solution hidden
Log in to view solution →
Q31
short answer
Solve: $x+ y+ z=5$; $2x-y+ z=9$; $x-2y+3z=16$.
Solution hidden
Log in to view solution →
Q32
short answer
Find the square root of $64x^4-16x^3+17x^2-2x+1$.
Solution hidden
Log in to view solution →
Q34
long answer
Show that in a triangle, the medians are concurrent.
Solution hidden
Log in to view solution →
Q35
short answer
Find the equation of the median of $\triangle ABC$ through $A$ where the vertices are $A(6,2)$, $B(-5,-1)$ and $C(1,9)$.
Solution hidden
Log in to view solution →
Q36
short answer
If the points $P(-1,-4)$, $Q(b,c)$ and $R(5,-1)$ are collinear and if $2b+c=4$, then find the values of $b$ and $c$.
Solution hidden
Log in to view solution →
Q37
short answer
Two ships are sailing in the sea on either sides of a lighthouse. The angle of elevation of the top of the lighthouse as observed from the ships are 30$^\circ$ and 45$^\circ$ respectively. If the lighthouse is 200 m high, find the distance between the two ships. $(\sqrt{3}=1.732)$
Solution hidden
Log in to view solution →
Q38
short answer
If the radii of the circular ends of a frustum which is 45 cm high are 28 cm and 7 cm, find the volume of the frustum.
Solution hidden
Log in to view solution →
Q39
short answer
A toy is in the shape of a cylinder surmounted by a hemisphere. The height of the toy is 25 cm. Find the total surface area of the toy if its common diameter is 12 cm.
Solution hidden
Log in to view solution →
Q40
short answer
Two dice are rolled. Find the probability that the sum of outcome is (i) equal to 4 (ii) greater than 10 (iii) less than 13.
Solution hidden
Log in to view solution →
Q41
short answer
If the equation $(1+m^2)x^2+2mcx+c^2-a^2=0$ has equal roots, then prove that $c^2=a^2(1+m^2)$.
Solution hidden
Log in to view solution →
Q42
long answer
Construct a $\triangle PQR$ whose base $PQ=4.5$ cm, $\angle R=35^\circ$ and the median from $R$ to $PQ$ is 6 cm.
OR
Draw a circle of diameter 6 cm from a point $P$, which is 8 cm away from its centre. Draw two tangents $PA$ and $PB$ to the circle and measure their lengths.
Solution hidden
Log in to view solution →
Q43
long answer
Draw the graph of $x^2+x-1=0$ and state the nature of their solution.
OR
Draw the graph of $y=$
Solution hidden
Log in to view solution →
Q44
long answer
Draw the graph of $x^2 + x - 12 = 0$ and state the nature of their solution. OR Draw the graph of $y = x^2 + 3x - 4$ and hence use it to solve $x^2 + 3x - 4 = 0$.
Solution hidden
Log in to view solution →